Lake Elsinore Unified School District Instructional Module To Enhance the Teaching of Envision Math CA Edition WORK IN PROGRESS Grade 3 Module 7 Understanding the Relationship Between Multiplication and Division Revised June 2015
Trimester 1 st Trimester Module/Topic (based on Dana Center) Exploring Equal Groups as Foundations for Multiplication and Division Developing Conceptual Understanding of Area 3rd Mathematics Sequence 2014-2015 Envision Lessons 6-1, 6-2, 6-3, 7-1, 7-4, 8-1, 8-2, 9-1, 9-2 Online: 4-1, 4-2, parts of 4-4 & 4-5 5-1, 5-5 6-2, 6-3 7-1, 7-2, 7-5, 7-6 8-1* 18-3, 18-4 Online 4-3 14-1, 14-2, 14-3, 14-4, 14 6 1-1, 1-2, 1-3, 1-6 Number & Base Ten 2-1*(intro round) Operations All Topic 3 & 4 lessons Developing Strategies 18-1(perimeter) for Addition and Online- All topic 1 & 3, 2-1, 2-2, Subtraction 2-3, 2-4, 2 5, 13-1 (perimeter) 12-1,12-2, 12-3, Understanding Fractions 18-5E (transition student edition pg.32-33) Online- 9-1 to 9-6, 11 6, 11 7, 11 8 Approximate Days 12 days 12 Days 15 days 12 days 2 nd Trimester Using Fractions in In Measurement and Data Solving Addition and Subtraction problems involving Measurement and Time 12-4, 16-2 Online- 9-7 & Review of topic 9* from unit 4, 16-1, 16-2 16-7,16-8, 16-9, 17-3, 17-4 Online - 12-1, 12-2, 12-3, 12-4, 12-5, 15-2, 15-3 7-2, 7-5a(transition student edition pg. 2 3) 8-1 to 8 6, 10-1, 10-2(review from 1) 10-3, 10-4, 10-5, 10-6 Understanding the 10-5a(transition student edition pg.6-7) Relationship Between Multiplication and Online only- Division 6-2, 6-3, 6-4, 8-1*(Rev. from 1), 8-2, 8-3, 8-4, 8 6, 8-7, 8-8 12 days 12 days 12 days
2 nd Trimester Investigating Patterns in Number and Operations/Rounding 2-1*(rev. from 3), 2-2, 6-5, 11-3, 11-4, 14-1, 20-5 Online 2-5*(rev. from 3),2-6, 2-7, 2 9, 5-2 to 5-6, 8-5, 9-8, 16-3, 16-4, 16 5 10 days Developing strategies for Multiplication and Division including Area 6-2, 14 3*(rev. from 1) 14-4, 18-3*,18-4*(rev. from 2) Online- 4-1* & 4-2*(rev. from 2) 4-3, 4-4, 4-5, 6-1, 9 8, 14-3*(rev. from 2), 14-5, 14-7 12 days Understanding Equivalent Fractions Comparing Fractions Solving Problems Involving Area 12-6, 12-7a (transition student edition pg.10-11), 12-7, 12-8a (transition student edition pg.12-13), 12-8 Online 5-7, 10 5 to 10 9 12-5, 12-7 Online 10-1 to 10-4, 10-8 9-3a(transition student edition pg.4-5) 18-5c(transition student edition pg.26-29), 18-5a(transition student edition pg.22-23), 6-2*(rev. from 1), 7-2, 7-3 Online- 4-3*(rev. from 2), 6-1*(rev. from 9),14-5*(rev. from 9), 14-8 to 14-10 10 days 10 days 12 days 3 rd Trimester Solving Problems Involving Shapes Using Multiplication and Division to solve Measurement Problems 5-5 to 5-8, 18-1, 18-2, 18-3a(transition student edition pg.20-21), 18-5d(transition student edition pg.30-31) Online 13-2 to 13 5, 11-1 to 11 5 14-3(rev. of 1 &9)All topic 10 review,17-3,17-4 17 6a(transition student edition pg.18-19) Online All of topic 15, 4-4, 4-5, 8-6*(rev. of 7) 12 days 12 days Demonstrating Computational Fluency in Problem Solving 7-2*(rev. of 1), 7-5, 14-4*(rev. of 9) 15-1, 15-2, 15-8a(transition student edition pg.14-15), 17-6 Online- 3-3 to 3-10*(rev. from 3), Review of topics 5,6, and 8 12 days
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 In the years prior to Grade 3 students gained an understanding of number and used strategies based on place value, properties of operations, and the relationship between addition and subtraction to add and subtract within 1000. Hey worked with standard units of measure for length and described attributes of shapes. Two major emphases of the Grade 3 year are the operations of multiplication and division and the concept of fractions. These concepts are introduced early in the year in order to build a foundation for students to revisit and extend their conceptual understanding with respect to these concepts as the year progresses. By the end of the year, students recall all products of two single- digit numbers. Third grade students develop understanding of fractions as numbers, and compare and reason about fraction sizes. This work with fractions is a cornerstone for developing reasoning skills and conceptual understanding of fraction size and fractions as part of the number system throughout this year and their future work with fractions and ratios. To continue the study of geometry, students describe and analyze shapes by their sides, angles, and definitions. In the final unit in this sequence of units, students generalize and apply strategies for computational fluency. This document reflects our current thinking related to the intent of the Common Core State Standards for Mathematics (CCSSM) and assumes 160 days for instruction, divided among 15 units. The number of days suggested for each unit assumes 45- minute class periods and is included to convey how instructional time should be balanced across the year. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the CCSSM; however, the order of the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited several times during the course; others may be only partially addressed in different units, depending on the focus of the unit. Strikethroughs in the text of the standards are used in some cases in an attempt to convey that focus, and comments are included throughout the document to clarify and provide additional background for each unit. Throughout Grade 3, students should continue to develop proficiency with the Common Core's eight Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 5. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 4. Model with mathematics. 8. Look for and express regularity in repeated reasoning. These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in those units. When using this document to help in planning your district's instructional program, you will also need to refer to the CCSSM document, relevant progressions documents for the CCSSM, and the appropriate assessment consortium framework. NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 1: Exploring equal groups as a foundation for multiplication and division Suggested number of days: 12 envision Lessons In Grade 2 students have added groups of objects by skip-counting and using repeated addition (2.OA.C.4). In this unit students connect these concepts to multiplication and division by interpreting and representing products and quotients. Students begin developing these concepts by working with numbers with which they are more familiar, such as 2s, 5s, and 10s, in addition to numbers that are easily skip counted, such as 3s and 4s. Since multiplication is a critical area for Grade 3, students will build on these concepts throughout the year, working towards fluency by the end of the year. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA A. Represent and solve problems involving multiplication and division. 1. Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 7. 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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8. 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. 1 NOTE: 1See Glossary, Table 2. C. Multiply and divide within 100. 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. Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. Comments In 3.OA.A.1 situations with discreet objects should be explored first when developing a conceptual understanding of multiplication, followed by measurement examples involving area models1 3.OA.A.2 will be readdressed in unit 7 in order to provide students the opportunity to develop computational strategies as they extend the range of numbers with which they compute. 3.OA.A.3 will be readdressed in unit 7 and finalized in unit 14 to include measurement quantities in order to provide students multiple opportunities to develop and practice these concepts. 3.OA.C.7 will be readdressed in unit 7 and unit 15 in order to provide students the opportunity to develop computational strategies as they extend the range of numbers with which they compute. Students use concrete objects or pictures to help conceptualize and solve problems (MP.1). They use arrays and other representations to model multiplication and division (MP.4) and contextualize 6-1, 6-2, 6-3,8-1,8-2, 9-1,9-2, 10-2, 14-3 Online - 4-1, 4-2, parts of 4-4 & 4-5, 5-1, 5-5, 6-2, 6-3, 7-1, 7-2, 7-5, 7-6, 8-1* NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 2: Developing conceptual understanding of area Suggested number of days: 12 envision Lessons This unit provides ample time, and should include multiple experiences, for students to explore the connections among counting tiles, skip counting the number of tiles in rows or columns, and multiplying the side lengths of a rectangle to determine area. Students understanding of these connections is critical content at this grade, and must occur early in the school year, thereby allowing time for understanding and fluency to develop across future units.2 Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA B. Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties of operations as strategies to multiply and divide. 2 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.) Note:2 Students need not use formal terms for these properties. Measurement and Data 3.MD C. Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 5. Recognize area as an attribute of plane figures and understand concepts of area measurement. 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. 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. 6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 7. Relate area to the operations of multiplication and addition. 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. Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 6. Attend to precision. 7. Look for and make use of structure. Comments 3.OA.B.5 will be readdressed in unit 9 with a focus on the distributive property and in unit 12 with a focus on the associative property. Students analyze the structure of multiplication and division (MP.7) through their work with arrays (MP.2) and work towards precisely expressing their understanding of the connection between area and multiplication (MP.6). 18-3, 18-4 Online 4-3, 14-1,14-2,14-3,14-4,14-6 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 3: Developing strategies for addition and subtraction Suggested number of days: 12 envision Lessons In Grade 2 students used addition and subtraction within 1000 using concrete objects and strategies. In this unit students increase the sophistication of computation strategies for addition and subtraction that will be finalized by the end of the year. This unit introduces the concept of rounding, which provides students with another strategy to judge the reasonableness of their answers in addition and subtraction situations. Perimeter provides a context in which students can practice both rounding and addition and subtraction (e.g. estimating the perimeter of a polygon). Common Core State Standards for Mathematical Content Number and Operations in Base Ten 3.NBT A. Use place value understanding and properties of operations to perform multi- digit arithmetic. 4 1. Use place value understanding to round whole numbers to the nearest 10 or 100. 2. 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. NOTE: 4A range of algorithms may be used. Measurement and Data 3.MD D. Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 8. 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. Common Core State Standards for Mathematical Practice 6. Attend to precision. 8. Look for and express regularity in repeated reasoning. Comments 3.NBT.A.1 introduces the concept of rounding, which is new to students and will be revisited in unit 8 in the context of multiplication. 