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Alabama Standards Alignment 2010 Alabama Course of Study (Common Core), produced by the State Department of Education Kindergarten through Precalculus Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other countries. P.O. Box 8036 • Wisconsin Rapids, WI 54495-8036 Phone: (800) 338-4204 • Fax: (715) 424-4242 www.renlearn.com Alabama Standards Alignment Standards List with Aligned Product Skills The Standards List with Aligned Product Skills Report is a standards-oriented document showing the entire list of standards for the subject and grade and the product skills aligned to those standards. This alignment report shows the breadth of standards coverage for the purpose and focus of this product. Note to Educator Kindergarten ................................................................................................... iii ....................................................................................................... 1 Grade 1 ............................................................................................................ 6 Grade 2 ............................................................................................................ 12 Grade 3 ............................................................................................................ 18 Grade 4 ............................................................................................................ 26 Grade 5 ............................................................................................................ 37 Grade 6 ............................................................................................................ 48 Grade 7 ............................................................................................................ 61 Grade 8 ............................................................................................................ 72 9-12 Analytic Math Algebra 2 .......................................................................................................... 85 Algebra 2 with Trig Algebra I Precalculus ................................................................................................ 97 .......................................................................................................... 110 Discrete Mathematics Geometry ................................................................................................. 82 ............................................................................................... 125 ........................................................................................................... 128 ............................ ............................................................................ 139 © 2012 by Renaissance Learning, Inc. All rights reserved. No portion of this document may be reproduced, by any process or technique, without the express written consent of Renaissance Learning, Inc. Any standards referenced in this document are the property of their respective copyright holders. ii P.O. Box 8036 Wisconsin Rapids, WI 54495-8036 Phone: (800) 338-4204 Fax: (715) 424-4242 www.renlearn.com Note to Educator: Thank you for your interest in Renaissance Learning technology. At Renaissance Learning, we recognize the impact that standards and assessment reform have on schools. We share the concerns of educators and administrators that students perform well and that teachers have the resources they need to support their efforts to address standards and assessments. Renaissance Learning provides alignment reports to customers to show how the skills within each product align to the skills within academic standards. The alignment report presents all of the academic math standards for a specific state/agency with the aligned Renaissance Learning product skills indented below each standard. Academic standards encompass the entire set of learning and expectations that teachers are responsible for. Renaissance Learning recognizes that teachers are the key to using products to address the entire set of standards. Renaissance Learning products are ideally suited to support teachers and academic standards. The Renaissance Learning alignment report supports teachers in this role by clearly identifying specific product skills that are aligned to the multiple skills within the standards. iii On the alignment report you will see the word “Core” next to some of the skills. Core skills are the most critical mathematics skills for a student to learn at a grade level. They are key building blocks in a student’s mathematics education. Students need to have proficiency with core skills to be successful in math at their grade levels and to progress in the grades that follow. Core skills are indicated on Renaissance Learning’s researchbased and empirically validated Core Progress math learning progression. Core Progress Math identifies the continuum of concepts and skills needed for success in math. The continuum begins with early numeracy and progresses to the level of mathematics required for college and careers. All of the skills in the Core Progress Math contribute to STAR Math Enterprise Assessment. Alignments for STAR Math Enterprise include alignments to the Common Core State Standards (CCSS) Mathematics Standards. In the alignment report, you will notice that most skills have a grade-level indicator at the end of the text. In addition to these grade-level skills, the report includes skills that are not grade specific and do not have a grade-level indicator. These skills are designated Overall Product Skills. They are skills that all students gain through use of the product. We hope this report answers your questions regarding the alignment of Renaissance Learning technology and materials to standards. The complete alignment strategy document is available. The Math document is number R39616. If you have any questions about the alignment report, please feel free to call us at (800) 338-4204. Sincerely, Renaissance Learning iv STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Kindergarten Kindergarten AL K.CC - Counting and Cardinality Know number names and the count sequence. AL K.CC.1 - Count to 100 by ones and by tens. Numbers and Operations Count on or count back with numbers 1 to 20 [Early Numeracy] Count on by ones from a number less than 100 [Grade 1] Core Skill Count by 5s or 10s to 100 starting from a multiple of 5 or 10, respectively [Grade 1] Core Skill AL K.CC.2 - Count forward beginning from a given number within the known sequence (instead of having to begin at 1). Numbers and Operations Count on or count back with numbers 1 to 20 [Early Numeracy] Count on by ones from a number less than 100 [Grade 1] Core Skill AL K.CC.3 - Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects). Numbers and Operations Recognize numbers 0-20 [Early Numeracy] Recognize that there are 0 objects [Early Numeracy] Read a whole number to 30 [Grade 1] Count objects to 20 [Grade 1] Core Skill Count to tell the number of objects. AL K.CC.4 - Understand the relationship between numbers and quantities; connect counting to cardinality. AL K.CC.4.a - When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. Numbers and Operations Count up to 15 objects [Early Numeracy] Count up to 20 objects [Early Numeracy] Count objects to 20 [Grade 1] Core Skill Count on by ones from a number less than 100 [Grade 1] Core Skill AL K.CC.4.b - Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. Numbers and Operations Count objects to 20 [Grade 1] Core Skill AL K.CC.4.c - Understand that each successive number name refers to a quantity that is one larger. Numbers and Operations Count on or count back with numbers 1 to 10 [Early Numeracy] Count on or count back with numbers 1 to 20 [Early Numeracy] 4/26/2012 - 1 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Count on by ones from a number less than 100 [Grade 1] Kindergarten Core Skill AL K.CC.5 - Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects. Numbers and Operations Count objects to 20 [Grade 1] Core Skill Compare numbers. AL K.CC.6 - Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.[Include groups with up to ten objects.] Numbers and Operations Compare sets of up to 5 objects [Early Numeracy] Compare sets of up to 10 objects using "more" or "fewer" [Early Numeracy] Identify two sets with equal numbers of up to 10 objects [Early Numeracy] Compare sets of objects using words [Grade 1] AL K.CC.7 - Compare two numbers between 1 and 10 presented as written numerals. Numbers and Operations Compare sets of up to 10 objects using "more" or "fewer" [Early Numeracy] AL K.OA - Operations and Algebraic Thinking Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. AL K.OA.8 - Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.[Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)] Numbers and Operations Add numbers with a sum up to 9 using pictures [Early Numeracy] Add numbers with a sum up to 10 using pictures [Early Numeracy] Subtract numbers with a minuend up to 9 using pictures [Early Numeracy] Subtract numbers with a minuend up to 10 using pictures [Early Numeracy] AL K.OA.9 - Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Numbers and Operations Determine the difference in number between two sets of objects [Early Numeracy] Add numbers with a sum up to 9 using pictures [Early Numeracy] 4/26/2012 - 2 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Kindergarten Add numbers with a sum up to 10 using pictures [Early Numeracy] Subtract numbers with a minuend up to 9 using pictures [Early Numeracy] Subtract numbers with a minuend up to 10 using pictures [Early Numeracy] Know basic addition facts to 10 plus 10 [Grade 1] Algebra Core Skill Determine the missing addend in an addition sentence using pictures [Early Numeracy] AL K.OA.10 - Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). Numbers and Operations Compose an equivalent number out of two numbers by using pictures [Early Numeracy] Decompose a number into two numbers by using pictures [Early Numeracy] Determine equivalent forms of a number, up to 10 [Grade 1] Core Skill Relate a picture model to a basic addition fact [Grade 1] Determine the basic addition fact shown by a picture model [Grade 1] Algebra Determine equivalent addition or subtraction expressions involving 1-digit numbers [Grade 1] AL K.OA.11 - For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation. Numbers and Operations Determine equivalent forms of a number, up to 10 [Grade 1] Core Skill AL K.OA.12 - Fluently add and subtract within 5. Numbers and Operations Add numbers with a sum up to 5 in horizontal format [Early Numeracy] Subtract numbers with a minuend up to 3 in horizontal format [Early Numeracy] Subtract numbers with a minuend up to 5 in horizontal format [Early Numeracy] Know basic addition facts to 10 plus 10 [Grade 1] Core Skill Know basic subtraction facts to 20 minus 10 [Grade 1] Core Skill AL K.NBT - Number and Operations in Base Ten Work with numbers 11-19 to gain foundations for place value. 4/26/2012 - 3 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Kindergarten AL K.NBT.13 - Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. AL K.MD - Measurement and Data Describe and compare measurable attributes. AL K.MD.14 - Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. Geometry and Measurement Sort objects according to one attribute [Early Numeracy] Determine the common attributes in a set of geometric shapes [Grade 1] Core Skill AL K.MD.15 - Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. Example: For example, directly compare the heights of two children and describe one child as taller/shorter. Geometry and Measurement Compare the sizes of two objects [Early Numeracy] Compare the weights of two objects [Early Numeracy] Compare the volumes of two objects [Early Numeracy] Compare objects using the vocabulary of measurement [Grade 1] Core Skill Classify objects and count the number of objects in each category. AL K.MD.16 - Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.[Limit category counts to be less than or equal to 10.] Geometry and Measurement Sort objects according to one attribute [Early Numeracy] Determine the group in which a geometric shape belongs [Grade 1] Determine the rule used to sort geometric shapes [Grade 1] Data Analysis, Statistics, and Probability Determine the rule used to sort objects [Grade 1] AL K.G - Geometry Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). AL K.G.17 - Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to. Geometry and Measurement Apply the vocabulary of position or direction [Grade 1] AL K.G.18 - Correctly name shapes regardless of their orientations or overall size. Geometry and Measurement Identify a circle, a triangle, a square, or a rectangle [Grade 1] 4/26/2012 Core Skill - 4 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Kindergarten AL K.G.19 - Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid"). Overall Product Skills Build comprehension of math vocabulary Analyze, compare, create, and compose shapes. AL K.G.20 - Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/"corners") and other attributes (e.g., having sides of equal length). Geometry and Measurement Determine the common attributes in a set of geometric shapes [Grade 1] Core Skill Determine the group in which a geometric shape belongs [Grade 1] Determine the rule used to sort geometric shapes [Grade 1] AL K.G.21 - Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes. AL K.G.22 - Compose simple shapes to form larger shapes. Example: For example, "Can you join these two triangles with full sides touching to make a rectangle?". Geometry and Measurement 4/26/2012 Compose a new shape out of two other shapes [Early Numeracy] - 5 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 1 Grade 1 AL 1.OA - Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. AL 1.OA.1 - Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Numbers and Operations Determine the missing portion in a partially screened (hidden) collection of up to 10 objects [Grade 1] Core Skill Relate a picture model to a basic addition fact [Grade 1] Determine the basic addition fact shown by a picture model [Grade 1] Relate a picture model to a basic subtraction fact [Grade 1] Algebra WP: Use basic addition or subtraction facts to solve problems [Grade 1] Core Skill Determine a missing addend in a basic addition-fact number sentence [Grade 1] Core Skill Determine a missing subtrahend in a basic subtraction-fact number sentence [Grade 1] Core Skill WP: Determine a missing addend or subtrahend in a basic addition- or subtraction-fact number sentence [Grade 1] Core Skill WP: Determine a basic addition- or subtraction-fact number sentence for a given situation [Grade 1] Core Skill AL 1.OA.2 - Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Numbers and Operations Add three 1-digit numbers [Grade 1] Algebra Determine a missing addend in an addition sentence with three 1digit numbers [Grade 1] Understand and apply properties of operations and the relationship between addition and subtraction. AL 1.OA.3 - Apply properties of operations as strategies to add and subtract. Example: Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.).[Students need not use formal terms for these properties.] Numbers and Operations Use turn-around facts to find the sum of two numbers [Early Numeracy] Add three 1-digit numbers [Grade 1] Complete an addition and subtraction fact family [Grade 1] 4/26/2012 - 6 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 1 AL 1.OA.4 - Understand subtraction as an unknown-addend problem. Example: For example, subtract 10 - 8 by finding the number that makes 10 when added to 8. Algebra Determine the missing addend in an addition sentence using pictures [Early Numeracy] Determine a missing addend in a basic addition-fact number sentence [Grade 1] Core Skill Determine a missing subtrahend in a basic subtraction-fact number sentence [Grade 1] Core Skill Add and subtract within 20. AL 1.OA.5 - Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). Numbers and Operations Count on or count back with numbers 1 to 20 [Early Numeracy] Add numbers with a sum up to 20 using pictures [Early Numeracy] Subtract numbers with a minuend up to 20 using pictures [Early Numeracy] Apply the relationship between addition and counting on [Grade 1] Apply the relationship between subtraction and counting back [Grade 1] AL 1.OA.6 - Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Numbers and Operations Add numbers with a sum up to 20 in horizontal format [Early Numeracy] Add numbers with a sum up to 20 in vertical format [Early Numeracy] Subtract numbers with a minuend up to 20 in horizontal format [Early Numeracy] Subtract numbers with a minuend up to 20 in vertical format [Early Numeracy] Apply the relationship between addition and counting on [Grade 1] Apply the relationship between subtraction and counting back [Grade 1] Know basic addition facts to 10 plus 10 [Grade 1] Core Skill Know basic subtraction facts to 20 minus 10 [Grade 1] Core Skill Complete an addition and subtraction fact family [Grade 1] 4/26/2012 - 7 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 1 Apply the inverse relationship between addition facts and subtraction facts [Grade 1] Work with addition and subtraction equations. AL 1.OA.7 - Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. Example: For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. Numbers and Operations Determine equivalent forms of a number, up to 10 [Grade 1] Core Skill Complete an addition and subtraction fact family [Grade 1] Algebra Determine equivalent addition or subtraction expressions involving 1-digit numbers [Grade 1] AL 1.OA.8 - Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. Example: For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = __ - 3, 6 + 6 = __. Numbers and Operations Add numbers with a sum up to 20 using pictures [Early Numeracy] Subtract numbers with a minuend up to 20 using pictures [Early Numeracy] Add numbers with a sum up to 20 in horizontal format [Early Numeracy] Add numbers with a sum up to 20 in vertical format [Early Numeracy] Subtract numbers with a minuend up to 20 in horizontal format [Early Numeracy] Subtract numbers with a minuend up to 20 in vertical format [Early Numeracy] Algebra Determine a missing addend in a basic addition-fact number sentence [Grade 1] Core Skill Determine a missing subtrahend in a basic subtraction-fact number sentence [Grade 1] Core Skill AL 1.NBT - Number and Operations in Base Ten Extend the counting sequence. AL 1.NBT.9 - Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. Numbers and Operations Read a whole number to 30 [Grade 1] Read a whole number from 31 to 100 [Grade 1] Count objects to 20 [Grade 1] Core Skill Count on by ones from a number less than 100 [Grade 1] Core Skill Understand place value. 4/26/2012 - 8 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 1 AL 1.NBT.10 - Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: AL 1.NBT.10.a - 10 can be thought of as a bundle of ten ones - called a "ten." Numbers and Operations Count objects grouped in tens and ones [Grade 1] Core Skill Recognize a number from a model of tens and ones to 100 [Grade 1] AL 1.NBT.10.b - The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. Numbers and Operations Determine the value of a digit in a 2-digit number [Grade 1] Core Skill AL 1.NBT.10.c - The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones). Numbers and Operations Identify the number of 10's in 10, 20, 30, 40, 50, 60, 70, 80, 90 [Grade 1] Represent a 2-digit number as tens and ones [Grade 1] Core Skill Determine the 2-digit number represented as tens and ones [Grade Core Skill 1] AL 1.NBT.11 - Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. Numbers and Operations Compare sets of up to 20 objects using "more" or "fewer" [Early Numeracy] Compare whole numbers to 100 using words [Grade 1] Core Skill Use place value understanding and properties of operations to add and subtract. AL 1.NBT.12 - Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. Numbers and Operations Add multiples of 10, sums to 90 [Grade 1] Add a 2-digit and a 1-digit number without regrouping [Grade 1] Core Skill Add two 2-digit numbers without regrouping [Grade 1] Core Skill Add two 2-digit numbers with regrouping, given a model [Grade 2] Add a 2-digit number to a 1-digit number with regrouping [Grade 2] AL 1.NBT.13 - Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. Numbers and Operations 4/26/2012 Determine ten more than or ten less than a given number [Grade 1 Core Skill - 9 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Overall Product Skills Grade 1 Use strategies to derive solutions for problems AL 1.NBT.14 - Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Numbers and Operations Add multiples of 10, sums to 90 [Grade 1] Subtract multiples of 10 up to 90 [Grade 1] AL 1.MD - Measurement and Data Measure lengths indirectly and by iterating length units. AL 1.MD.15 - Order three objects by length; compare the lengths of two objects indirectly by using a third object. Geometry and Measurement Order objects according to size [Early Numeracy] Compare objects using the vocabulary of measurement [Grade 1] Core Skill Compare and order objects by attributes of height or length [Grade 1] Core Skill AL 1.MD.16 - Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.[Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.] Geometry and Measurement Estimate length in nonstandard units [Grade 1] Tell and write time. AL 1.MD.17 - Tell and write time in hours and half-hours using analog and digital clocks. Geometry and Measurement Tell time to the hour [Grade 1] Tell time to the half hour [Grade 1] Core Skill Core Skill Represent and interpret data. AL 1.MD.18 - Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. Data Analysis, Statistics, and Probability Read a 2-category tally chart [Grade 1] Use a 2-category tally chart to represent groups of objects (1 symbol = 1 object) [Grade 1] Answer a question using information from a 2-category tally chart [Grade 1] Use a tally chart to represent data [Grade 2] 4/26/2012 - 10 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 1 AL 1.G - Geometry Reason with shapes and their attributes. AL 1.G.19 - Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes. Geometry and Measurement Determine the common attributes in a set of geometric shapes [Grade 1] Core Skill Determine the group in which a geometric shape belongs [Grade 1] AL 1.G.20 - Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.[Students do not need to learn formal names such as "right rectangular prism."] Geometry and Measurement Compose a new shape out of two other shapes [Early Numeracy] Decompose a shape into two new shapes [Early Numeracy] Decompose a plane shape composed of three or more simpler shapes [Grade 2] AL 1.G.21 - Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. Numbers and Operations 4/26/2012 Identify a shape divided into equal parts [Grade 1] - 11 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 2 Grade 2 AL 2.OA - Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. AL 2.OA.1 - Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Numbers and Operations Determine the missing portion in a partially screened (hidden) collection of up to 10 objects [Grade 1] Core Skill WP: Add or subtract numbers without regrouping [Grade 2] Core Skill WP: Add or subtract up to 2-digit numbers with one regrouping [Grade 2] Algebra WP: Determine a missing addend or subtrahend in a basic addition- or subtraction-fact number sentence [Grade 1] Core Skill WP: Determine a missing addend or a missing subtrahend involving 2-digit numbers [Grade 2] Add and subtract within 20. AL 2.OA.2 - Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. Numbers and Operations Apply the relationship between addition and counting on [Grade 1] Apply the relationship between subtraction and counting back [Grade 1] Know basic addition facts to 10 plus 10 [Grade 1] Core Skill Know basic subtraction facts to 20 minus 10 [Grade 1] Core Skill Complete an addition and subtraction fact family [Grade 1] Apply the inverse relationship between addition facts and subtraction facts [Grade 1] Algebra Determine equivalent addition or subtraction expressions involving 1-digit numbers [Grade 1] Work with equal groups of objects to gain foundations for multiplication. AL 2.OA.3 - Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. Numbers and Operations Count by 2s to 50 starting from a multiple of 2 [Grade 1] Core Skill Solve problems involving the concept of odd and even numbers [Grade 2] 4/26/2012 - 12 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 2 AL 2.OA.4 - Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. Numbers and Operations Use a repeated addition sentence to represent the sum of equal groups [Grade 2] Represent repeated addition as multiplication [Grade 2] Relate a multiplication sentence to an array model [Grade 2] AL 2.NBT - Number and Operations in Base Ten Understand place value. AL 2.NBT.5 - Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: AL 2.NBT.5.a - 100 can be thought of as a bundle of ten tens - called a "hundred." Numbers and Operations Determine the 3-digit number represented as hundreds, tens, and ones [Grade 2] Core Skill AL 2.NBT.5.b - The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). Numbers and Operations Determine the value of a digit in a 3-digit number [Grade 2] Determine which digit is in a specified place in a 3-digit whole number [Grade 2] Represent a 3-digit number as hundreds, tens, and ones [Grade 2] AL 2.NBT.6 - Count within 1000; skip-count by 5s, 10s, and 100s. Numbers and Operations Count on by ones from a number less than 100 [Grade 1] Core Skill Core Skill Complete a skip pattern starting from a multiple of 2, 5, or 10 [Grade 2] Core Skill Complete a skip pattern of 2, 5, or 10 starting from any number [Grade 2] Core Skill Count on by 100s from any number [Grade 2] Core Skill AL 2.NBT.7 - Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. Numbers and Operations Read a whole number to 1,000 [Grade 2] Core Skill Determine the word form of a whole number to 1,000 [Grade 2] Core Skill Model a number using hundreds, tens, and ones to 1,000 [Grade 2] Represent a 3-digit number as hundreds, tens, and ones [Grade 2] 4/26/2012 Core Skill - 13 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 2 AL 2.NBT.8 - Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. Numbers and Operations Compare whole numbers to 1,000 using words [Grade 2] Compare whole numbers to 1,000 using the symbols <, >, and = [Grade 2] Core Skill Use place value understanding and properties of operations to add and subtract. AL 2.NBT.9 - Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Numbers and Operations Determine a number pair that totals 100 [Grade 2] Core Skill Add two 2-digit numbers with regrouping, given a model [Grade 2] Add a 2-digit number to a 1-digit number with regrouping [Grade 2] Add two 2-digit numbers with regrouping [Grade 2] Core Skill Subtract a 1- or 2-digit number from a 2-digit number with one regrouping [Grade 2] Core Skill AL 2.NBT.10 - Add up to four two-digit numbers using strategies based on place value and properties of operations. Numbers and Operations Add two 2-digit numbers with regrouping, given a model [Grade 2] Add two 2-digit numbers with regrouping [Grade 2] Core Skill Add three 2-digit numbers with one regrouping, sum less than 100 [Grade 2] AL 2.NBT.11 - Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. Numbers and Operations Determine a number pair that totals 1,000 [Grade 2] Add two 3-digit numbers without regrouping [Grade 2] Add two 3-digit numbers with one regrouping [Grade 2] Core Skill Subtract a 3-digit number from a 3-digit number without regrouping [Grade 2] Subtract a 3-digit number from a 3-digit number with one regrouping [Grade 2] Core Skill Apply the inverse relationship between addition and subtraction with 2- and 3-digit numbers [Grade 2] 4/26/2012 - 14 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 2 AL 2.NBT.12 - Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900. Numbers and Operations Add multiples of 10, sums to 90 [Grade 1] Subtract multiples of 10 up to 90 [Grade 1] AL 2.NBT.13 - Explain why addition and subtraction strategies work, using place value and the properties of operations. Numbers and Operations Add two 2-digit numbers with regrouping, given a model [Grade 2] Overall Product Skills Develop collaboration skills by explaining mathematical reasoning and strategies AL 2.MD - Measurement and Data Measure and estimate lengths in standard units. AL 2.MD.14 - Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. Geometry and Measurement Select the appropriate measurement tool for length, weight, temperature, time, or capacity [Grade 1] Measure length in inches [Grade 2] Core Skill Measure length in centimeters [Grade 2] Core Skill AL 2.MD.15 - Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. AL 2.MD.16 - Estimate lengths using units of inches, feet, centimeters, and meters. Geometry and Measurement Estimate length in nonstandard units [Grade 1] AL 2.MD.17 - Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. Geometry and Measurement Compare and order objects by attributes of height or length [Grade 1] Core Skill Relate addition and subtraction to length. AL 2.MD.18 - Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. Geometry and Measurement Add or subtract compound units of length in inches and feet with no conversion [Grade 3] WP: Add or subtract compound units of length in inches and feet with no conversion [Grade 3] 4/26/2012 - 15 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 2 AL 2.MD.19 - Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2,..., and represent whole-number sums and differences within 100 on a number line diagram. Numbers and Operations Identify a number to 20 represented by a point on a number line [Grade 1] Core Skill Locate a number to 20 on a number line [Grade 1] Core Skill Relate a number-line model to a basic addition fact [Grade 1] Determine the basic addition fact shown by a number-line model [Grade 1] Determine the basic subtraction fact shown by a number-line model [Grade 1] Relate a number-line model to a basic subtraction fact [Grade 1] Work with time and money. AL 2.MD.20 - Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. Geometry and Measurement Tell time to the half hour [Grade 1] Core Skill Tell time to the quarter hour [Grade 2] Core Skill Tell time to 5-minute intervals [Grade 2] Core Skill AL 2.MD.21 - Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: Example: If you have 2 dimes and 3 pennies, how many cents do you have? Numbers and Operations Translate between a dollar sign and a cent sign [Grade 2] Core Skill Determine cent amounts that total a dollar [Grade 2] Core Skill WP: Add or subtract a collection of bills or coins [Grade 2] Represent and interpret data. AL 2.MD.22 - Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units. Geometry and Measurement Measure length in inches [Grade 2] Data Analysis, Statistics, and Probability Use a line plot to represent data [Grade 3] Core Skill AL 2.MD.23 - Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph. Data Analysis, Statistics, and Probability Read a picture graph [Grade 1] Use a picture graph to represent groups of objects (1 symbol = 1 object) [Grade 1] 4/26/2012 - 16 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 2 Read a bar graph [Grade 1] Use a bar graph to represent groups of objects [Grade 1] Answer a question using information from a bar graph [Grade 1] Determine a question that can be answered from a data representation [Grade 3] AL 2.G - Geometry Reason with shapes and their attributes. AL 2.G.24 - Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Geometry and Measurement Identify a circle, a triangle, a square, or a rectangle [Grade 1] Core Skill Determine the common attributes in a set of geometric shapes [Grade 1] Core Skill Determine the rule used to sort geometric shapes [Grade 1] Identify a parallelogram, a trapezoid, a pentagon, a hexagon, or an octagon [Grade 2] Determine attributes of a triangle or a quadrilateral from a model [Grade 3] Relate a model of a triangle or a quadrilateral to a list of attributes [Grade 3] AL 2.G.25 - Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. Geometry and Measurement Relate area to the number of square units [Grade 2] WP: Determine the area of a rectangular shape given a picture on a grid [Grade 3] AL 2.G.26 - Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Numbers and Operations Identify a shape divided into equal parts [Grade 1] Determine a pictorial model of a fraction of a whole [Grade 3] 4/26/2012 Core Skill - 17 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 3 Grade 3 AL 3.OA - Operations and Algebraic Thinking Represent and solve problems involving multiplication and division. AL 3.OA.1 - Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. Example: For example, describe a context in which a total number of objects can be expressed as 5 × 7. Numbers and Operations Relate a multiplication sentence to an array model [Grade 2] Recognize the number of equal groups using pictures [Grade 2] Use a multiplication sentence to represent an area or an array model [Grade 3] Core Skill AL 3.OA.2 - Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. Example: For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. Numbers and Operations Recognize the number of equal groups using pictures [Grade 2] Divide objects into equal groups using pictures [Grade 2] Determine the number of equal groups of objects excluding the remainder [Grade 2] Use a division sentence to represent objects divided into equal groups [Grade 3] Core Skill AL 3.OA.3 - Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Numbers and Operations Algebra WP: Multiply a 1-digit number by 2, 5, or 10 [Grade 2] WP: Divide objects into equal groups by sharing [Grade 2] Core Skill WP: Determine a multiplication or division sentence for a given situation [Grade 3] Core Skill AL 3.OA.4 - Determine the unknown whole number in a multiplication or division equation relating three whole numbers. Example: For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = __ ÷ 3, 6 × 6 = ?. Algebra Determine the missing multiplicand in a number sentence involving basic facts [Grade 3] Determine the missing dividend or divisor in a number sentence involving basic facts [Grade 3] Understand properties of multiplication and the relationship between multiplication and division. 