Investigations UNIT GUIDES: GRADE 4 GRADE. In Number, Data, And Space

Transcription

1 Investigations In Number, Data, And Space UNIT GUIDES: GRADE 4 GRADE 4

2 Investigations In Number, Data, And Space UNIT GUIDES: Grade 4 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 4

3 Contents Introduction Unit Guides: Grade 4 Unit 1: Factors, Multiples, and Arrays Unit 2: Describing the Shape of the Data Unit 3: Multiple Towers and Division Stories Unit 4: Size, Shape, and Symmetry Unit 5: Landmarks and Large Numbers Unit 6: Fraction Cards and Decimal Squares Unit 7: Moving Between Solids and Silhouettes Unit 8: How Many Packages? How Many Groups? Unit 9: Penny Jars and Plant Growth Unit Guides for Grade 4

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 4

5 Unit Guide for Grade 4, Unit 1: Factors, Multiples, and Arrays Multiplication and Division 1 Unit Guides for Grade 4, Unit 1

6 Unit Guide for Grade 4 Unit 1 Factors, Multiples, and Arrays Multiplication and Division 1 Unit Summary: Students deepen their understanding of the operation of multiplication. Students use rectangular arrays to represent the relationship between factors and multiples, use what they know to solve problems that increase in size, and focus on solving problems efficiently. They continue to develop fluency with multiplication combinations (facts up to 12 x 12). Materials: Factors, Multiples and Arrays (1 per person) Resource Master M2, One Centimeter Grid Paper (2-3 per person) 12 x 18 construction paper (1 per person) Student Activity Book p. 19, Multiple Turn Over Recording Sheet (1 per person) Resource Master M45, Multiple Turn Over (1 per pair) Multiple Cards (1 set per pair, use manufactured decks or Resource Masters M46-M49, see materials to Prepare, p. 55) Calculators (optional) Student Activity Book pp , Factors of 16 and 48 (1 per pair) Do the following activities from Factors, Multiples, and Arrays: 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, pp 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 4, Unit 1 1

7 2. Making Arrays (Session 1.2) In this Investigation, students use arrays to represent multiplication situations. In this session, they use what they know about multiplication to find all the arrays for given numbers. Read the Activity, Introducing Making Arrays, pp , to understand how students are introduced to arrays in Grade 4. Read the Activity, Making Arrays, pp , and choose one set of numbers to work on. (Make sure that each person in your group chooses a different pair of numbers from the list on p. 34.) Read the Teaching Note, Factors of a Number and Its Multiples, p. 34. What are the smallest and largest factors you found for your number? Could there be anything smaller or larger? Were there numbers that you knew wouldn t work? Why? How do you know you have all the arrays for a number? How does the array model support students in learning multiplication combinations? 3. Playing Multiple Turn Over (Session 2.3) Students have been working on their multiplication facts and identifying two factors that multiply together to make a multiple (for example, 8 x 6 = 48; 48 is a multiple of both 6 and 8). In this session, students are introduced to the game Multiple Turn Over to work on the relationship between factors and multiples and to continue to practice multiplication facts up to 12 x 12. Read the Activities, Introducing Multiple Turn Over, pp , Playing Multiple Turn Over, pp Play 3 or 4 rounds of the game with a partner. What knowledge of multiples are you using? For example, do you know that even numbers are multiples of 2, or that multiples of 5 end in 0 or 5? How did you determine the factors of each multiple? Did you just know them, and/or use reasoning? What strategies might students use in choosing a multiple on the board to cover? Read the Dialogue Box, Identifying Factors and Multiples in Multiple Turn Over, p What knowledge of factors and multiples are these students using as they explain their strategies? Unit Guide for Grade 4, Unit 1 2

