Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Positional notation wikipedia , lookup
Mathematics wikipedia , lookup
Large numbers wikipedia , lookup
Location arithmetic wikipedia , lookup
History of mathematics wikipedia , lookup
Secondary School Mathematics Curriculum Improvement Study wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Thumb-Area Student Achievement Model Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District March 25, 2010 Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 2 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Introduction When teachers plan instruction, they draw on many sources such as state assessment standards, local curriculum guides, textbook materials, and supplemental assessment resources. These documents serve as useful sources of information, and it is neither necessary nor desirable to replace them. Michigan’s Grade-Level Content Expectations (GLCEs) describe in detail many ways in which students can demonstrate their mastery of the mathematics curriculum. The GLCEs do not, however, describe the big ideas and enduring understandings that students must develop in order to achieve these expectations. The GLCEs describe products of student learning, but they do not describe the thinking that must take place within the minds of students as they learn. It is the purpose of this document to focus on the fundamental mathematical ideas that form the basis of elementary and middle school instruction. Although a variety of research materials were used in the development of this document, several sources were relied on quite heavily. In 2006, the National Council of Teachers of Mathematics (NCTM) released Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. The Focal Points describe big topics, or focus areas, for each grade level. In May, 2009, the Michigan Department of Education published the Michigan Focal Points Core and Extended Designations. In that document, the NCTM Focal Points were adjusted to align with Michigan’s GLCEs. The new core and extended designations for the MEAP reflect Michigan’s Focal Points. This document is structured around Michigan’s Focal Points and supporting documents, with significant content included from two other documents: Charles, Randall I. “Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics.” NCSM Journal of Mathematics Education Leadership. Spring-Summer, 2005. vol. 8, no. 1, pp. 9 – 24. “Chapter 4: Curricular Content.” Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education, 2008. pp. 15 – 25. Particular thanks go to Ruth Anne Hodges for her contributions to this project. references to research are cited throughout the document. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 3 of 40 Additional March 25, 2010 printed 6/25/2017 at 9:39:10 PM Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 4 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Focusing on Mathematics at Grade 1 Grade Grade 1 #1 Michigan Focal Point Related GLCE Topics Targeted Vocabulary Developing understanding of addition and subtraction and strategies for basic addition facts and related subtraction facts Count, write, and order numbers Add and subtract whole numbers Solve problems Work with money Count by/skip count Count backwards Quantity/amount/value Order Compare (i.e., same as, greater than, fewer than, equal to, bigger, smaller, smallest, largest) More / increase Less / decrease Addition-- add (+), sum, combine/join/put together, extend/ count on/more Subtraction—subtract (-), difference, compare, take away Fact family Number sentence Equals / equal to (=) Number path / line Grade 1 #2 Grade 1 #3 Developing an understanding of whole number relationships, including grouping tens and ones Explore place value Ones / tens Developing an understanding of linear measurement and facility in measuring lengths Estimate and measure length Length Measure Shorter/shortest, taller/tallest, longer/longest Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 5 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM National Math Panel Benchmarks By the end of Grade 4, students should be proficient with the addition and subtraction of whole numbers. By the end of Grade 5, students should By the end of Grade 6, students should By the end of Grade 7, students should be proficient with comparing fractions and decimals and common percents, and with the addition and subtraction of fractions and decimals. be proficient with multiplication and division of fractions and decimals. be proficient with all operations involving positive and negative fractions. be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids). be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area and analyze the properties of threedimensional shapes and solve problems involving surface area and volume. be proficient with multiplication and division of whole numbers. be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals. Geometry and Measurement Fluency with Fractions and Decimals Fluency with Whole Numbers By the end of Grade 3, students should be proficient with all operations involving positive and negative integers. be able to solve problems involving percent, ratio, and rate and extend this work to proportionality. be familiar with the relationship between similar triangles and the concept of the slope of a line. Taken from The National Mathematics Advisory Panel Final Report, 2008, p.20 Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 6 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Michigan Focal Points Leading to Proficiency with Addition and Subtraction Big Mathematical Ideas The same value can be represented in many equivalent ways You can only add or subtract things that are the same (that have the same unit) Kindergarten Grade 1 Grade 2 Grade 3 Focal Point #1: Focal Point #2: Focal Point #1: Focal Point #4: Representing, comparing, and ordering whole numbers and joining and separating sets Developing an understanding of whole number relationships, including grouping in tens and ones Developing an understanding of the base-ten numeration system and place-value concepts Developing an understanding of fractions and fraction equivalence Focal Point #1: Focal Point #2: Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts Developing quick recall of addition facts and related subtraction facts and fluency with multi-digit addition and subtraction Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 7 of 40 NATIONAL M ATH PANEL BENCHMARK: By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbers. March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #1: Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts Kindergarten Grade 1 Grade 2 Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts (NCTM-1st) Developing quick recall of addition facts and related subtraction facts and fluency with multidigit addition and subtraction (NCTM-2nd) Count, write, and order numbers Count, write, and order numbers Count, write, and order whole numbers Compose and decompose numbers Add and subtract whole numbers Add and subtract whole numbers Add and subtract numbers Solve problems using addition and subtraction of length, money, and time (M.PS.01.08) Measure, add, and subtract length Work with money Solve measurement problems involving length, money, and perimeter (M.PS.02.10 , M.TE.02.11) Tell time and solve time problems Record, add, and subtract money National Math Panel Benchmark: By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbers Key: bold, non-italic = Michigan Curriculum Focal Points non-bold, non-italic = GLCE topics associated with that focal point non-bold, italic = Cross over GLCE topics associated with another focal point Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 8 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #1: Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS Big Idea #1 (Numbers) The set of real numbers is infinite, and each real number can be associated with a unique point on the number line. Counting tells how many items there are altogether. The number of objects in a set is the last number named when the objects are counted. Counting a set in a different order does not change the total. There is a number word and a matching symbol that tell exactly how many items are in a group. Quantities are measured or counted using same-sized units. 