4/26/2012 - 18 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 3 AL 3.OA.5 - Apply properties of operations as strategies to multiply and divide. Example: Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.).[Students need not use formal terms for these properties.] Numbers and Operations Apply the distributive property to the multiplication of a 2-digit number by a 1- or 2-digit number [Grade 4] Apply the distributive property to multiply a multi-digit number by a 1-digit number [Grade 4] Core Skill AL 3.OA.6 - Understand division as an unknown-factor problem. Example: For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. Numbers and Operations Know basic multiplication facts to 10 x 10 [Grade 3] Core Skill Know basic multiplication facts for 11 and 12 [Grade 3] Know basic division facts to 100 ÷ 10 [Grade 3] Core Skill Know basic division facts for 11 and 12 [Grade 3] Complete a multiplication and division fact family [Grade 3] Algebra Determine the missing multiplicand in a number sentence involving basic facts [Grade 3] Determine the missing dividend or divisor in a number sentence involving basic facts [Grade 3] Multiply and divide within 100. AL 3.OA.7 - Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. Numbers and Operations Know basic multiplication facts to 10 x 10 [Grade 3] Core Skill Know basic division facts to 100 ÷ 10 [Grade 3] Core Skill WP: Multiply using basic facts to 10 x 10 [Grade 3] Core Skill WP: Divide using basic facts to 100 ÷ 10 [Grade 3] Core Skill Complete a multiplication and division fact family [Grade 3] Algebra Recognize equivalent multiplication or division expressions involving basic facts [Grade 3] Solve problems involving the four operations, and identify and explain patterns in arithmetic. 4/26/2012 - 19 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 3 AL 3.OA.8 - Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.[This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).] Numbers and Operations WP: Solve a 2-step problem involving addition and/or subtraction of multi-digit whole numbers [Grade 4] AL 3.OA.9 - Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. Example: For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. Algebra Determine an addition or subtraction number pattern given a rule [Grade 2] Determine the rule for an addition or subtraction number pattern [Grade 2] Core Skill Determine a rule for a table of related number pairs [Grade 3] Determine a rule that relates two variables [Grade 4] Core Skill AL 3.NBT - Number and Operations in Base Ten Use place value understanding and properties of operations to perform multi-digit arithmetic.[A range of algorithms may be used.] AL 3.NBT.10 - Use place value understanding to round whole numbers to the nearest 10 or 100. Numbers and Operations Round a 2- to 4-digit whole number to its greatest place [Grade 3] AL 3.NBT.11 - Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Numbers and Operations Add two 3-digit numbers without regrouping [Grade 2] Add 2- and 3-digit numbers with no more than one regrouping [Grade 2] Core Skill Add two 3-digit numbers with one regrouping [Grade 2] Core Skill Subtract a 3-digit number from a 3-digit number without regrouping [Grade 2] Subtract a 1- or 2-digit number from a 3-digit number with one regrouping [Grade 2] Core Skill Subtract a 3-digit number from a 3-digit number with one regrouping [Grade 2] Core Skill AL 3.NBT.12 - Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. Numbers and Operations 4/26/2012 Multiply a 1-digit whole number by a multiple of 10 to 100 [Grade 3] - 20 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 3 AL 3.NF - Number and Operations-Fractions[Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.] Develop understanding of fractions as numbers. AL 3.NF.13 - Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. Numbers and Operations Identify a unit fraction as part of a whole [Grade 2] Identify a fraction as part of a whole [Grade 2] AL 3.NF.14 - Understand a fraction as a number on the number line; represent fractions on a number line diagram. AL 3.NF.14.a - Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. Numbers and Operations Identify a fraction represented by a point on a number line [Grade Core Skill 3] Locate a fraction on a number line [Grade 3] Core Skill AL 3.NF.14.b - Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. Numbers and Operations Identify a fraction represented by a point on a number line [Grade Core Skill 3] Locate a fraction on a number line [Grade 3] Core Skill AL 3.NF.15 - Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. AL 3.NF.15.a - Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Numbers and Operations Identify equivalent fractions using models [Grade 3] Core Skill AL 3.NF.15.b - Recognize and generate simple equivalent fractions, (e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. Numbers and Operations Identify equivalent fractions using models [Grade 3] Core Skill Simplify a fraction [Grade 4] Determine a set of equivalent fractions [Grade 4] AL 3.NF.15.c - Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Example: Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. Numbers and Operations 4/26/2012 Relate a fraction equal to a whole to a pictorial model [Grade 2] - 21 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 3 AL 3.NF.15.d - Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Numbers and Operations Compare fractions using models [Grade 3] Core Skill Compare fractions with like denominators [Grade 3] Compare fractions with like numerators [Grade 3] AL 3.MD - Measurement and Data Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. AL 3.MD.16 - Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. Geometry and Measurement Tell time to the minute [Grade 3] Core Skill Calculate elapsed time within an hour, given two clocks, without regrouping [Grade 3] Calculate elapsed time within an hour, given two clocks, with regrouping [Grade 3] WP: Calculate elapsed time within an hour given two clocks [Grade 3] WP: Calculate elapsed time within an hour [Grade 3] AL 3.MD.17 - Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.[Excludes compound units such as cm³ and finding the geometric volume of a container. Excludes multiplicative comparison problems (problems involving notions of "times as much"; See Appendix A, Table 2.)] Overall Product Skills Build comprehension of math vocabulary Represent and interpret data. AL 3.MD.18 - Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. Example: For example, draw a bar graph in which each square in the bar graph might represent 5 pets. Data Analysis, Statistics, and Probability Use a pictograph to represent data (1 symbol = more than 1 object) [Grade 2] Core Skill Answer a question using information from a pictograph (1 symbol Core Skill = more than 1 object) [Grade 2] Use a bar graph with a y-axis scale by 2s to represent data [Grade Core Skill 2] 4/26/2012 - 22 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Answer a question using information from a bar graph with a yaxis scale by 2s [Grade 2] Grade 3 Core Skill Use a bar graph with a scale interval of 5 or 10 to represent data [Grade 3] Answer a question using information from a bar graph with a scale interval of 5 or 10 [Grade 3] AL 3.MD.19 - Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters. Geometry and Measurement Measure length in inches [Grade 2] Core Skill Measure length to the nearest half inch or quarter inch [Grade 3] Data Analysis, Statistics, and Probability Use a line plot to represent data [Grade 3] Geometric measurement: understand concepts of area and relate area to multiplication and to addition. AL 3.MD.20 - Recognize area as an attribute of plane figures and understand concepts of area measurement. AL 3.MD.20.a - A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area. Geometry and Measurement Relate area to the number of square units [Grade 2] AL 3.MD.20.b - A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. Geometry and Measurement Determine the area of a shape composed of rectangles given a picture on a grid [Grade 3] AL 3.MD.21 - Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Geometry and Measurement Determine the area of a shape composed of rectangles given a picture on a grid [Grade 3] WP: Determine the area of a rectangular shape given a picture on a grid [Grade 3] AL 3.MD.22 - Relate area to the operations of multiplication and addition. AL 3.MD.22.a - Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. Geometry and Measurement 4/26/2012 WP: Determine the area of a rectangular shape given a picture on a grid [Grade 3] - 23 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 3 AL 3.MD.22.b - Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. Geometry and Measurement Determine the area of a rectangle given the length and width [Grade 4] Core Skill WP: Determine the area of a rectangle [Grade 4] Core Skill AL 3.MD.22.c - Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. Numbers and Operations Apply the distributive property to the multiplication of a 2-digit number by a 1- or 2-digit number [Grade 4] Overall Product Skills Build comprehension of math vocabulary AL 3.MD.22.d - Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. Geometry and Measurement Determine the area of a shape composed of rectangles given a picture on a grid [Grade 3] Estimate the area of a shape composed of rectangles given a reference square [Grade 3] Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. AL 3.MD.23 - Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. Geometry and Measurement WP: Determine the perimeter of a rectangular shape given a grid with a reference unit [Grade 3] WP: Determine the perimeter of a shape given a model showing all side lengths [Grade 3] Determine the perimeter of a rectangle given a picture showing length and width [Grade 4] Core Skill WP: Determine the perimeter of a rectangle given a picture showing length and width [Grade 4] Core Skill WP: Determine the perimeter of a square or rectangle [Grade 4] Determine the missing side length of a rectangle given a side length and the perimeter [Grade 4] Core Skill AL 3.G - Geometry Reason with shapes and their attributes. 4/26/2012 - 24 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 3 AL 3.G.24 - Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. Geometry and Measurement Classify a triangle by its sides [Grade 4] Classify a quadrilateral [Grade 4] Relate a polygon to attributes or characteristics [Grade 4] AL 3.G.25 - Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. Example: For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. Numbers and Operations Identify a unit fraction as part of a whole [Grade 2] Identify a fraction as part of a whole [Grade 2] Determine a pictorial model of a fraction of a whole [Grade 3] 4/26/2012 Core Skill - 25 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 Grade 4 AL 4.OA - Operations and Algebraic Thinking Use the four operations with whole numbers to solve problems. AL 4.OA.1 - Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. Overall Product Skills Demonstrate and develop understanding of mathematical concepts and ideas Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. AL 4.OA.2 - Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. Overall Product Skills Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. AL 4.OA.3 - Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Numbers and Operations WP: Estimate a sum or difference of two 3- or 4-digit whole numbers using any method [Grade 3] Core Skill WP: Solve a 2-step problem involving addition and/or subtraction of multi-digit whole numbers [Grade 4] WP: Estimate a product of two whole numbers using any method [Grade 4] Core Skill WP: Solve a 2-step whole number problem using more than 1 operation [Grade 4] WP: Solve a 2-step problem involving whole numbers [Grade 5] Core Skill WP: Estimate a quotient using any method [Grade 5] Core Skill WP: Solve a multi-step problem involving whole numbers [Grade Core Skill 6] Gain familiarity with factors and multiples. AL 4.OA.4 - Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1100 is prime or composite. Numbers and Operations 4/26/2012 Determine if a number to 50 is prime or composite [Grade 5] - 26 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 Determine a complete list of whole number factor pairs for a number to 50 [Grade 5] Determine all the factors of a whole number to 50 [Grade 5] Determine the prime factorization of a number to 50 [Grade 5] Determine the common factors for two whole numbers to 50 [Grade 5] Determine the multiple(s) of a number [Grade 5] Determine common multiples for two whole numbers [Grade 5] Generate and analyze patterns. AL 4.OA.5 - Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Example: For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. Numbers and Operations Algebra Complete a skip pattern starting from a multiple of 2, 5, or 10 [Grade 2] Core Skill Complete a skip pattern of 2, 5, or 10 starting from any number [Grade 2] Core Skill Count on by 100s from any number [Grade 2] Core Skill Determine an addition or subtraction number pattern given a rule [Grade 2] Extend a number pattern [Grade 3] Determine a rule for a table of related number pairs [Grade 3] WP: Find the missing number in a table of paired values [Grade 3] Generate a table of paired numbers based on a rule [Grade 4] Core Skill AL 4.NBT - Number and Operations in Base Ten[Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.] Generalize place value understanding for multi-digit whole numbers. AL 4.NBT.6 - Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Example: For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division. Overall Product Skills Demonstrate and develop understanding of mathematical concepts and ideas AL 4.NBT.7 - Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Numbers and Operations 4/26/2012 Read a 4- or 5-digit whole number [Grade 3] Core Skill - 27 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 Determine the word form of a 4- or 5-digit whole number [Grade 3] Core Skill Determine the 4-digit whole number represented in thousands, hundreds, tens, and ones [Grade 3] Core Skill Represent a 4- or 5-digit whole number in expanded form [Grade 3] Core Skill Determine the 4- or 5-digit whole number represented in expanded form [Grade 3] Core Skill Compare 4- or 5-digit whole numbers using the symbols <, >, and = [Grade 3] Read a 6-digit whole number [Grade 4] Determine the word form of a 6-digit whole number [Grade 4] Convert between proper expanded form and improper expanded form up to a 5-digit whole number [Grade 4] Convert between standard form and improper expanded form up to a 5-digit whole number [Grade 4] AL 4.NBT.8 - Use place value understanding to round multi-digit whole numbers to any place. Numbers and Operations Round a 4- to 6-digit whole number to a specified place [Grade 4] Core Skill Use place value understanding and properties of operations to perform multi-digit arithmetic. AL 4.NBT.9 - Fluently add and subtract multi-digit whole numbers using the standard algorithm. Numbers and Operations Add 3- and 4-digit whole numbers with regrouping [Grade 3] Core Skill Subtract 3- and 4-digit whole numbers with regrouping [Grade 3] Core Skill Add up to 4-digit whole numbers in expanded form [Grade 4] Add a 5-digit or greater whole number and a 3-digit or greater whole number [Grade 4] Add three multi-digit whole numbers [Grade 4] Subtract a 3-digit or greater whole number from a 5-digit or greater whole number [Grade 4] WP: Add a 5-digit or greater whole number and a 3-digit or greater whole number [Grade 4] WP: Add three multi-digit whole numbers [Grade 4] WP: Subtract a 3-digit or greater whole number from a 5-digit or greater whole number [Grade 4] AL 4.NBT.10 - Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Numbers and Operations Multiply a 2-digit whole number by a 1-digit number [Grade 3] Apply the distributive property to the multiplication of a 2-digit number by a 1- or 2-digit number [Grade 4] 4/26/2012 - 28 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 Apply the distributive property to multiply a multi-digit number by a 1-digit number [Grade 4] Core Skill Multiply a 3- or 4-digit whole number by a 1-digit whole number [Grade 4] Core Skill Multiply a 2-digit whole number by a 2-digit whole number [Grade 4] Core Skill WP: Multiply a multi-digit whole number by a 1-digit whole number [Grade 4] Core Skill WP: Multiply a 2-digit whole number by a 2-digit whole number [Grade 4] Core Skill WP: Multiply a 3-digit whole number by a 2-digit whole number [Grade 4] AL 4.NBT.11 - Find whole-number quotients and remainders with up to four-digit dividends and onedigit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Numbers and Operations Divide a 2-digit whole number by a 1-digit whole number with no remainder in the quotient [Grade 4] Divide a 3-digit whole number by a 1-digit whole number with no remainder in the quotient [Grade 4] Divide a 2-digit whole number by a 1-digit whole number with a remainder in the quotient [Grade 4] Divide a 3-digit whole number by a 1-digit whole number with a remainder in the quotient [Grade 4] Core Skill WP: Divide a 3-digit whole number by a 1-digit whole number with no remainder in the quotient [Grade 4] Core Skill WP: Divide a 2-digit whole number by a 1-digit whole number with a remainder in the quotient [Grade 4] WP: Divide a 3-digit whole number by a 1-digit whole number with a remainder in the quotient [Grade 4] Core Skill AL 4.NF - Number and Operations-Fractions[Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.] Extend understanding of fraction equivalence and ordering. AL 4.NF.12 - Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Numbers and Operations Identify equivalent fractions using models [Grade 3] Core Skill Determine a set of equivalent fractions [Grade 4] Determine equivalent fractions not in simplest form [Grade 5] 4/26/2012 Core Skill - 29 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 AL 4.NF.13 - Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Numbers and Operations Compare fractions using models [Grade 3] Core Skill Compare fractions with like denominators [Grade 3] Compare fractions with like numerators [Grade 3] Compare fractions on a number line [Grade 4] Determine equivalent fractions not in simplest form [Grade 5] Core Skill Compare fractions with unlike denominators [Grade 5] Core Skill Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. AL 4.NF.14 - Understand a fraction a/b with a > 1 as a sum of fractions 1/b. AL 4.NF.14.a - Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Numbers and Operations Add fractions with like denominators no greater than 10 using models [Grade 4] Add fractions with like denominators no greater than 10 [Grade 4] Core Skill Add fractions with like denominators no greater than 10 and simplify the sum [Grade 4] Subtract fractions with like denominators no greater than 10 using models [Grade 4] Subtract fractions with like denominators no greater than 10 [Grade 4] Core Skill Subtract fractions with like denominators no greater than 10 and simplify the difference [Grade 4] AL 4.NF.14.b - Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Example: Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Numbers and Operations Identify a mixed number represented by a model [Grade 4] Identify an improper fraction represented by a model of a mixed number [Grade 4] AL 4.NF.14.c - Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. Numbers and Operations 4/26/2012 Add mixed numbers with like denominators and simplify the sum [Grade 5] - 30 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 Subtract mixed numbers with like denominators and simplify the difference [Grade 5] WP: Add or subtract mixed numbers with like denominators and simplify the sum or difference [Grade 5] Core Skill AL 4.NF.14.d - Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Numbers and Operations Add fractions with like denominators no greater than 10 using models [Grade 4] Add fractions with like denominators no greater than 10 and simplify the sum [Grade 4] WP: Add fractions with like denominators no greater than 10 and simplify the sum [Grade 4] Subtract fractions with like denominators no greater than 10 using models [Grade 4] Subtract fractions with like denominators no greater than 10 and simplify the difference [Grade 4] WP: Subtract fractions with like denominators no greater than 10 and simplify the difference [Grade 4] Add fractions with like denominators greater than 10 and simplify the sum [Grade 5] Subtract fractions with like denominators greater than 10 and simplify the difference [Grade 5] WP: Add or subtract fractions with like denominators and simplify the sum or difference [Grade 5] Core Skill WP: Add or subtract mixed numbers with like denominators and simplify the sum or difference [Grade 5] Core Skill AL 4.NF.15 - Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. AL 4.NF.15.a - Understand a fraction a/b as a multiple of 1/b. Example: For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). Numbers and Operations Multiply a whole number by a unit fraction using a model [Grade 5] Multiply a whole number by a unit fraction [Grade 5] AL 4.NF.15.b - Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. Example: For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.). Numbers and Operations Multiply a whole number by a unit fraction [Grade 5] Multiply a proper fraction by a whole number using a model [Grade 5] 4/26/2012 - 31 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 Multiply a proper fraction by a whole number [Grade 5] AL 4.NF.15.c - Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. Example: For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? Numbers and Operations WP: Multiply or divide a whole number by a unit fraction [Grade 5] Understand decimal notation for fractions, and compare decimal fractions. AL 4.NF.16 - Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. Example: For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.[Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.] Overall Product Skills Use strategies to derive solutions for problems AL 4.NF.17 - Use decimal notation for fractions with denominators 10 or 100. Example: For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. Numbers and Operations Determine the decimal number from a pictorial model of tenths or Core Skill hundredths [Grade 4] Identify a pictorial model of tenths or hundredths of a decimal number [Grade 4] Core Skill Identify a decimal number to tenths represented by a point on a number line [Grade 4] Core Skill Locate a decimal number to tenths on a number line [Grade 4] Core Skill Determine the decimal number equivalent to a fraction with a denominator of 10 or 100 [Grade 4] Determine a fraction equivalent to a decimal, using a denominator of 10 or 100 [Grade 4] Determine the decimal number equivalent to a fraction model [Grade 4] Core Skill Determine the fraction equivalent to a decimal number model [Grade 4] Core Skill AL 4.NF.18 - Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Numbers and Operations 4/26/2012 Read a decimal number through the hundredths place [Grade 4] Core Skill Identify a decimal number to tenths represented by a point on a number line [Grade 4] Core Skill Compare decimal numbers through the hundredths place [Grade 4] Core Skill - 32 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Order decimal numbers through the hundredths place [Grade 4] Grade 4 Core Skill AL 4.MD - Measurement and Data Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. AL 4.MD.19 - Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. Example: For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),... Geometry and Measurement Convert between inches, feet, and yards [Grade 3] Convert between centimeters and meters [Grade 3] Convert hours to minutes or minutes to seconds [Grade 3] Convert between customary units of capacity using whole numbers [Grade 4] Convert between customary units of weight using whole numbers [Grade 4] Convert between metric units of capacity using whole numbers [Grade 4] Convert between metric units of mass using whole numbers [Grade 4] Convert between millimeters and centimeters or meters using whole numbers [Grade 4] AL 4.MD.20 - Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Numbers and Operations Add two money amounts less than $1 in decimal notation [Grade 3] Subtract two money amounts less than $1 in decimal notation [Grade 3] Add or subtract cent amounts to or from whole dollar amounts [Grade 3] Determine money amounts that total $10 [Grade 3] Core Skill WP: Add or subtract money amounts less than $1 [Grade 3] WP: Add or subtract money amounts greater than $1 [Grade 4] WP: Solve a money problem involving 2 steps [Grade 4] WP: Multiply a money amount by a 1-digit number [Grade 4] WP: Divide a money amount by a 1-digit number [Grade 4] 4/26/2012 - 33 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 WP: Multiply a money amount by a 2-digit whole number [Grade 5] Geometry and Measurement Calculate elapsed time within an hour, given two clocks, with regrouping [Grade 3] WP: Calculate elapsed time within an hour given two clocks [Grade 3] WP: Add or subtract compound units of length requiring customary unit conversion using whole numbers [Grade 4] Calculate elapsed time exceeding an hour with regrouping [Grade Core Skill 4] WP: Calculate elapsed time exceeding an hour without regrouping hours [Grade 4] WP: Calculate elapsed time exceeding an hour with regrouping hours [Grade 4] Core Skill WP: Determine the end time given the start time and the elapsed time exceeding an hour [Grade 4] WP: Determine the start time given the end time and the elapsed time exceeding an hour [Grade 4] WP: Add time intervals involving hours and minutes [Grade 4] Solve an elapsed-time problem using a calendar [Grade 4] WP: Compare customary units of length, weight, or capacity using fractional amounts [Grade 5] WP: Compare metric units of length, mass, or capacity using decimal amounts [Grade 5] WP: Calculate elapsed time using a.m. and p.m. [Grade 5] WP: Use a calendar to solve a problem [Grade 5] AL 4.MD.21 - Apply the area and perimeter formulas for rectangles in real world and mathematical problems. Example: For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. Geometry and Measurement Determine the area of a shape composed of rectangles given a picture on a grid [Grade 3] WP: Determine the area of a rectangular shape given a picture on a grid [Grade 3] Determine the perimeter of a rectangle given a picture showing length and width [Grade 4] Core Skill WP: Determine the perimeter of a rectangle given a picture showing length and width [Grade 4] Core Skill WP: Determine the perimeter of a square or rectangle [Grade 4] Determine the missing side length of a rectangle given a side length and the perimeter [Grade 4] 4/26/2012 Core Skill - 34 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 Determine the area of a rectangle given a picture showing the length and width [Grade 4] Determine the area of a rectangle given the length and width [Grade 4] Core Skill WP: Determine the area of a rectangle [Grade 4] Core Skill Determine the missing side length of a rectangle given a side length and the area [Grade 4] Core Skill Represent and interpret data. AL 4.MD.22 - Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. Example: For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection. Data Analysis, Statistics, and Probability Read a line plot [Grade 3] Use a line plot to represent data [Grade 3] Answer a question using information from a line plot [Grade 3] Geometric measurement: understand concepts of angle and measure angles. AL 4.MD.23 - Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: AL 4.MD.23.a - An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles. Geometry and Measurement Associate degrees with a turn on a circle [Grade 4] AL 4.MD.23.b - An angle that turns through n one-degree angles is said to have an angle measure of n degrees. Overall Product Skills Demonstrate and develop understanding of mathematical concepts and ideas AL 4.MD.24 - Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. Geometry and Measurement Measure an angle to the nearest 5 degrees [Grade 5] Measure an angle, between two rays or in a shape, to the nearest degree [Grade 6] AL 4.MD.25 - Recognize angle measure as additive. When an angle is decomposed into nonoverlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. Geometry and Measurement 4/26/2012 Determine the measure of a missing angle using straight and right angle relationships [Grade 6] - 35 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 4 AL 4.G - Geometry Draw and identify lines and angles, and classify shapes by properties