Investigations UNIT GUIDES: GRADE 5 GRADE. In Number, Data, And Space

Transcription

1 Investigations In Number, Data, And Space UNIT GUIDES: GRADE 5 GRADE 5

2 Investigations In Number, Data, And Space UNIT GUIDES: Grade 5 The Investigations curriculum was developed by TERC, Cambridge, MA. The Investigations curriculum is published by Pearson Publishing. This material is based on work supported by the National Science Foundation ( NSF ) under Grant No. ESI Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Copyright TERC, Inc. All Rights Reserved. Limited reproduction permission: TERC grants permission to reproduce the Unit Guides as needed for use in a school or district. Reproduction for use in a school/district not your own is prohibited. Unit Guides for Grade 5

3 Contents Introduction Unit Guides: Grade 5 Unit 1: Number Puzzles and Multiple Towers Unit 2: Prisms and Pyramids Unit 3: Thousands of Miles, Thousands of Seats Unit 4: What s That Portion Unit 5: Measuring Polygons Unit 6: Decimals on Grids and Number lines Unit 7: How Many People? How Many Teams? Unit 8: Growth Patterns Unit 9: How Long Can You Stand on One Foot? Unit Guides for Grade 5

4 Unit Guides: Introduction Preparation to teach individual units in Investigations The Unit Guides for Investigations in Number, Data and Space are designed as study guides to help teachers become familiar and comfortable with the mathematical content, the activities, and the overall structure of each unit. The Unit Guides provide a structure for working through student activities, reading support material, looking at student work that is embedded in the materials, and discussing critical issues about mathematics and pedagogy. The expectation is that groups of teachers will work together on these Guides in preparation to teach a unit in their classroom. Below are some suggestions of how best to incorporate these Unit Guides into your professional development program. In order to get the most use from these Guides, it is important that groups of teachers from the same grade level work on a Guide together. This way, teachers can do the activities together and respond to the discussion questions. The Unit Guides are written with the assumption that teachers will read the Mathematics in This Unit essay before attending the Unit Guide session. A list of suggested materials and photocopied handouts for each unit is found at the beginning of each guide and should be prepared before the session begins. Unit Guides may be used independently by groups of teachers, with one teacher taking on the lead role, or you may have a Teacher Leader/Coach who can contribute some guidance or experience. The leader s role during the session is to act as a resource: answering questions, clarifying instructions, and prompting discussion when appropriate. In order to gain a real sense of the preparation, effort, and mathematics in the activities, it is imperative that teachers not just read through but actually do all of the activities suggested in the Unit Guides. As a follow up to a Unit Guide session, it is helpful for teachers to meet periodically while the unit is ongoing in their classrooms to support each other, seek and offer advice, and work together to evaluate sets of student work. Each Unit Guide is designed to take three hours, but could also be adapted and used flexibly in a variety of different settings. Unit Guides for Grade 5

5 Unit Guide for Grade 5, Unit 1: Number Puzzles and Multiple Towers Multiplication and Division 1 Unit Guide for Grade 5, Unit 1

6 Unit Guide for Grade 5, Unit 1 Number Puzzles and Multiple Towers Multiplication and Division 1 Unit Summary: Students refine and gain fluency in solving two-digit by two-digit multiplication problems, develop strategies for division problems with one- and two-digit divisors, and increase their knowledge of multiplication relationships by learning about prime factorization (e.g., 36 = 4 x 9 = (2 x 2) x 9 = 2 x 2 x 3 x 3). Materials: Number Puzzles and Multiples Towers (1 copy per person) Resource Master M15, Centimeter Grid Paper (2-3 per person) Student Activity Book pp. 1-2, Number Puzzles: 1 Clue. (1 per person) Color tiles (1 set available) 12 x 18 Construction paper (1 per person) Student Activity Book p. 35, Multiplication Cluster Problems (1 per person) Student Activity Book pp , Problems about Multiples of 21 (1 per person) Do the following activities from Number Puzzles and Multiple Towers: 1. Identify the mathematics in the unit To get an overview of the mathematics students will be doing in this unit, refer to these sections in the unit front matter. As you look at these sections, begin thinking about the main mathematical ideas students work on in this unit. Turn to pp. 8-9, Overview of This Unit. Look at the title of each Investigation and read the summary for each Investigation. Review the Mathematics in This Unit essay, pp Look at the Mathematical Emphases and Math Focus Points. (The emphases are numbered, and can be found above bulleted lists of Math Focus Points.) Read the Benchmarks in This Unit in the table on p. 15, Assessing the Benchmarks. What mathematical ideas and skills are students working on in this unit? What mathematics are students expected to know at the beginning of the unit? At the end? Unit Guide for Grade 5, Unit 1 1

7 2. Building and Using Arrays (Session 1.1) In this Investigation, students use arrays and number puzzles to learn about properties of numbers. In this session, students build rectangular arrays of 2-digit numbers and identify factors and multiples of these numbers. Read and do the Activity, Building Arrays, p Members of your group should work on different numbers, choosing from 24, 28, 36 or 42. How did you find arrays for 24 (28, 36 or 42)? What relationships do you notice among the arrays for (24)? Did you use the dimensions of one array to help you find the dimensions of another? Do you think you found all the arrays for your number? How do you know? Read the Activity, Number Puzzles: 1 Clue, pp Complete Student Activity Book pp. 1-2, Number Puzzles: 1 Clue. Read the Algebra Note, Models, p. 34. How do arrays show different properties (e.g. odd, even, prime, square, etc.) of numbers? How does the work of finding arrays and solving number puzzles help students better understand the operations of multiplication and division? 3. Solving 35 x 28 (Session 2.1) In this Investigation, students continue to develop strategies for solving 2-digit by 2-digit multiplication problems. In this session, students solve a two-digit multiplication problem and represent their solution. Working by yourself: Create a story problem for 35 x 28. Solve the problem, showing your solution clearly. Draw a quick representation for the problem, using cubes, arrays, groups, etc. With a partner: Create a poster that shows one (or both) of your strategies, including the representation. Share posters with the group. Read the Activity, Solving 35 x 28, pp Read the Teaching Notes, Story Contexts p. 71, and the Math Note, Clear and Concise Notation, p. 73. Unit Guide for Grade 5, Unit 1 2