0 represents a set with no objects. 0 added to or subtracted from a number does not change the number. All quantities have a unit. Sometimes the unit is unspoken (i.e., 23 ones). The counting numbers can be put in a path so that numbers are in order. The position of two numbers on the number path shows which number is larger. The smallest counting number is one, but a number path can start at any number. There is no largest counting number. If a number path is used for counting on or counting back, the numbers must be counted by the same unit all the way through. o 7, 8, 9, 10, 11, 12 is a valid number path because each number is one more than the number before it o 2, 4, 6, 8, 10, 12 is a valid number path because the difference between one number and the next is always 2 o 1, 2, 3, 6, 7, 15 is a list of numbers in order but can not be used for counting because the difference the difference between one number and the next keeps changing The number line uses segments of equal length to represent number. When using a number line to add, subtract, or count, count the “jumps” between numbers, not the tick marks. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 9 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM The distance between any two consecutive counting numbers on a given number line is the same. In other words, the distance between one counting number and the next is a unit of one. Big Idea #2 (The Base Ten Numeration System) The base ten numeration system is a scheme for recording numbers using digits 0-9, groups of ten, and place value. The same quantity can be expressed in many equivalent forms using different units: o 3 red teddy bears is the same number of objects as 3 teddy bears o 23 ones is equivalent to 2 tens and 3 ones o 24 tens is equivalent to 240 ones o 100 pennies has the same value as 1 dollar o 2 feet has the same length as 24 inches o 5+2=9–2 o 1/2 = 2/4 o 3x = 2x + x Rational numbers can be written as a sum of ones, tens, hundreds, and other types of ten-groups (powers of 10). o 23 = 2 tens + 3 ones = 20 + 3 o 874 = 8 hundreds + 7 tens + 4 ones = 800 + 70 + 4 o 12.3 = 1 ten + 2 ones + 3 tenths = 10 + 2 + 0.3 Teacher note: 10n, . . . 103, 102, 101, 100, 10-1, 10-2, 10-3, . . . 10-n Ones can only be added to ones, tens can only be added to tens, and so forth. Place value makes this easy to see, so that we can add and subtract faster. Sets of ten, one hundred and so forth must be perceived as single entities when interpreting numbers using place value (e.g., 1 hundred is one group; it is 10 tens or 100 ones). When there are no groups of a particular size, 0 is used as a place holder. Since adding 0 does not change the value of a number, the 0 is not used in expanded notation. o 302 = 300 + 2 In Base 10 notation, each place is ten times the value of the place to the right: o o o o o o 5 tens = 50 ones 2 hundreds = 20 tens 400 tens = 4 thousands 3 ones = 30 tenths 20 hundredths = 2 tenths 4 tenths = 400 thousandths Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 10 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Big Idea #3 (Equivalence) Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value. Only numbers that have the same unit can be added or subtracted. The sum or difference also has the same unit. Sometimes, it is useful to change a quantity to a different form in order to add or subtract quantities. o 4 teddy bears + 5 teddy bears = 9 teddy bears o 2 dogs + 3 cats is 2 dogs and 3 cats, but 2 pets + 3 pets = 5 pets o 2 ones + 4 ones = 6 ones o 2 tens + 4 ones = 2 tens and 4 ones, but 20 ones + 4 ones = 24 ones o 5 nickels + 8 nickels = 13 nickels o 5 dimes + 8 nickels = 5 dimes and 8 nickels, but 5 dimes + 4 dimes = 9 dimes. OR: 50 pennies + 40 pennies = 90 pennies o 1/2 + 2/2 = 3/2 o 1/2 + 2/3 can only be simplified by finding equivalent fractions: 3/6 + 4/6 = 7/6 Big Idea #5 (Operation Meanings & Relationships) The same number sentence (e.g. 12 – 4 = 8) can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation. Real-world situations for addition and subtraction: o Subtraction can be used to describe a situation where some items are taken away from a set: John had 12 pennies and gave 4 away; how many did he have left? o Subtraction can also be used to compare two quantities: John has 12 pennies, and Mary has 4. How many more pennies does John have than Mary? o Addition can be used to describe a situation where two sets of items are combined: John has 8 pennies, and Mary has 4. How many pennies do they have all together? o Addition can also be used to count on from a number: John had 8 pennies, and he found 4 more. How many does he have now? The real-world actions for addition and subtraction of whole numbers are the same for operations with fractions, decimals, and integers. Only numbers that have the same unit can be added or subtracted. The sum or difference also has the same unit. Sometimes, it is useful to change a quantity to a different form in order to add or subtract quantities. o 4 teddy bears + 5 teddy bears = 9 teddy bears o 2 dogs + 3 cats is 2 dogs and 3 cats, but 2 pets + 3 pets = 5 pets o 2 ones + 4 ones = 6 ones o 2 tens + 4 ones = 2 tens and 4 ones, but 20 ones + 4 ones = 24 ones o 5 nickels + 8 nickels = 13 nickels o 5 dimes + 8 nickels = 5 dimes and 8 nickels, but 5 dimes + 4 dimes = 9 dimes. OR: 50 pennies + 40 pennies = 90 pennies o 1/2 + 2/2 = 3/2 o 1/2 + 2/3 can only be simplified by finding equivalent fractions: 3/6 + 4/6 = 7/6 Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 11 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Big Idea #6 (Properties) For a given set of numbers there are relationships that are always true, and these are the rules that govern arithmetic and algebra. Sets can be joined (added) in any order. Addition facts can be “turned around.” o 3 + 5 = 8 and 5 + 3 = 8 (addition is commutative) o 3 + 6 + 4 = 6 + 4 + 3 = 10 + 3 = 13 (associative property applies because addition is commutative) Sets are separated (subtracted) by starting with the total number of objects. Subtraction facts do not “turn around” the same way as addition facts (subtraction is not commutative), but each subtraction fact has a related subtraction fact. o 8 – 5 5 – 8. However, 8 – 5 = 3 and 8 – 3 = 5. Adding zero to a number (or subtracting zero from a number) does not change the number. Big Idea #7 (Basic Facts & Algorithms) Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. Subtraction is the inverse of addition. Any addition problem has related subtraction problems, and any subtraction problem has related addition problems. Any subtraction calculation can be solved by adding up, and addition can be used to check subtraction. Any unknown addition or subtraction fact can be found by using known facts. o 5 is one more than 4, so 4 + 1 = 5. o 7 is one less than 8, so 8 – 1 is 7. That means that 8 – 7 = 1. o 5 + 5 is 10 and 5 + 3 is 8. 