of their lines and angles. AL 4.G.26 - Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. Geometry and Measurement Identify parallel, perpendicular, and intersecting lines [Grade 3] Classify an angle as equal to 90°, less than 90°, or greater than 90° [Grade 3] Classify an angle in a shape as equal to 90°, less than 90°, or greater than 90° [Grade 3] Classify an angle given a picture [Grade 4] Core Skill Classify an angle given its measure [Grade 4] Core Skill Identify a shape that has parallel or perpendicular sides [Grade 4] AL 4.G.27 - Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. Geometry and Measurement Classify a quadrilateral [Grade 4] Classify a triangle by its sides and angles [Grade 5] Know the properties of a triangle or a quadrilateral [Grade 6] AL 4.G.28 - Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. Geometry and Measurement Identify symmetry in a 2-dimensional shape [Grade 2] Determine lines of symmetry [Grade 4] 4/26/2012 Core Skill - 36 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 5 Grade 5 AL 5.OA - Operations and Algebraic Thinking Write and interpret numerical expressions. AL 5.OA.1 - Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Numbers and Operations Evaluate a numerical expression involving three operations, with no parentheses, using order of operations [Grade 5] Evaluate a numerical expression involving three operations, with parentheses, using order of operations [Grade 5] Evaluate a numerical expression of four or more operations, with parentheses, using order of operations [Grade 6] Algebra Core Skill Evaluate a numeric expression involving two operations [Grade 4] AL 5.OA.2 - Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Example: For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. Numbers and Operations Apply the distributive property to the multiplication of a 2-digit number by a 1- or 2-digit number [Grade 4] Apply the distributive property to multiply a multi-digit number by a 1-digit number [Grade 4] Core Skill Evaluate a numerical expression involving three operations, with no parentheses, using order of operations [Grade 5] Evaluate a numerical expression involving three operations, with parentheses, using order of operations [Grade 5] Analyze patterns and relationships. AL 5.OA.3 - Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. Example: For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. Algebra Generate a table of paired numbers based on a rule [Grade 4] Core Skill Extend a number pattern in a table of related pairs [Grade 4] Use a first quadrant graph to represent the values from a table generated in context [Grade 5] Core Skill Use a first quadrant graph to represent the values in an inputoutput table [Grade 6] AL 5.NBT - Number and Operations in Base Ten 4/26/2012 - 37 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 5 Understand the place value system. AL 5.NBT.4 - Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Numbers and Operations Determine the value of a digit in a 6-digit whole number [Grade 4] Determine the value of a digit in a decimal number to thousandths [Grade 5] Compare decimal numbers of differing places to thousandths [Grade 5] Core Skill Represent a decimal number in expanded form using powers of ten [Grade 6] Core Skill Determine the decimal number represented in expanded form using powers of ten [Grade 6] Core Skill AL 5.NBT.5 - Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Numbers and Operations Divide a multi-digit whole number by 10 or 100 with no remainder [Grade 4] Multiply a decimal number through thousandths by 10, 100, or 1,000 [Grade 5] Core Skill Determine the power of ten that relates a decimal number and a whole number [Grade 6] Determine the power of ten that relates two decimal numbers [Grade 6] Divide a decimal number by 10, 100, or 1,000 [Grade 6] Core Skill Relate division by a whole number power of ten to multiplication by the related decimal fraction power of ten [Grade 6] AL 5.NBT.6 - Read, write, and compare decimals to thousandths. AL 5.NBT.6.a - Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). Numbers and Operations Read a decimal number through the hundredths place [Grade 4] Core Skill Determine the word form of a decimal number through the hundredths place [Grade 4] Determine the value of a digit in a decimal number to thousandths [Grade 5] Determine a decimal number represented in expanded form [Grade 5] Represent a decimal number in expanded form [Grade 5] Represent a decimal number in expanded form using powers of ten [Grade 6] 4/26/2012 Core Skill - 38 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Determine the decimal number represented in expanded form using powers of ten [Grade 6] Grade 5 Core Skill AL 5.NBT.6.b - Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Numbers and Operations Compare decimal numbers through the hundredths place [Grade 4] Core Skill Order decimal numbers through the hundredths place [Grade 4] Core Skill Compare decimal numbers to thousandths represented in expanded form [Grade 5] Compare decimal numbers of differing places to thousandths [Grade 5] Core Skill Order decimal numbers of differing places to thousandths in ascending or descending order [Grade 5] Core Skill AL 5.NBT.7 - Use place value understanding to round decimals to any place. Numbers and Operations Round a decimal number to a specified place through hundredths [Grade 4] Core Skill Round a decimal number to a specified decimal place to thousandths [Grade 5] Perform operations with multi-digit whole numbers and with decimals to hundredths. AL 5.NBT.8 - Fluently multiply multi-digit whole numbers using the standard algorithm. Numbers and Operations Multiply a 1- or 2-digit whole number by a multiple of 10, 100, or Core Skill 1,000 [Grade 4] Multiply a 3- or 4-digit whole number by a 1-digit whole number [Grade 4] Core Skill Multiply a 2-digit whole number by a 2-digit whole number [Grade 4] Core Skill Multiply a 3-digit whole number by a 2-digit whole number [Grade 4] WP: Multiply a multi-digit whole number by a 1-digit whole number [Grade 4] Core Skill WP: Multiply a 2-digit whole number by a 2-digit whole number [Grade 4] Core Skill WP: Multiply a 3-digit whole number by a 2-digit whole number [Grade 4] Multiply a 3- or 4-digit whole number by a 3-digit whole number [Grade 5] WP: Multiply a 3- or 4-digit whole number by a 3-digit whole number [Grade 5] Find the product of three identical factors [Grade 6] 4/26/2012 - 39 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 5 AL 5.NBT.9 - Find whole-number quotients of whole numbers with up to four-digit dividends and twodigit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Numbers and Operations WP: Divide a 3-digit whole number by a 1-digit whole number with no remainder in the quotient [Grade 4] Core Skill Divide a multi-digit whole number by a 2-digit whole number, with no remainder and no zeros in the quotient [Grade 5] Divide a multi-digit whole number by a 2-digit whole number, with a remainder and no zeros in the quotient [Grade 5] Divide a multi-digit whole number by a 2-digit whole number, with no remainder and at least one zero in the quotient [Grade 5] Divide a multi-digit whole number by a 2-digit whole number, with a remainder and at least one zero in the quotient [Grade 5] Core Skill Divide a multi-digit whole number by a 2-digit whole number and Core Skill express the quotient as a mixed number [Grade 5] AL 5.NBT.10 - Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Numbers and Operations Add two decimal numbers through hundredths [Grade 4] Subtract two decimal numbers through hundredths [Grade 4] Subtract a decimal number from a whole number or a whole number from a decimal number [Grade 5] Multiply a money amount by a 2- or 3-digit whole number [Grade 5] Divide a decimal number by 10, 100, or 1,000 [Grade 6] Core Skill Divide a 1- to 3-digit whole number by a decimal number to tenths where the quotient is a whole number [Grade 6] Core Skill AL 5.NF - Number and Operations-Fractions Use equivalent fractions as a strategy to add and subtract fractions. AL 5.NF.11 - Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. Example: For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.). Numbers and Operations Add fractions with unlike denominators using a model and do not simplify the sum [Grade 5] Add fractions with unlike denominators and do not simplify the sum [Grade 5] Add fractions with unlike denominators that have factors in common and simplify the sum [Grade 5] 4/26/2012 Core Skill - 40 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Add fractions with unlike denominators that have no factors in common [Grade 5] Grade 5 Core Skill Subtract fractions with unlike denominators using a model and do not simplify the difference [Grade 5] Subtract fractions with unlike denominators and do not simplify the difference [Grade 5] Subtract fractions with unlike denominators that have factors in common and simplify the difference [Grade 5] Core Skill Subtract fractions with unlike denominators that have no factors in common [Grade 5] Core Skill Add mixed numbers with unlike denominators and simplify the sum [Grade 5] Subtract mixed numbers with unlike denominators and simplify the difference [Grade 5] WP: Add or subtract fractions with unlike denominators and simplify the sum or difference [Grade 6] Add mixed numbers with unlike denominators or a mixed number and a fraction with unlike denominators and simplify the sum [Grade 6] Subtract a mixed number from a whole number [Grade 6] Subtract mixed numbers with unlike denominators or a mixed number and a fraction and simplify the difference [Grade 6] Add and subtract three unlike-denominator fractions, mixed numbers, or fractions and mixed numbers, and simplify the answer [Grade 6] WP: Add or subtract mixed numbers with unlike denominators or a mixed number and a fraction with unlike denominators and simplify the sum or difference [Grade 6] AL 5.NF.12 - Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. Example: For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. Numbers and Operations WP: Add fractions with like denominators no greater than 10 and simplify the sum [Grade 4] WP: Subtract fractions with like denominators no greater than 10 and simplify the difference [Grade 4] WP: Add or subtract fractions with like denominators and simplify the sum or difference [Grade 5] Core Skill WP: Add or subtract fractions with unlike denominators that have Core Skill no factors in common [Grade 5] WP: Add or subtract mixed numbers with like denominators and simplify the sum or difference [Grade 5] 4/26/2012 Core Skill - 41 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 5 WP: Add or subtract mixed numbers with unlike denominators that have no factors in common [Grade 5] Round a fraction to a benchmark number of 0, 1/2, or 1 [Grade 5] Estimate a fraction sum using benchmark numbers 0, 1/2, and 1 [Grade 5] Estimate a fraction difference using benchmark numbers 0, 1/2, and 1 [Grade 5] WP: Estimate a fraction sum or difference using benchmark numbers 0, 1/2, and 1 [Grade 5] WP: Add or subtract fractions with unlike denominators and simplify the sum or difference [Grade 6] WP: Add or subtract mixed numbers with unlike denominators or a mixed number and a fraction with unlike denominators and simplify the sum or difference [Grade 6] Apply and extend previous understandings of multiplication and division to multiply and divide fractions. AL 5.NF.13 - Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Example: For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? Numbers and Operations WP: Use a mixed number to represent an amount in a sharing situation [Grade 4] Relate a sharing situation to a fraction [Grade 5] AL 5.NF.14 - Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. Numbers and Operations Multiply a whole number by a unit fraction using a model [Grade 5] Multiply a whole number by a unit fraction [Grade 5] Multiply a proper fraction by a whole number using a model [Grade 5] Multiply a proper fraction by a whole number [Grade 5] 4/26/2012 Multiply a fraction by a fraction [Grade 6] Core Skill Multiply a mixed number by a whole number [Grade 6] Core Skill Multiply a mixed number by a fraction [Grade 6] Core Skill Multiply a mixed number by a mixed number [Grade 6] Core Skill WP: Multiply or divide a fraction by a fraction [Grade 6] Core Skill - 42 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH WP: Multiply or divide two mixed numbers or a mixed number and a fraction [Grade 6] Grade 5 Core Skill AL 5.NF.14.a - Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. Example: For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.). Numbers and Operations Multiply a whole number by a unit fraction using a model [Grade 5] Multiply a whole number by a unit fraction [Grade 5] Multiply a proper fraction by a whole number using a model [Grade 5] Multiply a proper fraction by a whole number [Grade 5] AL 5.NF.14.b - Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. Geometry and Measurement Determine the area of a shape composed of rectangles given a picture on a grid [Grade 3] WP: Determine the area of a rectangular shape given a picture on a grid [Grade 3] Determine the area of a polygon on a grid [Grade 4] Core Skill Estimate the area of an irregular polygon on a grid [Grade 4] AL 5.NF.15 - Interpret multiplication as scaling (resizing), by: AL 5.NF.15.a - Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. Numbers and Operations Determine a complete list of whole number factor pairs for a number to 50 [Grade 5] Determine all the factors of a whole number to 50 [Grade 5] AL 5.NF.15.b - Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1. Numbers and Operations Multiply a whole number by a unit fraction using a model [Grade 5] Multiply a whole number by a unit fraction [Grade 5] Multiply a proper fraction by a whole number using a model [Grade 5] Multiply a proper fraction by a whole number [Grade 5] 4/26/2012 - 43 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Multiply a mixed number by a fraction [Grade 6] Grade 5 Core Skill AL 5.NF.16 - Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Numbers and Operations WP: Multiply or divide a whole number by a unit fraction [Grade 5] WP: Multiply or divide a fraction by a fraction [Grade 6] Core Skill WP: Multiply or divide two mixed numbers or a mixed number and a fraction [Grade 6] Core Skill AL 5.NF.17 - Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. AL 5.NF.17.a - Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. Example: For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. Numbers and Operations Divide a unit fraction by a whole number [Grade 5] AL 5.NF.17.b - Interpret division of a whole number by a unit fraction, and compute such quotients. Example: For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. Numbers and Operations Divide a whole number by a unit fraction using a model [Grade 5] Divide a whole number by a unit fraction [Grade 5] AL 5.NF.17.c - Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. Example: For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? Numbers and Operations WP: Multiply or divide a whole number by a unit fraction [Grade 5] AL 5.MD - Measurement and Data Convert like measurement units within a given measurement system. AL 5.MD.18 - Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. Geometry and Measurement Convert between customary units of length using fractional amounts [Grade 5] Convert between customary units of capacity using fractional amounts [Grade 5] Convert between customary units of weight using fractional amounts [Grade 5] 4/26/2012 - 44 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 5 Convert between metric units of capacity using decimal amounts [Grade 5] Convert between metric units of mass using decimal amounts [Grade 5] Convert between millimeters or centimeters and meters, or meters and kilometers using decimal amounts [Grade 5] WP: Compare customary units of length, weight, or capacity using fractional amounts [Grade 5] WP: Compare metric units of length, mass, or capacity using decimal amounts [Grade 5] WP: Add or subtract customary measures of weight requiring unit conversion [Grade 6] WP: Add or subtract metric measures of mass requiring unit conversion [Grade 6] WP: Multiply or divide customary measures of weight requiring unit conversion [Grade 6] WP: Multiply or divide metric measures of mass requiring unit conversion [Grade 6] Represent and interpret data. AL 5.MD.19 - Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. Example: For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. AL 5.MD.20 - Recognize volume as an attribute of solid figures and understand concepts of volume measurement. AL 5.MD.20.a - A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume. Geometry and Measurement Determine the volume of a rectangular prism given a diagram showing unit cubes [Grade 4] Overall Product Skills Build comprehension of math vocabulary AL 5.MD.20.b - A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. Geometry and Measurement Determine the volume of an object composed of rectangular prisms by counting units [Grade 5] Overall Product Skills Build comprehension of math vocabulary 4/26/2012 - 45 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 5 AL 5.MD.21 - Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. Geometry and Measurement Determine the volume of a rectangular prism given a diagram showing unit cubes [Grade 4] Determine the volume of an object composed of rectangular prisms by counting units [Grade 5] AL 5.MD.22 - Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. AL 5.MD.22.a - Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. Geometry and Measurement Determine the volume of a rectangular prism given a diagram showing unit cubes [Grade 4] AL 5.MD.22.b - Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. Geometry and Measurement Determine the volume of a rectangular prism given a diagram [Grade 5] Core Skill WP: Determine the volume of a rectangular prism given a diagram [Grade 5] Determine the volume of a rectangular prism [Grade 5] WP: Determine the volume of a rectangular prism [Grade 5] Core Skill AL 5.MD.22.c - Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. Geometry and Measurement Determine the volume of an object composed of rectangular prisms by counting units [Grade 5] Determine the volume of a prism with a right triangle base [Grade 6] AL 5.G - Geometry Graph points on the coordinate plane to solve real-world and mathematical problems. AL 5.G.23 - Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). Algebra 4/26/2012 Use a graph to determine the entries in an input-output table [Grade 6] - 46 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry and Measurement Grade 5 Determine a path or location on a grid using compass directions [Grade 4] Identify the location of an ordered pair in the first quadrant using compass directions [Grade 4] Locate a point by following compass directions on a grid [Grade 5] Use compass directions to describe a path to a point on a grid [Grade 5] AL 5.G.24 - Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. Algebra Use a first quadrant graph to represent the values from a table generated in context [Grade 5] Geometry and Measurement Determine the location of an ordered pair in the first quadrant [Grade 5] Core Skill Determine the ordered pair of a point in the first quadrant [Grade 5] Classify two-dimensional figures into categories based on their properties. AL 5.G.25 - Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. Example: For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. AL 5.G.26 - Classify two-dimensional figures in a hierarchy based on properties. Geometry and Measurement Identify a shape that has parallel or perpendicular sides [Grade 4] Classify a triangle by its sides [Grade 4] Relate a polygon to attributes or characteristics [Grade 4] Classify a triangle by its sides and angles [Grade 5] Know the properties of a triangle or a quadrilateral [Grade 6] 4/26/2012 - 47 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 Grade 6 AL 6.RP - Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. AL 6.RP.1 - Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Example: For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes.". Numbers and Operations WP: Determine a ratio using whole numbers less than 50 [Grade 6] WP: Determine the ratio of two whole numbers, at least one of which is larger than 50 [Grade 7] AL 6.RP.2 - Understand the concept of a unit rate a/b associated with a ratio a:b with b is not equal to 0, and use rate language in the context of a ratio relationship. Example: For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.".[Expectations for unit rates in this grade are limited to non-complex fractions.] Numbers and Operations WP: Determine a ratio using whole numbers less than 50 [Grade 6] WP: Determine a unit rate with a whole number value [Grade 6] Core Skill WP: Determine a unit rate [Grade 7] Core Skill AL 6.RP.3 - Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. AL 6.RP.3.a - Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. Numbers and Operations Determine if ratios, using whole numbers less than 50, are equivalent [Grade 6] Core Skill Determine ratios equivalent to a given ratio of two whole numbers, at least one of which is larger than 50 [Grade 7] Algebra WP: Extend a pattern to solve a problem [Grade 5] AL 6.RP.3.b - Solve unit rate problems including those involving unit pricing and constant speed. Example: For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? Numbers and Operations WP: Determine a part given a ratio and the whole where the whole is less than 50 [Grade 6] Core Skill WP: Determine a part given a ratio and another part where the whole is less than 50 [Grade 6] 4/26/2012 WP: Determine the whole given a ratio and a part where the whole is less than 50 [Grade 6] Core Skill WP: Determine a unit rate with a whole number value [Grade 6] Core Skill - 48 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra Grade 6 WP: Use a unit rate, with a whole number or whole cent value, to solve a problem [Grade 6] Core Skill WP: Determine a part, given part to whole ratio and the whole, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine a part, given part to part ratio and the whole, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine a part, given part to whole ratio and a part, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine a part, given part to part ratio and a part, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine the whole, given part to whole ratio and a part, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine the whole, given part to part ratio and a part, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine a unit rate [Grade 7] Core Skill WP: Use a unit rate to solve a problem [Grade 7] Core Skill WP: Solve a proportion [Grade 7] Core Skill AL 6.RP.3.c - Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. Numbers and Operations Calculate a percent of a whole number where the answer is a whole number [Grade 6] WP: Calculate the percent of a whole number where the answer is a whole number [Grade 6] Determine a percent of a whole number using less than 100% [Grade 7] Determine a percent of a whole number using more than 100% [Grade 7] Determine a whole number given a part and a percentage less than 100% [Grade 7] WP: Determine a percent of a whole number using less than 100% Core Skill [Grade 7] WP: Determine a whole number given a part and a percentage [Grade 7] Core Skill AL 6.RP.3.d - Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Geometry and Measurement Convert between customary units of length using fractional amounts [Grade 5] Convert between customary units of capacity using fractional amounts [Grade 5] Convert between customary units of weight using fractional amounts [Grade 5] 4/26/2012 - 49 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 Convert between metric units of capacity using decimal amounts [Grade 5] Convert between metric units of mass using decimal amounts [Grade 5] Convert between millimeters or centimeters and meters, or meters and kilometers using decimal amounts [Grade 5] Convert between Fahrenheit and Celsius temperature given a formula [Grade 6] Determine approximate conversions between metric and customary units of length [Grade 7] Determine approximate conversions between metric and customary units of capacity [Grade 7] Determine approximate conversions between metric and customary units of weight/mass [Grade 7] AL 6.NS - The Number System Apply and extend previous understandings of multiplication and division to divide fractions by fractions. AL 6.NS.4 - Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. Example: For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? Numbers and Operations Divide a whole number by a unit fraction using a model [Grade 5] Divide a whole number by a unit fraction [Grade 5] Divide a unit fraction by a whole number [Grade 5] Divide a whole number by a fraction, with a whole number quotient using a model [Grade 5] Divide a whole number by a fraction, with a whole number quotient [Grade 5] Divide a fraction by a whole number resulting in a fractional quotient [Grade 6] Core Skill Divide a fraction by a fraction [Grade 6] Core Skill Divide a whole number by a fraction resulting in a fractional quotient [Grade 6] Core Skill Divide a mixed number by a fraction [Grade 6] Divide a mixed number by a mixed number [Grade 6] 4/26/2012 WP: Multiply or divide a fraction by a fraction [Grade 6] Core Skill WP: Multiply or divide two mixed numbers or a mixed number and a fraction [Grade 6] Core Skill - 50 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 Compute fluently with multi-digit numbers and find common factors and multiples. AL 6.NS.5 - Fluently divide multi-digit numbers using the standard algorithm. Numbers and Operations Divide a multi-digit whole number by multiples of 100 or 1,000 [Grade 5] Divide a multi-digit whole number by a 2-digit whole number, with no remainder and no zeros in the quotient [Grade 5] Divide a multi-digit whole number by a 2-digit whole number, with a remainder and no zeros in the quotient [Grade 5] Divide a multi-digit whole number by a 2-digit whole number, with no remainder and at least one zero in the quotient [Grade 5] Divide a multi-digit whole number by a 2-digit whole number, with a remainder and at least one zero in the quotient [Grade 5] Core Skill Divide a multi-digit whole number by a 2-digit whole number and Core Skill express the quotient as a mixed number [Grade 5] Divide a whole number by a 1-digit whole number resulting in a decimal quotient through thousandths [Grade 6] Core Skill Divide a whole number by a 2-digit whole number resulting in a decimal quotient through thousandths [Grade 6] Core Skill WP: Divide a whole number by a 1- or 2-digit whole number resulting in a decimal quotient [Grade 6] Core Skill AL 6.NS.6 - Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Numbers and Operations Multiply a decimal number through thousandths by a whole number [Grade 6] Core Skill WP: Multiply a decimal number through thousandths by a whole number [Grade 6] Core Skill WP: Multiply a money expression by a decimal number [Grade 6] Multiply decimal numbers to thousandths using basic facts [Grade 6] Multiply decimal numbers less than one in hundredths or thousandths [Grade 6] Core Skill Multiply a decimal number greater than one by a decimal number to thousandths that has only 1 nonzero digit [Grade 6] Multiply decimal numbers greater than one where the product has Core Skill 2 or 3 decimal places [Grade 6] 4/26/2012 WP: Multiply two decimal numbers to thousandths [Grade 6] Core Skill Divide a decimal number through thousandths by a 1- or 2-digit whole number where the quotient has 2-5 decimal places [Grade 6] Core Skill WP: Divide a decimal number through thousandths by a 1- or 2digit whole number [Grade 6] Core Skill - 51 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 Divide a whole number or a decimal number by 0.1, 0.01, or 0.001 [Grade 6] Locate the decimal point in the quotient of a whole number, or a decimal number, divided by a decimal number [Grade 6] Divide a 1- to 3-digit whole number by a decimal number to tenths where the quotient is a whole number [Grade 6] Core Skill Divide a 1- to 3-digit whole number by a decimal number to tenths where the quotient is a decimal number to thousandths [Grade 6] Core Skill Divide a 2- or 3-digit whole number by a decimal number to hundredths or thousandths, rounded quotient if needed [Grade 6] Core Skill Divide a decimal number by a decimal number through thousandths, rounded quotient if needed [Grade 6] Core Skill WP: Divide a whole number by a decimal number through thousandths, rounded quotient if needed [Grade 6] Core Skill WP: Divide a decimal through thousandths by a decimal through thousandths, rounded quotient if needed [Grade 6] Core Skill AL 6.NS.7 - Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. Example: For example, express 36 + 8 as 4 (9 + 2). Numbers and Operations Determine the common factors for two whole numbers to 50 [Grade 5] Determine the greatest common factor of two whole numbers to 50 [Grade 5] Determine the multiple(s) of a number [Grade 5] Determine common multiples for two whole numbers [Grade 5] Determine the least common multiple of two whole numbers [Grade 5] Determine the greatest common factor of three numbers to 100 [Grade 6] Apply and extend previous understandings of numbers to the system of rational numbers. AL 6.NS.8 - Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. Numbers and Operations Compare two negative integers or a negative integer and a positive integer [Grade 6] Order negative integers or a mix of positive and negative integers [Grade 6] Identify a positive or negative rational number represented by a point on a number line [Grade 7] 4/26/2012 - 52 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 Locate a positive or negative rational number on a number line [Grade 7] Order rational numbers (positive and negative) [Grade 7] AL 6.NS.9 - Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. AL 6.NS.9.a - Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(3) = 3, and that 0 is its own opposite. Numbers and Operations Identify or locate an integer on a number line [Grade 6] Determine the opposite of an integer [Grade 7] Identify a positive or negative rational number represented by a point on a number line [Grade 7] Locate a positive or negative rational number on a number line [Grade 7] AL 6.NS.9.b - Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. Geometry and Measurement Determine the location of an ordered pair in the first quadrant [Grade 5] Determine the location of an ordered pair in any quadrant [Grade 6] AL 6.NS.9.c - Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. Numbers and Operations Identify or locate an integer on a number line [Grade 6] Identify a positive or negative rational number represented by a point on a number line [Grade 7] Locate a positive or negative rational number on a number line [Grade 7] Geometry and Measurement Determine the location of an ordered pair in the first quadrant [Grade 5] Determine the ordered pair of a point in the first quadrant [Grade 5] Determine the location of an ordered pair in any quadrant [Grade 6] Determine the ordered pair of a point in any quadrant [Grade 6] AL 6.NS.10 - Understand ordering and absolute value of rational numbers. Numbers and Operations Evaluate the absolute value of an integer [Grade 7] 4/26/2012 - 53 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 Compare rational numbers (positive and negative) [Grade 