3.NBT.A.2 will be finalized in unit 15 in order to give students time to reach fluency in addition and subtraction within 1000 by the end of the year. 3.MD.D.8 is the first time perimeter appears in the CCSS- M. Students are not expected to use formulas until Grade 4 (4.MD.A.3). 3.MD.D.8 will be addressed in full in unit 13 after students have been introduced to and worked with the concept of area. Students use precise language to make sense of their solution in the context of a problem and the magnitude of the numbers (MP.6). Students also generalize algorithms and strategies and look for shortcuts (MP.8). 1-1, 1-2, 1-3, 1-6, 2-1*(intro round) All topic 3&4, 18-1(perimeter) Online- All topic 1 & 3, 2-1, 2-2, 2-3, 2-4, 2-5, 13-1 (perimeter) NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 4: Understanding unit fractions Suggested number of days: 12 envision Lessons In previous grades students have had experience partitioning shapes into fair shares (1.G.A.3 and 2.G.A.3), using words to describe the quantity. In this unit students extend this understanding to partition shapes and number lines, representing these fair shares using fraction notation. Similar to how students view 1 as the building block of whole numbers, students learn to view unit fractions as building blocks understanding that every fraction is a combination of unit fractions.3 Common Core State Standards for Mathematical Content Geometry 3.G A. Reason with shapes and their attributes. 2. Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. 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. Number and Operations Fractions5 3.NF A. Develop understanding of fractions as numbers. 1. 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. 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. 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. NOTE: 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 6. Attend to precision. Comments The focus of 3.NF.A.1 and 3.NF.A.2a in this unit is on fractions between 0 and 1. Fractions greater than 1 will be introduced in unit 5. Students use number lines to represent fractions in a new way (MP.4). It is key for students to have meaningful conversations around this concept to develop precise language about the components of fractions and location on the number line (MP.3, MP.6). 12-1,12-2, 12-3, 18-5E (transition student edition pg.32-33) Online- 9-1 to 9-6, 11-6, 11-7, 11-8 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 5: Using fractions in measurement and data Suggested number of days: 12 envision Lessons In this unit students extend their work with measurement and data involving whole numbers to include fractional quantities. Measurement and data are used as a context to support students understanding of fractions as numbers. In students work with data, context is important, because data are not just numbers; they are numbers with meaning. Through experience with measurement, students realize fractions allow us to represent data much more accurately than just representing data with whole numbers.4 Common Core State Standards for Mathematical Content Number and Operations Fractions5 3.NF A. Develop understanding of fractions as numbers. 1. 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. 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. 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. NOTE: 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. Measurement and Data 3.MD B. Represent and interpret data. 4. 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. Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically. Comments 3.NF.A.1 is repeated here to include fractions greater than 1. Students use tools to generate measurement data (MP.5) and make connections among different representations of the quantities and their relation to the given data context (MP.2). 12-4, 16-2 Online- 9-7 & Review of topic 9* from unit 4, 16-1, 16-2 Unit 6: Solving addition and subtraction problems involving measurement Suggested number of days: 12 envision Lessons NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 The focus of this unit is to develop a conceptual understanding of measuring mass, liquid volume, intervals of time, and using measurement as a context for the development of fluency in addition and subtraction. Common Core State Standards for Mathematical Content Measurement and Data 3.MD A. Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. 1. 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. 2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). 6 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. 7 NOTE: 6 Excludes compound units such as cm3 and finding the geometric volume of a container. 7Excludes multiplicative comparison problems (problems involving notions of times as much ; see Glossary, Table 2). Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 5. Use appropriate tools strategically. 4. Model with mathematics. Comments 3.MD.A.1 is included here as an opportunity to model addition and subtraction situations with time as the context. 3.MD.A.2 is addressed in full in unit 14 to include multiplication and division situations. Students can apply the mathematics they know to persevere in solving problems arising in everyday life, society, and the workplace (MP.1, MP.4). Selecting and using appropriate tools supports the development of measurement concepts by asking students to reason about which tools are appropriate and how to use tools efficiently (MP.5). 16-7,16-8, 16-9, 17-3, 17-4 Online - 12-1, 12-2,12-3, 12-4, 12-5, 15-2,15-3 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 7: Understanding the relationship between multiplication and division Suggested number of days: 12 envision Lessons The emphasis of this unit is for students to develop a solid understanding of the connection between multiplication and division. Students recognize that multiplication strategies can be used to make sense of and solve division problems. This unit provides students a solid foundation in solving problems with equal groups and arrays, which is necessary to support future success with measurement problems.5 Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA A. Represent and solve problems involving multiplication and division. 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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8. 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. 1 NOTE: 1See Glossary, Table 2. B. Understand properties of multiplication and the relationship between multiplication and division. 6. Understand division as an unknown- factor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8. C. Multiply and divide within 100. 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. Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 7. Look for and make use of structure. Comments 3.OA.A.2 and 3.OA.C.7 are revisited in this unit to extend the range of numbers to include all numbers within 100 when multiplying and dividing. 3.OA.A.3 includes equal groups, arrays, and area problem types. Note that multiplicative compare problems are introduced in Grade 4 (4.OA.A.2).6 3.OA.C.7 is finalized in unit 15. This gives students the opportunity to develop and practice strategies in order to achieve fluency by the end of the year. Students make sense of and solve various types of multiplication and division problems (MP.1) by using the relationship between 7-2, 7-5a(transition student edition pg. 2-3) 8-1 to 8-6, 10-1,10-2 (review from 1) 10-3, 10-4, 10-5, 10-6 Online only- 6-2, 6-3, 6-4, 8-1*(Rev. from 1), 8-2, 8-3, 8-4, 8-6, 8-7, 8-8 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 8: Investigating patterns in number and operations Suggested number of days: 12 envision Lessons The focus of this unit is for students to identify arithmetic patterns in order to develop a deeper understanding of number and number relationships. In subsequent units, students will use the understanding of pattern developed in this unit to strengthen their computational strategies and skills. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA D. Solve problems involving the four operations, and identify and explain patterns in arithmetic. 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.3 NOTE: 3This 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). 9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. 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. Number and Operations in Base Ten 3.NBT A. Use place value understanding and properties of operations to perform multi- digit arithmetic. 4 1. Use place value understanding to round whole numbers to the nearest 10 or 100. 3. 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. NOTE: 4A range of algorithms may be used. Measurement and Data 3.MD B. Represent and interpret data. 3. 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. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. Comments 3.OA.D.8 will be revisited in unit 15 to address the use of equations and letters for unknown quantities. 3.NBT.A.1 is revisited in this unit to give students opportunities to make sense of rounding in multiplication situations. Students examine patterns in arithmetic (MP.7) and discuss what they discover (MP.3). 2-1*(rev. from 3), 2-2, 6-5, 11-3, 11-4, 14-1, 20-5 Online only 2-5*(rev. from 3),2-6, 2-7, 2-9, 5-2 to 5-6, 8-5, 9-8, 16-3, 16-4, 16-5 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 9: Developing strategies for multiplication and division Suggested number of days: 12 envision Lessons The focus for this unit is developing a conceptual understanding of decomposing multiplication problems through the use of the distributive property and the concept of area. Students are not required to use the properties explicitly, but are encouraged to discuss this concept and use area diagrams to support their reasoning. 7 Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA B. Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties of operations as strategies to multiply and divide. 2 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.) Note: 2Students need not use formal terms for these properties. Measurement and Data 3.MD C. Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 7. Relate area to the operations of multiplication and addition. 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. 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. Common Core State Standards for Mathematical Practice 5. Use appropriate tools strategically. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Comments 3.OA.B.5 will be revisited in unit 12 to address the associative property of multiplication. Students use area diagrams and tiling (MP.5) to model the distributive property and generalize this experience to calculations (MP.7, MP.8). 6-2, 14-3*(rev. from 1) 14-4, 18-3*,18-4*(rev. from 2) Online-4-1* & 4-2*(rev. from 2) 4-3, 4-4, 4-5, 6-1, 9-8, 14-3*(rev. from 2), 14-5, 14-7 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 10: Understanding equivalent fractions Suggested number of days: 12 envision Lessons In this unit students develop a conceptual understanding of equivalence. Multiple types of models and representations should be used to help students develop this understanding. Students will apply their understanding of equivalence in the next unit as they learn to compare fractions. Through repeated experience locating fractions on the number line, students will recognize that many fractions label the same point and use this to support their understanding of equivalency.8 Common Core State Standards for Mathematical Content Number and Operations Fractions5 3.NF A. Develop understanding of fractions as numbers. 3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. 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. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. 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. NOTE: 5Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. Common Core State Standards for Mathematical Practice 4. Model with mathematics. 