8 4. Factors of 16 and 48 (Session 3.3) In this Investigation students use what they know about factors of one number to find factors of other numbers. In this session, they explore the concept that the factors of a number are also factors of a multiple of that number. Read the Activity, Factors of 16 and 48, pp Work with a partner to complete Student Activity Book pp , Factors of 16 and 48, using any representations that might be useful. How did you explain why the factors of 16 are also the factors of 48? How are the factors of 16 and 48 related? That is, how many 3s fit in to 16? How many 3s fit in to 48? Why is that? How might students use arrays to demonstrate the relationship between the factors of 16 and 48? Read the section Finding Factors, p. 17, Algebra Connections in This Unit. Read the ion: Are the Factors of 16 also Factors of 48?, pp , to see examples of student s explanations. 5. Wrap Up How would you describe the relationship between the factors of 16 and 48? Is this relationship true with any other factors? How do you know? How does this activity support students in extending their knowledge of factors? 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 4, Unit 1 3

9 Other Key Features of Factors, Multiples and Arrays Algebra Connections in This Unit, pp Ten-Minute Math in This Unit, p. 18 Today s Number Quick Images Counting Around the Class Assessment Representing 8 x 6 (Session 1.5) Resource Master M32, Assessment Checklist Multiplication Combinations (Session 2.5) Teacher Note, pp End-of-Unit Assessment (Session 3.4) Teacher Note, pp Unit Guide for Grade 4, Unit 1 4

10 Unit Guide for Grade 4, Unit 2: Describing the Shape of the Data Data Analysis and Probability Unit Guide for Grade 4, Unit 2

11 Unit Guide for Grade 4, Unit 2 Describing the Shape of the Data Data Analysis and Probability Unit Summary: Students collect and represent data in a variety of ways. They describe data and compare two data sets. They interpret the data and draw conclusions based on the data using terms such as mode, median, range, and outlier. Students begin their study of probability by placing events on a likelihood line that goes from impossible to certain and work with mixtures of colored cubes to describe the probability of different outcomes. Materials: Describing the Shape of the Data (1 per person) Boxes of Raisins (At least 15, or 1 per person) Student Activity Book p. 7, Comparing the Heights of First and Fourth Graders (1 per pair) Student Activity Book pp , Mystery Data A-C (1 per person) Do the following activities from Describing the Shape of the Data: 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 4, Unit 2 1

12 2. Representing the Number of Raisins in A Box (Session 1.1) In this Investigation students record, organize and represent data. They learn to use the median to describe data sets. They also compare two sets of data. In this session, students record and organize data about the number of raisins in a box. Give each person at least one box of raisins. Count the raisins in your box, record the data as a list and then organize the list. (NOTE: if there are fewer than 15 people in your group, count more than 1 box a piece so you have at least 15 pieces of data.) How did you organize the data? What does your data show? That is, how can you describe what you found out about the number of raisins in a box? Read the Teacher Note, Data Terms and Representations, pp and the Dialogue Box, Describing the Shape of the Raison Data, pp How does this work support students in learning about what data collection is and how it can be used to answer questions? What features of data might students notice when they look at a representation? 3. Comparing the Heights of First and Fourth Graders (Session 1.4) Students measure their heights, record and represent the data, and then collect the heights of a classroom of 1 st Graders. They compare the heights of the two classes of students. Read the Activity, Comparing the Heights of Fourth and First Graders, pp Look at the representations of the heights of the two groups on p. 39 in Session 1.3. These are two examples of how students might represent the data. Use this data to answer the questions on Student Activity Book p. 7, Comparing the Heights of First and Fourth Graders. What statements did you make about the height of 1 st and 4 th graders? What aspects of the representations did you pay attention to? (For example, clumps, gaps, range, outliers, median) Read the Math Note, Describing Data, p. 43, and the Teacher Note, Focusing on the Shape of the Data, pp What aspects of data help students describe the shape of the data? How might students use these aspects of data to describe two groups? What are students learning about data analysis as they describe and compare data? Unit Guide for Grade 4, Unit 2 2