8 How are the strategies and representations your group used the same and different from those shown in the session? How do the story context and the representation help students answer the following questions about each strategy: What part of the problem has been solved with each step? What remains to be solved? Where is the 35 in your solution? Where is the 28? Where is the answer? How do the questions above help students achieve computational fluency in multiplication? Multiplication Cluster Problems (Session 2.4) Students are introduced to Cluster Problems, which are sets of problems that help students to use what they know to find the solution to harder problems. Read the Activity, Introduction to Cluster Problems, pp Complete Student Activity Book p. 35, Multiplication Cluster Problems. What are the relationships among the problems in each set of cluster problems? How did you use the problems in the cluster to solve the larger problem? How do cluster problems help students efficiently solve multiplication problems? How do they help students understand the operation of multiplication? Read the Teacher Note, Multiplication Strategies, pp Look again at the cluster problems on SAB p. 35. Which multiplication strategy is suggested by each set of clusters? 5. Using Multiple Towers (Session 3.2) In this Investigation, students develop various strategies for solving division problems. In this session, students list a sequence of multiples and use it to solve multiplication and division problems. Read the Activities, Introducing Multiple Towers, pp , and Using Multiple Towers, pp Use the multiple tower shown on p. 124 to solve the problems on Student Activity Book pp , Problems about Multiples of 21. What numbers and/or relationships on the Multiple Towers did you use to solve the problems? How does the solution to one problem help to solve the others? Unit Guide for Grade 5, Unit 1 3

9 Read the Teacher Note, The Relationship Between Multiplication and Division, p How is the relationship between multiplication and division highlighted in the Multiple Tower work? (That is, how would you describe the relationship between = and x 21 = 126?) How does this work help students solve multiplication and division problems? 6. Wrap Up Look back at the unit overview, pp How do the activities done during this unit study fit into the overall mathematical storyline of the unit? Other Key Features of Number Puzzles and Multiple Towers Algebra Connections in This Unit, pp Ten-Minute Math, p. 20 Quick Images Number Puzzles Assessment: Number Puzzles (Session 1.5) Resource Master M34, Assessment Checklist What Is the Answer? (Session 2.7) Teacher Note: Assessment, pp End-of-Unit Assessment (Session 3.8) Teacher Note, pp Unit Guide for Grade 5, Unit 1 4

10 Unit Guide for Grade 5, Unit 2: Prisms and Pyramids 3-D Geometry and Measurement Unit Guide for Grade 5, Unit 2

11 Unit Guide for Grade 5, Unit 2 Prisms and Pyramids 3-D Geometry and Measurement Unit Summary: Students investigate concepts of volume by finding the volume of prisms, pyramids, cylinders, and cones. They use patterns of open boxes and build prisms from cubes to develop a strategy for finding the volume of any rectangular prism. Using concrete materials, they also examine the 3- to-1 volume relationship between related (having the same base and height) prisms and pyramids, and related cylinders and cones. Geometry work includes naming geometric solids and their attributes. Materials: Prisms and Pyramids (1 copy per person) Student Activity Book pp.7-8, Volume of Boxes Resource Master M11, Three-Quarter-Inch Grid Paper (2-3 per person) Resource Master M22, Pattern for a Closed Box, (1 per person) Resource Masters M26-M30, Solid Patterns A-I (1 set per pair, see Materials to Prepare, p. 89) Scissors Tape Centimeter rulers (1 per pair) Centimeter cubes (20 per pair) Rice or sand Trays or buckets (to catch sand) Do the following activities from Prisms and Pyramids: 1. Identify the mathematics in the unit To get an overview of the mathematics students will be doing in this unit, refer to these sections in the unit front matter. As you look at these sections, begin thinking about the main mathematical ideas students work on in this unit. Turn to pp. 8-9, Overview of This Unit. Look at the title of each Investigation and read the summary for each Investigation. Review the Mathematics in This Unit essay, pp Look at the Mathematical Emphases and Math Focus Points. (The emphases are numbered, and can be found above bulleted lists of Math Focus Points.) Read the Benchmarks in This Unit in the table on p. 15, Assessing the Benchmarks. Unit Guide for Grade 5, Unit 2 1