6. Attend to precision. Comments [3.NF.A.3] The focus of this unit is around equivalence. Although the cluster heading includes comparison of fraction, fraction comparisons (3.NF.A.3d) will be addressed in unit 11. Students develop understanding of equivalence by modeling fractions (MP.4) and communicating their understanding of what it means for fractions 12-6, 12-7, 12-8, 12-8a (transition student edition pg.12-13) Online 5-7, 10-5 to 10-9 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 11: Comparing fractions Suggested number of days: 12 envision Lessons In this unit students build on their prior work with fractions to reason about fraction size and structure to compare quantities. This unit focuses on a single standard to provide time for students to develop conceptual understanding of fraction comparisons and practice reasoning about size. Students defend their reasoning and critique the reasoning of others using both visual models and their understanding of the structure of fractions. This reasoning is important to develop a solid understanding of fraction magnitudes.9 Common Core State Standards for Mathematical Content Number and Operations Fractions5 3.NF A. Develop understanding of fractions as numbers. 3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 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. NOTE: 5 Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 5. Use appropriate tools strategically. 7. Look for and make use of structure. Comments Students will use their understanding of structure (i.e., the role of the numerator and denominator) (MP.7) to reason about relative sizes of fractions (MP.3).10 Students use various tools to justify their comparisons, paying particular attention to the same- sized wholes (MP.5). 12-5, 12-7 Online 10-1 to 10-4 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 12: Solving problems involving area Suggested number of days: 12 envision Lessons The focus of this unit is to use area as a context to further develop multiplicative thinking. In this work, students bridge between concrete and abstract thinking, and use strategies to solve problems. This includes solving problems involving rectangular areas by multiplying side lengths and solving for an unknown number in related multiplication and division equations. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA A. Represent and solve problems involving multiplication and division. 4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8? = 48, 5 = 3, 6 6 =?. B. Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties of operations as strategies to multiply and divide. 2 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.) NOTE: 2Students need not use formal terms for these properties. Measurement and Data 3.MD C. Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 7. Relate area to the operations of multiplication and addition. 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. Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 6. Attend to precision. 8. Look for and express regularity in repeated reasoning. Comments 3.OA.B.5 introduces the associative property explicitly for the first time. This property is fundamental for developing higher- level computation strategies.11 In unit 9, students used various strategies to solve area problems. In 3.MD.C.7b students recognize that they can find area in real- world situations by multiplying side lengths without necessarily using a rectangular array. Students move in and out of context to solve these types of problems (MP.2) and use their repeated experience with area models to recognize that area problems can be solved using multiplication (MP.8). Students also explain precisely how an array corresponds to an expression (MP.6). 9-3a(transition student edition pg.4-5) 18-5c(transition student edition pg.26-29), 18-5a(transition student edition pg.22-23), 6-2*(rev. from 1), 7-2, 7-3 Online-4-3*(rev. from 2), 6-1*(rev. from 9), 14-5*(rev. from 9), 14-8 to 14-10 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 13: Solving problems involving shapes Suggested number of days: 12 envision Lessons The focus of this unit is reasoning with shapes and their attributes, including area and perimeter. The standards in this unit strongly support one another because perimeter, like area, is an attribute of shape. Prior work with area and perimeter allows students differentiate between the two measures in this unit.12 Common Core State Standards for Mathematical Content Measurement and Data 3.MD D. Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 8. 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 3.G A. Reason with shapes and their attributes. 1. 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. Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. Comments 3.MD.D.8 is addressed in full in this unit and focuses on distinguishing between linear and area measures and examining their relationship. Students look for and make use of structure (MP.7) as they determine categories and subcategories of shapes by identifying and reasoning about their attributes. Students make conjectures involving the attributes and measures of shapes and analyze various ways of approaching problems (MP.1, MP.3) 5-5 to 5-8, 18-1, 18-2, 18-3a (transition student edition pg.20-21) Online 13-2 to 13-5, 11-1 to 11-5 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 14: Using multiplication and division to solve measurement problems Suggested number of days: 12 envision Lessons This unit extends students work in unit 6 to include multiplication and division to solve problems involving measurement Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA A. Represent and solve problems involving multiplication and division. 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. 1 NOTE: 1See Glossary, Table 2. Measurement and Data 3.MD A. Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. 2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). 6 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. 7 NOTE: 6 Excludes compound units such as cm3 and finding the geometric volume of a container. 7Excludes multiplicative comparison problems (problems involving notions of times as much ; see Glossary, Table 2). Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically. Comments 3.OA.A.3 includes the use of all of the problem types Table 2 in CCSSM except for multiplicative compare problems which will be introduced in Grade 4.13 Students use strategies for multiplication and division to conceptualize and solve measurement problems (MP.1, MP.2). Students select appropriate tools and justify their selection for measuring different quantities (MP.5). 14-3(rev. of 1 &9) All topic 10 review, 17-3, 17-4 17-6a(transition student edition pg.18-19) Online All of topic 15, 4-4, 4-5, 8-6*(rev. of 7) NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Unit 15: Demonstrating computational fluency in problem solving Suggested number of days: 10 envision Lessons This is a culminating unit in which students focus on problem solving in order to demonstrate fluency with addition and subtraction to 1000 and demonstrate fluency for multiplication and division within 100. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 3.OA C. Multiply and divide within 100. 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. D. Solve problems involving the four operations, and identify and explain patterns in arithmetic. 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.3 NOTE: 3This 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). Number and Operations in Base Ten 3.NBT A. Use place value understanding and properties of operations to perform multi- digit arithmetic. 4 2. 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. NOTE: 4A range of algorithms may be used. Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 8. Look for and express regularity in repeated reasoning. Comments 3.OA.D.8 was introduced in unit 8 and is finalized in this unit to include the use of letters to represent unknown quantities in equations. Students demonstrate fluency in multiplication and division within 100 using various strategies and the properties of these operations (MP.8). They also represent these calculations and problem situations abstractly using letters (MP.2). 7-2*(rev. of 1), 7-5, 14-4*(rev. of 9) 15-1, 15-2, 17-6, 15-8a(transition student edition pg.14-15) Online- 3-3 to 3-10*(rev. from 3), Review of topics 5,6 and 8 NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Sequenced Units for the Common Core State Standards in Mathematics Grade 3 Suggested number of days: 10 envision Lessons NGA Center/CCSSO are the sole owners and developers of the Common Core Standards. Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Charles A. Dana Center at the University of Texas at Austin
Lesson Number 3rd Grade Module 7 at a Glance Some lessons may take more than one day. Lesson Focus 1 Solve for unknowns using known facts 2 Word problems using equal groups 3 Breaking apart to solve with known facts 4 Area model to solve with unknown facts 5 Division with measurement quantity 6 Inverse relationship of multiplication/division 7 EADMS assessment online or paper Mod7assessment
Standards 3 rd Grade Unit/Module Overview Unit 7- Understanding Relationships Between Multiplication and Division 3.OA A.2-3 A. Represent and solve problems involving multiplication and division. 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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8. 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. 3. OA B.6 B. Understand properties of multiplication and the relationship between multiplication and division. 6. Understand division as an unknown-factor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8. Model Lesson summary- Materials Provided: Book: Spaghetti and Meatballs for All! By Marilyn Burns Sentence strips Master cookie visual aide (3 Mod 7.2a) Graph paper Materials Needed: 1-inch tiles or Post-It notes or construction paper cut to size Toothpicks or uncooked spaghetti noodles (if not available students can draw chairs or use X s for seats Counters for modeling Paper or white boards to record student work Post-It notes or construction paper cut to size for each student Routines during the unit- Number strings or Number talks that include multiplication or division patterns. www.estimation180.com estimation warm ups Online lessons attached (homework) Reteach and Practice pages for lessons: 6-3, 6-4, 8-6,8-6 and 8-6 quick check Enrichment: 6-3, 6-4 Envision Workbook lessons needed (homework) Reteach and Practice pages for lessons: 10-1, 8-4 and 8-6