7] Order rational numbers (positive and negative) [Grade 7] AL 6.NS.10.a - Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. Example: For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right. Algebra Determine the graph of an inequality on a number line [Grade 7] Determine the graph of the solution set of a 1-step linear inequality [Grade 7] AL 6.NS.10.b - Write, interpret, and explain statements of order for rational numbers in real-world contexts. Example: For example, write -3 °C > -7 °C to express the fact that -3 °C is warmer than -7 ° C. Numbers and Operations Order numbers in decimal and fractional forms [Grade 6] Core Skill Compare two negative integers or a negative integer and a positive integer [Grade 6] Order negative integers or a mix of positive and negative integers [Grade 6] Order rational numbers (positive and negative) [Grade 7] AL 6.NS.10.c - Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. Example: For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars. Numbers and Operations Evaluate the absolute value of an integer [Grade 7] Identify a positive or negative rational number represented by a point on a number line [Grade 7] Locate a positive or negative rational number on a number line [Grade 7] AL 6.NS.10.d - Distinguish comparisons of absolute value from statements about order. Example: For example, recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars. Numbers and Operations Evaluate the absolute value of an integer [Grade 7] AL 6.NS.11 - Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. Geometry and Measurement Determine the location of a simple shape on the Cartesian plane given the coordinates of its vertices [Grade 7] Determine the coordinates of a missing point determined by geometric information [Grade 7] Determine a side length of a shape on the Cartesian plane [Grade 7] 4/26/2012 - 54 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 AL 6.EE - Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions. AL 6.EE.12 - Write and evaluate numerical expressions involving whole-number exponents. Numbers and Operations Determine the square of a whole number to 15 [Grade 6] Determine the repeated multiplication that is represented by a number raised to a power [Grade 7] Determine an exponential form of a whole number [Grade 7] Evaluate a numerical expression, with parentheses and exponents, using order of operations [Grade 7] Algebra Evaluate an algebraic expression involving whole number exponents [Grade 7] WP: Evaluate a variable expression involving exponents [Grade 7] AL 6.EE.13 - Write, read, and evaluate expressions in which letters stand for numbers. AL 6.EE.13.a - Write expressions that record operations with numbers and with letters standing for numbers. Example: For example, express the calculation "Subtract y from 5" as 5 - y. Algebra Use a variable expression with one operation to represent a verbal expression [Grade 5] Use a verbal expression to represent a variable expression with one operation [Grade 5] WP: Use a variable expression with one operation to represent a situation [Grade 5] Use a variable expression with two operations to represent a verbal expression [Grade 6] Use a verbal expression to represent a variable expression with two operations [Grade 6] WP: Use a variable expression with two operations to represent a situation [Grade 6] AL 6.EE.13.b - Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. Example: For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. Overall Product Skills Build comprehension of math vocabulary Demonstrate and develop understanding of mathematical concepts and ideas 4/26/2012 - 55 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 AL 6.EE.13.c - Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). Example: For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2. Numbers and Operations Evaluate a numerical expression of four or more operations, with parentheses, using order of operations [Grade 6] Algebra Evaluate a 1-variable expression, with two or three operations, using whole number substitution [Grade 6] Core Skill Evaluate a 2-variable expression, with two or three operations, using whole number substitution [Grade 6] Core Skill WP: Evaluate a 1- or 2-variable expression or formula using whole numbers [Grade 6] Core Skill WP: Generate a table of paired numbers based on a variable expression with two operations [Grade 6] Evaluate a 1-variable expression, with two or three operations, using integer substitution [Grade 7] Evaluate a 2-variable expression, with two or three operations, using integer substitution [Grade 7] Core Skill Evaluate an algebraic expression involving whole number exponents [Grade 7] WP: Evaluate a variable expression involving exponents [Grade 7] AL 6.EE.14 - Apply the properties of operations to generate equivalent expressions. Example: For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. Algebra Determine which property of addition or multiplication justifies a step in the simplification of an expression [Grade 6] AL 6.EE.15 - Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). Example: For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Reason about and solve one-variable equations and inequalities. AL 6.EE.16 - Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Algebra 4/26/2012 Determine some solutions to a 1-variable linear inequality [Grade 7] - 56 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 AL 6.EE.17 - Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Algebra WP: Use a variable expression with one operation to represent a situation [Grade 5] WP: Use a variable expression with two operations to represent a situation [Grade 6] Use a variable expression with two operations to represent a table of paired numbers [Grade 7] WP: Use a 2-variable expression to represent a situation [Grade 7] AL 6.EE.18 - Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. Algebra WP: Use a 2-variable equation to represent a situation involving a Core Skill direct proportion [Grade 6] WP: Use a 2-variable linear equation to represent a situation [Grade 6] Core Skill Solve a 1-step equation involving whole numbers [Grade 6] Core Skill Solve a 1-step linear equation involving integers [Grade 7] Core Skill Solve a 2-step linear equation involving integers [Grade 7] Core Skill WP: Use a 1-variable 1-step equation to represent a situation [Grade 7] Core Skill AL 6.EE.19 - Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Algebra Determine the graph of an inequality on a number line [Grade 7] Determine some solutions to a 1-variable linear inequality [Grade 7] WP: Use a 1-variable linear inequality to represent a situation [Grade 7] Represent and analyze quantitative relationships between dependent and independent variables. AL 6.EE.20 - Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. Example: For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Algebra 4/26/2012 Determine the variable expression with one operation for a table of paired numbers [Grade 5] Core Skill WP: Determine the variable expression with one operation for a table of paired numbers [Grade 5] Core Skill - 57 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 WP: Use a 2-variable equation to represent a situation involving a Core Skill direct proportion [Grade 6] WP: Use a 2-variable linear equation to represent a situation [Grade 6] Core Skill Use a 2-variable equation to construct an input-output table [Grade 6] Core Skill Use a 2-variable equation to represent a relationship expressed in a table [Grade 6] Core Skill Use a variable expression with two operations to represent a table of paired numbers [Grade 7] Use a table to represent a linear function [Grade 7] Determine the graph of a 1-operation linear function [Grade 7] Core Skill AL 6.G - Geometry Solve real-world and mathematical problems involving area, surface area, and volume. AL 6.G.21 - Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. Geometry and Measurement Use a formula to determine the area of a triangle [Grade 5] Core Skill Use a formula to determine the area of a parallelogram [Grade 5] WP: Determine the area of a triangle [Grade 5] Core Skill WP: Determine the area of a square or rectangle [Grade 5] Core Skill Determine the area of a complex shape [Grade 6] Core Skill WP: Determine the perimeter or the area of a complex shape [Grade 6] Core Skill Determine the area of a trapezoid [Grade 7] AL 6.G.22 - Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems. Geometry and Measurement Determine the volume of a rectangular prism given a diagram [Grade 5] Core Skill WP: Determine the volume of a rectangular prism given a diagram [Grade 5] Determine the volume of a rectangular prism [Grade 5] 4/26/2012 WP: Determine the volume of a rectangular prism [Grade 5] Core Skill Determine the volume of a rectangular or a triangular prism [Grade 7] Core Skill - 58 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 AL 6.G.23 - Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. Geometry and Measurement Determine the location of a simple shape on the Cartesian plane given the coordinates of its vertices [Grade 7] Determine a side length of a shape on the Cartesian plane [Grade 7] AL 6.G.24 - Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Geometry and Measurement Determine the surface area of a cube or a rectangular prism given a net [Grade 5] Determine the surface area of a rectangular prism [Grade 5] Core Skill WP: Find the surface area of a rectangular prism [Grade 5] Core Skill Determine the 3-dimensional shape that can be formed from a net [Grade 5] Determine a net of a 3-dimensional shape [Grade 5] Determine the surface area of a 3-dimensional shape given a net [Grade 7] Determine the surface area of a triangular prism [Grade 7] WP: Determine the surface area of a geometric solid [Grade 7] Core Skill AL 6.SP - Statistics and Probability Develop understanding of statistical variability. AL 6.SP.25 - Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. Example: For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages. Overall Product Skills Build comprehension of math vocabulary AL 6.SP.26 - Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. Overall Product Skills Build comprehension of math vocabulary AL 6.SP.27 - Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Overall Product Skills Build comprehension of math vocabulary Summarize and describe distributions. 4/26/2012 - 59 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 6 AL 6.SP.28 - Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Data Analysis, Statistics, and Probability Use a histogram to represent data [Grade 7] Core Skill Answer a question using information from a histogram [Grade 7] Core Skill Use a box-and-whisker plot to organize data [Grade 8] AL 6.SP.29 - Summarize numerical data sets in relation to their context, such as by: AL 6.SP.29.a - Reporting the number of observations. Overall Product Skills Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. AL 6.SP.29.b - Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. Overall Product Skills Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. AL 6.SP.29.c - Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. Data Analysis, Statistics, and Probability Determine the range from a graph [Grade 5] Determine the mean of a set of whole number data, whole number results [Grade 5] Determine the median of an odd number of data values [Grade 5] Core Skill Determine the range of a set of whole number data [Grade 5] Determine the mean of a set of whole number data [Grade 6] Determine the median of a set of whole number data [Grade 6] Determine the mean of a set of data [Grade 7] Determine the median of a set of data [Grade 7] AL 6.SP.29.d - Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. 4/26/2012 - 60 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 7 Grade 7 AL 7.RP - Ratios and Proportional Relationships Analyze proportional relationships and use them to solve real-world and mathematical problems. AL 7.RP.1 - Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. Example: For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. Numbers and Operations Divide a fraction by a fraction [Grade 6] Core Skill WP: Determine a unit rate with a whole number value [Grade 6] Core Skill WP: Use a unit rate, with a whole number or whole cent value, to solve a problem [Grade 6] Core Skill WP: Determine a unit rate [Grade 7] Core Skill WP: Use a unit rate to solve a problem [Grade 7] Core Skill AL 7.RP.2 - Recognize and represent proportional relationships between quantities. AL 7.RP.2.a - Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. Numbers and Operations WP: Determine the ratio of two whole numbers, at least one of which is larger than 50 [Grade 7] Determine ratios equivalent to a given ratio of two whole numbers, at least one of which is larger than 50 [Grade 7] Algebra Solve a proportion involving decimals [Grade 7] Core Skill WP: Solve a proportion [Grade 7] Core Skill AL 7.RP.2.b - Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Numbers and Operations WP: Determine a unit rate with a whole number value [Grade 6] Core Skill WP: Determine a unit rate [Grade 7] Core Skill AL 7.RP.2.c - Represent proportional relationships by equations. Example: For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. Algebra WP: Use a 2-variable equation to represent a situation involving a Core Skill direct proportion [Grade 6] AL 7.RP.2.d - Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Algebra Use a graph to represent the ordered pairs in a function table [Grade 7] Geometry and Measurement Determine the location of an ordered pair in any quadrant [Grade 6] 4/26/2012 - 61 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 7 Determine the ordered pair of a point in any quadrant [Grade 6] AL 7.RP.3 - Use proportional relationships to solve multistep ratio and percent problems. Example: Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Numbers and Operations Determine a percent of a whole number using less than 100% [Grade 7] Determine a percent of a whole number using more than 100% [Grade 7] Determine the percent a whole number is of another whole number, with a result less than 100% [Grade 7] Determine a whole number given a part and a percentage less than 100% [Grade 7] WP: Determine a percent of a whole number using less than 100% Core Skill [Grade 7] 4/26/2012 WP: Determine the percent a whole number is of another whole number, with a result less than 100% [Grade 7] Core Skill WP: Determine a whole number given a part and a percentage [Grade 7] Core Skill WP: Determine the percent of decrease applied to a number [Grade 7] Core Skill WP: Determine the percent of increase applied to a number [Grade 7] Core Skill WP: Determine the result of applying a percent of decrease to a value [Grade 7] Core Skill WP: Determine the result of applying a percent of increase to a value [Grade 7] Core Skill WP: Answer a question involving a fraction and a percent [Grade 7] Core Skill WP: Determine a part, given part to whole ratio and the whole, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine a part, given part to part ratio and the whole, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine a part, given part to whole ratio and a part, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine a part, given part to part ratio and a part, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine the whole, given part to whole ratio and a part, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine the whole, given part to part ratio and a part, where the whole is greater than 50 [Grade 7] Core Skill WP: Determine a unit rate [Grade 7] Core Skill WP: Use a unit rate to solve a problem [Grade 7] Core Skill - 62 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Determine a percent of a number given a percent that is not a whole percent [Grade 8] Grade 7 Core Skill Determine the percent one number is of another number [Grade 8] Core Skill Determine a number given a part and a decimal percentage or a percentage more than 100% [Grade 8] Core Skill WP: Determine a given percent of a number [Grade 8] Core Skill WP: Determine the percent one number is of another number [Grade 8] Core Skill WP: Determine a number given a part and a decimal percentage or a percentage more than 100% [Grade 8] Core Skill Solve a problem involving simple interest [Grade 8] Solve a problem involving annually compounded interest [Grade 8] WP: Find the result of two consecutive percentage changes applied to a given number [Grade 8] Algebra WP: Solve a proportion [Grade 7] Core Skill Geometry and Measurement Convert a rate from one unit to another with a change in one unit [Grade 7] Core Skill Convert a rate from one unit to another with a change in both units [Grade 7] Core Skill AL 7.NS - The Number System Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. AL 7.NS.4 - Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. AL 7.NS.4.a - Describe situations in which opposite quantities combine to make 0. Example: For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. Overall Product Skills Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. AL 7.NS.4.b - Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. Numbers and Operations Determine the opposite of an integer [Grade 7] Add integers using a number line [Grade 7] 4/26/2012 - 63 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 7 AL 7.NS.4.c - Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Numbers and Operations Subtract integers using a number line [Grade 7] AL 7.NS.4.d - Apply properties of operations as strategies to add and subtract rational numbers. Numbers and Operations Add integers [Grade 7] Core Skill Subtract integers [Grade 7] Core Skill Add or subtract signed fractions or mixed numbers [Grade 8] Add or subtract signed decimals [Grade 8] AL 7.NS.5 - Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. AL 7.NS.5.a - Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. Numbers and Operations Multiply integers [Grade 7] Core Skill WP: Multiply or divide integers [Grade 7] Core Skill AL 7.NS.5.b - Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then (p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts. Numbers and Operations Overall Product Skills Divide integers [Grade 7] Core Skill WP: Multiply or divide integers [Grade 7] Core Skill Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. AL 7.NS.5.c - Apply properties of operations as strategies to multiply and divide rational numbers. Numbers and Operations Multiply a fraction by a fraction [Grade 6] Core Skill Multiply a mixed number by a fraction [Grade 6] Core Skill Divide a fraction by a fraction [Grade 6] Core Skill Multiply a decimal number through thousandths by a whole number [Grade 6] Core Skill Multiply decimal numbers greater than one where the product has Core Skill 2 or 3 decimal places [Grade 6] 4/26/2012 Divide a decimal number by 10, 100, or 1,000 [Grade 6] Core Skill Divide a decimal number by a decimal number through thousandths, rounded quotient if needed [Grade 6] Core Skill Multiply integers [Grade 7] Core Skill - 64 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Divide integers [Grade 7] Grade 7 Core Skill Multiply or divide signed fractions or mixed numbers [Grade 8] Multiply or divide signed decimals [Grade 8] AL 7.NS.5.d - Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Numbers and Operations Convert a mixed number to a decimal number [Grade 6] Convert a fraction to a repeating decimal number [Grade 6] Convert a percentage to a decimal number greater than 1 [Grade 7] Convert a mixed number to a percentage [Grade 7] AL 7.NS.6 - Solve real-world and mathematical problems involving the four operations with rational numbers.[Computations with rational numbers extend the rules for manipulating fractions to complex fractions.] Numbers and Operations WP: Multiply or divide a fraction by a fraction [Grade 6] Core Skill WP: Multiply or divide two mixed numbers or a mixed number and a fraction [Grade 6] Core Skill WP: Solve a 2-step problem involving fractions [Grade 6] Core Skill WP: Multiply a decimal number through thousandths by a whole number [Grade 6] Core Skill WP: Divide a decimal number through thousandths by a 1- or 2digit whole number [Grade 6] Core Skill WP: Divide a whole number by a decimal number through thousandths, rounded quotient if needed [Grade 6] Core Skill WP: Divide a decimal through thousandths by a decimal through thousandths, rounded quotient if needed [Grade 6] Core Skill WP: Solve a 2-step problem involving decimals [Grade 6] Core Skill WP: Determine the percent of decrease applied to a number [Grade 7] Core Skill WP: Determine the percent of increase applied to a number [Grade 7] Core Skill WP: Determine the result of applying a percent of decrease to a value [Grade 7] Core Skill WP: Answer a question involving a fraction and a percent [Grade 7] Core Skill WP: Answer a question involving a fraction and a decimal [Grade 7] 4/26/2012 WP: Solve a multi-step problem involving decimal numbers [Grade 7] Core Skill WP: Solve a multi-step problem involving fractions or mixed numbers [Grade 7] Core Skill - 65 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 7 WP: Add and subtract using integers [Grade 7] Core Skill WP: Multiply or divide integers [Grade 7] Core Skill AL 7.EE - Expressions and Equations Use properties of operations to generate equivalent expressions. AL 7.EE.7 - Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Algebra Determine which property of addition or multiplication justifies a step in the simplification of an expression [Grade 6] Simplify an algebraic expression by combining like terms [Grade 8] Core Skill Use the distributive property to simplify an algebraic expression [Grade 8] AL 7.EE.8 - Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. Example: For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05.". Algebra WP: Evaluate a variable expression [Grade 7] Overall Product Skills Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. AL 7.EE.9 - Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. Example: For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. Numbers and Operations WP: Solve a multi-step problem involving whole numbers [Grade Core Skill 6] WP: Solve a multi-step problem involving decimal numbers [Grade 7] Core Skill WP: Solve a multi-step problem involving fractions or mixed numbers [Grade 7] Core Skill WP: Estimate the result of dividing or multiplying a whole number by a fraction [Grade 7] Algebra 4/26/2012 Solve a 1-step equation involving rational numbers [Grade 8] Core Skill Solve a 2-step equation involving rational numbers [Grade 8] Core Skill WP: Use a 1-variable equation with rational coefficients to represent a situation involving two operations [Grade 8] Core Skill - 66 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH WP: Use a 2-variable equation with rational coefficients to represent a situation [Grade 8] Grade 7 Core Skill AL 7.EE.10 - Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. AL 7.EE.10.a - Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. Example: For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? Numbers and Operations Algebra WP: Determine the result of applying a percent of decrease to a value [Grade 7] Core Skill WP: Determine the result of applying a percent of increase to a value [Grade 7] Core Skill Solve a 2-step linear equation involving integers [Grade 7] Core Skill Solve a 2-step equation involving rational numbers [Grade 8] Core Skill WP: Use a 1-variable equation with rational coefficients to represent a situation involving two operations [Grade 8] Core Skill WP: Use a 2-variable equation with rational coefficients to represent a situation [Grade 8] Core Skill WP: Solve a problem involving a 1-variable, 2-step equation [Grade 8] Core Skill AL 7.EE.10.b - Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Example: For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. Algebra Determine some solutions to a 1-variable linear inequality [Grade 7] Solve a 1-step linear inequality [Grade 7] Determine the graph of the solution set of a 1-step linear inequality [Grade 7] WP: Use a 1-variable linear inequality to represent a situation [Grade 7] WP: Use a 2-step linear inequality in one variable to represent a situation [Grade 8] WP: Solve a problem involving a 2-step linear inequality in one variable [Grade 8] Core Skill Determine the graph of the solutions to a 2-step linear inequality in one variable [Grade 8] Core Skill AL 7.G - Geometry 4/26/2012 - 67 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 7 Draw, construct, and describe geometrical figures and describe the relationships between them. AL 7.G.11 - Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Geometry and Measurement Determine a length given a scale [Grade 6] Determine the scale for a drawing or map question [Grade 7] WP: Solve a problem involving a map or scale drawing [Grade 7] WP: Solve a problem involving scale [Grade 8] AL 7.G.12 - Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. AL 7.G.13 - Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Geometry and Measurement Visualize a 2-dimensional shape [Grade 7] Relate a 3-dimensional shape to its top and side views [Grade 7] Solve a problem involving 3-dimensional visualization [Grade 8] Identify a cross section of a 3-dimensional shape [Grade 8] Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. AL 7.G.14 - Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Geometry and Measurement Determine the circumference of a circle in terms of pi [Grade 7] Solve a problem involving the circumference of a circle [Grade 7] Core Skill Determine the area of a circle in terms of pi [Grade 7] Determine the area of a circle using 3.14 for pi [Grade 7] Determine the area of a circle using 22/7 for pi [Grade 7] WP: Determine the area of a circle using 3.14 for pi [Grade 7] Solve a problem given the area of a circle [Grade 7] Core Skill AL 7.G.15 - Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. Geometry and Measurement Determine the measure of a missing angle using straight and right angle relationships [Grade 6] Determine the missing angle measure in a triangle given two other angle measures [Grade 7] Determine the missing angle measure in a quadrilateral given three other angle measures [Grade 7] Identify vertical, adjacent, complementary, or supplementary angles [Grade 7] 4/26/2012 - 68 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 7 Determine the measure of a missing angle using angle relationships [Grade 7] Identify angle relationships formed by parallel lines cut by a transversal [Grade 8] Determine an angle measure formed by parallel lines cut by a transversal [Grade 8] AL 7.G.16 - Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Geometry and Measurement Determine the volume of a prism with a right triangle base [Grade 6] Determine the surface area of a 3-dimensional shape made from cubes [Grade 6] Determine the area of a trapezoid [Grade 7] Determine the volume of a rectangular or a triangular prism [Grade 7] Core Skill Determine the surface area of a 3-dimensional shape given a net [Grade 7] WP: Determine the surface area of a geometric solid [Grade 7] Core Skill Determine the surface area of a pyramid or a cone [Grade 8] WP: Determine the surface area of a pyramid or a cone [Grade 8] Solve a problem involving the surface area or the volume of a pyramid or a cone [Grade 8] AL 7.SP - Statistics and Probability Use random sampling to draw inferences about a population. AL 7.SP.17 - Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. Data Analysis, Statistics, and Probability Determine if a sample is likely to be representative of the larger population [Grade 8] AL 7.SP.18 - Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. Example: For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. Data Analysis, Statistics, and Probability Use a proportion to make an estimate, related to a population, based on a sample [Grade 7] Core Skill Draw informal comparative inferences about two populations. 