12 What mathematical ideas and skills are students working on in this unit? What mathematics are students expected to know at the beginning of the unit? At the end? 2. Finding Volume (Session 1.2) In this Investigation, students study the volume of rectangular prisms by looking at patterns of boxes and by building rectangular prisms from connecting cubes. In this session, students develop strategies for finding the volume of rectangular prisms. Read the ion, Strategies for Finding Volume p. 32 to see an example of how students may find the volume of any rectangular prism. Read the Activity, Finding Volume, pp Complete Student Activity Book pp. 7-8, Volume of Boxes, including drawing the patterns on three-quarter inch grid paper. How did you determine the number of cubes in the box? (Did you build the rectangular prism, or use some aspects of the pattern or the picture?) What difficulties might students encounter drawing their own patterns for problems 3-5? Read the Teacher Note, Strategies for Finding the Number of Cubes in 3-D Arrays, pp , and the Dialogue Box, Understanding Multiplication and Arrays, p What strategies might students use to find the volume of a rectangular prism? What confusions might students have as they work on finding the volume of boxes? What might you do to support students? 3. Finding Cubic Centimeters (Session 2.1) and Measuring the Space Inside Our Classroom (Session 2.3 and 2.4) In this Investigation, students move to measuring standard units of volume (e.g. cubic centimeters). They build models of cubic inches and feet as well as cubic centimeters and meters. In this session, students find the volume of a box using cubic centimeters. Use Resource Master, M22, Pattern for a Closed Box to make the box. Show how you could use 20 centimeter cubes (or fewer) to determine the volume, in cubic centimeters of the box. Share strategies with the group. Read the Activity Finding Cubic Centimeters, pp to see how this task is introduced with students. Read the ion, How Many Cubic Centimeters?, pp Unit Guide for Grade 5, Unit 2 2

13 What strategies did you use to find the number of centimeter cubes that will fill the box? What is the purpose of limiting the number of centimeter cubes students can use? How were your strategies the same or different than those of the students in the ion? In this Investigation, students also measure the volume of their classroom. Read the Math Note, Relating Cubic Units to Linear Units, p. 65, and the Dialogue Box, Choosing a Volume Unit to Measure the Classroom, p How might students approach the task of finding the volume of their classroom? What challenges might they encounter? How might this activity (volume of the classroom) support their understanding of volume and their ability to measure volume? 4. Comparing Solids and Their Volumes (Session 3.2) In this Investigation, students build models of prisms, pyramids, cylinders, and cones. They find the volume of these solids in cubic centimeters. In this session you will build solids out of paper, compare their volumes, and identify any relationships you find. Using Resource Masters M26-M30, each pair should make a set of solids, as described in the Activity, Making Solids from Patterns, p. 93. Read the Activity, Comparing Solids and Their Volumes, pp Do this activity, recording the relationships each pair finds on a piece of chart paper. How did the volume of each larger solid compare with the smaller solid in the pair? How do the bases and heights in each pair of solids compare? Read the Math Notes, Measurement Imprecisions, and A Three-to-One Relationship, p. 96. Read the Dialogue Box, Exploring the Three-to-One Relationship, pp What should students notice about the relationship between the volumes of the solids in each pair? What measuring imprecisions might interfere with their understanding of this relationship? How can you support students in making sense of these imprecisions? Unit Guide for Grade 5, Unit 2 3

14 5. Wrap Up Look back at the unit overview, pp How do the activities done during this unit study fit into the overall mathematical storyline of the unit? Other Key Features of Prisms and Pyramids Ten-Minute Math, p. 16 Quick Images: 3-D Estimation and Number Sense Assessments: Finding the Volume of Rectangular Prisms (Session 1.5) Teacher Note, pp Measuring Volume in Cubic Centimeters (Session 2.4) Resource Master M25, Assessment Checklist End-of-Unit Assessment (Session 3.5) Teacher Note, pp Unit Guide for Grade 5, Unit 2 4

15 Unit Guide for Grade 5, Unit 3: Thousands of Miles, Thousands of Seats Addition, Subtraction, and the Number System Unit Guide for Grade 5, Unit 3

16 Unit Guide for Grade 5, Unit 3 Thousands of Miles, Thousands of Seats Addition, Subtraction, and the Number System Unit Summary: Students study place value in large numbers by building a 10,000 chart and by adding multiples of 10 to, and subtracting multiples of 10 from, 4- and 5-digit numbers. Students finalize their study of subtraction by refining and gaining fluency in solving subtraction problems, including a study of the US algorithm for subtraction. Using a context of the capacities of stadiums and arenas, they solve addition and subtraction problems involving 4- and 5- digit numbers. In this unit, students also demonstrate fluency with the division facts up to Materials: Thousands of Miles, Thousands of Seats (1 copy per person) Student Activity Book, pp. 7-8 How Many Steps to 10,000? (1 per person) Student Activity Book pp , Related Problems (1 per person) Student Activity Book p. 35, The U.S. Algorithm, p. 2 of 3 (1 per person) Student Activity Book p. 51, Stadium and Arena Capacities (1 per person) Student Activity Book p. 59, Rock On! page 1 of 4 (1 per person) Do the following activities from Thousands of Miles, Thousand of Seats: 1. Identify the mathematics in the unit To get an overview of the mathematics students will be doing in this unit, refer to these sections in the unit front matter. As you look at these sections, begin thinking about the main mathematical ideas students work on in this unit. Turn to pp. 8-9, Overview of This Unit. Look at the title of each Investigation and read the summary for each Investigation. Review the Mathematics in This Unit essay, pp Look at the Mathematical Emphases and Math Focus Points. (The emphases are numbered, and can be found above bulleted lists of Math Focus Points.) Read the Benchmarks in This Unit in the table on p. 15, Assessing the Benchmarks. What mathematical ideas and skills are students working on in this unit? What mathematics are students expected to know at the beginning of the unit? At the end? Unit Guide for Grade 5, Unit 3 1