Connecting Mathematical Practices and Content Grade 3 The Standards for Mathematical Practice (MP) are developed throughout each grade and, together with the content standards, prescribe that students experience mathematics as a rigorous, coherent, useful, and logical subject that makes use of their ability to make sense of mathematics. The MP standards represent a picture of what it looks like for students to understand and do mathematics in the classroom and should be integrated into every mathematics lesson for all students Although the description of the MP standards remains the same at all grades, the way these standards look as students engage with and master new and more advanced mathematical ideas does change. Below are some examples of how the MP standards may be integrated into tasks appropriate for Grade 3 students. Standards for Mathematical Practice MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP. 3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. Explanation and Examples from Mathematics Framework In third grade, mathematically proficient students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Students may use concrete objects, pictures, or drawings to help them conceptualize and solve problems, such as Jim purchased 5 packages of muffins. Each package contained 3 muffins. How many muffins did Jim purchase? or Describe another situation where there would be 5 groups of 3 or 5 3. Students may check their thinking by asking themselves, Does this make sense? Students listen to other students strategies and are able to make connections between various methods for a given problem. Students recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities. For example, students apply their understanding of the meaning of the equal sign as the same as to interpret an equation with an unknown. When given 4? = 40, they might think: 4 groups of some number is the same as 40 4 times some number is the same as 40 I know that 4 groups of 10 is 40 so the unknown number is 10 The missing factor is 10 because 4 times10 equals 40. Teachers might ask, How do you know or What is the relationship between the quantities? to reinforce students reasoning and understanding. Students may construct arguments using concrete referents, such as objects, pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions that the teacher facilities by asking questions such as How did you get that? and Why is that true? Students explain their thinking to others and respond to others thinking. For example, after investigating patterns on the 100s chart, students might explain why the pattern makes sense. Students represent problem situations in multiple ways using numbers, words (mathematical language), drawing pictures, and objects. They might also represent a problem by acting it out or by creating charts, lists, graphs, or equations. For example, students use various contexts and a variety of models (e.g., circles, squares, rectangles, fraction bars, and number lines) to represent and develop understanding of fractions. Students use models to represent both equations and story problems and can explain their thinking. They evaluate their results in the context of the situation and reflect on whether the results make sense. Students should be encouraged to answer questions, such as What math drawing or diagram could you make and label to represent the problem? or What are some ways to represent the quantities? Connecting Mathematical Practices and Content -Grade 3
Connecting Mathematical Practices and Content Grade 3 MP.5 Use appropriate tools strategically MP.6 Attend to precision. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. Mathematically proficient students consider the available tools (including drawings or estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper to find all the possible rectangles that have a given perimeter. They compile the possibilities into an organized list or a table and determine whether they have all the possible rectangles. Students should be encouraged to answer questions such as, Why was it helpful to use? Students develop mathematical communication skills as they use clear and precise language in their discussions with others and in their own reasoning. They are careful to specify units of measure and to state the meaning of the symbols they choose. For instance, when calculating the area of a rectangle they record the answer in square units. Students look closely to discover a pattern or structure. For instance, students use properties of operations (e.g., commutative and distributive properties) as strategies to multiply and divide. Teachers might ask, What do you notice when? or How do you know if something is a pattern? Students notice repetitive actions in computations and they look for shortcut methods. For instance, students may use the distributive property as a strategy to work with products of numbers they do know to solve products they do not know. For example, to find the product of 7 8, students might decompose 7 into 5 and 2 and then multiply 5 8 and 2 8 to arrive at 40 + 16 or 56. Third grade students continually evaluate their work by asking themselves, Does this make sense? Students should be encouraged to answer questions, such as What is happening? Connecting Mathematical Practices and Content -Grade 3
Instructional Strategies Used in K-7 Instructional Modules Taken from the CA Mathematics Framework and 5 Practices for Orchestrating Productive Mathematics Discussions by Peg Smith and Kay Stein POSE THE PROBLEM INDEPENDENT THINK-PAIR-SHARE TABLE TALK WHOLE GROUP CONSENSUS MONITOR SELECT SEQUENCE CONNECT DISPLAY CAROUSEL- MUSEUM WALK Simply pose the problem, without suggesting or allowing other students to suggest any particular mathematical strategy to solve the problem. Students work independently and quietly, often for the purpose of letting students think about their own reasoning and informal assessment. Students get time to think quietly, then share their thoughts with a partner and listen to their partners thinking. THINK-PAIR-SHARE with more than 2 students. Focus is on pulling the whole class together. Students share their individual ideas and come to an agreement within the group to share with the whole class. Teacher pays close attention to students mathematical thinking and solution strategies as they work on a task, for the purpose of using their observations to decide what and whom to focus on during the class discussion that follows. The teacher, through monitoring, selects student work samples or strategies to display or have students present. The teacher purposefully chooses the order in which student strategies are displayed and/or discussed, often beginning with the more concrete strategies moving to more abstract. The teacher helps students draw connections between their solutions/strategies and others solutions/strategies for the purpose of connecting their thinking to the mathematics we want them to learn The teacher shows student work to the rest of the class for the purpose of allowing students to analyze the students strategies. Each group posts sample work on the wall while students rotate around the room to analyze other students work. A leader from each group may, but does not need to stand near his/her own group s work.
Questions for investigative lessons What makes you say that? Can you explain your reasoning? Did your model help you solve the problem? Did you learn anything new from group s presentation? Does anything you heard said or saw modeled, give you reason to reconsider your reasoning? Can you see the problem differently after group s presentation? What do you notice is happening in this strategy? What does each of the numbers in this strategy represent? Why do you think he/she circled that, labeled that, put that over here etc? Do you agree or disagree with this strategy, why? How is this strategy like the one that we just saw? How is this strategy different than the one we just saw? Where do you see Student A's work in Student B's work? Have we ever seen a similar strategy? When, and why do you think it seems to work in this problem, too? Why do you think he added (or subtracted, multiplied, divided) in this part? Is there another way to say what he/she just said? Who did it the right way? (All ways are worth looking at, even those that do not yield a correct answer) Would the strategy you used work with other numbers?
Exit Ticket Exit Ticket Exit Ticket
Boleto de Salida Boleto de Salida Boleto de Salida
Technology Tools 3 rd grade modules Virtual Manipulatives Mathplayground.com Schoolkitmath for ipad Explain Anything for ipad Tiny Fractions for ipad Educreations for ipad Geoboard for ipad Oh No Fractions for ipad Scholastic Sushi Monster for ipad Math Zombies for ipad Math Friendzy for ipad Videos Estimation180.com Envision math Pearson Success net or Pearson Realize Learnzillion lessons and video lessons You tube videos links given throughout modules Interactive Games Illuminations.nctm.org games and lessons Gamequarium.com A+math.com Envision Pearson success net or Pearson Realize student games can be assigned Lessons Learnzillion.com Illuminations.nctm.org
Problem Solving Recording Sheet Lesson 3 rd Teaching Tool Name Date Show your understanding more than one way drawing, words, and numbers. Draw a picture to represent the problem Use numbers to show the problem and solution Explain how you solved the problem in words
Grade 3 Module7, Lesson 1 Lesson Focus Using known multiplication / division facts to solve for unknown facts. Lesson Purpose Content Standards Practice Standards Introduce Book: Spaghett i and Meatballs for All! By Marilyn Burns To use multiplication and division facts in context. 3.OA A.2-3, B.6: A. Represent and solve problems involving multiplication and division. 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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8. 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. B. Understand properties of multiplication and the relationship between multiplication and division. 6. Understand division as an unknown-factor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. POSE THE PROBLEM Read aloud the book Spaghetti and Meatballs for All! By Marilyn Burns (Read to Page 6 where it shows the diagram of the tables and chairs) then pose problem below: Mr. and Mrs. Comfort are having a dinner party for their family. You are going to be helping them set up the tables and the chairs to fit everyone. PLC Notes
Investigate Materials 1-inch tiles, Post-It notes or construction paper cut to size AND toot hpicks or uncooked spaghetti, students can just draw X s for the chairs of toothpicks and spaghetti is not avalable Provide pairs of students with sticky notes, construction paper cut to size of sticky note or tiles AND students can draw chairs by putting an X (toothpicks or spaghetti as chairs is optional). Model that the sticky note, paper or tile represents the table and X s, the toothpicks or broken pieces of spaghetti represents the chairs. Have students create the table arrangement with 8 tables and 4 people at each table. How many people are the Comforts prepared for? What math facts can you use to represent that seating arrangement? Are there any other facts that you can use to represent that same arrangement? Continue reading the story. Pause each time there is a new table arrangement. Allow students the opportunity to rearrange their tables to model the new arrangement. Allow for discussion time, ask probing questions: How many guests can be seated now? What facts can you use to represent this seating arrangement? What if more guests come? Teachers remember to write equations in differing formats: 32 = 8 x 4 8 x 4 = 32 4 x 8 = 32 32 4 + 8 32 4 + 8 Optional Practice Summarize At the end of the story, ask students if there are any other possible table arrangements for the 32 guests. Give students a chance to work with their partners to explore and record their discoveries. Teacher monitors and selects different pairs of students to share their work. ***UA notes: by using the post-it and X s students can experience moving the tables and chairs around discovering different combinations of seating arrangements and discussing in a group what works and what doesn t. Suppose there are only 12 guests coming to dinner. What different table arrangements are possible? Which arrangements use the fewest number of tables? SEQUENCE Teacher will sequence student answers to be shared with class. Purpose of sequence is to bring out multiple strategies using multiplication and division. Look for different representations of answers and different ways of explaining their work: Possible answers: 4 groups of 4 tables, 3 groups of 3 tables, What strategy did you use to come up with that arrangement? What math fact represents your arrangement? Are there equal groups? Are the same number of guests at each table group? Is there an arrangement that uses fewer tables? Homework envision work book 10-1.