3 more than 10 is 13, so 5 + 8 = 13. [ The light gray points are related to the same Big Idea and topic, but are addressed at a later grade level.] Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 12 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #1: Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts INSTRUCTIONAL IMPLICATIONS In Chapter 6 of their book Learning and Teaching Early Math: The Learning Trajectories Approach, Douglas H. Clements and Julie Sarama (2009) 1 address three topics of increasing sophistication: arithmetic combinations (“facts”), place value, and multi-digit addition and subtraction. Each of these topics builds on composition of number, a central idea in the development of number sense. In this section, we provide a summary of the research implications regarding arithmetic combinations. A robust understanding of arithmetic combinations involves both counting-based strategies and part-whole relationships. Counting-based strategies rely on an understanding of contextual situations that can be modeled with objects or pictures, as described in Table 1 on the next page. Both the situation (context) and the nature of the unknown value impact the difficulty of the problem. For some situations, which value is unknown does not significantly affect the difficulty of the problem; for other situations, different unknowns lead to problems with large differences in difficulty. In “Add to” situations, for example2, Result unknown problems are easy, change unknown problems are moderately difficult, and start unknown are the most difficult. This is due in large part to the increasing difficulty children have modeling, or “act outing,” each type (p. 62). In other words, students find it much easier to answer the question “3 and 2 more is how many?” (result unknown) than to determine “how many do you start with if 2 more makes 5?” (start unknown). Therefore, in sequencing arithmetic instruction, teachers should find where a child is in the learning trajectory and tailor instructional tasks to move the child forward. In addition to counting strategies that can be modeled within the different problem types described earlier, children also develop arithmetic combinations through their understanding of part-whole relationships. To start, children separate a group of objects in various ways. For example, a child could separate 7 objects into a pile of 5 and a pile of 2, or 6 and 1, or 0 and 7. These separations are represented as addition problems so that, eventually, a child can produce all of the number combinations composing a given number. For example, 6 is represented as 5+1, as 4+2, and so forth. Clements, Douglas H. and Sarama, Julie (2009) Learning and Teaching Early Math: The Learning trajectories Approach. Routledge: New York, NY. 2 The authors refer to this situation as “Join” or “Change Plus.” 1 Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 13 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Table 1 Add to Take from Put together/ Take apart Compare Result Unknown Change Unknown Start Unknown Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2+3=? Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2+?=5 Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ?+3=5 Five apples were on the table. I ate two apples. How many apples are on the table now? 5–2=? Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5–?=3 Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ?–2=3 Total Unknown Addend Unknown Both Addends Unknowns Three red apples and two green apples are on the table. How many apples are on the table? 3+2=? Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ? = 5, 5 – 3 = ? Grandma has five flowers. How many can she put in her red vase and how many in her blue vase? 5 = 0 + 5, 5 = 5 + 0 5 = 1 + 4, 5 = 4 + 1 5 = 2 + 3, 5 = 3 + 2 Difference Unknown Bigger Unknown Smaller Unknown (“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? (“How many fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? 2 + ? = 5, 5 – 2 = ? (Version with “more”): (Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? (Version with “fewer”): (Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? 2 + 3 = ?, 3 + 2 = ? Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 5 – 3 = ?, ? + 3 = 5 “Table 1” taken from page 65 of the 3/10/2010 draft of the Common Core State Standards for Mathematics . Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 14 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM What Research Says about Basic Facts and Fluency According to Clements and Sarama (2009), World-wide research shows that the way most people in the U.S. think about arithmetic combinations and children’s learning of them, and the language they use may harm more than help . . . For example, we hear about “memorizing facts” and “recalling your facts.” This is misleading. . . children move through a long developmental progression to reach the point where they can compose numbers. Further they also should learn about arithmetic properties, patterns, and relationships as they do so, and that knowledge, along with intuitive magnitude and other knowledge and skills, ideally is learned simultaneously and in an integrated fashion with knowledge of arithmetic combinations. (pp. 82-83) Research suggests that producing basic combinations is not just a simple “lookup” process. Retrieval is an important part of the process, but many brain systems help. . . So, when children really know 8 – 3 = 5, they also know that 3 + 5 = 8, 8 – 5 = 3, and so forth, and all these “facts” are related. (p. 83) So what does this mean? The authors highlight several implications of the research (p. 83): 1. Children need considerable practice, distributed across time. 2. Because counting strategies did not activate the same brain systems, we need to guide children to move to more sophisticated composition strategies. 3. Practice should not be “meaningless drill” but should occur in a context of making sense of the situation and the number relationships. 4. Children who are strong in calculations know and use multiple strategies, so teachers should not teach “one correct procedure.” Many rote approaches to basic fact recall, such as timed tests, flash cards, or more emphasis on easier arithmetic problems than harder problems, were found to have no positive – and sometimes negative – effect. Instead, teachers should use strategies that emphasis reasoning strategies along with flexible use of number combinations. By developing the understanding and skills described below, students can become fluent, flexible users of addition and subtraction. 1. Understand commutativity: 3 + 2 has the same value as 2 + 3. 2. Understand associativity: 5 + 3 + 2 can be seen as 8 + 2 or 5 + 5. 3. Find all the pairs of numbers “hiding inside” other numbers (decompose numbers). For example, 6 = 0 + 6; 6 = 1+ 5; 6 = 2 + 4; 6 = 3 + 3; 6 = 4 + 2; 6 = 5 + 1; 6 = 6 + 0. 4. Use special patterns such as “doubles” (4 + 4) and “doubles-plus-one” (7 + 8). 5. Understand that adding one to a number is the next number (the n + 1 rule). This should be understood before “doubles-plus-one.” 6. Know that adding 0 does not change a number (the n + 0 rule). 