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Unit 1 Operations with Rational Numbers Math 7 Unit Goals – Stage 1 Number of Days: 34 days 9/6/16 – 10/21/16 Unit Description: Students continue to build an understanding of the number line from their work in 6th grade. Students convert between the fraction form to decimal form of a rational number using long division. They learn to add, subtract, multiply, and divide integers, while maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. Students will then apply their understanding to perform the four operations on positive and negative rational numbers in the context of real-world situations. Materials: number lines*, two-color counters* (* Paper tool available at LBUSD Curriculum Intranet Instructional Tools Middle School Mathematical Tools) Standards for Mathematical Practice Transfer Goals SMP 1 Make sense of problems and Students will be able to independently use their learning to… persevere in solving them. • Make sense of never-before-seen problems and persevere in solving them. SMP 2 Reason abstractly and • Construct viable arguments and critique the reasoning of others. quantitatively. Making Meaning SMP 3 Construct viable arguments and UNDERSTANDINGS ESSENTIAL critique the reasoning of others. Students will understand that… QUESTIONS SMP 4 Model with mathematics. Students will keep • Integer operations can be represented in multiple ways using manipulatives, SMP 5 Use appropriate tools considering… number lines, and real-world situations. strategically. • How are adding and • Adding any two rational numbers, p + q, can be represented as the number SMP 6 Attend to precision. subtracting integers located a distance |q| from p, in the positive or negative direction depending SMP 7 Look for and make use of related? on whether q is positive or negative. structure. • How are multiplying • The distance between two rational numbers on a number line is the absolute SMP 8 Look for and express regularity and dividing integers value of their difference. in repeated reasoning. related? • Rational numbers can be divided as long as the divisor is not zero; every • When in the real-world quotient of rational numbers is a rational number. Standards for Mathematical Content would you add, • Applying previous understandings of properties of operations as strategies to Clusters Addressed subtract, multiply, or add, subtract, multiply, and divide positive and negative rational numbers will [m] 7.NS.A Apply and extend previous divide rational help solve real-world and mathematical problems. understandings of numbers? operations with fractions to Acquisition add, subtract, multiply, and KNOWLEDGE SKILLS divide rational numbers. Students will know… Students will be skilled at and/or be able to… • The definition of academic vocabulary words, • Add, subtract, multiply, and divide positive and [m] 7.EE.B Solve real-life and such as additive inverse, integers, opposites, negative rational numbers to solve real-world and mathematical problems and rational number. mathematical problems. using numerical and • The sum of opposite numbers is zero. • Interpret rational numbers on a number line, including algebraic expressions and representing addition and subtraction expressions. • Any subtraction problem can be written as an equations. equivalent addition problem. • Apply properties of operations as strategies to add, subtract, multiply, and divide rational numbers. • The decimal form of a rational number terminates in 0s or eventually repeats. • Convert a fraction to a decimal using long division. LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 1 Posted 6/9/16 Unit 1 Operations with Rational Numbers Math 7 Assessed Grade Level Standards Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Standards for Mathematical Content [m] 7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.1 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. a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. 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. 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. d. Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 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. 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. c. Apply properties of operations as strategies to multiply and divide rational numbers. 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. 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. [m] 7.EE.B Solve real‐life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.3 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. 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. Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters LONG BEACH UNIFIED SCHOOL DISTRICT 2 Posted 6/9/16 2016-2017 Unit 1 Operations with Rational Numbers Math 7 Evidence of Learning – Stage 2 Assessment Evidence Unit Assessment Students will complete selected response and constructed response items to indicate level of mastery/understanding of the unit standards as outlined in this guide. [m] 7.NS.A • The student interprets rational number values on a number line, including representations of addition and subtraction expressions. • The student applies properties of operations as strategies to add and subtract rational numbers. • The student applies properties of operations as strategies to multiply and divide rational numbers. • The student converts from a fractional form of rational numbers to a decimal form of rational numbers. • The student solve real-world and mathematical problems involving the four operations with rational numbers. [m] 7.EE.B • The student applies properties of operations to evaluate numeric expressions, including converting between different forms of numbers. For selected content, students will need to… • Solve complex problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. • Clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Other Evidence Formative Assessment Opportunities • Opening Tasks • Informal teacher observations • Checking for understanding using active participation strategies • Exit slips/summaries • Modeling Lessons (SMP 4) • • • • • Tasks Formative Assessment Lessons (FAL) Quizzes / Chapter Tests Big Ideas Math Performance Tasks SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website – “M” Mathematics – Curriculum Documents LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 3 Posted 6/9/16 Unit 1 Operations with Rational Numbers Math 7 Learning Plan – Stage 3 Days Learning Target I will explore operations with rational numbers by participating in the Opening Task. 1 day I will add integers by… 4-5 days Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Course 2 Expectations (Activities and Lessons) OPENING TASK—Number Line Reasoning This Opening Task is a Solo-Team-Teach activity. Pass out the activity, Number Line Reasoning, and have students work independently. If students are struggling, you may play the interactive video “The Number Line.” Next, have students work as a group to come to a consensus and then have a class discussion about students’ solutions utilizing Talk Moves to share the different methods used. This task is a gateway into the entire unit. • Using manipulatives to create conjectures about the rules for adding integers. (SMP 5, 7, 8) • Using a horizontal and vertical number line diagram. • Showing that a number and its opposite have a sum of zero (called additive inverses). • Explaining 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. (SMP 3) • Interpreting sums of integers by describing realworld contexts. (SMP 2) • Answering questions such as… o Why do additive inverses have a sum of zero? o How can you add integers using multiple methods? o How is adding integers similar to adding whole numbers? How is it different? • • • Section 1.1 (Activities 2 and 3; Examples 1, 2, 3, and 4) Section 1.2 (Activities 1, 2, 3, and 4; Examples 1, 2, and 3) STEM Video: Freezing Solid Curriculum Intranet Conceptual Understanding: • Number Line Reasoning • The Number Line Interactive Video Conceptual Understanding: • Adding Integers with Number Lines • Adding Two-Digit Numbers Task • Additive Inverse Video • myPD Course #2572: Adding Integers With Two Color Counters Procedural Skills and Fluency: • Beat the Computer: Integer Addition • Tic-Tac-Toe: Adding Integers Application: • NBC Learn Video – Science of Golf: Math of Golf Scoring • STEM Video Performance Task: Melting Matters LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 4 Posted 6/9/16 Unit 1 Operations with Rational Numbers Math 7 Learning Plan – Stage 3 Days Learning Target I will subtract integers by… Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Course 2 Expectations (Activities and Lessons) • Using manipulatives to create conjectures about the • Section 1.3 rules for subtracting integers. (SMP 5, 7, 8) (Activities 1, 2, 3, and 4; Examples 1, 2, and 3) • Showing that the distance between two integers on a number line is the absolute value of their difference. • Explaining subtraction of integers as adding the additive inverse, p − q = p + ( − q ) . (SMP 3) • Interpreting differences of integers by describing real-world contexts. (SMP 2) • Answering questions such as… o Why do you add the inverse when subtracting? o How can you subtract integers using multiple methods? o How is subtracting integers similar to subtracting whole numbers? How is it different? 