Grade 3 Module7, Lesson 2 Lesson Focus Use multiplication in word problems using equal groups. PLC Notes Lesson Purpose Content Standards Compare equal group representations to solve problem using facts for 6 and 7. 3.OA A.2-3, B.6: A. Represent and solve problems involving multiplication and division. 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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8. 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. B. Understand properties of multiplication and the relationship between multiplication and division. Practice Standards Introduce 34T Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Joe and Mindy are preparing for a party. They each have the same number of cookies. Joe wanted to serve his cookies on plates (Show the image on the projector) Mindy had a tray that she wanted to serve her cookies on (Show the image on the projector). Do both of these arrangements show the same number of cookies? Technology tool: Explain Everything on ipad: Yellow and red counters to make arrays
Investigate Materials Counters for modeling THINK-PAIR-SHARE Independently thinks about how to determine if the two arrangements model the same number of cookies. With a partner model, discuss, and come to a consensus on whether or not they model the same number of cookies and what they would do to make them show the same number. Monitor students to see their strategies, listen for language (equal groups, array, multiply, add, subtract). Select student work that shows different arguments and solutions. ***UA notes: allowing students to solve the problem in various ways allows access for all students. Students that use just counting strategies can still access the problem at their level while observing others strategies that are incorporating multiplication or division strategies. Optional Practice One of Joe s dogs ate a plate of the cookies. Can you show how many he has left more than one way? Summarize 34T Teacher will sequence student answers to be shared with class. Purpose of sequence is to bring out multiple strategies using 6 and 7multiplication facts. Repeated addition, breaking the groups/array apart, compares visually without clear computation, and fact families. Look for different representations of answers and different ways of explaining their work: Possible answers: Joe needs one more plate. Mindy needs to take off a row of her cookies. I see that Mindy has more because she has seven equal groups, Joe only has six. What strategy did you use to come up with that arrangement? What math fact represents your arrangement? Are there equal groups? How did you find the total amount of cookies each person has? Homework envision workbook 8-3 reteach and practice
.PE B
Grade 3 Module7, Lesson 3 Lesson Focus Use knowledge of multiplication facts to solve division problems. PLC Notes Lesson Purpose Content Standards To understand the relationship between multiplication and division. 3.OA A.2-3, B.6: A. Represent and solve problems involving multiplication and division. 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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8. 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. B. Understand properties of multiplication and the relationship between multiplication and division. Practice Standards Introduce Counters Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. OPTIONAL: show You Tube video showing various round tables set for four: https://www.youtube.com/watch?v=_gtouy44kes POSE THE PROBLEM You are in charge of ordering tables for the annual pie baking contest. There are going to be 28 people attending the event. You know that 4 people can sit at each table. How many tables will you need to order so that everyone will have a place to sit?
Investigate Materials Counters for modeling Paper to record their work THINK-PAIR-SHARE Independently think about what you know and what you need to know. How will you solve the problem? With a partner model, discuss, and come to a consensus on how to solve the problem. Draw a picture, write an equation, and explain in words how you solved the problem. MONITOR Monitor students to see their strategies, listen for language (equal groups, array, multiply, divide). Select student work that shows different arguments and solutions. ***UA notes: By providing the You Tube video or pictures of tables that seat four allows students with limited English to access the problem and get a visual of what needs to be created. Optional Practice You found another company that rents tables that can seat 7 people at each table. If you use this company, how many tables will you have to order? Summarize SEQUENCE Teacher will sequence student answers to be shared with class. Purpose of sequence is to bring out multiple strategies such as fair share, division, multiplication facts, drawing a picture. Look for different representations of answers and different ways of explaining their work. Homework What are we trying to find, the number of groups or the size of each group? What do you know? What do you need to know? How many total people are coming to the pie baking contest? In what ways does dividing remind you or our work with multiplication? Can multiplication help us solve division problems? How? What division equation represents this problem? Is there a multiplication equation that can be used to represent this problem? envision 6-3 practice and reteach (attached) Optional: envision Enrichment 6-3 (attached)
Name 4 as a Factor 3mod7.3-1a If you know a 2s multiplication fact, you can find a 4s multiplication fact. Reteaching 6-3 4s Facts 4 0 = 0 4 5 = 20 4 1 = 4 4 6 = 24 4 2 = 8 4 7 = 28 4 3 = 12 4 8 = 32 4 4 = 16 4 9 = 36 You can double a 2s fact or add a 2s fact by itself to find a 4s fact. Find 4 3 by doubling a 2s fact. a. Find a 2s fact with 3 as a factor. 2 3 = 6 b. Double it. 2 6 = 12 When you double an array of 2 1, you get an array of 4 1. (2) (2) (1) (1) = (4) Find 4 3 by adding a 2s fact by itself. a. Find a 2s fact with 3 as a factor. 2 3 = 6 b. Add the fact to itself. 6 + 6 = 12 (1) Find each product. 1. 4 6 2. 8 4 3. 4 5 4. 9 4 5. 4 1 6. 4 3 7. 4 7 8. 12 2 9. 0 4 10. 4 4 11. Reason How can you use 2 8 to find 4 8? R 6 3
Name 4 as a Factor Reteaching 6-3 If you know a 2s multiplication fact, you can find a 4s multiplication fact. 4s Facts 4 0 0 4 5 20 4 1 4 4 6 24 4 2 8 4 7 28 4 3 12 4 8 32 4 4 16 4 9 36 You can double a 2s fact or add a 2s fact by itself to find a 4s fact. When you double an array of 2 1, you get an array of 4 1. (2) (2) (1) (1) (4) (1) Find 4 3 by doubling a 2s fact. a. Find a 2s fact with 3 as a factor. 2 3 6 b. Double it. 2 6 12 Find 4 3 by adding a 2s fact by itself. a. Find a 2s fact with 3 as a factor. 2 3 6 b. Add the fact to itself. 6 6 12 Find each product. 1. 4 6 2. 8 4 3. 4 5 4. 9 4 5. 4 1 24 32 20 36 6. 4 3 7. 4 7 8. 12 2 9. 0 4 10. 4 4 12 28 24 0 16 11. Reason How can you use 2 8 to find 4 8? Answers will vary. Possible answer: Multiply 2 8 16 and add 16 to itself to to get 32. 4 R 6 3 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Name 4 as a Factor 3mod7.3-2a Find the product. Practice 6-3 1. 2 4 2. 4 5 3. 3 4 4. 4 4 5. 5 8 6. 4 6 7. 1 4 8. 4 3 9. 0 4 10. 4 7 11. 10 4 12. 1 4 13. 2 4 14. 4 9 15. 8 4 16. What multiplication fact can you double to find 4 7? 17. Each square table can seat 4 people. How many people can be seated at 8 square tables? 18. Jillian sold 4 books of raffle tickets. Each book had 9 tickets. How many tickets did Jillian sell all together? 19. The soccer team has practice 4 times each week during the season. If the season is 10 weeks long, how many practices does the team have? 20. Writing to Explain If you know that 4 5 = 20, how can you use the Commutative (Order) Property of Multiplication to find 5 4? 21. Aaron changed the tires on 5 cars. Each car had 4 tires. How many tires did Aaron change? A 12 B 16 C 20 D 24 P 6 3