7. Use the Break-Apart-to-Make-Tens (BAMT) strategy. Before developing this strategy, children a. Have a solid knowledge of numerals and counting b. Understand the structure of the teens numbers (10 + another number) c. Can solve addition and subtraction of numbers with totals less than 10 d. Can find “break-apart” partners of numbers less than or equal to 10 Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 15 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM e. Use the 10s structure to solve addition and subtraction problems involving teen numbers (10 + 2 = 12; 18 – 8 = 10) f. Add and subtract with three addends using 10s (4 + 6 + 3 = 10 + 3 = 13; 15 – 5 – 9 = 10 – 9 = 1) When students have reached this point in the learning progression, they are ready to develop understanding of the BAMT strategy. Essentially, students break apart one number in order to make a 10 and some ones. For example, 9 + 4 becomes 9 + 1 + 3 = 10 + 3 = 13. For more information on the four phases to develop this strategy, see Clements and Sarama (2009), pp. 86-87. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 16 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #1: Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts RELATED GLCES Number and Operations Count, write, and order numbers N.ME.01.01 Count to 110 by 1’s, 2’s, 5’s, and 10’s, starting from any number in the sequence; count to 500 by 100’s and 10’s; use ordinals to identify position in a sequence, e.g., 1st, 2nd, 3rd. N.ME.01.02 Read and write numbers to 110 and relate them to the quantities they represent. N.ME.01.03 Order numbers to 110; compare using phrases such as “same as”, “more than”, “greater than”, “fewer than”, use = symbol. Arrange small sets of numbers in increasing or decreasing order, e.g., write the following from smallest to largest: 21, 16, 35, 8. N.ME.01.04 Identify one more than, one less than, 10 more than, and 10 less than for any number up to 100. N.ME.01.05 Understand that a number to the right of another number on the number line is bigger and that a number to the left is smaller. N.ME.01.06 Count backward by 1’s starting from any number between 1 and 100. Add and subtract whole numbers N.ME.01.08 List number facts (partners inside of numbers) for 2 through 10, e.g., 8 = 7 + 1 = 6 + 2 = 5 + 3 = 4 + 4; 10 = 8 + 2 = 2 + 8. N.MR.01.09 Compare two or more sets in terms of the difference in number of elements. N.MR.01.10 Model addition and subtraction for numbers through 30 for a given contextual situation using objects or pictures; explain in words; record using numbers and symbols; solve. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 17 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM N.MR.01.11 Understand the inverse relationship between addition and subtraction, e.g., subtraction “undoes” addition: if 3 + 5 = 8, we know that 8 – 3 = 5 and 8 – 5 = 3; recognize that some problems involving combining, “taking away”, or comparing can be solved by either operation. N.FL.01.12 Know all the addition facts up to 10 + 10, and solve the related subtraction problems fluently. N.MR.01.13 Apply knowledge of fact families to solve simple open sentences for addition and subtraction, such as: + 2 = 7 and 10 - = 6. N.FL.01.014 Add three one-digit numbers. N.FL.01.15 Calculate mentally sums and differences involving: a two-digit number and a one-digit number without regrouping; a two-digit number and a multiple of 10. N.FL.01.16 Compute sums and differences through 30 using number facts and strategies, but no formal algorithm. Measurement Solve problems M.PS.01.08 Solve one-step word problems using addition and subtraction of length, money and time, including “how much more/less”, without mixing units. Work with money M.UN.01.04 Identify the different denominations of coins and bills. M.UN.01.05 Match one coin or bill of one denomination to an equivalent set of coins/bills of other denominations, e.g., 1 quarter = 2 dimes and 1 nickel. M.UN.01.06 Tell the amount of money: in cents up to $1, in dollars up to $100. Use the symbols $ and ¢. M.PS.01.07 Add and subtract money in dollars only or in cents only. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 18 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM FROM THE 3/10/2010 DRAFT OF THE COMMON CORE STANDARDS Number—Operations and the Problems They Solve 1-NOP Addition and subtraction 1. Understand the properties of addition. a. Addition is commutative. For example, if 3 cups are added to a stack of 8 cups, then the total number of cups is the same as when 8 cups are added to a stack of 3 cups; that is, 8 + 3 = 3 + 8. b. Addition is associative. For example, 4 + 3 + 2 can be found by first adding 4 + 3 = 7 then adding 7 + 2 = 9, or by first adding 3 + 2 = 5 then adding 4 + 5 = 9. c. 0 is the additive identity. 2. Explain and justify properties of addition and subtraction, e.g., by using representations such as objects, drawings, and story contexts. Explain what happens when: a. The order of addends in a sum is changed in a sum with two addends. b. 0 is added to a number. c. A number is subtracted from itself. d. One addend in a sum is increased by 1 and the other addend is decreased by 1. Limit to two addends. 3. Understand that addition and subtraction have an inverse relationship. For example, if 8 + 2 = 10 is known, then 10 – 2 = 8 and 10 – 8 = 2 are also known. 4. Understand that when all but one of three numbers in an addition or subtraction equation are known, the unknown number can be found. Limit to cases where the unknown number is a whole number. 5. Understand that addition can be recorded by an expression (e.g., 6 + 3), or by an equation that shows the sum (e.g., 6 + 3 = 9). Likewise, subtraction can be recorded by an expression (e.g., 9 – 5), or by an equation that shows the difference (e.g., 9 – 5 = 4). Describing situations and solving problems with addition and subtraction 6. Understand that addition and subtraction apply to situations of adding-to, taking-from, putting together, taking apart, and comparing. (Table- page14). 7. Solve word problems involving addition and subtraction within 20, e.g., by using objects, drawings and equations to represent the problem. Students should work with all of the addition and subtraction situations shown in the Table on page 14, solving problems with unknowns in all positions, and representing these situations with equations that use a symbol for the unknown (e.g., a question mark or a small square). Grade 1 students need not master the more difficult problem types. 8. Solve word problems involving addition of three whole numbers whose sum is less than or equal to 20. Number—Base Ten 1-NBT Adding and subtracting in base ten 5. Calculate mentally, additions and subtractions within 20. a. Use strategies that include counting on; making ten (for example, 7 + 6 = 7 + 3 + 3 = 10 + 3 = 13); and decomposing a number (for example, 17 – 9 = 17 – 7 – 2 = 10 – 2 = 8). 6. Demonstrate fluency in addition and subtraction within 10. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 19 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM 7. Understand that in adding or subtracting two-digit numbers, one adds or subtracts like units (tens and tens, ones and ones) and sometimes it is necessary to compose or decompose a higher value unit. 8. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count. 9. Add one-digit numbers to two-digit numbers, and add multiples of 10 to one-digit and two-digit numbers. 10. Explain addition of two-digit numbers using concrete models or drawings to show composition of a ten or a hundred. 