17 2. How Many Steps to 10,000? (Session 1.3) Related Problems (Session 1.4) In this Investigation, students use place-value relationships and multiples of 10, 100, and 1,000, to add and subtract large numbers. In these sessions, students consider the relationship between addition and subtraction as they find how far a number is from 10,000, and by solving related problems. Complete Student Activity Book pp. 7-8, How Many Steps to 10,000?, and pp , Related Problems. As you re working, talk with a neighbor about what strategies you re using, and how you re using place value and understanding the operations of addition and subtraction to solve the problems. Read the Algebra Notes, The Relationship Between Addition and Subtraction, p. 37, and Changing the Numbers in a Subtraction Problem p. 49. How do the two activities (How Many Steps to 10,000? and Related Problems) support students in understanding the structure of our base-ten number system? How do the related problems help students consider the difference between addition and subtraction? How does this work support students as they add and subtract large numbers? How does the information in the Algebra Notes help support you as you work with students in subtraction? 3. Naming Subtraction Strategies (Session 2.1) In this Investigation, students practice and study various strategies for subtraction. In this session, students solve a subtraction problem, and then as a class, analyze and classify different subtraction strategies. Solve = with each other how you solved this problem and how your strategies are similar or different. Read the Activity, Solving a Subtraction Problem, pp , and the ion, Subtraction Strategies, pp Read the Math Note, Using Minus Instead of Take Away, p. 59, Is the Number Line a Strategy?, p. 60, and the Teaching Note The U.S. Algorithm, p. 60. Read the Dialogue Box, Classifying and Naming Subtraction Strategies, pp Unit Guide for Grade 5, Unit 3 2

18 How does analyzing and naming subtraction strategies support students in becoming fluent with subtraction? How does the curriculum (the math focus points of the lesson, the ongoing assessment section, the text of the discussion pp , the Dialogue Box, etc,) support you in understanding which math ideas to emphasize with students as you guide them in comparing and naming subtraction strategies? 4. Studying the U.S. Algorithm (Session 2.4) As part of their study of subtraction, students study the U.S. algorithm, including understanding the notation used, and the ideas of place value involved. Read the Activity, Examining the U.S. Algorithm, pp Solve the problems on Student Activity Book p. 35, The U.S. Algorithm, p. 2 of 3. What math is highlighted for you in this activity? What do you expect students will learn? Read the Dialogue Box, Working With the U.S. Algorithm, pp What moves does the teacher make during this discussion? How does that help students make sense of the notation? 5. Rock On! continued (Session 3.4) In this Investigation, students solve addition and subtraction problems involving 4- and 5- digit numbers. Use the data on Student Activity Book p. 51, Stadium and Arena Capacities, to solve the problems on Student Activity Book p. 59, Rock On!. your solution to problem #1 with a partner. Read the Activity, Rock On!, pp Do your addition and subtraction strategies change when you solve problems with 4- and 5- digit numbers? Why or why not? Read the ion, Strategies for Large Numbers, pp Go through each of the student work samples. How are these students making sense of solving subtraction problems with large numbers? Unit Guide for Grade 5, Unit 3 3

19 6. Wrap Up Look back at the unit overview, pp How do the activities done during this unit study fit into the overall mathematical storyline of the unit? Other Key Features of Thousands of Miles, Thousand of Seats Algebra Connections in This Unit, pp Ten-Minute Math in This Unit, p. 20 Practicing Place Value Estimation and Number Sense: Closest Estimate Assessment Numbers on the 10,000 Chart, (Session 1.2) Resource Master M3, Assessment Checklist Subtraction Problems (Session 2.5) Teacher Note pp Division Facts (Session 3.1) Teacher Note p. 131 Numbers to 100,000 (Sessions ) Resource Master M3, Assessment Checklist End-of-Unit Assessment (Session 3.5) Teacher Note pp Unit Guide for Grade 5, Unit 3 4

20 Unit Guide for Grade 5, Unit 4: What s That Portion? Fractions and Percents 1 Unit Guide for Grade 5, Unit 4

21 Unit Guide for Grade 5, Unit 4 What s That Portion? Fractions and Percents 1 Unit Summary: Students study the relationship among fractions and between fractions and percents and use this knowledge to find equivalent fractions, order fractions, and add commonly used fractions. They use a variety of contexts and models, including area, number lines, and rotation, to further understand the meaning of fractions and model their strategies. Materials: What s That Portion? (1 copy per person) Student Activity Book, p. 5, Name the Shaded Portion (1 per person) Student Activity Book, p. 10, Fraction and Percent Equivalents (1 per person) Student Activity Book, pp , Which Is Greater? (1 per person) Student Activity Book p. 37, Clock Fractions (1 per pair) Student Activity Book, p. 38, Clock Fractions Addition Problems, p. 1 of 2 (1 per pair) Student Activity Book, p. 51, Fraction Tracks, (1 per person) Student Activity Book, pp , Fraction Problems (1 per person) Do the following activities from What s That Portion?: 1. Identify the mathematics in the unit To get an overview of the mathematics students will be doing in this unit, refer to these sections in the unit front matter. As you look at these sections, begin thinking about the main mathematical ideas students work on in this unit. Turn to pp. 8-9, Overview of This Unit. Look at the title of each Investigation and read the summary for each Investigation. Review the Mathematics in This Unit essay, pp Look at the Mathematical Emphases and Math Focus Points. (The emphases are numbered, and can be found above bulleted lists of Math Focus Points.) Read the Benchmarks in This Unit in the table on p. 15, Assessing the Benchmarks. What mathematical ideas and skills are students working on in this unit? What mathematics are students expected to know at the beginning of the unit? At the end? Unit Guide for Grade 5, Unit 4 1