Name 4 as a Factor Find the product. 1. 2 4 2. 4 5 3. 3 4 4. 4 4 5. 5 8 6. 4 6 7. 1 4 8. 4 3 9. 0 4 10. 4 7 11. 10 4 12. 1 4 13. 2 4 14. 4 9 Practice 6-3 8 20 12 16 40 24 4 12 0 28 15. 8 4 40 4 8 36 32 16. What multiplication fact can you double to find 4 7? 2 7 18. Jillian sold 4 books of raffle tickets. Each book had 9 tickets. How many tickets did Jillian sell all together? 36 tickets 17. Each square table can seat 4 people. How many people can be seated at 8 square tables? 32 people 19. The soccer team has practice 4 times each week during the season. If the season is 10 weeks long, how many practices does the team have? 40 practices 20. Writing to Explain If you know that 4 5 = 20, how can you use the Commutative (Order) Property of Multiplication to find 5 4? The Commutative Property of Multiplication states the order of the factors does not change the product, so 5 4 = 20. 21. Aaron changed the tires on 5 cars. Each car had 4 tires. How many tires did Aaron change? A 12 B 16 C 20 D 24 P 6 3 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Name At the Bakery Julia, Ralph, and Tom went to a bakery to buy cookies for a class party. Use the clues to complete the two charts below. Enrichment 6-3 CLUES T T T Julia Ralph Tom Cookies That Were Bought Oatmeal Peanut Butter Cookies Cookies Chocolate Chip Cookies Total Number of Cookies Bought Oatmeal Cookies Peanut Butter Cookies Chocolate Chip Cookies Total Cookies Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Name At the Bakery Julia, Ralph, and Tom went to a bakery to buy cookies for a class party. Use the clues to complete the two charts below. Enrichment 6-3 CLUES T T T Julia Ralph Tom Cookies That Were Bought Oatmeal Peanut Butter Cookies Cookies 6 36 24 3 12 48 Chocolate Chip Cookies 12 24 48 Total Number of Cookies Bought Oatmeal Cookies Peanut Butter Cookies Chocolate Chip Cookies Total Cookies 66 63 84 213 cookies Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Grade 3 Module7, Lesson 4 Lesson Focus Using known multiplication / division facts to solve for unknown facts. Lesson Purpose Content Standards Practice Standards Introduce Counters Make connections between division and multiplication. Recognize inverse relationship of operations. 3.OA A.2-3, B.6: A. Represent and solve problems involving multiplication and division. 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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8. 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. B. Understand properties of multiplication and the relationship between multiplication and division. 6. Understand division as an unknown-factor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Read the book The Doorbell Rang by Pat Hutchins. Stop to ask questions after each new person arrives: What is happening to the number of cookies each person is getting? Why? POSE THE PROBLEM You won 48 tickets playing a video game. Your two friends were helping you play the game, so they want a fair share of the tickets. How many tickets will each of you receive? PLC Notes Technology tool: Explain Everything red and yellow counters
Investigate Materials INDEPENDENT Give students time to think about what they know and what they need to know. Counters should be available for students that may need them. The number is large and doesn t lend itself to using manipulatives, but some students may still require them. THINK-PAIR-SHARE Think about possible ways to solve the problem. Get with a partner and work together to determine how many tickets each child will get. Select students to orally share their answer. Write the answer they get on the board. Ask if any group came up with a different answer. Once you have all the possible answers on the board, ask students to share their strategies for finding the answer. Then have them get in groups of 4 and discuss which answer is correct. Skip this step if all pairs came up with the same answer. Extension Problem: Uh-oh, you forgot that your friends, Mikey, Tyrone, and Billy, gave you some money to play the game. When they heard that you won, they also felt they should get some of the tickets. How many tickets will you each get now? INDEPENDENT: Have students solve this problem independently. Let students know that you will be calling on them to explain their thinking to the class. After students have been given ample time, call several students up to explain their thinking. ***UA notes: Look for students that have shown the problem in different ways (with counters, with an array, showing fair share/equal groups, using a multiplication equation, using division) Fast finishers can work on the optional practice. Will this way work all the time? Is there a faster way? Do you have as many tickets now? Why not? What if you needed to share the tickets with more friends? Can you use multiplication to solve this problem? What equation can you use to solve this problem? Optional Practice You won $36 dollars in a raffle. Your two sisters and your brother gave you the money to buy the raffle ticket, so they want you to split the money equally with them. How much money will you each get? What if you won $72? What if you won $64? What if you had to share it with more people? What would happen to the amount of money you won?
Summarize 34T SEQUENCE and DISPLAY Teacher will sequence student answers to be shared with class. Purpose of sequence is to bring out multiple strategies. Look for different representations in answers and different ways of explaining their work: Could you use multiplication to help you find your answer? What division equation can you write to show this problem? What do you notice about the 3 numbers in the equation? Homework envision 6-4 reteach and practice (attached) Optional: envision Enrichment 6-4 (attached)
Name 6 and 7 as Factors 3mod7.4-1a You can use multiplication facts that you already know to find other multiplication facts. You can use a 3s fact to find a 6s fact. Find the 3s fact and then add the product to itself. Find 6 9. a. Find the 3s fact with 9: 3 9 = 27. b. Add the product to itself: 27 + 27 = 54. Reteaching 6-4 You can use a 2s and a 5s fact to find a 7s fact. Find 7 5. a. Find the 2s fact with 5: 2 5 = 10. b. Find the 5s fact with 5: 5 5 = 25. c. Add the products: 10 + 25 = 35. Find each product. 1. 2 7 2. 6 7 3. 7 9 4. 6 4 5. 6 8 6. 7 7 7. 6 2 8. 8 7 9. 3 7 10. 6 6 11. 5 6 12. 7 4 13. 6 9 14. 7 3 15. 7 6 16. Construct Arguments Harold says, To find 6 8, I can use the facts for 5 4 and 1 4. Do you agree? Explain. R 6 4
Name 6 and 7 as Factors Reteaching 6-4 You can use multiplication facts that you already know to find other multiplication facts. You can use a 3s fact to find a 6s fact. Find the 3s fact and then add the product to itself. Find 6 9. a. Find the 3s fact with 9: 3 9 = 27. b. Add the product to itself: 27 + 27 = 54. You can use a 2s and a 5s fact to find a 7s fact. Find 7 5. a. Find the 2s fact with 5: 2 5 = 10. b. Find the 5s fact with 5: 5 5 = 25. c. Add the products: 10 + 25 = 35. Find each product. 1. 2 7 2. 6 7 3. 7 9 4. 6 4 5. 6 8 6. 7 7 7. 6 2 8. 8 7 9. 3 7 10. 6 6 11. 5 6 14 42 63 24 49 12 56 21 36 12. 7 4 13. 6 9 14. 7 3 15. 7 6 30 28 54 21 42 48 16. Construct Arguments Harold says, To find 6 8, I can use the facts for 5 4 and 1 4. Do you agree? Explain. Sample answer: No, 5 4 and 1 4 is 5 groups of 4 plus 1 group of 4. That is 6 groups of 4, not 6 groups of 8. R 6 4 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Name 6 and 7 as Factors 3mod7.4-2a Find the product. You may draw pictures to help. Practice 6-4 1. 5 6 2. 6 3 3. 6 8 4. 3 7 5. 7 10 6. 7 4 7. 6 4 8. 5 7 9. 7 8 10. 6 6 11. 7 6 12. 10 6 13. 10 7 14. 7 7 15. 2 6 16. Reason What multiplication fact can be found by using the arrays for 2 9 and 5 9? 17. The chicken eggs that Raul s science class is watching take 3 weeks to hatch. There are 7 days in each week. How many days will it be until the eggs hatch? 18. Emily has 7 apples. She cut each apple into 6 slices. How many slices in all does she have? 19. At a picnic there are 6 tables set up. Each table can seat 8 people. How many people can be seated at the tables all together? 20. How could you use 5 6 = 30 to find the product 6 6? 21. Barry takes 7 minutes to ride his bicycle one mile. At this rate, how long would Barry take to ride his bicycle 4 miles? A 21 minutes B 24 minutes C 27 minutes D 28 minutes P 6 4