4/26/2012 - 69 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 7 AL 7.SP.19 - Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. Example: For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. AL 7.SP.20 - Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. Example: For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. Data Analysis, Statistics, and Probability Determine the mean of a set of data [Grade 7] Determine the mode(s) of a set of data [Grade 7] Determine the median of a set of data [Grade 7] WP: Use the mean of a data set to solve a problem [Grade 7] Answer a question using information from two box-and-whisker plots [Grade 8] Compare the medians, the modes, or the ranges of the data in a double stem-and-leaf plot [Grade 8] Investigate chance processes and develop, use, and evaluate probability models. AL 7.SP.21 - Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. Overall Product Skills Build comprehension of math vocabulary AL 7.SP.22 - Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. Example: For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. Data Analysis, Statistics, and Probability Determine an experimental probability given a list of results [Grade 6] Make a prediction based on an experimental probability [Grade 6] AL 7.SP.23 - Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. AL 7.SP.23.a - Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. Example: For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. Data Analysis, Statistics, and Probability 4/26/2012 Determine the probability of a single event [Grade 6] - 70 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 7 Determine the probability of the complement of a single event [Grade 6] Make a prediction based on a theoretical probability [Grade 6] Make a prediction involving the probability of compound events [Grade 8] AL 7.SP.23.b - Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. Example: For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? Data Analysis, Statistics, and Probability Determine an experimental probability given a list of results [Grade 6] Make a prediction based on an experimental probability [Grade 6] Compare predictions from experimental and theoretical probability [Grade 6] AL 7.SP.24 - Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. AL 7.SP.24.a - Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. Data Analysis, Statistics, and Probability Determine the probability of an event consisting of mutually exclusive events [Grade 8] Make a prediction involving the probability of compound events [Grade 8] AL 7.SP.24.b - Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event. Data Analysis, Statistics, and Probability Determine all possible outcomes of a compound event using a list [Grade 5] Determine all possible outcomes of a compound event using a tree diagram [Grade 5] AL 7.SP.24.c - Design and use a simulation to generate frequencies for compound events. Example: For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? Overall Product Skills 4/26/2012 Use strategies to derive solutions for problems - 71 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 8 Grade 8 AL 8.NS - The Number System Know that there are numbers that are not rational, and approximate them by rational numbers. AL 8.NS.1 - Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Numbers and Operations Convert a repeating decimal to a fraction or a mixed number [Grade 8] Identify rational or irrational numbers [Grade 8] AL 8.NS.2 - Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi²). Example: For example, by truncating the decimal expansion of the square root of 2, show that the square root of 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. Numbers and Operations Identify a positive or negative rational number represented by a point on a number line [Grade 7] Locate a positive or negative rational number on a number line [Grade 7] Determine the two closest integers to a given square root [Grade 8] Core Skill Approximate the location of a square root on a number line [Grade 8] Core Skill Determine the square root of a whole number to the nearest tenth [Grade 8] Compare rational numbers and/or irrational numbers in various forms [Grade 8] Order rational numbers and irrational numbers [Grade 8] AL 8.EE - Expressions and Equations Work with radicals and integer exponents. AL 8.EE.3 - Know and apply the properties of integer exponents to generate equivalent numerical expressions. Example: For example, 3² × (3 to the -5 power) = (3 to the -3 power) = 1/3³ = 1/27. Numbers and Operations Determine the exponential notation that represents a repeated multiplication [Grade 7] Evaluate a whole number power of a whole number [Grade 7] Evaluate an integer raised to a whole number power [Grade 8] Core Skill Evaluate a zero or negative power of an integer [Grade 8] Evaluate a numerical expression involving integer exponents and/or integer bases [Grade 8] 4/26/2012 Core Skill - 72 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra Grade 8 Evaluate an algebraic expression involving negative integer exponents [Grade 8] Evaluate a fraction raised to an integer power [Algebra 1] Apply the product of powers property to a monomial numerical expression [Algebra 1] Apply the power of a power property to a monomial numerical expression [Algebra 1] Apply the quotient of powers property to monomial numerical expressions [Algebra 1] AL 8.EE.4 - Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that the square root of 2 is irrational. Numbers and Operations Determine the cube of a whole number to 15 [Grade 6] Determine the whole number that can be squared to make a given number [Grade 7] Evaluate the positive square root of a perfect square [Grade 7] Determine the square root of a perfect-square fraction or decimal [Grade 8] Core Skill Determine both square roots of a perfect square [Grade 8] Determine the two closest integers to a given square root [Grade 8] Core Skill Determine the square root of a whole number to the nearest tenth [Grade 8] Algebra Simplify a monomial numerical expression involving the square root of a whole number [Algebra 1] Core Skill AL 8.EE.5 - Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. Example: For example, estimate the population of the United States as 3 × (10 to the 8th power) and the population of the world as 7 × (10 to the 9th power), and determine that the world population is more than 20 times larger. Numbers and Operations Convert a whole number greater than 10 to scientific notation [Grade 7] Convert a whole number greater than 10 from scientific notation to standard form [Grade 7] 4/26/2012 Convert a number less than 1 to scientific notation [Grade 8] Core Skill Convert a number less than 1 from scientific notation to standard form [Grade 8] Core Skill - 73 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 8 AL 8.EE.6 - Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Numbers and Operations Convert a whole number greater than 10 to scientific notation [Grade 7] Convert a whole number greater than 10 from scientific notation to standard form [Grade 7] Convert a number less than 1 to scientific notation [Grade 8] Core Skill Convert a number less than 1 from scientific notation to standard form [Grade 8] Core Skill Understand the connections between proportional relationships, lines, and linear equations. AL 8.EE.7 - Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Algebra Determine the slope of a line given its graph or a graph of a line with a given slope [Grade 8] Core Skill WP: Interpret the meaning of the slope of a graphed line [Grade 8] Core Skill Determine the slope-intercept form or the standard form of a linear equation [Algebra 1] Core Skill AL 8.EE.8 - Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Analyze and solve linear equations and pairs of simultaneous linear equations. AL 8.EE.9 - Solve linear equations in one variable. AL 8.EE.9.a - Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Algebra Solve a 1-variable linear equation with the variable on both sides [Algebra 1] Overall Product Skills Use strategies to derive solutions for problems Core Skill AL 8.EE.9.b - Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Algebra 4/26/2012 Solve a proportion involving decimals [Grade 7] Core Skill Solve a 1-step linear equation involving integers [Grade 7] Core Skill Solve a 2-step linear equation involving integers [Grade 7] Core Skill - 74 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 8 Use the distributive property to simplify an algebraic expression [Grade 8] Solve a 1-step equation involving rational numbers [Grade 8] Core Skill Solve a 2-step equation involving rational numbers [Grade 8] Core Skill Solve a 1-variable linear equation that requires simplification and has the variable on one side [Algebra 1] Solve a 1-variable linear equation with the variable on both sides [Algebra 1] Core Skill AL 8.EE.10 - Analyze and solve pairs of simultaneous linear equations. AL 8.EE.10.a - Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Algebra Determine the graph of a linear equation given in slope-intercept, point-slope, or standard form [Algebra 1] Core Skill Solve a system of linear equations in two variables by graphing [Algebra 1] AL 8.EE.10.b - Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. Algebra Solve a system of linear equations in two variables by graphing [Algebra 1] Solve a system of linear equations in two variables by substitution [Algebra 1] Solve a system of linear equations in two variables by elimination [Algebra 1] Determine the number of solutions to a system of linear equations [Algebra 1] Solve a system of linear equations in two variables using any method [Algebra 1] Core Skill Determine if a given ordered pair is a solution to a system of linear inequalities [Algebra 1] AL 8.EE.10.c - Solve real-world and mathematical problems leading to two linear equations in two variables. Example: For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Algebra Determine if an ordered pair is a solution to a 2-variable linear inequality [Algebra 1] WP: Determine a system of linear equations that represents a given situation [Algebra 1] Core Skill WP: Solve a mixture problem that can be represented by a system of linear equations [Algebra 1] 4/26/2012 - 75 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 8 WP: Solve a motion problem that can be represented by a system of linear equations [Algebra 1] Solve a number problem that can be represented by a linear system of equations [Algebra 1] Determine if a given ordered pair is a solution to a system of linear inequalities [Algebra 1] AL 8.F - Functions Define, evaluate, and compare functions. AL 8.F.11 - Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Algebra Use a table to represent a linear function [Grade 7] Use a graph to represent the ordered pairs in a function table [Grade 7] Determine the graph of a 1-operation linear function [Grade 7] Core Skill Determine the graph of a 2-operation linear function [Grade 8] Core Skill Determine if a relation is a function [Algebra 1] AL 8.F.12 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Algebra Use a table to represent a linear function [Grade 7] Use a graph to represent the ordered pairs in a function table [Grade 7] Overall Product Skills Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. AL 8.F.13 - Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Example: For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Algebra Determine the graph of a 1-operation linear function [Grade 7] Core Skill Determine the graph of a 2-operation linear function [Grade 8] Core Skill Determine if a function is linear or nonlinear [Algebra 1] Determine whether a graph or a table represents a linear or nonlinear function [Algebra 1] Use functions to model relationships between quantities. Algebra Use a table to represent a linear function [Grade 7] Determine if a relation is a function [Algebra 1] 4/26/2012 - 76 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 8 AL 8.F.14 - Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Algebra Determine the table of values that represents a linear equation with rational coefficients in two variables [Grade 8] Core Skill Determine a linear equation in two variables that represents a table of values [Grade 8] Core Skill Determine the slope of a line given its graph or a graph of a line with a given slope [Grade 8] Core Skill WP: Interpret the meaning of the slope of a graphed line [Grade 8] Core Skill WP: Interpret the meaning of the y-intercept of a graphed line [Grade 8] Core Skill WP: Determine a linear graph that can represent a situation [Grade 8] WP: Determine a graph that can represent a situation involving a varying rate of change [Grade 8] Determine the slope of a line given two points on the line [Algebra 1] Core Skill Determine an equation of a line given the slope and y-intercept of Core Skill the line [Algebra 1] Determine an equation that represents a graphed line [Algebra 1] Core Skill Determine an equation for a line given the slope of the line and a point on the line that is not the y-intercept [Algebra 1] Core Skill Determine an equation of a line given two points on the line [Algebra 1] Core Skill AL 8.F.15 - Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Algebra Determine the graph of a 2-operation linear function [Grade 8] Core Skill Determine the slope of a line given its graph or a graph of a line with a given slope [Grade 8] Core Skill WP: Determine a graph that can represent a situation involving a varying rate of change [Grade 8] AL 8.G - Geometry Understand congruence and similarity using physical models, transparencies, or geometry software. AL 8.G.16 - Verify experimentally the properties of rotations, reflections, and translations: AL 8.G.16.a - Lines are taken to lines, and line segments to line segments of the same length. 4/26/2012 - 77 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 8 AL 8.G.16.b - Angles are taken to angles of the same measure. AL 8.G.16.c - Parallel lines are taken to parallel lines. AL 8.G.17 - Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. Geometry and Measurement Identify corresponding parts of congruent shapes [Grade 7] Identify congruent shapes given side and angle measures [Grade 7] Determine a missing dimension given two congruent shapes [Grade 7] AL 8.G.18 - Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Geometry and Measurement Determine the result of a reflection, a rotation, or a translation on the Cartesian plane [Grade 6] Determine the transformation that generates the image of a figure in the Cartesian plane [Grade 6] Determine the coordinates of a translated, a rotated, or a reflected shape on the Cartesian plane [Grade 7] AL 8.G.19 - Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Geometry and Measurement Determine the coordinates of a translated, a rotated, or a reflected shape on the Cartesian plane [Grade 7] Relate the coordinates of a preimage or an image to a translation described using mapping notation [Geometry] Relate the coordinates of a preimage or an image to a dilation centered at the origin [Geometry] Determine the coordinates of the image of a figure after two transformations of the same type [Geometry] Determine the coordinates of the image of a figure after two transformations of different types [Geometry] AL 8.G.20 - Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Example: For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Geometry and Measurement Identify angle relationships formed by parallel lines cut by a transversal [Grade 8] Determine an angle measure formed by parallel lines cut by a transversal [Grade 8] 4/26/2012 - 78 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 8 Identify angle relationships formed by multiple lines and transversals [Geometry] Overall Product Skills Determine the measure of an angle formed by parallel lines and one or more transversals [Geometry] Core Skill Identify a triangle similarity postulate that justifies a similarity statement [Geometry] Core Skill Identify similar triangles using triangle similarity postulates or theorems [Geometry] Core Skill Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. Understand and apply the Pythagorean Theorem. AL 8.G.21 - Explain a proof of the Pythagorean Theorem and its converse. Overall Product Skills Demonstrate and develop understanding of mathematical concepts and ideas AL 8.G.22 - Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Geometry and Measurement Determine the length of the hypotenuse of a right triangle using the Pythagorean theorem [Grade 8] Core Skill Determine the length of a leg of a right triangle using the Pythagorean theorem [Grade 8] Core Skill WP: Use the Pythagorean theorem to find a length or a distance [Grade 8] Core Skill Solve for the length of a side of a triangle using the Pythagorean theorem [Geometry] AL 8.G.23 - Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Geometry and Measurement Determine a distance on the Cartesian plane using the Pythagorean theorem [Grade 8] Determine the distance between two points [Geometry] Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. AL 8.G.24 - Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Geometry and Measurement Determine the volume of a cylinder [Grade 7] WP: Determine the volume of a cylinder [Grade 7] WP: Solve a problem involving the volume of a geometric solid [Grade 7] Core Skill Determine the volume of a pyramid or a cone [Grade 8] WP: Determine the volume of a pyramid or a cone [Grade 8] 4/26/2012 - 79 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Grade 8 Solve a problem involving the volume of a right pyramid or a right cone [Geometry] Core Skill Determine the volume of a sphere or hemisphere [Geometry] Core Skill WP: Determine the volume of a sphere or hemisphere [Geometry] AL 8.SP - Statistics and Probability Investigate patterns of association in bivariate data. AL 8.SP.25 - Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Data Analysis, Statistics, and Probability Use a scatter plot to organize data [Grade 8] Determine if a scatter plot shows a positive relationship, a negative relationship, or no relationship between the variables [Grade 8] Answer a question using information from a scatter plot [Grade 8] AL 8.SP.26 - Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Data Analysis, Statistics, and Probability Approximate a trend line for a scatter plot [Grade 8] AL 8.SP.27 - Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Algebra WP: Interpret the meaning of the slope of a graphed line [Grade 8] Core Skill WP: Interpret the meaning of the y-intercept of a graphed line [Grade 8] Core Skill AL 8.SP.28 - Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Example: For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Data Analysis, Statistics, and Probability Use a frequency table to represent 2 related data sets [Grade 6] Answer a question using information from a frequency table representing 2 related data sets [Grade 6] 4/26/2012 - 80 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Overall Product Skills 4/26/2012 Grade 8 Develop understanding and ability to analyze mathematical relationships including those between and among arithmetic operations. - 81 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH 9-12 Analytic Math 9-12 Analytic Math AL N - Number and Quantity AL N-VM - Vector and Matrix Quantities Represent and model with vector quantities. AL N-VM.1 - Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, v, v), including the use of eigen-values and eigen-vectors. Algebra Determine the component form of a vector represented on a graph [Algebra 2] AL N-VM.2 - Solve problems involving velocity and other quantities that can be represented by vectors, including navigation (e.g., airplane, aerospace, oceanic). Algebra WP: Add or subtract vectors [Algebra 2] AL N-VM.3 - Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Find the dot product and the cross product of vectors. Algebra Add or subtract vectors on a coordinate plane [Algebra 2] Add or subtract vectors component-wise [Algebra 2] AL N-VM.4 - Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum, including vectors in complex vector spaces. Algebra Determine the magnitude of a vector [Algebra 2] Determine the direction of a vector [Algebra 2] Determine a unit vector in the same direction as a given vector [Algebra 2] WP: Add or subtract vectors [Algebra 2] AL N-VM.5 - Understand vector subtraction v - w as v + (-w), where (-w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise, including vectors in complex vector spaces. Algebra Add or subtract vectors on a coordinate plane [Algebra 2] Add or subtract vectors component-wise [Algebra 2] Perform operations on matrices and use matrices in applications. AL N-VM.6 - Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network, including linear programming. Algebra 4/26/2012 Represent a system of linear equations as a single matrix equation [Algebra 2] - 82 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH 9-12 Analytic Math AL N-VM.7 - Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled, including rotation matrices. Algebra Multiply a matrix by a scalar [Algebra 2] AL N-VM.8 - Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. Solve matrix equations using augmented matrices. Algebra Determine the determinant of a matrix [Algebra 2] Determine the inverse of a matrix [Algebra 2] Core Skill AL N-VM.9 - Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors, including matrices larger than 2 × 2. Algebra Combine matrix operations [Algebra 2] Multiply matrices [Algebra 2] AL N-VM.10 - Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. Solve matrix application problems using reduced row echelon form. Algebra Determine the determinant of a matrix [Algebra 2] AL N-CN - Complex Numbers Use complex numbers in polynomial identities and equations. AL N-CN.11 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Understand the importance of using complex numbers in graphing functions on the Cartesian or complex plane. Algebra Determine the graph of a given quadratic function [Algebra 1] Core Skill Solve a quadratic equation by graphing the associated quadratic function [Algebra 1] Determine a polynomial of lowest degree that has given roots or a graph of a given shape [Algebra 2] AL N-L - Limits Understand limits of functions. AL N-L.12 - Calculate the limit of a sequence, of a function, and of an infinite series. Algebra Find the sum of an infinite geometric series if it exists [Algebra 2] AL A - Algebra AL A-SSE - Seeing Structure in Expressions 4/26/2012 - 83 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH 9-12 Analytic Math AL A-SSE.13 - Use the laws of Boolean Algebra to describe true/false circuits. Simplify Boolean expressions using the relationships between conjunction, disjunction, and negation operations. AL A-SSE.14 - Use logic symbols to write truth tables. AL A-APR - Arithmetic With Polynomials and Rational Functions AL A-APR.15 - Reduce the degree of either the numerator or denominator of a rational function by using partial fraction decomposition or partial fraction expansion. Algebra Decompose rational functions with quadratic denominators into sums of rational functions with linear denominators [Algebra 2] AL F - Functions AL F-TF - Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. AL F-TF.16 - Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Algebra WP: Determine a trigonometric function that represents a situation [Algebra 2] Apply trigonometry to general triangles. AL F-TF.17 - Prove the Law of Sines and the Law of Cosines and use them to solve problems. Understand Law of Sines = 2r, where r is the radius of the circumscribed circle of the triangle. Apply the Law of Tangents. Algebra Solve a problem involving the law of sines [Algebra 2] Geometry and Measurement Solve a problem involving the law of cosines [Algebra 2] WP: Solve a problem involving the law of sines or the law of cosines [Algebra 2] AL F-TF.18 - Apply Euler's and deMoivre's formulas as links between complex numbers and trigonometry. 4/26/2012 - 84 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 Algebra 2 AL N - Number and Quantity AL N-CN - The Complex Number System Perform arithmetic operations with complex numbers. AL N-CN.1 - Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Algebra Extend polynomial identities to the complex numbers [Algebra 2] Raise i to a power [Algebra 2] AL N-CN.2 - Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Algebra Raise i to a power [Algebra 2] Add or subtract complex numbers [Algebra 2] Multiply complex numbers [Algebra 2] Simplify an expression involving a complex denominator [Algebra 2] Core Skill Use complex numbers in polynomial identities and equations.[(Polynomials with real coefficients.)] AL N-CN.3 - Solve quadratic equations with real coefficients that have complex solutions. Algebra Solve a quadratic equation with complex solutions [Algebra 2] Core Skill AL N-CN.4 - Extend polynomial identities to the complex numbers. Example: For example, rewrite x² + 4 as (x + 2i)(x - 2i). Algebra Extend polynomial identities to the complex numbers [Algebra 2] AL N-CN.5 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Algebra Determine the graph of a given quadratic function [Algebra 1] Core Skill Solve a quadratic equation by graphing the associated quadratic function [Algebra 1] Determine a polynomial of lowest degree that has given roots or a graph of a given shape [Algebra 2] AL A - Algebra AL A-SSE - Seeing Structure in Expressions Interpret the structure of expressions[(Polynomial and rational.)] AL A-SSE.6 - Interpret expressions that represent a quantity in terms of its context. Algebra Evaluate a numerical expression involving one or more exponents Core Skill and multiple forms of rational numbers [Algebra 1] 4/26/2012 - 85 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 AL A-SSE.6.a - Interpret parts of an expression, such as terms, factors, and coefficients. Overall Product Skills Build comprehension of math vocabulary AL A-SSE.6.b - Interpret complicated expressions by viewing one or more of their parts as a single entity. Example: For example, interpret P(1+r) to the n power as the product of P and a factor not depending on P. Numbers and Operations Evaluate a numerical expression involving integer exponents and/or integer bases [Grade 8] Core Skill AL A-SSE.7 - Use the structure of an expression to identify ways to rewrite it. Example: For example, see (x to the 4th power) - (y to the 4th power) as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²). Algebra Simplify a monomial algebraic expression that includes fractional exponents and/or nth roots [Algebra 2] Write expressions in equivalent forms to solve problems Algebra Simplify an algebraic expression by combining like terms [Grade 8] Core Skill Use the distributive property to simplify an algebraic expression [Grade 8] Simplify a polynomial expression by combining like terms [Algebra 1] Simplify a monomial algebraic radical expression [Algebra 1] Core Skill Simplify a rational expression involving polynomial terms [Algebra 1] Core Skill Simplify a numerical expression that includes fractional exponents and/or nth roots [Algebra 2] Decompose rational functions with quadratic denominators into sums of rational functions with linear denominators [Algebra 2] Simplify a monomial algebraic expression that includes fractional exponents and/or nth roots [Algebra 2] Identify an equivalent form of an exponential expression by using the properties of exponents [Algebra 2] Identify equivalent logarithmic expressions using the properties of Core Skill logarithms [Algebra 2] AL A-SSE.8 - Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. Example: For example, calculate mortgage payments. Algebra Find the sum of a finite geometric series [Algebra 2] Core Skill Find the sum of a finite geometric series given in summation notation [Algebra 2] WP: Solve a problem that can be represented by a finite geometric Core Skill series [Algebra 2] 4/26/2012 - 86 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 AL A-APR - Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials[(Beyond quadratic.)] AL A-APR.9 - Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Algebra Multiply two monomial algebraic expressions [Grade 8] Core Skill Simplify a polynomial expression by combining like terms [Algebra 1] Add polynomial expressions [Algebra 1] Core Skill Subtract polynomial expressions [Algebra 1] Core Skill Multiply a polynomial by a monomial [Algebra 1] Multiply two binomials of the form (x +/- a)(x +/- b) [Algebra 1] Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra 1] Core Skill Multiply two binomials of the form (ax +/- by)(cx +/- dy) [Algebra 1] Square a binomial [Algebra 1] Multiply two nonlinear binomials [Algebra 1] Multiply a trinomial by a binomial [Algebra 1] Multiply polynomial expressions [Algebra 2] Understand the relationship between zeros and factors of polynomials AL A-APR.10 - Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x). Algebra Divide a polynomial expression by a monomial [Algebra 1] Core Skill Divide a polynomial expression by a binomial [Algebra 1] Determine factors of a polynomial by applying the remainder theorem [Algebra 2] AL A-APR.11 - Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Algebra Determine the graph of a given quadratic function [Algebra 1] Core Skill Determine the solution(s) of an equation given in factored form [Algebra 1] Solve a quadratic equation by factoring [Algebra 1] Core Skill Identify the end behavior, the intercepts, or the zeros of a polynomial function [Algebra 2] Core Skill Use polynomial identities to solve problems 4/26/2012 - 87 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 AL A-APR.12 - Prove polynomial identities and use them to describe numerical relationships. Example: For example, the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate Pythagorean triples. Algebra Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra 1] Core Skill Factor the difference of two squares [Algebra 1] Factor a polynomial of degree 3 or higher [Algebra 2] WP: Solve a problem modeled by a polynomial equation [Algebra 2] Multiply polynomial expressions [Algebra 2] AL A-APR.13 - Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.[(The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)] Algebra Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra 1] Core Skill Rewrite rational expressions[(Linear and quadratic denominators.)] AL A-APR.14 - Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q (x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Algebra Divide a polynomial expression by a monomial [Algebra 1] Core Skill Divide a polynomial expression by a binomial [Algebra 1] Divide polynomial expressions [Algebra 2] AL A-APR.15 - Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Algebra Multiply rational expressions [Algebra 1] Core Skill Divide rational expressions [Algebra 1] Determine the LCD of two rational expressions [Algebra 1] Add or subtract two rational expressions with like denominators [Algebra 1] Add or subtract two rational expressions with unlike monomial denominators [Algebra 1] Add or subtract two rational expressions with unlike polynomial denominators [Algebra 1] Core Skill AL A-CED - Creating Equations 4/26/2012 - 88 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 Create equations that describe numbers or relationships[(Equations using all available types of expressions, including simple root functions.)] AL A-CED.16 - Create equations and inequalities in one variable and use them to solve problems. [Include equations arising from linear and quadratic functions, and simple rational and exponential functions.] Algebra WP: Use a 1-variable linear inequality to represent a situation [Grade 7] WP: Use a 1-variable equation with rational coefficients to represent a situation involving two operations [Grade 8] Core Skill WP: Solve a problem involving a 1-variable, 2-step equation [Grade 8] Core Skill Solve a 2-step linear inequality in one variable [Grade 8] Core Skill WP: Use a 2-step linear inequality in one variable to represent a situation [Grade 8] WP: Solve a problem involving a 2-step linear inequality in one variable [Grade 8] Core Skill Solve a 1-variable linear equation with the variable on both sides [Algebra 1] Core Skill WP: Determine a linear equation that can be used to solve a percent problem [Algebra 1] WP: Solve a direct- or inverse-variation problem [Algebra 1] Solve a 1-variable linear inequality with the variable on one side [Algebra 1] Solve a 1-variable linear inequality with the variable on both sides Core Skill [Algebra 1] WP: Solve a problem involving a rational equation [Algebra 1] AL A-CED.17 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra WP: Use a 2-variable equation with rational coefficients to represent a situation [Grade 8] Core Skill WP: Determine an equation representing a direct variation or an inverse variation [Algebra 1] Determine the graph of a linear equation given in slope-intercept, point-slope, or standard form [Algebra 1] Core Skill WP: Solve a mixture problem that can be represented by a system of linear equations [Algebra 1] WP: Solve a motion problem that can be represented by a system of linear equations [Algebra 1] Solve a number problem that can be represented by a linear system of equations [Algebra 1] 4/26/2012 - 89 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 Determine the equation of a circle from given information [Algebra 2] Determine the graph of a circle given the equation in standard form [Algebra 2] Determine the graph of a circle or an ellipse given the equation in Core Skill general form [Algebra 2] Determine the graph of a hyperbola given the equation in standard form [Algebra 2] Determine the equation of a parabola using given information [Algebra 2] Core Skill Determine the equation of an ellipse using given information [Algebra 2] AL A-CED.18 - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Algebra WP: Determine a system of linear equations that represents a given situation [Algebra 1] Core Skill WP: Determine a system of linear inequalities that represents a given situation [Algebra 1] WP: Determine possible solutions to a problem that can be represented by a system of linear inequalities [Algebra 1] WP: Solve a linear programming problem [Algebra 2] WP: Solve a problem that can be represented by a system of linear Core Skill equations in three variables [Algebra 2] WP: Solve a problem that can be modeled with a quadratic inequality [Algebra 2] AL A-CED.19 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm's law V = IR to highlight resistance R. Algebra Rewrite an equation to solve for a specified variable [Algebra 1] Core Skill AL A-REI - Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning[(Simple rational and radical.)] AL A-REI.20 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Algebra Solve a radical equation that leads to a linear equation [Algebra 1] Core Skill Solve a radical equation that leads to a quadratic equation [Algebra 1] Core Skill WP: Solve a problem involving a radical function [Algebra 1] 4/26/2012 - 90 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 Determine the excluded values of a rational algebraic expression [Algebra 1] Solve a proportion that generates a linear or quadratic equation [Algebra 1] Solve a rational equation involving terms with monomial denominators [Algebra 1] Core Skill Solve a rational equation involving terms with polynomial denominators [Algebra 1] Core Skill Represent and solve equations and inequalities graphically[(Combine polynomial, rational, radical, absolute value, and exponential functions.)] AL A-REI.21 - Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Algebra Solve a system of linear equations in two variables by graphing [Algebra 1] Determine the graph of a 2-variable absolute value inequality [Algebra 2] Relate a graph to a polynomial function given in factored form [Algebra 2] Core Skill Determine an exponential function that represents a graph [Algebra 2] Relate a logarithmic function to its graph [Algebra 2] AL F - Functions AL F-IF - Interpreting Functions Interpret functions that arise in applications in terms of the context[(Emphasize selection of appropriate models.)] AL F-IF.22 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.[Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.] Algebra Determine the slope of a line given its graph or a graph of a line with a given slope [Grade 8] Core Skill Determine the x- or y-intercept of a line given its graph [Grade 8] Core Skill 4/26/2012 WP: Answer a question using the graph of a quadratic function [Algebra 1] Core Skill Identify the vertex, axis of symmetry, or direction of the graph of a quadratic function [Algebra 2] Core Skill - 91 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 Relate changes in the values of a, b, and/or c to the graph of the equation y = a(x - b)^2 + c [Algebra 2] Core Skill Identify the end behavior, asymptotes, excluded values, or behavior near excluded values of a rational function [Algebra 2] Core Skill Identify the asymptotes, growth or decay, or half-life of an exponential function [Algebra 2] Identify the domain, range, asymptotes, or intercepts of a logarithmic function [Algebra 2] Determine the amplitude, frequency, period, or phase shift of a sine or a cosine function [Algebra 2] AL F-IF.23 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Algebra Determine the domain or range of a function [Algebra 1] WP: Determine a reasonable domain or range for a function in a given situation [Algebra 1] Core Skill Identify the domain or range of a radical function [Algebra 2] Identify the domain, range, asymptotes, or intercepts of a logarithmic function [Algebra 2] AL F-IF.24 - Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Algebra WP: Interpret the meaning of the slope of a graphed line [Grade 8] Core Skill WP: Determine a graph that can represent a situation involving a varying rate of change [Grade 8] Determine the slope of a line given two points on the line [Algebra 1] Core Skill WP: Interpret an interest rate, rate of change, initial amount, frequency of compounding and other parameters of an exponential function [Algebra 2] Core Skill Analyze functions using different representations[(Focus on using key features to guide selection of appropriate type of model function.)] AL F-IF.25 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. AL F-IF.25.a - Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Algebra Determine the graph of a 2-variable absolute value equation [Algebra 1] Determine the graph of a radical function [Algebra 1] 4/26/2012 - 92 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 Determine the graph of a piecewise-defined function [Algebra 2] Relate a graph to a square or cube root function [Algebra 2] AL F-IF.25.b - Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Algebra Relate a graph to a polynomial function given in factored form [Algebra 2] Core Skill Identify the end behavior, the intercepts, or the zeros of a polynomial function [Algebra 2] Core Skill Determine a polynomial function which goes through specified points on a graph [Algebra 2] Core Skill AL F-IF.25.c - Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Algebra Determine the graph of an exponential function [Algebra 1] Identify the asymptotes, growth or decay, or half-life of an exponential function [Algebra 2] Determine an exponential function that represents a graph [Algebra 2] Relate a logarithmic function to its graph [Algebra 2] Identify the domain, range, asymptotes, or intercepts of a logarithmic function [Algebra 2] Determine the graph of a sine, cosine or tangent function [Algebra Core Skill 2] Determine the amplitude, frequency, period, or phase shift of a sine or a cosine function [Algebra 2] AL F-IF.26 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Algebra Determine the domain or range of a function [Algebra 1] Evaluate a function written in function notation for a given value [Algebra 1] Core Skill Convert a quadratic function between vertex and standard form of the equation [Algebra 2] Compare characteristics of two linear, two quadratic, or two exponential functions represented in different ways [Algebra 2] AL F-IF.27 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Algebra Determine the graph of a given quadratic function [Algebra 1] Core Skill Solve a quadratic equation by graphing the associated quadratic function [Algebra 1] 4/26/2012 - 93 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 Compare characteristics of two linear, two quadratic, or two exponential functions represented in different ways [Algebra 2] AL F-BF - Building Functions Build a function that models a relationship between two quantities[(Include all types of functions studied.)] AL F-BF.28 - Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Algebra WP: Apply function operations or compositions of functions to represent a situation [Algebra 2] Build new functions from existing functions[(Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types.)] AL F-BF.29 - Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Algebra Determine the equation of a function resulting from a translation and/or scaling of a given function [Algebra 2] Core Skill AL F-BF.30 - Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. Example: For example, f(x) =2 x³ or f(x) = (x+1)/(x-1) for x is not equal to 1. Algebra Determine if the inverse of a function is a function [Algebra 2] Determine the equation of the inverse of a linear, rational root, or polynomial function [Algebra 2] Core Skill AL F-LE - Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems[(Logarithms as solutions for exponentials.)] AL F-LE.31 - For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Algebra Convert between a simple exponential equation and its corresponding logarithmic equation [Algebra 2] Solve an exponential equation using logarithms [Algebra 2] Core Skill AL S - Statistics and Probability AL S-ID - Interpreting Categorical and Quantitative Data Summarize, represent, and interpret data on a single count or measurement variable 4/26/2012 - 94 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 AL S-ID.32 - Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. AL S-IC - Making Inferences and Justifying Conclusions Understand and evaluate random processes underlying statistical experiments AL S-IC.33 - Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Data Analysis, Statistics, and Probability Determine if a sample is likely to be representative of the larger population [Grade 8] AL S-IC.34 - Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. Example: For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? Data Analysis, Statistics, and Probability Determine if a sample is likely to be representative of the larger population [Grade 8] Make inferences and justify conclusions from sample surveys, experiments, and observational studies AL S-IC.35 - Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Data Analysis, Statistics, and Probability Determine if a question is likely to produce a biased survey result [Grade 8] AL S-IC.36 - Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. Data Analysis, Statistics, and Probability Analyze the effect that changing elements in a data set has on the mean, the median, or the range [Grade 8] AL S-IC.37 - Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. AL S-IC.38 - Evaluate reports based on data. AL S-MD - Using Probability to Make Decisions Use probability to evaluate outcomes of decisions AL S-MD.39 - Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Data Analysis, Statistics, and Probability Determine the probability of three or more independent events [Grade 8] Determine the probability of three or more dependent events [Grade 8] Determine the probability of an event consisting of mutually exclusive events [Grade 8] 4/26/2012 - 95 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 Make a prediction involving the probability of compound events [Grade 8] AL S-MD.40 - Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Overall Product Skills 4/26/2012 Use strategies to derive solutions for problems - 96 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig Algebra 2 with Trig AL N - Number and Quantity AL N-CN - The Complex Number System Perform arithmetic operations with complex numbers. AL N-CN.1 - Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Algebra Raise i to a power [Algebra 2] AL N-CN.2 - Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Algebra Raise i to a power [Algebra 2] Add or subtract complex numbers [Algebra 2] Multiply complex numbers [Algebra 2] Simplify an expression involving a complex denominator [Algebra 2] Core Skill Use complex numbers in polynomial identities and equations.[(Polynomials with real coefficients.)] AL N-CN.3 - Solve quadratic equations with real coefficients that have complex solutions. Algebra Solve a quadratic equation with complex solutions [Algebra 2] Core Skill AL N-CN.4 - Extend polynomial identities to the complex numbers. Example: For example, rewrite x² + 4 as (x + 2i)(x - 2i). Algebra Extend polynomial identities to the complex numbers [Algebra 2] AL N-CN.5 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Algebra Determine the graph of a given quadratic function [Algebra 1] Core Skill Solve a quadratic equation by graphing the associated quadratic function [Algebra 1] Determine a polynomial of lowest degree that has given roots or a graph of a given shape [Algebra 2] AL A - Algebra AL A-SSE - Seeing Structure in Expressions Interpret the structure of expressions[(Polynomial and rational.)] AL A-SSE.6 - Interpret expressions that represent a quantity in terms of its context. AL A-SSE.6.a - Interpret parts of an expression, such as terms, factors, and coefficients. Overall Product Skills Build comprehension of math vocabulary 4/26/2012 - 97 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig AL A-SSE.6.b - Interpret complicated expressions by viewing one or more of their parts as a single entity. Example: For example, interpret P(1+r) to the n power as the product of P and a factor not depending on P. AL A-SSE.7 - Use the structure of an expression to identify ways to rewrite it. Example: For example, see (x to the 4th power) - (y to the 4th power) as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²). Algebra Simplify a monomial algebraic expression that includes fractional exponents and/or nth roots [Algebra 2] Write expressions in equivalent forms to solve problems Algebra Simplify an algebraic expression by combining like terms [Grade 8] Core Skill Use the distributive property to simplify an algebraic expression [Grade 8] Simplify a polynomial expression by combining like terms [Algebra 1] Simplify a monomial algebraic radical expression [Algebra 1] Core Skill Simplify a rational expression involving polynomial terms [Algebra 1] Core Skill Simplify a numerical expression that includes fractional exponents and/or nth roots [Algebra 2] Decompose rational functions with quadratic denominators into sums of rational functions with linear denominators [Algebra 2] Simplify a monomial algebraic expression that includes fractional exponents and/or nth roots [Algebra 2] Identify an equivalent form of an exponential expression by using the properties of exponents [Algebra 2] Identify equivalent logarithmic expressions using the properties of Core Skill logarithms [Algebra 2] AL A-SSE.8 - Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. Example: For example, calculate mortgage payments. Algebra Find the sum of a finite geometric series [Algebra 2] Core Skill Find the sum of a finite geometric series given in summation notation [Algebra 2] WP: Solve a problem that can be represented by a finite geometric Core Skill series [Algebra 2] AL A-APR - Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials[(Beyond quadratic.)] 4/26/2012 - 98 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig AL A-APR.9 - Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Algebra Multiply two monomial algebraic expressions [Grade 8] Core Skill Simplify a polynomial expression by combining like terms [Algebra 1] Add polynomial expressions [Algebra 1] Core Skill Subtract polynomial expressions [Algebra 1] Core Skill Multiply a polynomial by a monomial [Algebra 1] Multiply two binomials of the form (x +/- a)(x +/- b) [Algebra 1] Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra 1] Core Skill Multiply two binomials of the form (ax +/- by)(cx +/- dy) [Algebra 1] Square a binomial [Algebra 1] Multiply two nonlinear binomials [Algebra 1] Multiply a trinomial by a binomial [Algebra 1] Multiply polynomial expressions [Algebra 2] Understand the relationship between zeros and factors of polynomials AL A-APR.10 - Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x). Algebra Divide a polynomial expression by a monomial [Algebra 1] Core Skill Divide a polynomial expression by a binomial [Algebra 1] Determine factors of a polynomial by applying the remainder theorem [Algebra 2] AL A-APR.11 - Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Algebra Determine the graph of a given quadratic function [Algebra 1] Core Skill Determine the solution(s) of an equation given in factored form [Algebra 1] Solve a quadratic equation by factoring [Algebra 1] Core Skill Identify the end behavior, the intercepts, or the zeros of a polynomial function [Algebra 2] Core Skill Use polynomial identities to solve problems 4/26/2012 - 99 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig AL A-APR.12 - Prove polynomial identities and use them to describe numerical relationships. Example: For example, the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate Pythagorean triples. Algebra Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra 1] Core Skill Factor the difference of two squares [Algebra 1] Factor a polynomial of degree 3 or higher [Algebra 2] WP: Solve a problem modeled by a polynomial equation [Algebra 2] Multiply polynomial expressions [Algebra 2] AL A-APR.13 - Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.[The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.] Algebra Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra 1] Core Skill Find a specified term of a binomial expression raised to a positive Core Skill integer power [Algebra 2] Rewrite rational expressions[(Linear and quadratic denominators.)] AL A-APR.14 - Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q (x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Algebra Divide a polynomial expression by a monomial [Algebra 1] Core Skill Divide a polynomial expression by a binomial [Algebra 1] Divide polynomial expressions [Algebra 2] AL A-APR.15 - Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Algebra Multiply rational expressions [Algebra 1] Core Skill Divide rational expressions [Algebra 1] Determine the LCD of two rational expressions [Algebra 1] Add or subtract two rational expressions with like denominators [Algebra 1] Add or subtract two rational expressions with unlike monomial denominators [Algebra 1] Add or subtract two rational expressions with unlike polynomial denominators [Algebra 1] 4/26/2012 Core Skill - 100 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig AL A-CED - Creating Equations Create equations that describe numbers or relationships[(Equations using all available types of expressions, including simple root functions.)] AL A-CED.16 - Create equations and inequalities in one variable and use them to solve problems. [Include equations arising from linear and quadratic functions, and simple rational and exponential functions.] Algebra WP: Use a 1-variable 1-step equation to represent a situation [Grade 7] Core Skill WP: Use a 1-variable linear inequality to represent a situation [Grade 7] WP: Use a 1-variable equation with rational coefficients to represent a situation involving two operations [Grade 8] Core Skill WP: Solve a problem involving a 1-variable, 2-step equation [Grade 8] Core Skill Solve a 2-step linear inequality in one variable [Grade 8] Core Skill WP: Use a 2-step linear inequality in one variable to represent a situation [Grade 8] WP: Solve a problem involving a 2-step linear inequality in one variable [Grade 8] Core Skill Solve a 1-variable linear equation with the variable on both sides [Algebra 1] Core Skill WP: Determine a linear equation that can be used to solve a percent problem [Algebra 1] WP: Solve a direct- or inverse-variation problem [Algebra 1] Solve a 1-variable linear inequality with the variable on one side [Algebra 1] Solve a 1-variable linear inequality with the variable on both sides Core Skill [Algebra 1] WP: Solve a problem involving a rational equation [Algebra 1] AL A-CED.17 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra WP: Use a 2-variable equation to represent a situation involving a Core Skill direct proportion [Grade 6] WP: Use a 2-variable equation with rational coefficients to represent a situation [Grade 8] Core Skill WP: Determine an equation representing a direct variation or an inverse variation [Algebra 1] Determine the graph of a linear equation given in slope-intercept, point-slope, or standard form [Algebra 1] Core Skill WP: Solve a mixture problem that can be represented by a system of linear equations [Algebra 1] 4/26/2012 - 101 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig WP: Solve a motion problem that can be represented by a system of linear equations [Algebra 1] Solve a number problem that can be represented by a linear system of equations [Algebra 1] Determine the equation of a circle from given information [Algebra 2] Determine the graph of a circle given the equation in standard form [Algebra 2] Determine the graph of a circle or an ellipse given the equation in Core Skill general form [Algebra 2] Determine the graph of a hyperbola given the equation in standard form [Algebra 2] Determine the equation of a parabola using given information [Algebra 2] Core Skill Determine the equation of an ellipse using given information [Algebra 2] AL A-CED.18 - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Algebra WP: Determine a system of linear equations that represents a given situation [Algebra 1] Core Skill WP: Determine a system of linear inequalities that represents a given situation [Algebra 1] WP: Determine possible solutions to a problem that can be represented by a system of linear inequalities [Algebra 1] WP: Solve a linear programming problem [Algebra 2] WP: Solve a problem that can be represented by a system of linear Core Skill equations in three variables [Algebra 2] WP: Solve a problem that can be modeled with a quadratic inequality [Algebra 2] AL A-CED.19 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm's law V = IR to highlight resistance R. Algebra Rewrite an equation to solve for a specified variable [Algebra 1] Core Skill AL A-REI - Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning[(Simple rational and radical.)] AL A-REI.20 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Algebra 4/26/2012 Solve a radical equation that leads to a linear equation [Algebra 1] Core Skill - 102 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig Solve a radical equation that leads to a quadratic equation [Algebra 1] Core Skill WP: Solve a problem involving a radical function [Algebra 1] Determine the excluded values of a rational algebraic expression [Algebra 1] Solve a proportion that generates a linear or quadratic equation [Algebra 1] Solve a rational equation involving terms with monomial denominators [Algebra 1] Core Skill Solve a rational equation involving terms with polynomial denominators [Algebra 1] Core Skill Represent and solve equations and inequalities graphically[(Combine polynomial, rational, radical, absolute value, and exponential functions.)] AL A-REI.21 - Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Algebra Solve a system of linear equations in two variables by graphing [Algebra 1] Determine the graph of a 2-variable absolute value inequality [Algebra 2] Relate a graph to a polynomial function given in factored form [Algebra 2] Core Skill Determine an exponential function that represents a graph [Algebra 2] Relate a logarithmic function to its graph [Algebra 2] AL F - Functions AL F-IF - Interpreting Functions Interpret functions that arise in applications in terms of the context[(Emphasize selection of appropriate models.)] AL F-IF.22 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.[Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.] Algebra Determine the slope of a line given its graph or a graph of a line with a given slope [Grade 8] Core Skill Determine the x- or y-intercept of a line given its graph [Grade 8] Core Skill 4/26/2012 - 103 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig WP: Answer a question using the graph of a quadratic function [Algebra 1] Core Skill Identify the vertex, axis of symmetry, or direction of the graph of a quadratic function [Algebra 2] Core Skill Relate changes in the values of a, b, and/or c to the graph of the equation y = a(x - b)^2 + c [Algebra 2] Core Skill Identify the end behavior, asymptotes, excluded values, or behavior near excluded values of a rational function [Algebra 2] Core Skill Identify the asymptotes, growth or decay, or half-life of an exponential function [Algebra 2] Identify the domain, range, asymptotes, or intercepts of a logarithmic function [Algebra 2] Determine the amplitude, frequency, period, or phase shift of a sine or a cosine function [Algebra 2] AL F-IF.23 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Algebra Determine the domain or range of a function [Algebra 1] WP: Determine a reasonable domain or range for a function in a given situation [Algebra 1] Core Skill Identify the domain or range of a radical function [Algebra 2] Identify the domain, range, asymptotes, or intercepts of a logarithmic function [Algebra 2] AL F-IF.24 - Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Algebra WP: Interpret the meaning of the slope of a graphed line [Grade 8] Core Skill WP: Determine a graph that can represent a situation involving a varying rate of change [Grade 8] Determine the slope of a line given two points on the line [Algebra 1] Core Skill WP: Interpret an interest rate, rate of change, initial amount, frequency of compounding and other parameters of an exponential function [Algebra 2] Core Skill Analyze functions using different representations[(Focus on using key features to guide selection of appropriate type of model function.)] AL F-IF.25 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Algebra 4/26/2012 Determine the graph of a 2-operation linear function [Grade 8] Core Skill - 104 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig Determine the graph of a given quadratic function [Algebra 1] Core Skill Determine the graph of an exponential function [Algebra 1] Determine the graph of a radical function [Algebra 1] Determine the graph of a rational function [Algebra 1] Determine the graph of a piecewise-defined function [Algebra 2] Identify the vertex, axis of symmetry, or direction of the graph of a quadratic function [Algebra 2] Core Skill Determine the graph of a rational function involving linear or quadratic denominators [Algebra 2] Core Skill Determine an exponential function that represents a graph [Algebra 2] Relate a logarithmic function to its graph [Algebra 2] Determine the graph of the inverse of a function [Algebra 2] AL F-IF.25.a - Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Algebra Determine the graph of a 2-variable absolute value equation [Algebra 1] Determine the graph of a radical function [Algebra 1] Determine the graph of a piecewise-defined function [Algebra 2] Relate a graph to a square or cube root function [Algebra 2] AL F-IF.25.b - Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Algebra Relate a graph to a polynomial function given in factored form [Algebra 2] Core Skill Identify the end behavior, the intercepts, or the zeros of a polynomial function [Algebra 2] Core Skill Determine a polynomial function which goes through specified points on a graph [Algebra 2] Core Skill AL F-IF.25.c - Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Algebra Determine the graph of an exponential function [Algebra 1] Identify the asymptotes, growth or decay, or half-life of an exponential function [Algebra 2] Determine an exponential function that represents a graph [Algebra 2] Relate a logarithmic function to its graph [Algebra 2] Identify the domain, range, asymptotes, or intercepts of a logarithmic function [Algebra 2] 4/26/2012 - 105 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig Determine the graph of a sine, cosine or tangent function [Algebra Core Skill 2] Determine the amplitude, frequency, period, or phase shift of a sine or a cosine function [Algebra 2] AL F-IF.26 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Algebra Evaluate a function written in function notation for a given value [Algebra 1] Core Skill Convert a quadratic function between vertex and standard form of the equation [Algebra 2] Compare characteristics of two linear, two quadratic, or two exponential functions represented in different ways [Algebra 2] AL F-IF.27 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Algebra Determine the graph of a given quadratic function [Algebra 1] Core Skill Solve a quadratic equation by graphing the associated quadratic function [Algebra 1] Compare characteristics of two linear, two quadratic, or two exponential functions represented in different ways [Algebra 2] AL F-BF - Building Functions Build a function that models a relationship between two quantities[(Include all types of functions studied.)] AL F-BF.28 - Write a function that describes a relationship between two quantities. AL F-BF.28.a - Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Algebra WP: Apply function operations or compositions of functions to represent a situation [Algebra 2] Build new functions from existing functions[(Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types.)] AL F-BF.29 - Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.[Include recognizing even and odd functions from their graphs and algebraic expressions for them.] Algebra 4/26/2012 Determine the equation of a function resulting from a translation and/or scaling of a given function [Algebra 2] Core Skill - 106 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig AL F-BF.30 - Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. Example: For example, f(x) =2 x³ or f(x) = (x+1)/(x-1) for x is not equal to 1. Algebra Determine if the inverse of a function is a function [Algebra 2] Determine the equation of the inverse of a linear, rational root, or polynomial function [Algebra 2] Core Skill AL F-LE - Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems[(Logarithms as solutions for exponentials.)] AL F-LE.31 - For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Algebra Convert between a simple exponential equation and its corresponding logarithmic equation [Algebra 2] Solve an exponential equation using logarithms [Algebra 2] Core Skill AL F-TF - Trigonometric Functions Extend the domain of trigonometric functions using the unit circle AL F-TF.32 - Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Overall Product Skills Demonstrate and develop understanding of mathematical concepts and ideas AL F-TF.33 - Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Algebra Determine an angle coterminal with a given angle [Algebra 2] AL F-TF.34 - Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions. Algebra Determine the sine, cosine or tangent of an angle given in degrees or radians [Algebra 2] Determine the exact value of a sine, cosine or tangent function given an angle in degrees or radians [Algebra 2] Determine the value of an inverse sine, cosine, or tangent expression [Algebra 2] WP: Determine a trigonometric function that represents a situation [Algebra 2] WP: Solve a problem involving a sine, cosine, or tangent function Core Skill [Algebra 2] Geometry and Measurement 4/26/2012 Determine a sine, cosine, or tangent ratio in a right triangle [Geometry] - 107 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig Determine a length using a sine, cosine, or tangent ratio in a right triangle [Geometry] Determine the measure of an angle using a sine, cosine, or tangent ratio in a right triangle [Geometry] WP: Determine a length in a right triangle using a sine, cosine, or tangent ratio [Geometry] WP: Determine the measure of an angle in a right triangle using a sine, cosine, or tangent ratio [Geometry] Relate a trigonometric function to the sides of a right triangle [Algebra 2] Model periodic phenomena with trigonometric functions AL F-TF.35 - Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Algebra Determine the amplitude, frequency, period, or phase shift of a sine or a cosine function [Algebra 2] Prove and apply trigonometric identities AL F-TF.36 - Prove the Pythagorean identity sin²(theta) + cos²(theta) = 1 and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle. Algebra Solve a problem involving the Pythagorean identity sin^2(theta) + cos^2(theta) = 1 [Algebra 2] AL S - Statistics and Probability AL S-ID - Interpreting Categorical and Quantitative Data Summarize, represent, and interpret data on a single count or measurement variable AL S-ID.37 - Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. AL S-IC - Making Inferences and Justifying Conclusions Understand and evaluate random processes underlying statistical experiments AL S-IC.38 - Understand statistics as a process for making inferences about population parameters based on a random sample from that population. AL S-IC.39 - Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. Example: For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? Make inferences and justify conclusions from sample surveys, experiments, and observational studies 4/26/2012 - 108 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra 2 with Trig AL S-IC.40 - Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. AL S-IC.41 - Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. Data Analysis, Statistics, and Probability Determine if a sample is likely to be representative of the larger population [Grade 8] AL S-IC.42 - Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. AL S-IC.43 - Evaluate reports based on data. AL S-MD - Using Probability to Make Decisions Use probability to evaluate outcomes of decisions[(Include more complex situations.)] AL S-MD.44 - Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). AL S-MD.45 - Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Overall Product Skills 4/26/2012 Use strategies to derive solutions for problems - 109 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I Algebra I AL N - Number and Quantity AL N-RN - The Real Number System Extend the properties of exponents to rational exponents. AL N-RN.1 - Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Example: For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)³ = 5 to the power (1/3 times 3) to hold, so (5 to the 1/3 power) cubed must equal 5. Algebra Evaluate a numerical expression involving one or more exponents Core Skill and multiple forms of rational numbers [Algebra 1] Apply the product of powers property to a monomial algebraic expression [Algebra 1] Core Skill Apply the power of a power property to a monomial algebraic expression [Algebra 1] Core Skill Apply the power of a product property to a monomial algebraic expression [Algebra 1] Core Skill Apply the quotient of powers property to monomial algebraic expressions [Algebra 1] Core Skill Apply the power of a quotient property to monomial algebraic expressions [Algebra 1] Core Skill Simplify a numerical expression that includes fractional exponents and/or nth roots [Algebra 2] AL N-RN.2 - Rewrite expressions involving radicals and rational exponents using the properties of exponents. Algebra Simplify a monomial numerical expression involving the square root of a whole number [Algebra 1] Core Skill Multiply monomial numerical expressions involving radicals [Algebra 1] Divide monomial numerical expressions involving radicals [Algebra 1] Simplify a monomial algebraic radical expression [Algebra 1] Core Skill Multiply monomial algebraic radical expressions [Algebra 1] Divide monomial algebraic radical expressions [Algebra 1] Simplify a numerical expression that includes fractional exponents and/or nth roots [Algebra 2] Use properties of rational and irrational numbers. 