13 4. Mystery Data (Session 2.5) Students describe and construct theories about three sets of mystery data that represent the heights or lengths of the members of a group of living things. Read the ion, Describing Mystery Data A, pp Work through the questions about Mystery Data A on Student Activity Book p. 25 with your group. Read the Activity, Mystery Data B and C, pp Work with a partner on Student Activity Book pp , Mystery Data B and Mystery Data C. How did you find the median for Mystery Data B? How did you make a line plot from the information in Mystery Set C? What was your reasoning to determine the living things represented in each graph? Read the Teacher Note, Finding and Using the Median, pp and the Dialogue Box, What Does the Median Tell You?, pp Note: The answers to the mystery data sets can be found in the Teacher Note, About the Mystery Data, pp What does it mean to find the median of a line plot? What does it tell you about the data? How might students find the median of a data set? What do they need to understand about the median? 5. Creating Events for the Likelihood Line (Session 3.1) In Investigation 3, students learn to describe probability using words such as impossible and certain, and the numbers 0 to 1. They conduct probability experiments and describe and compare results. In this session, students develop categories that represent a range from certain to impossible, and place events on a likelihood line. Read the Activity, Introducing Probability and the Likelihood Line, pp , to see how the work on probability is introduced to students. Read the Activity, Creating Events for the Likelihood Line, pp Work with a partner to make a line and then place two events on the line under each category. Read the ion, Characterizing Events from Impossible to Certain, pp , and the Teacher Note, Impossible, Certain, and Everything in Between, pp What did you consider as you placed events along the line? How might this activity support students in understanding probability? Unit Guide for Grade 4, Unit 2 3

14 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 Describing the Shape of the Data Ten-Minute Math in This Unit, p. 16 Today s Number Quick Survey Assessment Comparing Numbers of Cavities (Session 1.5) Teacher Note, pp Collecting and Comparing Data (Session 2.2) Resource Master M10, Assessment Checklist End-of-Unit Assessment (Session 3.5) Teacher Note, pp Unit Guide for Grade 4, Unit 2 4

15 Unit Guide for Grade 4, Unit 3: Multiple Towers and Division Stories Multiplication and Division 2 Unit Guide for Grade 4, Unit 3

16 Unit Guide for Grade 4, Unit 3 Multiple Towers and Division Stories Multiplication and Division 2 Unit Summary: Students develop strategies for solving multiplication problems with two-digit numbers and deepen their understanding of the operation of division by focusing on the relationship between multiplication and division. Using story contexts and multiple towers, students continue their investigation of the relationship between numbers and their factors. Students practice multiplying by 10 and multiples of 10, break problems into smaller parts that can be multiplied easily, and find the multiples of two-digit numbers. They gain fluency with all multiplication combinations to 12 x 12. Students solve, represent, and discuss division story problems, including some that have a remainder. Materials: Multiple Towers and Division Stories (1 per person) Student Activity Book p. 1, Mr. Jones and the Bagels (1 per person) Array Cards (one set per pair, use manufactured sets or Resource Masters M17-M37, see Materials to Prepare, p. 25) Resource Masters M38-M39, Small Array/Big Array (one per pair) Student Activity Book p. 7, Small Array, Big Array Recording Sheet (1 per pair) Student Activity Book pp , What Do You Do With the Extras? (1 per person) Adding Machine Tape (1 five-foot strip per pair, see Materials to Prepare, p. 95) Student Activity Book pp , Multiplication Cluster Problems (1 per person) Do the following activities from Multiple Towers and Division Stories: 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 4, Unit 3 1

17 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. Mr. Jones and the Bagels (Session 1.1) Students begin this unit by solving multiplication problems and consider ways to break apart the problems in order to make them easier to solve. Read the Activity, Mr. Jones and the Bagels, pp and do the problems on Student Activity Book p. 1, Mr. Jones and the Bagels. Write and share equations to represent each problem. Look at and discuss the representations in the sample student work on pp How did these problems encourage you to break apart the problem 14 x 12? 3. Small Array/Big Array (Session 1.3) Students worked with arrays as a model for multiplication in Unit 1, Factors, Multiples and Arrays, and they return to them in this unit as they work on breaking apart larger multiplication problems. In this session, students are introduced to and play a game Small Array/Big Array. Students find ways to show how two or more small arrays can be combined to equal one larger array. Read the Activity Introducing Small Array/Big Array, pp As a group, go over the rules of the game (Resource Masters M38-39, Small Array/Big Array), before you break into pairs to play the game. As you play the game, record each match on Student Activity Book p. 7, Small Array/Big Array Recording Sheet. Read the Teacher Note, Visualizing Arrays, pp When part of an array is covered, how do you determine what other array(s) will complete the product of the full array? How can arrays be useful in understanding multiplication and solving multiplication problems? 4. Problems with Remainders (Session 2.2) In this Investigation students solve division problems both with a remainder and without. They match division problems with a context and consider strategies for solving division problems that involve making groups of the divisor. In this session, students solve division problems with a remainder. Unit Guide for Grade 4, Unit 3 2