22 2. Grid Patterns as Percents and Fractions (Session 1.2) and Fraction and Percent Equivalents (Session 1.3) In this Investigation, students use 10 x 10 grids to identify equivalent fractions and percents and to consider the relationship between fractions and percents. In this session, they use what they know about landmark fractions and percents to determine percent equivalents for many fractions. Read the Activity, Grid Patterns as Percents and Fractions, pp Represent and fill in the equivalent fractions and percents on Student Activity Book p. 5, Name the Shaded Portion. How did you identify the percent? How does the grid support students in identifying percent and fraction equivalents? Read the Activity, What Percent is 3/8?, pp Read the Activity, Fraction and Percent Equivalents, pp Complete Student Activity Book p. 10, Fraction and Percents Equivalents. Fill in as many percent equivalents as you can, paying attention to which ones are easier and which ones are harder to fill in. Read the Teaching Note, Finding Equivalent Fractions p. 39, and the Math Note, Fractions as Division p. 39. How might students find the percent and fraction equivalents? How might they use grids? What relationships did you notice in the chart that helped you fill in all of the percent equivalents? What relationships do you want students to see? 3. Which is Greater? (Session 2.2) In this Investigation, students develop and use a variety of strategies to compare and order fractions. In this session, students compare pairs of fractions by using fraction equivalents, the relationship of a fraction to 1/2 or 1, and representations such as rectangles. Read the Activity, Introducing Which is Greater?, pp Complete Student Activity Book pp , Which is Greater? As you work, consider the strategies you are using to determine which fraction is greater. (Note: Try comparing these fractions using reasoning about fractions, relationships to landmarks, 10 x 10 grids, and percent equivalents.) Pick 1 or 2 of the problems to discuss and share strategies used to compare fractions. What strategies (using grids, relationships to landmarks, reasoning about fractions, percent equivalents, etc.) do you want students to use to determine which fraction is greater? What is the important mathematical idea that is highlighted in Problem 8? Unit Guide for Grade 5, Unit 4 2

23 Read the Teacher Note, Strategies for Comparing Fractions, pp the strategies students are using in this Teacher Note. Follow the reasoning of each student. What are these students noticing about fractions? How might these conjectures support their work in identifying, comparing and adding fractions? 4. Clock Fractions (Session 3.1) In this Investigation, students use different models and contexts to add and subtract fractions. In this session, students use a clock face to represent fractions as parts of a rotation around a circle. They find equivalent fractions and add fractions using this model. Read the Activity, Clock Fractions, pp With a partner, complete Student Activity Book p. 37, Clock Fractions. Read the Activity, Adding Clock Fractions, pp , and with a partner, complete Student Activity Book p. 38, Clock Fractions Addition Problems, p. 1 of 2. How might the clock (a rotational model) support students in understanding and adding fractions? Read the Teacher Note, Adding and Subtracting Fractions, pp How do the representations (rectangles and clocks) support students in adding fractions? 5. Introducing Fraction Tracks and Labeling Fraction Tracks (Session 3.4) Students use their knowledge of fraction equivalents and relationships to label a linear model for fractions. Read the Activity, Introducing Fraction Tracks, pp to see how a linear model of fractions is introduced to students. Read the Activity, Labeling Fraction Tracks, pp and complete Student Activity Book p. 51, Fraction Tracks. Read the Math Note Halfway on the Thirds or Fifths Tracks, p How did you use knowledge of equivalent fractions to label the dots? What other knowledge of fractions did you use to label the dots? Read the Dialogue Box, Equivalents on the Fraction Tracks, pp Unit Guide for Grade 5, Unit 4 3

24 What do these students understand about fractions and equivalent fractions? How does the teacher support students to develop an understanding about equivalent fractions? Fractions Problems (Session 3.7) Based on the work students have done in this unit, students solve a set of addition and subtraction fractions problems. Solve the problems on Student Activity Book pp , Fractions Problems p. 1 of 3. (If time allows, go on to pages 2 and 3.) Note: Try using strategies that you worked on in this Unit Guide, instead of the procedure of finding common denominators. What strategies did you use to solve these problems? Did you use any representations or models? Did you use knowledge of equivalent fractions? In what ways does this unit support students in adding and subtracting fractions? 7. Wrap Up Look back at the unit overview, pp How do the activities done during this unit study fit into the overall mathematical storyline of the unit? Unit Guide for Grade 5, Unit 4 4

25 Other Key Features of What s That Portion? Ten-Minute Math in This Unit, p. 16 Estimation and Number Sense Guess My Rule Assessment Solving Problems with Fractions and Percents (Session 1.5) Resource Master M11, Assessment Checklist Using Fractions and Percents (Session 2.6) Teacher Note: Assessment, pp Adding Fractions (Session 3.3) Resource Master M24, Assessment Checklist End-of-Unit Assessment (Session 3.10) Teacher Note, pp Unit Guide for Grade 5, Unit 4 5