Name 6 and 7 as Factors Find the product. You may draw pictures to help. 1. 5 6 2. 6 3 3. 6 8 4. 3 7 5. 7 10 6. 7 4 7. 6 4 8. 5 7 9. 7 8 10. 6 6 11. 7 6 30 18 48 21 70 28 24 35 56 36 12. 10 6 13. 10 7 16. Reason What multiplication fact can be found by using the arrays for 2 9 and 5 9? 14. 7 7 15. 2 6 42 60 70 49 12 7 9 17. The chicken eggs that Raul s science class is watching take 3 weeks to hatch. There are 7 days in each week. How many days will it be until the eggs hatch? 21 days Practice 6-4 18. Emily has 7 apples. She cut each apple into 6 slices. How many slices in all does she have? 42 slices 19. At a picnic there are 6 tables set up. Each table can seat 8 people. How many people can be seated at the tables all together? 48 people 20. How could you use 5 6 = 30 to find the product 6 6? Sample answer: Add the product 1 6 to 30. 30 + 6 = 36, so 6 6 = 36. 21. Barry takes 7 minutes to ride his bicycle one mile. At this rate, how long would Barry take to ride his bicycle 4 miles? A 21 minutes B 24 minutes C 27 minutes D 28 minutes P 6 4 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Name Shelve That Idea Janice wants to fill her shelves with books. Her bookcase has 5 shelves above her television and 2 shelves below the television. She needs 9 books to fill each shelf. 1. Draw the correct number of books on each shelf. Enrichment 6-4 Shelf 1 Shelf 2 Shelf 3 Shelf 4 Shelf 5 Shelf 6 Shelf 7 2. What multiplication sentence shows the number of books in all? 3. How many books does she need to fill the shelves above the television? Write the multiplication sentence. 4. How many books does she need to fill the shelves below the television? Write the multiplication sentence. 5. Writing to Explain Explain how Janice s bookshelves show breaking apart to multiply by 7. E 6 4 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Name Shelve That Idea Janice wants to fill her shelves with books. Her bookcase has 5 shelves above her television and 2 shelves below the television. She needs 9 books to fill each shelf. 1. Draw the correct number of books on each shelf. Enrichment 6-4 Shelf 1 Shelf 2 Shelf 3 Shelf 4 Shelf 5 Shelf 6 Shelf 7 2. What multiplication sentence shows the number of books in all? 7 9 = 63 3. How many books does she need to fill the shelves above the television? Write the multiplication sentence. 45 books; 5 9 = 45 4. How many books does she need to fill the shelves below the television? Write the multiplication sentence. 18 books; 2 9 = 18 5. Writing to Explain Explain how Janice s bookshelves show breaking apart to multiply by 7. You can use 5s and 2s facts to multiply by 7. E 6 4 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Grade 3 Module7, Lesson 5 Lesson Focus Division with measurement quantity. Introduce use of variable for the unknown factor. PLC Notes Lesson Purpose Content Standards Use division facts in measurement quantities with a variable to represent the unknown number. 3.OA A.3 A. Represent and solve problems involving multiplication and division. 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. Practice Standards Introduce Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. POSE THE PROBLEM Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Allow for various use of tools :Scratchpaper, problem solving recording sheet, dryerase board, sentence strip to create number lines and hundred chart and multiplication chart. Jimmy was participating in a long race. Runners are allowed to get water as they are running the 27 mile course. If you need to set up the water stations every three miles, how many water stations will there be? WHOLE GROUP Let s think together about what this problem is asking. Use your think pad to note What is the total distance? How far apart is each water stop? INDEPENDENT Give students time to think about what they know and what they need to know.
Investigate THINK-PAIR-SHARE Think about possible ways to solve the problem. Get with a partner and work together to determine how many water stations will be needed along the course of the marathon. Student may use drawings or the sentence strip to visually represent the races 27 miles. Note to teacher since this does not place a station at the final two miles of the course, think about possible reasons. MONITOR and ***UA notes: Misconceptions may arise as students attempt to do the division problem without completing the model. Redirect as needed to think about the runner and where he could stop for water. They would start at the starting line which would be 0 on the number line and reach their first water station at mile 3. You are looking for s, the number of stations needed. Optional Practice What would change if you needed to put water stations every two miles? Show the stations on your model. Summarize 34T SEQUENCE and DISPLAY Teacher will sequence student answers to be shared with class. Purpose of sequence is to bring out multiple strategies. Discuss how reasoning with the marathon context helps you reach a solution that makes sense. Look for different representations in answers and different ways of explaining their work: Possible Exit Ticket: Introduce the idea of a variable in an equation. What would this problem look like if written in an equation with the unknown variable? **Note the variable is introduced here, it will need to be used in routines over time to build understanding. Homework envision quick check 8-6 (attached)
Name Quick Check 8-6 1. Laurie has n flowers and Denise has 32 flowers. Denise has 4 times the number of flowers that Laurie has. How many flowers does Laurie have? 4 n 32 A n 7 flowers B n 8 flowers C n 9 flowers D n 10 flowers 2. Which value for n will make the equation true? 9 n 5 A n 63 B n 54 C n 45 D n 14 3. Writing to Explain Chris has 8 times the number of games that Sue has. If Chris has 16 games and Sue has n games, how many games does Sue have? Write an equation to represent the situation. Solve for the unknown value of n and explain how you found this value. Q 8 6 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Name Quick Check 8-6 1. Laurie has n flowers and Denise has 32 flowers. Denise has 4 times the number of flowers that Laurie has. How many flowers does Laurie have? 4 n 32 A n 7 flowers B n 8 flowers C n 9 flowers D n 10 flowers 2. Which value for n will make the equation true? 9 n 5 A n 63 B n 54 C n 45 D n 14 3. Writing to Explain Chris has 8 times the number of games that Sue has. If Chris has 16 games and Sue has n games, how many games does Sue have? Write an equation to represent the situation. Solve for the unknown value of n and explain how you found this value. See students samples at the right. Q 8 6 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Grade 3 Module7, Lesson 6 Lesson Focus Using known multiplication /division facts to solve for unknown facts. PLC Notes Lesson Purpose Content Standards Practice Standards Introduce Counters Make connections between division and multiplication. Recognize inverse relationship of operations. 3.OA A.2-3, B.6: A. Represent and solve problems involving multiplication and division. 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. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8. 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. B. Understand properties of multiplication and the relationship between multiplication and division. 6. Understand division as an unknown-factor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. POSE THE PROBLEM Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Belinda is planting a flower garden. She purchased 54 pots of daisies, pansies, roses, tulips, carnations, and lilies. If she purchased the same number of each type of flower, how many pots of each flower does Belinda have? Technology tool: Explain everything ipad App yellow and red counters or Schoolkit Math ipad app.
Investigate Materials INDEPENDENT Give students time to think about what they know and what they need to know. Counters should be available for students that may need them. The number is large and doesn t lend itself to using manipulatives, but some students may still require them. Have students solve this problem independently. Give them a recording sheet to show the problem in pictures, words, and numbers. Let students know that you will be calling on them to explain their thinking to the class. After students have been given ample time, call several students up to explain their thinking. Look for students that have shown the problem in different ways (with counters, with an array, showing fair share/equal groups, using a multiplication equation, using division) Fast finishers can work on the optional practice. Will this way work all the time? Is s way faster than s way? Can we write your equation a different way? What do you notice about the relationship between the numbers in the equation? Can you write the equation as a multiplication problem? Optional Practice Think about other things you might buy. Can you write your own word problem to help us have more practice with these skills? Use a total of 64, or 56. Show envision online lesson video 8-6 and do guided practice.