11. Add two-digit numbers to two-digit numbers using strategies based on place value, properties of operations, and/or the inverse relationship between addition and subtraction; explain the reasoning used. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 20 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #2: Developing an understanding of whole number relationships, including grouping in tens and ones Kindergarten Grade 1 Grade 2 Representing, comparing, and ordering whole numbers and joining and separating sets (NCTM -K) Developing an understanding of whole number relationships, including grouping in tens and ones (NCTM-1st) Developing an understanding of the baseten numeration system and place-value concepts (NCTM-2nd) Count, write, and order numbers Count, write, and order numbers Count, write, and order whole numbers Compose and decompose numbers Explore place value Understand place value Work with unit fractions Add and subtract numbers Explore number patterns Key: bold, non-italic = Michigan Curriculum Focal Points non-bold, non-italic = GLCE topics associated with that focal point non-bold, italic = Cross over GLCE topics associated with another focal point Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 21 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #2: Developing an understanding of whole number relationships, including grouping in tens and ones BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS Big Idea #1 (Numbers) The set of real numbers is infinite, and each real number can be associated with a unique point on the number line. Counting tells how many items there are altogether. The number of objects in a set is the last number named when the objects are counted. Counting a set in a different order does not change the total. There is a number word and a matching symbol that tell exactly how many items are in a group. Quantities are measured or counted using same-sized units. 0 represents a set with no objects. 0 added to or subtracted from a number does not change the number. All quantities have a unit. Sometimes the unit is unspoken (i.e., 23 ones). The counting numbers can be put in a path so that numbers are in order. The position of two numbers on the number path shows which number is larger. The smallest counting number is one, but a number path can start at any number. There is no largest counting number. If a number path is used for counting on or counting back, the numbers must be counted by the same unit all the way through. o 7, 8, 9, 10, 11, 12 is a valid number path because each number is one more than the number before it o 2, 4, 6, 8, 10, 12 is a valid number path because the difference between one number and the next is always 2 o 1, 2, 3, 6, 7, 15 is a list of numbers in order but can not be used for counting because the difference the difference between one number and the next keeps changing The number line is a linear (length) model, not a set (object) model. When using a number line to add, subtract, or count, count the “jumps” between numbers, not the tick marks. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 22 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM The distance between any two consecutive counting numbers on a given number line is the same. In other words, the distance between one counting number and the next is a unit of one. Big Idea #2 (The Base Ten Numeration System) The base ten numeration system is a scheme for recording numbers using digits 0-9, groups of ten, and place value. The same quantity can be expressed in many equivalent forms using different units: o 3 red teddy bears is the same number of objects as 3 teddy bears o 23 ones is equivalent to 2 tens and 3 ones o 24 tens is equivalent to 240 ones o 100 pennies has the same value as 1 dollar o 2 feet has the same length as 24 inches o 5+2=9–2 o 1/2 = 2/4 o 3x = 2x + x Any real number can be written as a sum of ones, tens, hundreds, tenths, hundredths, and other types of ten-groups (powers of 10). You can add the value of the digits together to get the value of the number. o 23 = 2 tens + 3 ones = 20 + 3 o 874 = 8 hundreds + 7 tens + 4 ones = 800 + 70 + 4 o 12.3 = 1 ten + 2 ones + 3 tenths = 10 + 2 + 0.3 o -78 = - (7 tens + 8 ones) or -7 tens + -8 ones Teacher note: 10n, . . . 103, 102, 101, 100, 10-1, 10-2, 10-3, . . . 10-n Ones can only be added to ones; tens can only be added to tens, and so forth. Place value makes this easy to see, so that we can add and subtract faster. Sets of ten, one hundred and so forth must be perceived as single entities when interpreting numbers using place value (e.g., 1 hundred is one group; it is 10 tens or 100 ones). When there are no groups of a particular size, 0 is used as a place holder. Since adding 0 does not change the value of a number, the 0 is not used in expanded notation. o 302 = 300 + 2 In Base 10 notation, each place is ten times the value of the place to the right: o 5 tens = 50 ones o 2 hundreds = 20 tens o 400 tens = 4 thousands o 3 ones = 30 tenths o 20 hundredths = 2 tenths o 4 tenths = 400 thousandths The value of each digit depends on its position within the number. Decimal place value is an extension of whole number place value. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 23 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM The base-ten numeration system extends infinitely to very large and very small numbers (e.g., millions and millionths). Big Idea #3 (Equivalence) Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value. The same quantity can be expressed in many equivalent forms using different units: o 3 red teddy bears is the same number of objects as 3 teddy bears o 23 ones is equivalent to 2 tens and 3 ones o 24 tens is equivalent to 240 ones o 100 pennies has the same value as 1 dollar o 2 feet has the same length as 24 inches o 5+2=9–2 o 1/2 = 2/4 o 3x = 2x + x Big Idea #4 (Comparison) Numbers, expressions, and measures can be compared by their relative values. [ The light gray points are related to the same Big Idea and topic, but are addressed at a later grade level.] Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 24 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #2: Developing an understanding of whole number relationships, including grouping in tens and ones INSTRUCTIONAL IMPLICATIONS Preschool children begin to understand the process of making groups with equal numbers of objects. Such grouping, and the knowledge of the special grouping into tens, appears not to be related to counting skill. However, experience with additive composition does appear to contribute to knowledge of grouping and place value. Teachers often believe that their students understand place value because they can, for example, put digits into “tens and one charts.” However, ask these students what the “1” in “16” means and they are as likely to say “one” (and mean 1 singleton) as they are to say “one ten.” (Clements and Sarama (2009), pp. 88-893) There is more to understanding place value than naming the place value positions or aligning ones and tens for column addition. Children need concrete experiences to make sense of what it means to group objects by tens and ones. However, these experiences must be connected to, not separate from, the language and symbolism of mathematics. In their book Learning and Teaching Early Math: The Learning Trajectories Approach, Douglas H. Clements and Julie Sarama (2009) identify several learning experiences that help children to make sense of place value and grouping: Solving simple addition problems in early