4/26/2012 - 110 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I AL N-RN.3 - Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Algebra Simplify an algebraic expression by combining like terms [Grade 8] Core Skill Multiply two monomial algebraic expressions [Grade 8] Core Skill AL N-Q - Quantities Reason quantitatively and use units to solve problems.[(Foundation for work with expressions, equations, and functions.)] AL N-Q.4 - Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Algebra WP: Solve a direct- or inverse-variation problem [Algebra 1] WP: Solve a multi-step problem involving dimensional analysis [Algebra 2] Geometry and Measurement Convert a rate from one unit to another with a change in both units [Grade 7] Core Skill AL N-Q.5 - Define appropriate quantities for the purpose of descriptive modeling. AL N-Q.6 - Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Algebra WP: Determine the degree of accuracy of a computation [Algebra 2] AL A - Algebra AL A-SSE - Seeing Structure in Expressions Interpret the structure of expressions[(Linear, exponential, quadratic.)] AL A-SSE.7 - Interpret expressions that represent a quantity in terms of its context. AL A-SSE.7.a - Interpret parts of an expression, such as terms, factors, and coefficients. Overall Product Skills Build comprehension of math vocabulary AL A-SSE.7.b - Interpret complicated expressions by viewing one or more of their parts as a single entity. Example: For example, interpret P(1+r) to the n power as the product of P and a factor not depending on P. AL A-SSE.8 - Use the structure of an expression to identify ways to rewrite it. Example: For example, see (x to the 4th power) - (y to the 4th power) as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²). Algebra 4/26/2012 Simplify a monomial algebraic expression that includes fractional exponents and/or nth roots [Algebra 2] - 111 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Write expressions in equivalent forms to solve problems[(Quadratic and exponential.)] Algebra Simplify an algebraic expression by combining like terms [Grade 8] Algebra I Core Skill Use the distributive property to simplify an algebraic expression [Grade 8] Simplify a monomial algebraic radical expression [Algebra 1] Core Skill Simplify a numerical expression that includes fractional exponents and/or nth roots [Algebra 2] Simplify a monomial algebraic expression that includes fractional exponents and/or nth roots [Algebra 2] Identify an equivalent form of an exponential expression by using the properties of exponents [Algebra 2] Solve an exponential equation using the properties of exponents [Algebra 2] Identify equivalent logarithmic expressions using the properties of Core Skill logarithms [Algebra 2] AL A-SSE.9 - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. AL A-SSE.9.a - Factor a quadratic expression to reveal the zeros of the function it defines. Algebra Factor trinomials that result in factors of the form (x +/- a)(x +/b) [Algebra 1] Factor trinomials that result in factors of the form (ax +/- b)(cx +/- Core Skill d) [Algebra 1] Factor trinomials that result in factors of the form (ax +/- by)(cx +/- dy) [Algebra 1] Factor the difference of two squares [Algebra 1] Factor a perfect-square trinomial [Algebra 1] Solve a quadratic equation by factoring [Algebra 1] Core Skill AL A-SSE.9.b - Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Algebra Complete the square of a quadratic 1-variable equation [Algebra 2] AL A-SSE.9.c - Determine a quadratic equation when given its graph or roots. Algebra Solve a quadratic equation by factoring [Algebra 1] Solve a quadratic equation using the quadratic formula [Algebra 1] 4/26/2012 Core Skill Core Skill - 112 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I AL A-SSE.9.d - Use the properties of exponents to transform expressions for exponential functions. Example: For example the expression 1.15 to the t power can be rewritten as ((1.15 to the 1/12 power) to the 12t power) is approximately equal to (1.012 to the 12t power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Algebra Evaluate a numerical expression involving one or more exponents Core Skill and multiple forms of rational numbers [Algebra 1] Apply properties of exponents to monomial algebraic expressions [Algebra 1] WP: Evaluate an exponential growth or an exponential decay function [Algebra 1] Solve a problem involving exponential growth or exponential decay [Algebra 1] AL A-APR - Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials[(Linear and quadratic.)] AL A-APR.10 - Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Algebra Multiply two monomial algebraic expressions [Grade 8] Core Skill Simplify a polynomial expression by combining like terms [Algebra 1] Add polynomial expressions [Algebra 1] Core Skill Subtract polynomial expressions [Algebra 1] Core Skill Multiply a polynomial by a monomial [Algebra 1] Multiply two binomials of the form (x +/- a)(x +/- b) [Algebra 1] Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra 1] Core Skill Multiply two binomials of the form (ax +/- by)(cx +/- dy) [Algebra 1] Square a binomial [Algebra 1] Multiply two nonlinear binomials [Algebra 1] Multiply a trinomial by a binomial [Algebra 1] Multiply polynomial expressions [Algebra 2] AL A-CED - Creating Equations Create equations that describe numbers or relationships[(Linear, quadratic, and exponential (integer inputs only); for Standard 13, linear only.)] 4/26/2012 - 113 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I AL A-CED.11 - Create equations and inequalities in one variable and use them to solve problems. [Include equations arising from linear and quadratic functions, and simple rational and exponential functions.] Algebra WP: Use a 1-variable 1-step equation to represent a situation [Grade 7] Core Skill WP: Use a 1-variable equation with rational coefficients to represent a situation involving two operations [Grade 8] Core Skill WP: Use a 2-step linear inequality in one variable to represent a situation [Grade 8] WP: Determine a linear equation that can be used to solve a percent problem [Algebra 1] WP: Solve a direct- or inverse-variation problem [Algebra 1] Solve a 1-variable linear inequality with the variable on one side [Algebra 1] Solve a 1-variable linear inequality with the variable on both sides Core Skill [Algebra 1] AL A-CED.12 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra WP: Use a 2-variable equation with rational coefficients to represent a situation [Grade 8] Core Skill WP: Determine an equation representing a direct variation or an inverse variation [Algebra 1] Determine the graph of a linear equation given in slope-intercept, point-slope, or standard form [Algebra 1] Core Skill WP: Solve a mixture problem that can be represented by a system of linear equations [Algebra 1] WP: Solve a motion problem that can be represented by a system of linear equations [Algebra 1] Solve a number problem that can be represented by a linear system of equations [Algebra 1] AL A-CED.13 - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Algebra WP: Determine a system of linear equations that represents a given situation [Algebra 1] Core Skill WP: Determine a system of linear inequalities that represents a given situation [Algebra 1] WP: Determine possible solutions to a problem that can be represented by a system of linear inequalities [Algebra 1] WP: Solve a linear programming problem [Algebra 2] 4/26/2012 - 114 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I WP: Solve a problem that can be represented by a system of linear Core Skill equations in three variables [Algebra 2] AL A-CED.14 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm's law V = IR to highlight resistance R. Algebra Rewrite an equation to solve for a specified variable [Algebra 1] Core Skill AL A-REI - Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning[(Master linear; learn as general principle.)] AL A-REI.15 - Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Overall Product Skills Develop collaboration skills by explaining mathematical reasoning and strategies Solve equations and inequalities in one variable[(Linear inequalities; literal that are linear in the variables being solved for; quadratics with real solutions.)] AL A-REI.16 - Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Algebra Solve a 1-step equation involving rational numbers [Grade 8] Core Skill Solve a 2-step equation involving rational numbers [Grade 8] Core Skill Solve a 2-step linear inequality in one variable [Grade 8] Core Skill Solve a 1-variable linear equation with the variable on both sides [Algebra 1] Core Skill Solve a 1-variable linear inequality with the variable on one side [Algebra 1] Solve a 1-variable linear inequality with the variable on both sides Core Skill [Algebra 1] Solve a 1-variable compound inequality [Algebra 1] AL A-REI.17 - Solve quadratic equations in one variable. AL A-REI.17.a - Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form. Algebra Complete the square of a quadratic 1-variable equation [Algebra 2] AL A-REI.17.b - Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Algebra 4/26/2012 Solve a quadratic equation by taking the square root [Algebra 1] - 115 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I Solve a quadratic equation by factoring [Algebra 1] Core Skill Solve a quadratic equation using the quadratic formula [Algebra 1] Core Skill Use the discriminant to determine the number of real solutions [Algebra 1] WP: Use a given quadratic equation to solve a problem [Algebra 1] Complete the square of a quadratic 1-variable equation [Algebra 2] Solve a quadratic equation with complex solutions [Algebra 2] Core Skill Solve systems of equations[(Linear-linear and linear-quadratic.)] AL A-REI.18 - Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Algebra Solve a system of linear equations in two variables by substitution [Algebra 1] Solve a system of linear equations in two variables by elimination [Algebra 1] Solve a system of linear equations in two variables using any method [Algebra 1] Core Skill AL A-REI.19 - Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Algebra Solve a system of linear equations in two variables by graphing [Algebra 1] Solve a system of linear equations in two variables by substitution [Algebra 1] Solve a system of linear equations in two variables by elimination [Algebra 1] Solve a system of linear equations in two variables using any method [Algebra 1] Core Skill AL A-REI.20 - Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3. Algebra Solve a system consisting of a linear equation and a nonlinear equation in two variables [Algebra 2] Core Skill Represent and solve equations and inequalities graphically[(Linear and exponential; learn as general principle.)] AL A-REI.21 - Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Algebra 4/26/2012 Determine the graph of a linear equation given in slope-intercept, point-slope, or standard form [Algebra 1] Core Skill - 116 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Determine the graph of a given quadratic function [Algebra 1] Algebra I Core Skill Determine the graph of a rational function [Algebra 1] Relate a graph to a square or cube root function [Algebra 2] Determine the graph of a vertically or horizontally oriented parabola [Algebra 2] Core Skill AL A-REI.22 - Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Algebra Solve a system of linear equations in two variables by graphing [Algebra 1] Determine the graph of a 2-variable absolute value inequality [Algebra 2] Relate a graph to a polynomial function given in factored form [Algebra 2] Core Skill Determine an exponential function that represents a graph [Algebra 2] Relate a logarithmic function to its graph [Algebra 2] AL A-REI.23 - Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Algebra Solve a 2-variable linear inequality for the dependent variable [Algebra 1] Determine the graph of a 2-variable linear inequality [Algebra 1] Determine the graph of the solution set of a system of linear inequalities in two variables [Algebra 1] AL F - Functions AL F-IF - Interpreting Functions Understand the concept of a function and use function notation[(Learn as general principle; focus on linear and exponential and on arithmetic and geometric sequences.)] AL F-IF.24 - Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Algebra 4/26/2012 Determine the domain or range of a function [Algebra 1] - 117 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I AL F-IF.25 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Algebra WP: Determine a reasonable domain or range for a function in a given situation [Algebra 1] Core Skill Evaluate a function written in function notation for a given value [Algebra 1] Core Skill AL F-IF.26 - Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n greater than or equal to 1. Algebra Evaluate a function written in function notation for a given value [Algebra 1] Core Skill Identify a possible formula for function of n, where n is a whole number [Algebra 2] Determine the recursive formula for an arithmetic sequence [Algebra 2] Interpret functions that arise in applications in terms of the context[(Linear, exponential, and quadratic.)] AL F-IF.27 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.[Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.] Algebra Determine the graph of an exponential function [Algebra 1] Determine the graph of a rational function [Algebra 1] Identify the end behavior, asymptotes, excluded values, or behavior near excluded values of a rational function [Algebra 2] Core Skill AL F-IF.28 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Algebra WP: Determine a reasonable domain or range for a function in a given situation [Algebra 1] Core Skill AL F-IF.29 - Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Algebra WP: Interpret an interest rate, rate of change, initial amount, frequency of compounding and other parameters of an exponential function [Algebra 2] Core Skill Analyze functions using different representations[(Linear, exponential, quadratic, absolute value, step, piecewise-defined.)] 4/26/2012 - 118 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I AL F-IF.30 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. AL F-IF.30.a - Graph linear and quadratic functions and show intercepts, maxima, and minima. Algebra Determine the graph of a linear equation given in slope-intercept, Core Skill point-slope, or standard form [Algebra 1] Determine the graph of a given quadratic function [Algebra 1] Core Skill Solve a quadratic equation by graphing the associated quadratic function [Algebra 1] Determine if the graph of a relation is modeled by a linear or quadratic function or represents exponential growth or decay [Algebra 2] AL F-IF.30.b - Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Algebra Determine the graph of a 2-variable absolute value equation [Algebra 1] Determine the graph of a radical function [Algebra 1] Determine the graph of a piecewise-defined function [Algebra 2] Relate a graph to a square or cube root function [Algebra 2] AL F-IF.30.c - Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Algebra Determine the graph of an exponential function [Algebra 1] Determine an exponential function that represents a graph [Algebra 2] AL F-IF.31 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Algebra Evaluate a function written in function notation for a given value [Algebra 1] Core Skill AL F-IF.31.a - Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Algebra Solve a quadratic equation by factoring [Algebra 1] Core Skill Complete the square of a quadratic 1-variable equation [Algebra 2] AL F-IF.31.b - Use the properties of exponents to interpret expressions for exponential functions. Example: For example, identify percent rate of change in functions such as y = (1.02) to the t power, y = (0.97) to the t power, y = (1.01) to the 12t power, y = (1.2) to the t/10 power, and classify them as representing exponential growth or decay. Algebra 4/26/2012 Identify an equivalent form of an exponential expression by using the properties of exponents [Algebra 2] - 119 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I WP: Determine an exponential function that represents a situation such as exponential growth or decay [Algebra 2] Solve an exponential equation using the properties of exponents [Algebra 2] WP: Solve an exponential growth or decay problem [Algebra 2] Core Skill AL F-IF.32 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Algebra Compare characteristics of two linear, two quadratic, or two exponential functions represented in different ways [Algebra 2] AL F-BF - Building Functions Build a function that models a relationship between two quantities[(For standards 33 and 34, linear, exponential, and quadratic.)] AL F-BF.33 - Write a function that describes a relationship between two quantities. AL F-BF.33.a - Determine an explicit expression, a recursive process, or steps for calculation from a context. Algebra Determine the recursive formula for an arithmetic sequence [Algebra 2] Determine the recursive formula for a geometric sequence [Algebra 2] AL F-BF.33.b - Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Algebra WP: Evaluate an exponential growth or an exponential decay function [Algebra 1] Identify the asymptotes, growth or decay, or half-life of an exponential function [Algebra 2] WP: Determine an exponential function that represents a situation such as exponential growth or decay [Algebra 2] AL F-BF.34 - Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Algebra WP: Solve a problem that can be represented by an arithmetic sequence [Algebra 2] Find a specific term of a geometric sequence [Algebra 2] Determine the common ratio for a geometric sequence [Algebra 2] Determine the recursive formula for a geometric sequence [Algebra 2] 4/26/2012 - 120 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I Determine the explicit formula for a geometric sequence [Algebra 2] WP: Solve a problem that can be represented by a geometric sequence [Algebra 2] Core Skill Build new functions from existing functions[(Linear, exponential, quadratic, and absolute value; for standard 36a, linear only.)] AL F-BF.35 - Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.[Include recognizing even and odd functions from their graphs and algebraic expressions for them.] Algebra Determine the resulting change in a linear function, f(x), when replacing x with x + b, where b is an integer [Algebra 2] AL F-BF.36 - Find inverse functions. AL F-BF.36.a - Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. Example: For example, f(x) =2 x³ or f(x) = (x+1)/(x-1) for x is not equal to 1. Algebra Determine values of the inverse of a function using a table or a graph [Algebra 2] Determine if the inverse of a function is a function [Algebra 2] Determine the equation of the inverse of a linear, rational root, or polynomial function [Algebra 2] Core Skill AL F-LE - Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems AL F-LE.37 - Distinguish between situations that can be modeled with linear functions and with exponential functions. AL F-LE.37.a - Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Algebra Determine if the graph of a relation is modeled by a linear or quadratic function or represents exponential growth or decay [Algebra 2] AL F-LE.37.b - Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Algebra Determine if a table or an equation represents a direct variation, an inverse variation, or neither [Algebra 1] WP: Determine an equation representing a direct variation or an inverse variation [Algebra 1] 4/26/2012 - 121 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I AL F-LE.37.c - Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Algebra Identify the asymptotes, growth or decay, or half-life of an exponential function [Algebra 2] WP: Determine an exponential function that represents a situation such as exponential growth or decay [Algebra 2] Determine if the graph of a relation is modeled by a linear or quadratic function or represents exponential growth or decay [Algebra 2] AL F-LE.38 - Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Algebra Determine the graph of a line for a given table of values [Grade 8] WP: Determine a linear graph that can represent a situation [Grade 8] Determine the graph of a line using given information [Algebra 1] Core Skill Determine the graph of an exponential function [Algebra 1] Determine an exponential function that represents a graph [Algebra 2] WP: Determine an exponential function that represents a situation such as exponential growth or decay [Algebra 2] Determine if the graph of a relation is modeled by a linear or quadratic function or represents exponential growth or decay [Algebra 2] WP: Solve a problem that can be represented by an arithmetic sequence [Algebra 2] WP: Solve a problem that can be represented by a geometric sequence [Algebra 2] Core Skill AL F-LE.39 - Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Interpret expressions for functions in terms of the situation they model[(Linear and exponential of form f(x) = b to the x power + k.)] AL F-LE.40 - Interpret the parameters in a linear or exponential function in terms of a context. Algebra WP: Interpret an interest rate, rate of change, initial amount, Core Skill frequency of compounding and other parameters of an exponential function [Algebra 2] AL S - Statistics and Probability AL S-ID - Interpreting Categorical and Quantitative Data Summarize, represent, and interpret data on a single count or measurement variable 4/26/2012 - 122 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I AL S-ID.41 - Represent data with plots on the real number line (dot plots, histograms, and box plots). Data Analysis, Statistics, and Probability Use a histogram to represent data [Grade 7] Core Skill Use a box-and-whisker plot to organize data [Grade 8] AL S-ID.42 - Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Data Analysis, Statistics, and Probability Determine the quartiles of a data set [Grade 8] Analyze the effect that changing elements in a data set has on the mean, the median, or the range [Grade 8] Determine the median of the data in a frequency table or a bar graph [Grade 8] Determine the mean of the data in a frequency table or a bar graph [Grade 8] AL S-ID.43 - Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Data Analysis, Statistics, and Probability Analyze the effect that changing elements in a data set has on the mean, the median, or the range [Grade 8] Summarize, represent, and interpret data on two categorical and quantitative variables[(Linear focus, discuss general principle.)] AL S-ID.44 - Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Data Analysis, Statistics, and Probability Determine the median of the data in a frequency table or a bar graph [Grade 8] Determine the mean of the data in a frequency table or a bar graph [Grade 8] AL S-ID.45 - Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. AL S-ID.45.a - Fit a function to the data; use functions fitted to data to solve problems in the context of the data.[Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.] Data Analysis, Statistics, and Probability Use a scatter plot to organize data [Grade 8] Determine if a scatter plot shows a positive relationship, a negative relationship, or no relationship between the variables [Grade 8] Answer a question using information from a scatter plot [Grade 8] 4/26/2012 - 123 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Algebra I AL S-ID.45.b - Informally assess the fit of a function by plotting and analyzing residuals. AL S-ID.45.c - Fit a linear function for a scatter plot that suggests a linear association. Data Analysis, Statistics, and Use a scatter plot to organize data [Grade 8] Probability Determine if a scatter plot shows a positive relationship, a negative relationship, or no relationship between the variables [Grade 8] Interpret linear models AL S-ID.46 - Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Algebra WP: Interpret the meaning of the slope of a graphed line [Grade 8] Core Skill WP: Interpret the meaning of the y-intercept of a graphed line [Grade 8] Core Skill AL S-ID.47 - Compute (using technology) and interpret the correlation coefficient of a linear fit. AL S-ID.48 - Distinguish between correlation and causation. AL S-CP - Conditional Probability and the Rules of Probability Understand independence and conditional probability and use them to interpret data[(Link to data from simulations or experiments.)] AL S-CP.49 - Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not"). Data Analysis, Statistics, and Probability Answer a question using information from a Venn diagram with three circles [Grade 8] Use a Venn diagram to organize summarized data [Grade 8] AL S-CP.50 - Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Data Analysis, Statistics, and Probability Determine the probability of three or more independent events [Grade 8] Determine the probability of an event consisting of mutually exclusive events [Grade 8] 4/26/2012 - 124 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Discrete Mathematics Discrete Mathematics AL N - Number and Quantity AL N.1 - Analyze topics from elementary number theory, including perfect numbers and prime numbers, to determine properties of integers. AL N.2 - Determine characteristics of sequences, including the Fibonacci sequence, the triangular numbers, and pentagonal numbers. Example: Write a sequence of the first 10 triangular numbers and hypothesize a formula to find the nth triangular number. Algebra Find a specific term of an arithmetic sequence [Algebra 2] Determine the recursive formula for an arithmetic sequence [Algebra 2] Determine the explicit formula for an arithmetic sequence [Algebra 2] WP: Solve a problem that can be represented by an arithmetic sequence [Algebra 2] Find a specific term of a geometric sequence [Algebra 2] Determine the common ratio for a geometric sequence [Algebra 2] Determine the recursive formula for a geometric sequence [Algebra 2] Determine the explicit formula for a geometric sequence [Algebra 2] Identify a given sequence as arithmetic, geometric, or neither [Algebra 2] WP: Solve a problem that can be represented by a geometric sequence [Algebra 2] Core Skill AL N.3 - Use the recursive process and difference equations to create fractals, population growth models, sequences, series, and compound interest models. Algebra Determine the recursive formula for an arithmetic sequence [Algebra 2] Determine the recursive formula for a geometric sequence [Algebra 2] AL N.4 - Convert between base ten and other bases. AL A - Algebra AL A.5 - Determine results of operations upon 3 × 3 and larger matrices, including matrix addition and multiplication of a matrix by a matrix, vector, or scalar. Algebra Multiply a matrix by a scalar [Algebra 2] Add or subtract matrices [Algebra 2] Combine matrix operations [Algebra 2] 4/26/2012 - 125 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Discrete Mathematics Multiply matrices [Algebra 2] AL A.6 - Analyze determinants and inverses of 2 × 2, 3 × 3, and larger matrices to determine the nature of the solution set of the corresponding system of equations, including solving systems of equations in three variables by echelon row reduction and matrix inverse. Algebra Determine the determinant of a matrix [Algebra 2] Represent a system of linear equations as a single matrix equation [Algebra 2] Determine the inverse of a matrix [Algebra 2] Core Skill AL A.7 - Solve problems through investigation and application of existence and nonexistence of Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits. Example: Show why a 5 × 5 grid has no Hamilton circuit. AL A.7.a - Develop optimal solutions of application-based problems using existing and student-created algorithms. AL A.8 - Apply algorithms, including Kruskal's and Prim's, relating to minimum weight spanning trees, networks, flows, and Steiner trees. AL A.8.a - Use shortest path techniques to find optimal shipping routes. AL A.9 - Determine a minimum project time using algorithms to schedule tasks in order, including critical path analysis, the list-processing algorithm, and student-created algorithms. AL G - Geometry AL G.10 - Use vertex-coloring techniques and matching techniques to solve application-based problems. Example: Use graph-coloring techniques to color a map of the western states of the United States so no adjacent states are the same color, including determining the minimum number of colors needed and why no fewer colors may be used. AL G.11 - Solve application-based logic problems using Venn diagrams, truth tables, and matrices. Data Analysis, Statistics, and Answer a question using information from a Venn diagram with Probability three circles [Grade 8] Use a Venn diagram to organize summarized data [Grade 8] AL S - Statistics and Probability AL S.12 - Use combinatorial reasoning and counting techniques to solve application-based problems. Example: Determine the probability of a safe opening on the first attempt given the combination uses the digits 2, 4, 6, and 8 with the order unknown. Answer: The probability of the safe opening on the first attempt is 1/24. Algebra Evaluate an expression involving factorial, permutation, or combination notation [Algebra 2] Data Analysis, Statistics, and Probability Determine the number of combinations possible in a given situation [Grade 8] 4/26/2012 - 126 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Discrete Mathematics AL S.13 - Analyze election data to compare election methods and voting apportionment, including determining strength within specific groups. 