Summarize 34T SEQUENCE and DISPLAY Teacher will sequence student answers to be shared with class. Purpose of sequence is to bring out multiple strategies. Look for different representations in answers and different ways of explaining their work: Could you use multiplication to help you find your answer? What division equation can you write to show this problem? What do you notice about the 3 numbers in the equation? Exit ticket optional: Envision 8-6 online quiz Homework envision 8-6 reteach and practice (attached)
Name Making Sense of Multiplication and Division Equations Reteaching 8-6 3mod7.6-1a Remember that an equation is a number sentence that uses an equal sign (=) to show that the value to its left is the same as the value to its right. 2 3 = 6 is an example of a multiplication equation. Some equations have letters in them or unknowns. 10 = 40 n This equation means: 10 is equal to 40 divided by some number. You can find the value of n that makes the equation true or equal on each side by thinking of multiplication or division facts. Think: You know that 40 10 = 4, so n = 4. In 1 8, write a basic fact that is related to each equation. Then find the value for n that makes the equation true. 1. 81 = 9 n 2. n 4 = 0 3. 7 = 49 n 4. 16 n = 4 5. 8 = 56 n 6. n 5 = 15 7. 6 = 48 n 8. 5 n = 40 9. Critique Reasoning Alex decides that 21 3 = 7 is NOT a true equation. Is Alex correct? Explain. R 8 6
Name Making Sense of Multiplication and Division Equations Reteaching 8-6 Remember that an equation is a number sentence that uses an equal sign (=) to show that the value to its left is the same as the value to its right. 2 3 = 6 is an example of a multiplication equation. Some equations have letters in them or unknowns. 10 = 40 n This equation means: 10 is equal to 40 divided by some number. You can find the value of n that makes the equation true or equal on each side by thinking of multiplication or division facts. Think: You know that 40 10 = 4, so n = 4. In 1 8, write a basic fact that is related to each equation. Then find the value for n that makes the equation true. 1. 81 = 9 n 9 9 81 n 9 5. 8 = 56 n 56 7 8 n 7 2. n 4 = 0 0 4 0 n 0 6. n 5 = 15 3 5 15 n 3 3. 7 = 49 n 49 7 7 n 7 7. 6 = 48 n 48 8 6 n 8 4. 16 n = 4 16 4 4 n 4 8. 5 n = 40 5 8 40 n 8 9. Critique Reasoning Alex decides that 21 3 = 7 is NOT a true equation. Is Alex correct? Explain. Alex is not correct. 21 7 3, so the equation is true. R 8 6 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
Name Making Sense of Multiplication and Division Equations Practice 8-6 3mod7.6-2a In 1 8, decide if the two sides are equal. If yes, write =. If no, write (not equal). 1. 54 9 6 2. 10 5 2 3. 25 5 7 4. 16 4 5 5. 9 1 1 6. 45 5 9 7. 14 2 7 8. 81 9 8 In 9 16, find the value for n that makes the equation true. 9. 30 = 6 n 10. 3 = n 7 11. 80 =10 n 12. n 6 = 7 13. 20 n = 5 14. 36 n = 6 15. n = 9 2 16. 56 = 8 n For 17 and 18, use the given equation to solve the problem. 17. Together Karen and Mary have n bouquets of roses in their window display. There are 9 roses in each bouquet and 36 roses in all. How many bouquets are in the display? n 9 = 36 18. Hector found an equal number of shells at the beach on 7 different days. If Hector found 63 shells in all, how many shells did he find each day? 63 n = 7 19. Model Bruce has 35 pencils on his desk arranged in groups with 7 pencils in each group. How many groups of pencils are on his desk? Write an equation using n for the unknown value. Solve for n. 20. Which value for n makes the equation n 8 = 1 true? A n = 1 C n = 2 B n = 4 D n = 8 P 8 6
Name Making Sense of Multiplication and Division Equations In 1 8, decide if the two sides are equal. If yes, write. If no, write (not equal). 1. 54 9 6 2. 10 5 2 3. 25 5 7 5. 9 1 1 6. 45 5 9 7. 14 2 7 In 9 16, find the value for n that makes the equation true. Practice 8-6 4. 16 4 5 8. 81 9 8 9. 30 6 n n 5 10. 3 n 7 n 21 11. 80 10 n n 8 12. n 6 7 n 42 13. 20 n 5 n 4 14. 36 n 6 n 6 15. n 9 2 n 18 16. 56 8 n n 7 For 17 and 18, use the given equation to solve the problem. 17. Together Karen and Mary have n bouquets of roses in their window display. There are 9 roses in each bouquet and 36 roses in all. How many bouquets are in the display? n 9 36 4 bouquets 18. Hector found an equal number of shells at the beach on 7 different days. If Hector found 63 shells in all, how many shells did he find each day? 63 n 7 9 shells 19. Model Bruce has 35 pencils on his desk arranged in groups with 7 pencils in each group. How many groups of pencils are on his desk? Write an equation using n for the unknown value. Solve for n. 35 n 7; n 5 groups 20. Which value for n makes the equation n 8 1 true? A n 1 C n 2 B n 4 D n 8 P 8 6 Copyright Pearson Education, Inc., or its affiliates. All Rights Reserved. 3
3Mod7.7 (206893) Item Rubrics Scoring rubric for question 4: Points Description Sample Response 1 Correct 20 0 Incorrect Scoring rubric for question 5: Points Description Sample Response 1 Correct 4 0 Incorrect Scoring rubric for question 6: Points Description Sample Response 1 Correct 12 0 Incorrect Scoring rubric for question 7: Points Description Sample Response 1 Correct 15 0 Incorrect Scoring rubric for question 10: Points 2 1 0 Description The student demonstrates thorough understanding of using multiplication to find the total number of objects in equal groups by multiplying the number of packages times the number of erasers per package (4 8 = 32 and 5 8 = 40). The student demonstrates a partial understanding of using multiplication to find the total number of objects in equal groups by correctly multiplying the number of packages times the number of erasers per package for one of two entries in the table (4 8 = 32 or 5 8 = 40). The student demonstrates inconsistent or no understanding of using multiplication to find the total number of objects in equal groups. The student does not accurately multiply the number of packages times the number of erasers per package for either of the two entries in the table (4 8 = 32 or 5 8 = 40). Sample Response Part A: 32 erasers Part B: 40 erasers EADMS 2015 Adrylan Communications, Inc. INSPECT 2015 Key Data Systems. Page 1 3Mod7.7 ID: 206893
CCSS Math 3Mod7.7 Test Booklet ID: 206893 Answer Key EADMS 2015 Adrylan Communications, Inc. 3Mod7.7 ID: 206893 Generated: 6/13/2015 Page 1 of 3
1. Correct Answer: B,C 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.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. TEACHER READS: Read the question to yourself and select the best answer(s). 2. Correct Answer: A,C 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.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. TEACHER READS: Read the question to yourself and select the best answer. 3. Correct Answer: B,C 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.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. TEACHER READS: Read the question to yourself and select the best answer. 4. Open Ended 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.1 > Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. TEACHER READS: Read and complete the task that follows. 5. Open Ended 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.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. TEACHER READS: Read and complete the task that follows. EADMS 2015 Adrylan Communications, Inc. 3Mod7.7 ID: 206893 Generated: 6/13/2015 Page 2 of 3
6. Open Ended 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.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. TEACHER READS: Read and complete the task that follows. 7. Open Ended 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.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. TEACHER READS: Read and complete the task that follows. 8. Correct Answer: C 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.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. TEACHER READS: Read the question to yourself and select the best answer. 9. Correct Answer: A 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.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. TEACHER READS: Read the question to yourself and select the best answer. 10. Open Ended 3rd Grade > Operations and Algebraic Thinking > Operations and Algebraic Thinking > MA.3.OA.A.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. TEACHER READS: Read and complete the task that follows. EADMS 2015 Adrylan Communications, Inc. 3Mod7.7 ID: 206893 Generated: 6/13/2015 Page 3 of 3
Lake Elsinore USD Assessment CCSS Math 3rd Grade ID: 206893 Module 7 assesment Directions: Read the question. Fill in the bubble next to the corresponding question number on your answer sheet. Sample Question Sample Answer Sheet Sample Item Not Available 1. A B C D 2. A B C D 3. A B C D 4. A B C D 5. A B C D
1 Danny is having a birthday party. He is going to have a total of 32 people at his party. He is going to have 8 tables set up and would like the same number of people at each table. Select from A D, which expression or expressions Danny could use to help him find the correct number of people at each table. Select TWO that apply. A 32 8 = B 32 8 = C 8 = 32 D 8 = 32 2 Jacob has 21 toy cars. He is putting an equal number of cars into 3 groups. Select all options below that can be used to find the number of cars Jacob put in each group. A 3 = 21 B 3 = 21 C 21 3 = EADMS 2015 Adrylan Communications, Inc. INSPECT 2015 Key Data Systems. Page 1 Module 7 assesment ID: 206893 Generated: 6/2015
3 Justin has 56 gumballs. He is putting an equal number of gumballs into 4 bags. Select all options below that can be used to find the number of gumballs Justin will put in each bag. A 56 4 = B 56 4 = C 4 = 56 D 4 = 56 4 Mrs. Lawler broke her students into groups for an activity. The image shows how she separated her students. Use multiplication to find the total number of students in Mrs. Lawler's class. Enter the total number of students in Mrs. Lawler's class. students EADMS 2015 Adrylan Communications, Inc. INSPECT 2015 Key Data Systems. Page 2 Module 7 assesment ID: 206893 Generated: 6/2015
5 Layla has cut 32 cloth squares for a quilt. The squares will be sewn together in an array. Each row will have 8 squares. How many rows can Layla make? 6 Third graders are making life size portraits on pieces of paper that are 5 feet long. The art teacher has a roll of paper that is 60 feet long. If the teacher cuts up the roll of paper, how many pieces will she get? EADMS 2015 Adrylan Communications, Inc. INSPECT 2015 Key Data Systems. Page 3 Module 7 assesment ID: 206893 Generated: 6/2015
7 Kenneth is replacing the knobs on his 5 dresser drawers. Each drawer needs 3 knobs. How many knobs will Kenneth need for the whole dresser? 8 Zoe's book club has pizza while they discuss their book. Each pizza has 8 slices. If there are 6 people in the book club and they order 3 pizzas, how many slices will each person eat? A 48 slices B 24 slices C 4 slices D 2 slices EADMS 2015 Adrylan Communications, Inc. INSPECT 2015 Key Data Systems. Page 4 Module 7 assesment ID: 206893 Generated: 6/2015