years lays the foundation for understanding place value Students should group sets of objects into groups of tens and ones while discussing the ideas of place value When grouping objects, make links to number words (twenty-two, that’s 2 tens and 2 ones) and to the written numerals (“22”) It takes many experiences to fix “ten” in students’ minds as a benchmark. Any activity where students collect or count objects sorted into tens can form this link: o Roll two number cubes and take that many pennies. If you have 10 or more pennies, you must trade 10 pennies for 1 dime before your turn is over. Take turns, continuing until someone has 100 cents (10 dimes). o The Michigan Mathematics Program Improvement project (MMPI) describes similar Trading Games using Base 10 blocks. For more information, visit www.michiganmathematics.org (Chapter 2, Page 4, Activities 6 and 7). Use number language that symbolizes composing and decomposing. For example, when reading “52” say “fifty-two, that’s 5 tens and 2 ones.” Experiences such as the trading games described above, along with use of number language that emphasizes place value, can help students to see “ten” as a unit in and of itself, laying the foundation for counting by tens, and later, adding groups of ten. Clements, Douglas H. and Sarama, Julie (2009). Learning and Teaching Early Math: The Learning Trajectories Approach. Routledge: New York, NY. 3 Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 25 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Even with the strategies above, some children still view the digits in a multidigit number as if they were singletons. “Secret code cards” can help to make explicit that the value of each digit depends on its position in the number. The MMPI project has templates for place value number cards that can be used in this way (www.michiganmathematics.org, Chapter 2, Page 7, Activity 15). It is common to use manipulatives and pictures in early elementary instruction. However, it is essential not simply to do “hands-on activities,” but to connect those activities to the fundamental ideas – and symbolism – of mathematics. Clements and Sarama (2009) describe one process by which this happens: High-quality instruction often uses manipulatives or other objects to demonstrate and record quantities. Further, such manipulatives are used consistently enough that they becomes tools for thinking. They are discussed to explicate the placevalue ideas. They are used to solve problems, including arithmetic problems. Finally, they are replaced by symbols (p. 90). This process of using concrete objects to form strong mental images connected to concepts, connecting those images to pictures and symbols, and eventually withdrawing the objects, is an example of the research-based Concrete-Representational-Abstract strategy4. In addition to connecting manipulatives to concepts and symbolism, it is also important to select robust models that apply to a variety of mathematical contexts. When done well, mathematical modeling is one of three “powerful practices” in mathematics and science.5 When selecting manipulatives with which to model the base 10 place value system, choose the model appropriate to your student’s needs: 1. Bustable and proportional. Some models, like bundled straws or linked unifix cubes, are proportional in that 10 ones is the same physical size as 1 ten. Additionally, the model is “bustable,” meaning that 1 ten can be taken apart into 10 ones. 2. Tradable and proportional. Base 10 blocks are an example of a tradable and proportional model. 1 ten is the same size as 10 ones, but a ten cannot be broken into 10 ones – it must be traded for 10 ones. 3. Tradable and non-proportional. Money, such as pennies and dimes, is an example of a tradable and non-proportional model. One dime has the same value as 10 pennies, but the dime appears physically smaller than the 10 pennies. For examples of a variety of place value manipulatives, along with strategies to use them, refer to the MMPI project (www.michiganmathematics.org, Chapter 2). 4 “Concrete-Representational-Abstract Instructional Approach.” The Access Center: Improving Outcomes for All Students K-8. Accessed on 3-22-2010 from http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp Carpenter, Thomas P. and Romberg, Thomas A. (2004) Powerful Practices in Mathematics and Science. Learning Point Associates: Naperville, IL. 5 Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 26 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #2: Developing an understanding of whole number relationships, including grouping in tens and ones RELATED GLCES Number and Operations Explore place value N.ME.01.07 Compose and decompose numbers through 30, including using bundles of tens and units, e.g., recognize 24 as 2 tens and 4 ones, 10 and 10 and 4, 20 and 4, and 24 ones. Count, write, and order numbers – see Focal Point #1 N.ME.01.01 Count to 110 by 1’s, 2’s, 5’s, and 10’s, starting from any number in the sequence; count to 500 by 100’s and 10’s; use ordinals to identify position in a sequence, e.g., 1st, 2nd, 3rd. N.ME.01.02 Read and write numbers to 110 and relate them to the quantities they represent. N.ME.01.03 Order numbers to 110; compare using phrases such as “same as”, “more than”, “greater than”, “fewer than”; use = symbol. Arrange small sets of numbers in increasing or decreasing order, e.g., write the following from smallest to largest: 21, 16, 35, 8. N.ME.01.04 Identify one more than, one less than, 10 more than, and 10 less than for any number up to 100. N.ME.01.05 Understand that a number to the right of another number on the number line is bigger and that a number to the left is smaller. N.ME.01.06 Count backward by 1’s starting from any number between 1 and 100. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 27 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM FROM THE 3/10/2010 DRAFT OF THE COMMON CORE STANDARDS Number—Base Ten 1-NBT Numbers up to 100 1. Read and write numbers to 100. 2. Starting at any number, count to 100 or beyond. 3. Understand that when comparing two-digit numbers, if one number has more tens, it is greater; if the amount of tens is the same in each number, then the number with more ones is greater. 4. Compare and order two-digit numbers based on meanings of the tens and ones digits, using > and < symbols to record the results of comparisons. Adding and subtracting in base ten 5. Calculate mentally, additions and subtractions within 20. b. Use strategies that include counting on; making ten (for example, 7 + 6 = 7 + 3 + 3 = 10 + 3 = 13); and decomposing a number (for example, 17 – 9 = 17 – 7 – 2 = 10 – 2 = 8). 6. Demonstrate fluency in addition and subtraction within 10. 7. Understand that in adding or subtracting two-digit numbers, one adds or subtracts like units (tens and tens, ones and ones) and sometimes it is necessary to compose or decompose a higher value unit. 8. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count. 9. Add one-digit numbers to two-digit numbers, and add multiples of 10 to one-digit and twodigit numbers. 10. Explain addition of two-digit numbers using concrete models or drawings to show composition of a ten or a hundred. 11. Add two-digit numbers to two-digit numbers using strategies based on place value, properties of operations, and/or the inverse relationship between addition and subtraction; explain the reasoning used. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 28 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #3: Developing an understanding of linear measurement and facility in measuring lengths Kindergarten Grade 1 Measurement Measurement Ordering objects by measurable attributes (NCTM-K) Developing an understanding of linear measurement and facility in measuring lengths (NCTM2nd) Explore other measurement attributes Grade 2 Estimate and measure length Solve problems using addition and subtraction of length, money, and time (M.PS.01.08) Solve measurement problems involving length, money, and perimeter (M.PS.01.10, M.TE.02.11) Geometry Geometry Describing shapes and space (NCTM-K) Composing and decomposing geometric shapes (NCTM-1st) Create, explore, and describe shapes Identify and describe shapes Explore geometric patterns Work with unit fractions Key: bold, non-italic = Michigan Curriculum Focal Points non-bold, non-italic = GLCE topics associated with that focal point non-bold, italic = Cross over GLCE topics associated with another focal point Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 29 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #3: Developing an understanding of linear measurement and facility in measuring lengths BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS Big Idea #17 (Measurement) Some attributes of objects are measurable and can be quantified using unit amounts. Measurement involves a selected attribute of an object (length, area, mass, volume, capacity) and a comparison of the object being measured against a unit of the same attribute. The larger the unit of measure, the fewer units it takes to measure the object. A given measurement can be expressed in many equivalent forms of different units of the same attribute or dimension: o 2 feet = 24 inches o 1 cubic yard = 27 cubic feet The magnitude of the attribute to be measured and the accuracy needed determines the appropriate measurement unit. A figure or object can be constructed from or decomposed into figures of the same dimension. The measurement of a given attribute of the object is equal to the sums of the measurements of the components of the object for that attribute: o if a polygon is decomposed into other polygons, the area of the original polygon is equal to the sum of the areas of the component polygons o the perimeter of a polygon can be found by adding together the lengths of the sides o if an angle is composed from smaller angles, the measure of the total angle is equal to the sums of the measures of the component angles o if a box is composed from smaller boxes, the total volume of the box is equal to the sum of the volumes of the component boxes Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 30 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #3: Developing an understanding of linear measurement and facility in measuring lengths INSTRUCTIONAL IMPLICATIONS This may seem obvious, but if children are going to compare and order objects by measurable attributes, they must do so with actual objects. The Concrete-Representational-Abstract (CRA) strategy, which has a strong research base, suggests that drawing pictures is a good way to connect the concrete to the representational6. Children first compare and order real objects, drawing pictures of what happened. Eventually, children can compare pictures of objects. In their book Learning and Teaching Early Math: The Learning Trajectories Approach, Douglas H. Clements and Julie Sarama (2009) suggest these instructional tasks for five- and six-yearolds exploring length: What’s the Missing Step? (p.171) Children see stairs made from connecting cubes from 1 to 6. They cover their eyes and the teacher hides one step. They uncover their eyes and identify the missing step, telling how they knew. X-Ray Vision 1 (p. 171) Children place Counting Cards, 1 to 6 or more, in order, face down. Then they take turns pointing to the cards and using their “x-ray vision” to tell which card it is. Length Riddles (p.171) Ask questions such as, “You write with me and I am 7 cubes long. What am I?” As children develop (around age 7), provide fewer cues (e.g., only the length) and only one unit per child so they have to iterate (repeatedly “lay down”) a single unit to measure. Measure with physical or drawn units (p.171) Focus on long, thin units such as toothpicks cut to one inch sections. Explicit emphasis should be given to the linear nature of the unit. That is children should learn that, when measuring with, say, centimeter cubes, it is the length of one edge that is the linear unit – not the area of a face or volume of the cube. It is around age 7 that children relate size and number of units explicitly and can make statements such as, “If you measure with centimeters instead of inches, you’ll need more of them, because each one is smaller.”7 The authors also note that children may be able to draw a line to a given length before they measure objects accurately, and they suggest this instructional task: “Concrete-Representational-Abstract Instructional Approach.” The Access Center: Improving Outcomes for All Students K-8. Accessed on 3-22-2010 from http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp 7 Clements, Douglas H. and Sarama, Julie. (2009) Learning and Teaching early Math: The Learning Trajectories Approach. Routledge: New York, NY. p. 172. 6 Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 31 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Use line-drawing activities to emphasize how you start at the 0 (zero point) and discuss how, to measure objects, you have to align the object to that point. Similarly, explicitly discuss what the intervals and the number represent, connecting these to end-to-end length measuring with physical units. (p. 172) Children reach the “length measurer” stage around age 8. At this stage, the child sees the length of a bent path as the sum of its parts (not the distance between the endpoints). He is able to measure length. He knows the need for identical units and understands the relationship between different units. He can use partitions of a unit, use the zero point on rulers, understands accumulation of distance, and begins to estimate. Here is an example of an instructional task appropriate for children at this level: Children create units of units, such as a “footstrip” consisting of traces of their feet glued to a roll of adding-machine tape. They measure in different-sized units (e.g., 15 paces or 3 footstrips each of which has 5 paces) and accurately relate these units. They also discuss how to deal with leftover space, to count it as a whole unit or as part of a unit. (p. 172). Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 32 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 Focal Point #3: Developing an understanding of linear measurement and facility in measuring lengths RELATED GLCES Measurement Estimate and measure length M.UN.01.01 Measure the lengths of objects in non-standard units, e.g., pencil length, shoe lengths, to the nearest whole unit. M.UN.01.02 Compare measured lengths using the words shorter, shortest, longer, longest, taller, tallest, etc. FROM THE 3010-2010 DRAFT OF THE COMMON CORE STANDARDS Measurement and Data 1-MD Length measurement 1. Order three objects by length; compare the length of two objects indirectly by using a third object. 