26 Unit Guide for Grade 5, Unit 5: Measuring Polygons 2-D Geometry and Measurement Unit Guide for Grade 5, Unit 5

27 Unit Guide for Grade 5, Unit 5 Measuring Polygons 2-D Geometry and Measurement Unit Summary: Students create polygons using Power Polygon pieces and discuss, apply, and evaluate definitions of these polygons. They focus on properties of quadrilaterals and also study similarity of 2-D shapes. Measurement work includes finding measures of angles using known angles and finding perimeter and area of rectangles. LogoPaths, a Logo programming environment designed for Investigations students in Grades 3 5, is introduced in this unit. It allows students to explore geometrical relationships, especially focusing on angle, length and perimeter, patterns in sides and angles, and characteristics of specific shapes. Materials: Measuring Polygons (1 per person) Student Activity Book pp. 1-2, Triangles: Two the Same, One Different (1 per pair) Resource Master, M9-M10, Shape Cards (1 set per pair, see Materials to Prepare, p. 21) Student Activity Book pp , Angles in Power Polygons, pp. 1-2 of 3 (1 per pair) Student Activity Book p. 27, Building a Sequence of Squares (1 per pair) Student Activity Book p. 49, Building Similar Polygons (1 per pair) Power Polygons (a handful of various shapes for each pair) Color tiles (1 set per group of 6) Do the following activities from Measuring Polygons: 1. Identify the mathematics in the unit To get an overview of the mathematics students will be doing in this unit, refer to these sections in the unit front matter. As you look at these sections, begin thinking about the main mathematical ideas students work on in this unit. Turn to pp. 8-9, Overview of This Unit. Look at the title of each investigation and read the summary for each investigation. Review the Mathematics in This Unit essay, pp Look at the Mathematical Emphases and Math Focus Points. (The emphases are numbered, and can be found above bulleted lists of Math Focus Points.) Read the Benchmarks in This Unit in the table on p. 15, Assessing the Benchmarks. Unit Guide for Grade 5, Unit 5 1

28 What mathematical ideas and skills are students working on in this unit? What mathematics are students expected to know at the beginning of the unit? At the end? 2. Triangles: Two the Same, One Different (Session 1.1) In this Investigation, students classify polygons and find measures of angles. In this session, students sort triangles by attributes that include angle size and side length. They use geometric vocabulary to describe different types of triangles. Read the Activity, Triangles: Two the Same, One Different, pp Sort out the triangles from the set of Shape Cards and then work with a partner to complete Student Activity Book pp. 1-2, Triangles: Two the Same, One Different. Read the Teacher Note, Classification of Triangles and Quadrilaterals, pp What attributes were you paying attention to as you worked with the triangles? How does this activity support students to learn about the classification of triangles? How does it support classification of other polygons? 3. Angles in Power Polygons (Session 1.5) In this session, students use what they know about 90 degree angles to find other angle measures. Begin by working with a partner on finding the measure of the angles in Power Polygon piece D. Have a discussion with your group and share strategies for finding the measure of each of the angles. Read the Activity, Introducing Angles in Power Polygons, pp the method of putting the same pieces around a point, and why that might be helpful to know. Read the Activity, Angles in Power Polygons, pp and work with a partner to complete Student Activity Book pp , Angles in Power Polygons, pp. 1-2 of 3. Pick several of the shapes and share strategies for finding the measure of the angles in the polygons, including different reasoning to find the measure of the same angle. Read the Dialogue Box, Finding Angle Measures of Power Polygons, pp Unit Guide for Grade 5, Unit 5 2

29 What kind of reasoning do these students use to find the measure of unknown angles? 4. Building a Sequence of Squares (Session 2.1) In this Investigation, students find the perimeter and area of a series of related squares. They consider how a change in the dimensions of a square changes the perimeter and the area. Read the Activity, Building a Sequence of Squares, pp With a partner, complete Student Activity Book p. 27, Building a Sequence of Squares. How are you determining the perimeter and area of each square? What patterns do you notice in the growth of the area and the perimeter? Read the ion, Observations About Perimeter and Area, pp Read the Math Note, Is there a relationship between area and perimeter?, p. 72. What are students learning about area, perimeter, and the relationship between them? 5. Building Similar Polygons (Session 3.1) In this Investigation, students investigate mathematically similar polygons. In this session, students build figures that are mathematically similar to some of the Power Polygon pieces. They examine how many of the same polygon pieces it takes to build the larger, similar figures. Read the Activity, Introducing Similarity, pp Read the Math Notes, Similar Shapes and Defining Shape on p Read the Activity, Building Similar Polygons, p Work with a partner to complete Student Activity Book p. 49, Building Similar Polygons. Read the Teaching Note, Visually Checking for Similarity, p. 110, and the Math Note, Building Similar Polygons, p How do you know your shape is similar to the one you started with? What characteristics of the shape are you paying attention to? What do you notice about the number of smaller polygons it takes to compose the larger, similar one? Read the Teacher Note, Similar Shapes, p What are students learning about similar shapes and similarity in this unit?? Unit Guide for Grade 5, Unit 5 3

30 6. Wrap Up Look back at the unit overview, pp 8-9. How do the activities done during this unit study fit into the overall mathematical storyline of the unit? Other Key Features of Measuring Polygons LogoPaths Software Teacher Note, Introducing and Managing LogoPaths, pp Teacher Note, About the Mathematics in the LogoPaths Software, p. 141 Ten-Minute Math in This Unit, p. 16 Quick Images: 2-D Quick Survey Assessment Quadrilaterals and Angles (Session 1.7) Teacher Note, pp Perimeter and Area of Rectangles (Session 2.6) Teacher Note, pp End-of-Unit Assessment (Session 3.5) Teacher Note, pp Unit Guide for Grade 5, Unit 5 4

31 Unit Guide for Grade 5, Unit 6: Decimals on Grids and Number Lines Decimals, Fractions, and Percents 2 Unit Guide for Grade 5, Unit 6