4/26/2012 - 127 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry Geometry AL G - Geometry AL G-CO - Congruence Experiment with transformations in the plane AL G-CO.1 - Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. AL G-CO.2 - Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Geometry and Measurement Relate the coordinates of a preimage or an image to a translation described using mapping notation [Geometry] Determine the coordinates of a preimage or an image given a reflection across a horizontal line, a vertical line, the line y = x, or the line y = -x [Geometry] Relate the coordinates of a preimage or an image to a dilation centered at the origin [Geometry] Determine the angle of rotational symmetry of a figure [Geometry] Determine the coordinates of the image of a figure after two transformations of the same type [Geometry] Determine the coordinates of the image of a figure after two transformations of different types [Geometry] AL G-CO.3 - Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Geometry and Measurement Determine the coordinates of a preimage or an image given a reflection across a horizontal line, a vertical line, the line y = x, or the line y = -x [Geometry] Determine the angle of rotational symmetry of a figure [Geometry] AL G-CO.4 - Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Geometry and Measurement Determine the angle of rotational symmetry of a figure [Geometry] AL G-CO.5 - Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Geometry and Measurement 4/26/2012 Relate the coordinates of a preimage or an image to a translation described using mapping notation [Geometry] - 128 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry Determine the coordinates of a preimage or an image given a reflection across a horizontal line, a vertical line, the line y = x, or the line y = -x [Geometry] Determine the angle of rotational symmetry of a figure [Geometry] Determine the coordinates of the image of a figure after two transformations of the same type [Geometry] Determine the coordinates of the image of a figure after two transformations of different types [Geometry] Understand congruence in terms of rigid motions[(Build on rigid motions as a familiar starting point for development of concept of geometric proof.)] AL G-CO.6 - Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Geometry and Measurement Relate the coordinates of a preimage or an image to a translation described using mapping notation [Geometry] Determine the coordinates of a preimage or an image given a reflection across a horizontal line, a vertical line, the line y = x, or the line y = -x [Geometry] Determine the coordinates of the image of a figure after two transformations of the same type [Geometry] Determine the coordinates of the image of a figure after two transformations of different types [Geometry] AL G-CO.7 - Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Geometry and Measurement Identify a triangle congruence postulate that justifies a congruence Core Skill statement [Geometry] Identify congruent triangles using triangle congruence postulates or theorems [Geometry] Overall Product Skills Core Skill Use strategies to derive solutions for problems AL G-CO.8 - Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Geometry and Measurement Identify a triangle congruence postulate that justifies a congruence Core Skill statement [Geometry] Identify congruent triangles using triangle congruence postulates or theorems [Geometry] Core Skill Prove geometric theorems[(Focus on validity of underlying reasoning while using variety of ways of writing proofs.)] 4/26/2012 - 129 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry AL G-CO.9 - Prove theorems about lines and angles.[Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.] Geometry and Measurement Identify angle relationships formed by parallel lines cut by a transversal [Grade 8] Determine an angle measure formed by parallel lines cut by a transversal [Grade 8] Identify angle relationships formed by multiple lines and transversals [Geometry] Determine the measure of an angle formed by parallel lines and one or more transversals [Geometry] Core Skill Identify parallel lines using angle relationships [Geometry] Determine the measure of an angle in a figure involving parallel and/or perpendicular lines [Geometry] Solve a problem involving a point on the bisector of an angle [Geometry] AL G-CO.10 - Prove theorems about triangles.[Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.] Geometry and Measurement Identify a triangle congruence postulate that justifies a congruence Core Skill statement [Geometry] Identify congruent triangles using triangle congruence postulates or theorems [Geometry] Core Skill AL G-CO.11 - Prove theorems about parallelograms.[Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.] Geometry and Measurement Determine a length or an angle measure using general properties of parallelograms [Geometry] Make geometric constructions[(Formalize and explain processes.)] AL G-CO.12 - Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). [Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.] Geometry and Measurement Determine a length using a perpendicular bisector of a chord [Geometry] AL G-CO.13 - Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 4/26/2012 - 130 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry AL G-SRT - Similarity, Right Triangles, and Trigonometry Understand similarity in terms of similarity transformations AL G-SRT.14 - Verify experimentally the properties of dilations given by a center and a scale factor: Geometry and Measurement Determine the scale factor relating two similar polygons [Geometry] Relate the coordinates of a preimage or an image to a dilation centered at the origin [Geometry] AL G-SRT.14.a - A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. AL G-SRT.14.b - The dilation of a line segment is longer or shorter in the ratio given by the scale factor. AL G-SRT.15 - Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Geometry and Measurement Determine the scale factor relating two similar polygons [Geometry] Identify similar triangles using triangle similarity postulates or theorems [Geometry] Core Skill AL G-SRT.16 - Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity AL G-SRT.17 - Prove theorems about triangles.[Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.] Geometry and Measurement Determine the length of the hypotenuse of a right triangle using the Pythagorean theorem [Grade 8] Core Skill Determine the length of a leg of a right triangle using the Pythagorean theorem [Grade 8] Core Skill Solve for the length of a side of a triangle using the Pythagorean theorem [Geometry] Identify a triangle similarity postulate that justifies a similarity statement [Geometry] Core Skill Identify similar triangles using triangle similarity postulates or theorems [Geometry] Core Skill AL G-SRT.18 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Geometry and Measurement 4/26/2012 Determine the length of a side or the measure of an angle in similar triangles [Geometry] - 131 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry Determine a length given the perimeters of similar triangles or the lengths of corresponding interior line segments [Geometry] Determine a length in a triangle using a midsegment [Geometry] Determine a length using similar triangles formed by the altitude to the hypotenuse of a right triangle [Geometry] WP: Determine a length using similarity [Geometry] Define trigonometric ratios and solve problems involving right triangles Geometry and Measurement Determine a length using the properties of a 45-45-90 degree triangle or a 30-60-90 degree triangle [Geometry] Core Skill Core Skill Solve a problem using multiple non-trigonometric right-triangle relationships [Geometry] WP: Determine a length using the properties of a 45-45-90 degree triangle or a 30-60-90 degree triangle [Geometry] AL G-SRT.19 - Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Geometry and Measurement Determine a sine, cosine, or tangent ratio in a right triangle [Geometry] AL G-SRT.20 - Explain and use the relationship between the sine and cosine of complementary angles. Algebra Determine the sine, cosine or tangent of an angle given in degrees or radians [Algebra 2] Determine the exact value of a sine, cosine or tangent function given an angle in degrees or radians [Algebra 2] WP: Solve a problem involving a sine, cosine, or tangent function Core Skill [Algebra 2] Geometry and Measurement Determine a sine, cosine, or tangent ratio in a right triangle [Geometry] WP: Determine the measure of an angle in a right triangle using a sine, cosine, or tangent ratio [Geometry] AL G-SRT.21 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Algebra WP: Solve a problem involving a sine, cosine, or tangent function Core Skill [Algebra 2] Geometry and Measurement WP: Use the Pythagorean theorem to find a length or a distance [Grade 8] Core Skill Determine a length in a complex figure using the Pythagorean theorem [Geometry] Core Skill WP: Solve a problem involving a complex figure using the Pythagorean theorem [Geometry] 4/26/2012 - 132 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry Determine a length using a sine, cosine, or tangent ratio in a right triangle [Geometry] WP: Determine a length in a right triangle using a sine, cosine, or tangent ratio [Geometry] WP: Determine the measure of an angle in a right triangle using a sine, cosine, or tangent ratio [Geometry] Apply trigonometry to general triangles AL G-SRT.22 - Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Geometry and Measurement Approximate the area of a regular polygon using trigonometry [Geometry] Determine the area of a triangle using the formula A = 1/2absin(c) [Algebra 2] AL G-SRT.23 - Prove the Laws of Sines and Cosines and use them to solve problems. Algebra Solve a problem involving the law of sines [Algebra 2] Geometry and Measurement Solve a problem involving the law of cosines [Algebra 2] AL G-SRT.24 - Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Algebra Solve a problem involving the law of sines [Algebra 2] Geometry and Measurement Solve a problem involving the law of cosines [Algebra 2] WP: Solve a problem involving the law of sines or the law of cosines [Algebra 2] AL G-C - Circles Understand and apply theorems about circles Geometry and Measurement Solve a problem involving intersecting chords, tangents, and/or secants of a circle [Geometry] Determine the radius, center, or diameter of a circle given an equation [Geometry] Determine the area of a segment of a circle [Geometry] WP: Determine a length or an area involving a sector of a circle [Geometry] Core Skill Determine the measure of an arc or an angle given the area of a sector of a circle [Geometry] Core Skill AL G-C.25 - Prove that all circles are similar. 4/26/2012 - 133 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry AL G-C.26 - Identify and describe relationships among inscribed angles, radii, and chords.[Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.] Geometry and Measurement Determine the measure of an arc or a central angle using the relationship between the arc and the central angle [Geometry] Core Skill AL G-C.27 - Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. AL G-C.28 - Construct a tangent line from a point outside a given circle to the circle. Find arc lengths and areas of sectors of circles[(Radian introduced only as unit of measure.)] Geometry and Measurement Determine the measure of an arc or a central angle using the Core Skill relationship between the arc and the central angle [Geometry] Determine the area of a sector of a circle [Geometry] Core Skill AL G-C.29 - Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Geometry and Measurement Determine the measure of an arc or a central angle using the relationship between the arc and the central angle [Geometry] Core Skill Determine the area of a sector of a circle [Geometry] Core Skill AL G-GPE - Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section AL G-GPE.30 - Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Geometry and Measurement Determine an equation of a circle [Geometry] Determine the radius, center, or diameter of a circle given an equation [Geometry] Use coordinates to prove simple geometric theorems algebraically[(Include distance formula; relate to Pythagorean Theorem.)] AL G-GPE.31 - Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, the square root of 3) lies on the circle centered at the origin and containing the point (0, 2). Geometry and Measurement Determine the midpoint of a line segment given the coordinates of Core Skill the endpoints [Geometry] Determine the area of a right triangle or a rectangle given the coordinates of the vertices of the figure [Geometry] Core Skill Determine if lines through points with given coordinates are parallel or perpendicular [Geometry] 4/26/2012 - 134 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry Determine the coordinates of a point through which a line must pass in order to be parallel or perpendicular to a given line [Geometry] AL G-GPE.32 - Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Algebra Determine if two lines are perpendicular or parallel given the equations of the lines [Algebra 1] Geometry and Measurement Determine if lines through points with given coordinates are parallel or perpendicular [Geometry] Determine the coordinates of a point through which a line must pass in order to be parallel or perpendicular to a given line [Geometry] AL G-GPE.33 - Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Geometry and Measurement Determine the midpoint of a line segment given the coordinates of Core Skill the endpoints [Geometry] Solve a problem involving the midpoint formula [Geometry] AL G-GPE.34 - Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Geometry and Measurement Solve a problem involving the distance formula [Geometry] Core Skill Determine the area of a right triangle or a rectangle given the coordinates of the vertices of the figure [Geometry] Core Skill Use coordinates to prove simple geometric theorems algebraically. AL G-GPE.35 - Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. Geometry and Measurement Solve a problem involving perimeters or areas of two different shapes [Grade 8] Determine the ratio of the perimeters or areas of similar shapes [Grade 8] Determine the area of a right triangle or a rectangle given the coordinates of the vertices of the figure [Geometry] Core Skill Determine the area of a quadrilateral [Geometry] Core Skill Solve a problem involving regular polygons and/or circles that have the same area or perimeter [Geometry] AL G-GMD - Geometric Measurement and Dimension Explain volume formulas and use them to solve problems 4/26/2012 - 135 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry AL G-GMD.36 - Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.[Use dissection arguments, Cavalieri's principle, and informal limit arguments.] AL G-GMD.37 - Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Geometry and Measurement Solve a problem involving the surface area or the volume of a pyramid or a cone [Grade 8] Determine the volume of a sphere or hemisphere [Geometry] Core Skill WP: Determine the volume of a sphere or hemisphere [Geometry] Determine the volume of a complex solid figure [Geometry] WP: Solve a problem involving the volume of a complex solid figure [Geometry] AL G-GMD.38 - Determine the relationship between surface areas of similar figures and volumes of similar figures. Geometry and Measurement Solve a problem involving the surface area of a cone or a pyramid that has a rectangle or right triangle as a base [Geometry] Solve a problem involving the volume of a right pyramid or a right cone [Geometry] Core Skill Determine the radius, diameter, or circumference of a sphere given its surface area [Geometry] WP: Solve a problem involving the surface area of a sphere [Geometry] Determine the volume of a sphere or hemisphere [Geometry] Core Skill Solve a problem involving the surface areas of similar solid figures [Geometry] Solve a problem involving the volumes of similar solid figures [Geometry] Visualize relationships between two-dimensional and three-dimensional objects AL G-GMD.39 - Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Geometry and Measurement Identify a cross section of a 3-dimensional shape [Grade 8] AL G-MG - Modeling with Geometry Apply geometric concepts in modeling situations AL G-MG.40 - Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). AL G-MG.41 - Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). 4/26/2012 - 136 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry AL G-MG.42 - Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). AL S - Statistics and Probability AL S-CP - Conditional Probability and the Rules of Probability Understand independence and conditional probability and use them to interpret data[(Link to data from simulations or experiments.)] AL S-CP.43 - Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Data Analysis, Statistics, and Probability Determine the probability of an event consisting of mutually exclusive events [Grade 8] AL S-CP.44 - Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Data Analysis, Statistics, and Probability Determine if a sample is likely to be representative of the larger population [Grade 8] AL S-CP.45 - Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Example: For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Use the rules of probability to compute probabilities of compound events in a uniform probability model AL S-CP.46 - Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. Data Analysis, Statistics, and Probability Determine the probability of an event consisting of mutually exclusive events [Grade 8] AL S-CP.47 - Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. AL S-CP.48 - Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P (A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. AL S-CP.49 - Use permutations and combinations to compute probabilities of compound events and solve problems. Data Analysis, Statistics, and Probability 4/26/2012 Determine the number of permutations possible in a given situation [Grade 8] - 137 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Geometry Determine the number of combinations possible in a given situation [Grade 8] AL S-MD - Using Probability to Make Decisions Use probability to evaluate outcomes of decisions[(Introductory; apply counting rules.)] AL S-MD.50 - Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). AL S-MD.51 - Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Overall Product Skills 4/26/2012 Use strategies to derive solutions for problems - 138 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Precalculus Precalculus AL N - Number and Quantity AL N-CN - The Complex Number System Represent complex numbers and their operations on the complex plane. AL N-CN.2 - Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. Algebra Identify a complex number represented as a vector on a coordinate plane [Algebra 2] AL N-CN.3 - Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. Example: For example, (-1 + the square root of 3i)³ = 8 because (-1 + the square root of 3i) has modulus 2 and argument 120°. Algebra Add or subtract complex numbers [Algebra 2] Multiply complex numbers [Algebra 2] AL N-CN.4 - Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. AL N-L - Limits Understand limits of functions. AL N-L.5 - Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. AL N-L.5.a - Apply limits in problems involving convergence and divergence. AL N-VM - Vector and Matrix Quantities Represent and model with vector quantities. AL N-VM.6 - Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v). Algebra Determine the component form of a vector represented on a graph [Algebra 2] AL N-VM.7 - Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. Algebra 4/26/2012 Determine the component form of a vector represented on a graph [Algebra 2] - 139 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Precalculus AL N-VM.8 - Solve problems involving velocity and other quantities that can be represented by vectors. Algebra WP: Add or subtract vectors [Algebra 2] Perform operations on vectors. AL N-VM.9 - Add and subtract vectors. AL N-VM.9.a - Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Algebra Add or subtract vectors on a coordinate plane [Algebra 2] Add or subtract vectors component-wise [Algebra 2] AL N-VM.9.b - Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. Algebra Determine the magnitude of a vector [Algebra 2] Determine the direction of a vector [Algebra 2] Add or subtract vectors on a coordinate plane [Algebra 2] Add or subtract vectors component-wise [Algebra 2] Determine a unit vector in the same direction as a given vector [Algebra 2] WP: Add or subtract vectors [Algebra 2] AL N-VM.9.c - Understand vector subtraction v - w as v + (-w), where -w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. Algebra Add or subtract vectors on a coordinate plane [Algebra 2] Add or subtract vectors component-wise [Algebra 2] AL N-VM.10 - Multiply a vector by a scalar. AL N-VM.10.a - Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v subscript x, v subscript y) = (cv subscript x, cv subscript y). Algebra Multiply a matrix by a scalar [Algebra 2] Determine the direction of a vector [Algebra 2] Add or subtract vectors component-wise [Algebra 2] AL N-VM.10.b - Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v is not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). Algebra 4/26/2012 Multiply a matrix by a scalar [Algebra 2] - 140 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Precalculus Determine the magnitude of a vector [Algebra 2] Perform operations on matrices and use matrices in applications. AL N-VM.11 - Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. Algebra Represent a system of linear equations as a single matrix equation [Algebra 2] AL N-VM.12 - Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. Algebra Multiply a matrix by a scalar [Algebra 2] AL N-VM.13 - Add, subtract, and multiply matrices of appropriate dimensions. Algebra Add or subtract matrices [Algebra 2] Multiply matrices [Algebra 2] AL N-VM.14 - Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. Algebra Multiply matrices [Algebra 2] AL N-VM.15 - Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. Algebra Determine the determinant of a matrix [Algebra 2] Determine the inverse of a matrix [Algebra 2] Core Skill AL N-VM.16 - Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. Algebra Multiply matrices [Algebra 2] AL N-VM.17 - Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. Algebra Multiply matrices [Algebra 2] Determine the determinant of a matrix [Algebra 2] AL A - Algebra AL A-REI - Reasoning with Equations and Inequalities Solve systems of equations AL A-REI.18 - Represent a system of linear equations as a single matrix equation in a vector variable. Algebra 4/26/2012 Represent a system of linear equations as a single matrix equation [Algebra 2] - 141 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Precalculus AL A-REI.19 - Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). Algebra Determine the inverse of a matrix [Algebra 2] Core Skill AL F - Functions AL F-CS - Conic Sections Understand the graphs and equations of conic sections. AL F-CS.20 - Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. Example: Graph x² - 6x + y2 - 12y + 41 = 0 or y² 4x + 2y + 5 = 0. Algebra Determine the graph of a given quadratic function [Algebra 1] Core Skill Determine a polynomial of lowest degree that has given roots or a graph of a given shape [Algebra 2] Determine the graph of a circle given the equation in standard form [Algebra 2] Relate a graph of an ellipse centered at the origin to its equation [Algebra 2] Core Skill Determine the graph of a circle or an ellipse given the equation in Core Skill general form [Algebra 2] Determine the graph of a hyperbola given the equation in standard form [Algebra 2] Determine the standard form of the equation for a conic section given its general form equation [Algebra 2] Determine which conic section a given equation represents [Algebra 2] Determine the equation of a parabola using given information [Algebra 2] Core Skill Determine the equation of a hyperbola using given information [Algebra 2] AL F-CS.20.a - Formulate equations of conic sections from their determining characteristics. Example: Write the equation of an ellipse with center (5, -3), a horizontal major axis of length 10, and a minor axis of length 4. Answer: (x - 5)²/25 + (y + 3)²/4 = 1. Algebra Determine the standard form of the equation for a conic section given its general form equation [Algebra 2] Determine which conic section a given equation represents [Algebra 2] AL F-IF - Interpreting Functions Analyze functions using different representations[(Logarithmic and trigonometric functions.)] 4/26/2012 - 142 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Precalculus AL F-IF.21 - Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Algebra Determine the graph of a rational function [Algebra 1] Identify the end behavior, asymptotes, excluded values, or behavior near excluded values of a rational function [Algebra 2] Core Skill Determine the graph of a rational function involving linear or quadratic denominators [Algebra 2] Core Skill AL F-BF - Building Functions Build a function that models a relationship between two quantities AL F-BF.22 - Compose functions. Example: For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. Algebra Determine the composition of two functions [Algebra 2] WP: Apply function operations or compositions of functions to represent a situation [Algebra 2] Build new functions from existing functions AL F-BF.23 - Determine the inverse of a function and a relation. Algebra Determine values of the inverse of a function using a table or a graph [Algebra 2] Determine if the inverse of a function is a function [Algebra 2] Determine the equation of the inverse of a linear, rational root, or polynomial function [Algebra 2] Core Skill AL F-BF.24 - Verify by composition that one function is the inverse of another. Algebra Determine values of the inverse of a function using a table or a graph [Algebra 2] Determine the composition of two functions [Algebra 2] Determine if the inverse of a function is a function [Algebra 2] AL F-BF.25 - Read values of an inverse function from a graph or a table, given that the function has an inverse. Algebra Determine values of the inverse of a function using a table or a graph [Algebra 2] AL F-BF.26 - Produce an invertible function from a non-invertible function by restricting the domain. AL F-BF.27 - Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Algebra 4/26/2012 Convert between a simple exponential equation and its corresponding logarithmic equation [Algebra 2] - 143 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Precalculus Solve an exponential equation using logarithms [Algebra 2] Core Skill Solve a logarithmic equation [Algebra 2] Core Skill Evaluate a logarithmic expression using the change of base formula [Algebra 2] WP: Solve a problem that can be modeled using a logarithmic equation [Algebra 2] WP: Solve an exponential growth or decay problem [Algebra 2] Overall Product Skills Core Skill Demonstrate and develop understanding of mathematical concepts and ideas AL F-BF.28 - Compare effects of parameter changes on graphs of transcendental functions. Example: Explain the relationship of the graph y = e to the (x-2) power to the graph y = e to the x power. AL F-TF - Trigonometric Functions Recognize attributes of trigonometric functions and solve problems involving trigonometry. AL F-TF.29 - Determine the amplitude, period, phase shift, domain, and range of trigonometric functions and their inverses. Algebra Determine the value of an inverse sine, cosine, or tangent expression [Algebra 2] Determine the amplitude, frequency, period, or phase shift of a sine or a cosine function [Algebra 2] Determine the graph of an inverse sine, cosine, or tangent function [Algebra 2] AL F-TF.30 - Use the sum, difference, and half-angle identities to find the exact value of a trigonometric function. AL F-TF.31 - Utilize parametric equations by graphing and by converting to rectangular form. AL F-TF.31.a - Solve application-based problems involving parametric equations. AL F-TF.31.b - Solve applied problems that include sequences with recurrence relations. Extend the domain of trigonometric functions using the unit circle AL F-TF.32 - Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi-x, pi+x, and 2pi-x in terms of their values for x, where x is any real number. Algebra Determine the exact value of a sine, cosine or tangent function given an angle in degrees or radians [Algebra 2] Geometry and Measurement WP: Determine a length using the properties of a 45-45-90 degree triangle or a 30-60-90 degree triangle [Geometry] Determine a sine, cosine, or tangent ratio in a right triangle [Geometry] 4/26/2012 - 144 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Precalculus AL F-TF.33 - Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Algebra Determine an angle coterminal with a given angle [Algebra 2] Model periodic phenomena with trigonometric functions AL F-TF.34 - Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. Algebra Determine the graph of an inverse sine, cosine, or tangent function [Algebra 2] Overall Product Skills Demonstrate and develop understanding of mathematical concepts and ideas AL F-TF.35 - Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. Algebra WP: Determine a trigonometric function that represents a situation [Algebra 2] WP: Solve a problem involving a sine, cosine, or tangent function Core Skill [Algebra 2] Prove and apply trigonometric identities AL F-TF.36 - Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. AL G - Geometry AL G-GPE - Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section AL G-GPE.37 - Derive the equation of a parabola given a focus and directrix. Algebra Determine the equation of a parabola using given information [Algebra 2] Core Skill AL G-GPE.38 - Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Algebra Determine the equation of an ellipse using given information [Algebra 2] Determine the equation of a hyperbola using given information [Algebra 2] Explain volume formulas and use them to solve problems AL G-GPE.39 - Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. Geometry and Measurement 4/26/2012 Determine the volume of an oblique prism or an oblique cylinder [Geometry] - 145 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Precalculus Determine the volume of a right pyramid or a right cone [Geometry] Solve a problem involving the volume of a right pyramid or a right cone [Geometry] Core Skill Determine the volume of an oblique pyramid or an oblique cone [Geometry] Determine the volume of a sphere or hemisphere [Geometry] Core Skill WP: Determine the volume of a sphere or hemisphere [Geometry] WP: Solve a problem involving the volume of a complex solid figure [Geometry] Solve a problem involving the volumes of similar solid figures [Geometry] WP: Solve a problem involving two solid figures that have the same surface area or volume [Geometry] AL S - Statistics and Probability AL S-MD - Using Probability to Make Decisions Calculate expected values and use them to solve problems AL S-MD.40 - Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. AL S-MD.41 - Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. AL S-MD.42 - Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. Example: For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. AL S-MD.43 - Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. Example: For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Use probability to evaluate outcomes of decisions AL S-MD.44 - Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. AL S-MD.44.a - Find the expected payoff for a game of chance. Example: For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. 4/26/2012 - 146 of 147 - STAR Math Enterprise $ODEDPD $ODEDPD&RXUVHRI6WXG\&RPPRQ&RUH Precalculus AL S-MD.44.b - Evaluate and compare strategies on the basis of expected values. Example: For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. 4/26/2012 - 147 of 147 -