2. Understand that the length of an object can be expressed numerically by using another object as a length unit (such as a paper-clip, yardstick, or inch length on a ruler). The object to be measured is partitioned into as many equal parts as possible with the same length as the length unit. The length measurement of the object is the number of length units that span it with no gaps or overlaps. For example, “I can put four paperclips end to end along the pencil, so the pencil is four paperclips long.” 3. Measure the length of an object by using another object as a length unit. Time measurement 4. Tell time from analog clocks in hours and half- or quarter-hours. Representing and interpreting data 5. Organize, represent, and interpret data with several 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. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 33 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 GLCEs not related to a focal point Key: Builds on previous grade(s) Related to topics within or beyond mathematics Later grade at which topic relates to a focal point Kindergarten Explore concepts of time Grade 2 Grade 1 Tell time Grade 2 Grade 2 Understand meaning of multiplication and division Grades 3, 4, 5 Create and describe shapes Grade 2 Understand the concept of area Grade 3 Create and describe patterns involving geometric objects Read thermometers Use pictographs Use coordinate systems Create, interpret, and solve problems involving pictographs Key: bold, non-italic = Michigan Curriculum Focal Points non-bold, non-italic = GLCE topics associated with that focal point non-bold, italic = Cross over GLCE topics associated with another focal point Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 34 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Grade 1 GLCEs not related to a focal point Approximately 70% - 80% of Tier 1 instruction should relate to the grade-level Focal Points identified previously. No more than 20% - 30% of Tier 1 instruction should be devoted to the following GLCEs, which are not related to a focal point. Measurement Tell time M.UN.01.03 Tell time on a twelve-hour clock face to the hour and half-hour. Geometry Create and describe shapes G.GS.01.01 Create common two-dimensional and three-dimensional shapes, and describe their physical and geometric attributes, such as color and shape. G.LO.01.02 Describe relative position of objects on a plane and in space, using words such as above, below, behind, in front of. Create and describe patterns involving geometric objects G.SR.01.03 Create and describe patterns, such as repeating patterns and growing patterns using number, shape, and size. G.SR.01.04 Distinguish between repeating and growing patterns. G.SR.01.05 Predict the next element in a simple repeating pattern. G.SR.01.06 Describe ways to get to the next element in simple repeating patterns. Data and Probability Use pictographs D.RE.01.01 Collect and organize data to use in pictographs. D.RE.01.02 Read and interpret pictographs. D.RE.01.03 Make pictographs of given data using both horizontal and vertical forms of graphs; scale should be in units of one and include symbolic representations, e.g., represents one child. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 35 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM FROM THE 3/10/2010 DRAFT OF THE COMMON CORE STANDARDS Geometry 1-G Shapes, their attributes, and spatial reasoning 1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size) for a wide variety of shapes. 2. Understand that shapes can be joined together (composed) to form a larger shape or taken apart (decomposed) into a collection of smaller shapes. Composing multiple copies of some shapes creates tilings. In this grade, “circles,” “rectangles,” and other shapes include their interiors as well as their boundaries. 3. Compose two-dimensional shapes to create a unit, using cutouts of rectangles, squares, triangles, half-circles, and quarter-circles. Form new shapes by repeating the unit. 4. Compose three-dimensional shapes to create a unit, using concrete models of cubes, right rectangular prisms, right circular cones, and right circular cylinders. Form new shapes by repeating the unit. Students do not need to learn formal names such as “right rectangular prism.” 5. Decompose circles and rectangles into two and four equal parts. Describe the parts using the words halves, fourths, and quarters, and using the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the parts. Understand that decomposing into more equal shares creates smaller shares. 6. Decompose two-dimensional shapes into rectangles, squares, triangles, half-circles, and quarter-circles, including decompositions into equal shares. Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 36 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Suggested First Grade Vocabulary Taken from Huron County Mathematics Curriculum Framework January 3, 2006 Number and Operations add addend addition altogether backward base ten block between combination compare* counting back counting on counting up difference digit doubles doubles plus one eight equal (=) equation estimate even* fact family facts fair share fifth first five forward* four fourth fraction greater than (>) half hundreds left less less than (<) mental math minus more most nine number line number sentences numbers odd* one one less one more one-fourth one-half ones one-third operation order ordinal numbers* pattern place place value (hundreds, ten, ones) plus quantity record same same as second sequence* sets seven six sixth skip count strategy subtract subtraction sum taking away ten tens third three turn-arounds two unit whole zero * Instructional term on which student might not be assessed Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 37 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Measurement add addition afternoon analog clock* between calendar capacity Celsius cent sign (¢) centimeter cents circumference clock coin(s) colder cooler cup date degree difference digital clock* dime distance dollar bills dollar sign ($) equals estimate evening gallon gram half-hour height hotter hour hour hand inch increase last month last week last year length liter longer longest measure measurement meter metric minute minute hand money month morning next month next week next year nickel night non-standard unit* penny pint pound quart quarter ruler scale second hand shorter shortest solution standard unit subtraction sum taller tallest temperature thermometer* time today tomorrow unit volume warmer week weight width year yesterday * Instructional term on which student might not be assessed Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 38 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM Geometry above/below angle balance box category change circle closed color cone construct corner cube curve cylinder decrease* dot edge enlarge* face flip folding front/back geoboard group growing hexagon inside/outside large left line next to objects on open out oval overlap parallel part pattern perpendicular rectangle reduce* repeating right same shape same size segment shape shrinking* side size slide small solid sort sphere square symmetry tangrams texture thick/thin tile top trapezoid triangle turn under/over up record represent results same sort survey table tally tally marks traits Data and Probability bar graph chance chart collect compare estimate experiment fewer graph growth hypothesis /guess interpret least less line graph more most organize pictograph pictograph graph picture graph predict * Instructional term on which student might not be assessed Finding Focus for Mathematics Instruction – Grade 1 Huron Intermediate School District 39 of 40 March 25, 2010 printed 6/25/2017 at 9:39:10 PM