32 Unit Guide for Grade 5, Unit 6 Decimals on Grids and Number Lines Decimals, Fractions, and Percents 2 Unit Summary: Students focus on decimals and how the number system extends to numbers less than one. Students use their understanding of fractions and percents and the relationship to decimals to solve computation problems involving decimals. Students use contexts (time and precipitation) and models (area and number lines) to order and add decimals. Materials: Decimals on Grids and Number Lines (1 per person) Student Activity Book pp pp. 1-4 of 6, Decimals on Hundredths and Thousandths Grids (1 per person) Student Activity Book p. 25 p. 3 of 3, Decimal Problems, (1 per person) Student Activity Book p. 45, The Jeweler s Gold (1 per person) 12 x 18 construction paper (1 per pair) Resource Master M24, Close to 1, (1 per pair) Student Activity Book p. 57, Close to 1 Recording Sheet, (1 per person) Decimal Cards Set A (1 deck per pair or group, use manufactured sets Resource Master M11, see Materials to Prepare, p. 21) Decimal Cards Set B (1 deck per pair or group, use manufactured sets Resource Master M13, see Materials to Prepare, p. 21) Do the following activities from Decimals on Grids and Number Lines: 1. Identify the mathematics in the unit To get an overview of the mathematics students will be doing in this unit, refer to these sections in the unit front matter. As you look at these sections, begin thinking about the main mathematical ideas students work on in this unit. Turn to pp. 8-9, Overview of This Unit. Look at the title of each Investigation and read the summary for each Investigation. Review the Mathematics in This Unit essay, pp Look at the Mathematical Emphases and Math Focus Points. (The emphases are numbered, and can be found above bulleted lists of Math Focus Points.) Read the Benchmarks in This Unit in the table on p. 15, Assessing the Benchmarks. Unit Guide for Grade 5, Unit 6 1

33 What mathematical ideas and skills are students working on in this unit? What mathematics are students expected to know at the beginning of the unit? At the end? 2. Introducing Thousandths and Representing Hundredths and Thousandths (Session 1.2) In this Investigation, students use representations (grids and number lines), and their understanding of fractions and the number system to order and compare decimals to the thousandths. In this session, they develop meaning for decimal notation for thousandths, and represent decimals on grids divided into tenths, hundredths, and thousandths. Read the Activity, Introducing Thousandths, pp (Note: The questions posed about 1/4 of the garden are based on Student Activity Book p. 7, Hundredths and Thousandths. The mini of this page is found on p. 32) Read the Activity, Representing Hundredths and Thousandths, pp , and complete Student Activity Book pp. 9-12, Decimals on Hundredths and Thousandths Grids, pp. 1-4 of 6. How did you know how much of the grid to shade in? How does this model support students in determining equivalent decimals, fractions and percents? Read the Teacher Note, Extending Place Value to Thousandths, pp What is the same and what is different about the place value of numbers as they extend into tenths, hundredths and thousandths? What are some common misunderstandings students have about decimals? How does this work with grids help students understand place value and decimals. 3. Assessment: Decimal Problems (Session 1.5) In this session, students compare the value of decimals and fractions. Read the Activity, Assessment: Decimal Problems, pp and solve the problems on Student Activity Book p. 25, Decimal Problems, p. 3 of 3. Use grids and/or reasoning about decimals to solve these problems. How did you solve each problem? What knowledge of fractions and decimals did you use? Read the Dialogue Box, Putting Decimals in Order, pp Unit Guide for Grade 5, Unit 6 2

34 What kinds of confusions do these students bring to their work with decimals? How might these decimal problems support students in understanding the value of decimals? 4. Jewelers Gold (Session 2.2) In this Investigation, students consider representations, place value, and contexts as they add and subtract decimals. In this session, students work in groups to add numbers that involve tenths, hundredths and thousandths. They create a poster explaining their solution and discuss and compare strategies. Read and do the Activity, The Jeweler s Gold, pp Read the task on the mini of Student Activity Book, p.45. Solve the problem with a partner (or individually) and make a poster that shows your strategy. Note: Try a different addition strategy than lining up the decimal points. What strategies are the members of your group using to add the decimals? Make sure each member of the group understands each strategy. When you added, were you paying attention to the value of each decimal? Did you use any representations? How could this activity support students to add decimals accurately? Read the Teacher Note, Adding Decimals, pp , and the Dialogue Box, Adding Decimals, the Jeweler s Gold, pp How is the addition of decimals similar or different from the addition of whole numbers? What important ideas about addition and about decimals do students need to understand to be able to add decimals? 5. Close to 1 (Session 2.5) Students use two or more addends to add decimals to the thousandths in a new game, Close to 1. Read the Activity, Introducing Close to 1, pp Play the game Close to 1 with a partner or small group, using Decimal Cards Sets A and B, Resource Master M24 (rules), and Student Activity Book p. 57 (recording sheet.) Unit Guide for Grade 5, Unit 6 3

35 What strategies are you using to find combinations that are exactly, or close to 1? Are any of your strategies similar to those you might use to find numbers close to 100 or 1,000? How might this game support students to gain a better understanding of decimals and the ability to add them together? Read the Dialogue Box, Student Strategies for Adding Decimals, pp Wrap Up What moves does the teacher make in this discussion to help deepen student understanding of adding decimals? (Consider which students were asked to share strategies, and the questions the teacher asked.) Look back at the unit overview, pp How do the activities done during this unit study fit into the overall mathematical storyline of the unit? Other Key Features of Decimals on Grids and Number Lines Ten-Minute Math in This Unit, p. 16 Practicing Place Value Estimation and Number Sense Assessment Decimal Problems (Session 1.5) Resource Master M18, Assessment Checklist Comparing and Ordering Decimals (Session 1.10) Teacher Note: Assessment, pp End-of-Unit Assessment (Session 2.8) Teacher Note, pp Unit Guide for Grade 5, Unit 6 4