4-5 days Curriculum Intranet Conceptual Understanding: • Subtracting Integers Using Two-Color Counters PowerPoint • Subtracting Two-Digit Numbers Task • Which One Doesn’t Belong? Integer Operations on the Number Line Procedural Skills and Fluency: • Comic Strip Activity: Adding and Subtracting Integers • Game Closet: Top This • Math-O: Adding and Subtracting Integers • Tic-Tac-Toe: Adding and Subtracting Integers Application: • Cliff Diving Task • Differences of Integers Task • Modeling: How Much Did the Temperature Drop? (SMP 4) 2-3 days I will check my understanding of operations with rational numbers by participating in the FAL. FORMATIVE ASSESSMENT LESSON (choose one) • Adding and Subtracting Directed Numbers (SMP 1, 2, 3, 5, 6, 7, 8) • Using Positive and Negative Numbers in Context (SMP 1, 2, 3, 4, 6, 7, 8) LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 5 Posted 6/9/16 Unit 1 Operations with Rational Numbers Math 7 Learning Plan – Stage 3 Days Learning Target I will multiply integers by… • • • • Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Course 2 Expectations (Activities and Lessons) Using manipulatives to create conjectures about the • Section 1.4 rules for multiplying integers. (SMP 5, 7, 8) (Activities 1, 2, 3, and 4; Examples 1, 2, 3, and 4) Explaining why the product of two negative numbers • Section 2.4 must be positive using properties, i.e. (–1)(–1) = 1. (Only Activity 1) (SMP 3) Interpreting products of integers in real-world contexts. (SMP 2) Answering questions such as… o How can you multiply integers using multiple methods? o How is multiplying integers similar to multiplying whole numbers? How is it different? 4-5 days LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 Curriculum Intranet Conceptual Understanding: • Multiplying Two-Digit Numbers Task • myPD Course #2574: Multiplying Integers with Two Color Counters Procedural Skills and Fluency: • Beat the Computer: Integer Multiplication • Game Closet: Right on Target • How Do They Fit?: Add, Subtract, Multiply Integer Operations • Product Game: Integers Application: • Multiply to Make −64 Task • The Running Videos Task 6 Posted 6/9/16 Unit 1 Operations with Rational Numbers Math 7 Learning Plan – Stage 3 Days Learning Target I will divide integers by… • • • • Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Course 2 Expectations (Activities and Lessons) Using manipulatives to create conjectures about the • Section 1.5 rules for dividing integers. (SMP 5, 7, 8) (Activities 1, 2, 3, and 4; Examples 1, 2, 3, and 4) Explaining that integers can be divided as long as the divisor is not zero and know that every quotient of integers is a rational number. (SMP 3) Interpreting quotients of integers in real-world contexts. (SMP 2) Answering questions such as… o Compare and contrast multiplying and dividing integers. o How can you divide integers using multiple methods? o How is dividing integers similar to dividing whole numbers? How is it different? 4-5 days LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 7 Curriculum Intranet Conceptual Understanding: • Dividing Positive and Negative Numbers Video • Dividing Two-Digit Numbers Task • Missing Factor Model for Dividing Integers • myPD Course #2467: Dividing Integers with Two Color Counters Procedural Skills and Fluency: • Beat the Computer: Integer Division • Connect 4: Multiplying and Dividing Integers • Math-O: Mixed Operations with Integers • Substituting Negative Values into Algebraic Expressions Posted 6/9/16 Unit 1 Operations with Rational Numbers Math 7 Learning Plan – Stage 3 Days Learning Target I will apply operations of rational numbers by… • • • • • 9-10 days Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Course 2 Expectations (Activities and Lessons) Converting a fraction into a decimal using long • Section 2.1 division. (Examples 1 and 2) Adding, subtracting, multiplying, and dividing • Section 2.2 positive and negative decimals, fractions, and mixed (Activities 1, 2, and 3; numbers. Examples 1, 2, 3, and 4) Solving real-world and mathematical problems • Section 2.3 involving the four operations with positive and (Activities 1 and 2; negative rational numbers. Examples 1, 2, 3, and 4) Applying properties and order of operations as • Section 2.4 strategies to add, subtract, multiply, and divide (Activities 2, 3, and 4; positive and negative rational numbers. Examples 1, 2, 3, and 4) Answering questions such as… • STEM Video: Carpenter or o When would you need to add, subtract, multiply, Joiner or divide rational numbers in a real-world context? o How is adding rational numbers similar to adding integers? How is it different? Curriculum Intranet Conceptual Understanding: • Converting Rational Numbers Video • Decimal Expansions of Fractions Task • Operations on the Number Line Task • Which One Doesn’t Belong? Rational Numbers Procedural Skills and Fluency: • Converting Between Fractions and Decimals Task • Decimal Maze • Product Game: Multiplying Rational Numbers Application: • Differences and Distances Task • Who is the Better Batter? Task • STEM Video Performance Task: Precisely Perfect 1-2 days I will prepare for the unit assessment on operations with rational numbers by… • Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. • • • • Ch. 1 Study Help (p. 20) Ch. 1 Review (p. 35 – 37) Ch. 2 Study Help (p. 56) Ch. 2 Review (p. 71 – 73) Application: • Bottling Performance Task • Downloads Performance Task • Elevator Task Unit Assessment 1 day At this point, all standards addressed in the Math 7 SBAC Interim Assessment Block – Number System have been covered. This block may now be administered. LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 8 Posted 6/9/16