36 Unit Guide for Grade 5, Unit 7: How Many People? How Many Teams? Multiplication and Division 2 Unit Guide for Grade 5, Unit 7

37 Unit Guide for Grade 5, Unit 7 How Many People? How Many Teams? Multiplication and Division 2 Unit Summary: Students find and study equivalent expressions for multiplication and division problems (e.g., = 28 2). Students practice solving larger multiplication problems (3 digit x 2 digit) accurately and efficiently and study the U.S. algorithm for multiplication. Students gain fluency solving division problems with one-, two-, and three-digit divisors. Materials: How Many People? How Many Teams? (1 copy per person) Student Activity Book, p. 3, Tripling and Thirding, (1 copy per person) Student Activity Book pp , Multiplication: How Did I Solve It? (one per person) Student Activity Book p. 52, Field Day: Third to Fifth Graders (one per person) Student Activity Book p. 53, Field Day Refreshments (one per person) Do the following activities from How Many People? How Many Teams?: 1. Identify the mathematics in the unit To get an overview of the mathematics students will be doing in this unit, refer to these sections in the unit front matter. As you look at these sections, begin thinking about the main mathematical ideas students work on in this unit. Turn to pp. 8-9, Overview of This Unit. Look at the title of each Investigation and read the summary for each Investigation. Review the Mathematics in This Unit essay, pp Look at the Mathematical Emphases and Math Focus Points. (The emphases are numbered, and can be found above bulleted lists of Math Focus Points.) Read the Benchmarks in This Unit in the table on p. 15, Assessing the Benchmarks. What mathematical ideas and skills are students working on in this unit? What mathematics are students expected to know at the beginning of the unit? At the end? Unit Guide for Grade 5, Unit 7 1

38 2. Tripling and Thirding (Session 1.2) In this Investigation, students find equivalent multiplication expressions and equivalent division expressions. In this session, students investigate what happens when one factor in a multiplication expression is tripled and the other is divided by three. Read the Activity, Tripling and Thirding, pp As you read, do the activities from the text fill in the missing number in the equations, create a story problem, and a representation, when asked. Complete Student Activity Book p. 3, Tripling and Thirding. Read the Algebra Note, Why Study Equivalent Expressions?, p. 33 Share the story problems and representations you created. Explain where the numbers are in each representation (e.g. If using 2 x 9 = 6 x 3, where is the 2 x 9? The 6 x 3?). How does this work deepen students understanding of multiplication? Of early algebraic ideas? 3. ion: Reviewing Strategies for Multiplication and Multiplication: How Did I Solve It? (Session 2.1) In this Investigation, students review and practice strategies for solving multiplication problems. In this session, students examine multiplication strategies by completing the solution to a multiplication problem when given the first step of the solution. Read the ion, Reviewing Strategies, pp Briefly discuss the student strategies shown on p. 50, talking through each solution to be certain you understand each student s strategy. Read the Activity, Multiplication: How Did I Solve It?, pp Complete Student Activity Book pp , Multiplication: How Did I Solve It? (Carefully follow the directions on the top of p. 14.) What strategies did you use to solve the problem? Choose one solution from Student Activity Book p. 14 and explain how you completed the solution, given your partner s start. What part of the problem was solved in the first step? What was left to solve? How does this work support students in becoming fluent in multiplication? 4. Solving A Division Problem (Session 3.1) In this Investigation, students refine strategies for solving division problems. In this session, students solve and represent a division problem and share their solutions. Read the Activity, Solving a Division Problem, pp Do the activity as described at the bottom of p. 69 (solve the problem, write a story problem, and create a representation). Unit Guide for Grade 5, Unit 7 2

39 Read the Teaching Notes on p. 69, Division Posters, and Story Contexts and Representations. Read the Algebra Note, Inverse Relationship, p. 70. What strategies did people in your group use to solve the division problem? How did your story problem and representation show your solution? (i.e. Where is the 374? Where is the 12? Where is the remainder?) Read the Teacher Note, Division Strategies, p. 125 and the Dialogue Box, Naming Division Strategies, pp Using the Teacher Note as a guide, how would you categorize the strategy you used? What are the general approaches students take in the Dialogue Box? Can you describe what is the same and different among some of the strategies? 5. Field Day Refreshments (Session 4.1) In this Investigation, students use all four operations to solve a variety of multistep problems in real world contexts. In this session, students use information from two charts to solve a complex problem involving all operations. Read the Activity, Field Day Refreshments, pp Use the information on Student Activity Book page 52, Field Day: Third to Fifth Graders, to solve the problems on Student Activity Book p. 53, Field Day Refreshments. Share different strategies people in your group used to solve these problems including what part of the problem was solved first, second, etc, and how you kept track of the work. Read the Dialogue Box, Understanding and Keeping Track of Multistep Problems, pp What is challenging for these students in this Dialogue Box? What do you think will challenge your students about these problems? How can you support students in solving these problems, without providing too much structure? 6. Wrap Up Look back at the unit overview, pp How do the activities done during this unit study fit into the overall mathematical storyline of the unit? Unit Guide for Grade 5, Unit 7 3