18 Read the Activity, Problems with Remainders, pp , and solve the problems on Student Activity Book pp , What Do You Do With the Extras? Once you have solved the problems, compare your answers with a partner. How does making groups of 5 help you to solve 44 8? What did you do with the remainders in each situation? How did the situation (in the story problem) affect your answer? Read the Dialogue Box, What Do You Do With the Extras?, pp How do students explain the remainder in each situation? How might you follow up on this discussion? 5. Multiplication Cluster Problems (Session 4.2) In this Investigation, students develop strategies for solving multiplication problems. In this activity, students consider ways to break apart multiplication problems in order to use familiar, smaller problems. Read the Activities, Introducing Multiplication Cluster Problems, pp , and Multiplication Cluster Problems, pp Choose two sets of problems from Student Activity Book pp , Multiplication Cluster Problems, to solve, and do more if there is time. What is the relationship between the problems in the set and the final problem to be solved? How do the smaller problems help you to find the solution to the final problem? How are you sure your answer to the final problem is correct? Read the Teacher Note, Multiplication Clusters and the Properties of Multiplication, pp As students are solving Multiplication Cluster Problems, what math ideas are they working on? 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 4, Unit 3 3

19 Other Key Features of Multiple Towers and Division Stories Algebra Connections in This Unit, pp Ten-Minute Math In this Unit, p. 20 Quick Images Counting Around the Class Assessment Solving 18 x7 (Session 1.5) Teacher Note, pp Writing and Solving a Division Problem (Session 2.6) Teacher Note, pp Multiplication Combinations (Session 3.4) Teaching Notes, p. 117 Teacher Note, pp End-of-Unit Assessment (Session 4.5) Teacher Note, pp Unit Guide for Grade 4, Unit 3 4

20 Unit Guide for Grade 4, Unit 4: Size, Shape, and Symmetry 2-D Geometry and Measurement Unit Guide for Grade 4, Unit 4

21 Unit Guide for Grade 4, Unit 4 Size, Shape, and Symmetry 2-D Geometry and Measurement Unit Summary: Students define and categorize polygons by identifying sets of shapes that have a common attribute and use 90 degrees as a reference for finding the measurement of other angles. They continue their measurement work from earlier grades by measuring distance and perimeter, using both U.S. and metric units and finding the area of polygons in square units. 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: Size, Shape and Symmetry (1 per person) Student Activity Book p. 2, Using Measurement Benchmarks and Measurement Tools (1 per person) Student Activity Book p. 29, All or Some Quadrilaterals (1 per pair) Resource Masters, M19-M20, Shape Cards (1 set per pair) See Materials to Prepare, p. 53 Scissors (1 per pair) Power Polygons (1-2 sets) Student Activity Book pp , Building Angles (1per pair) Student Activity Book pp , Area of Polygons (1 per person) Rulers (1 per pair) Yardsticks/meter sticks (1 per group of 4) Do the following activities from Perimeter, Angles, and Area: 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. Unit Guide for Grade 4, Unit 4 1

22 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? 2. Using Measurement Benchmarks and Estimating Length (Session 1.1) In this Investigation, students estimate length in U.S. standard and metric units, including lengths up to 100 feet. They practice using measurement tools accurately, and review perimeter. In this session, students work to establish measurement benchmarks. They identify something that is familiar that is the same amount as a particular unit of measurement (e.g., the width of my fingernail is about 1 cm). Read the Activity, Reviewing Linear Measurements and Measurement Tools, pp , to see how students begin to create some measurement benchmarks. Read the Activity, Using Measurement Benchmarks and Estimating Length, pp and work with a partner to complete Student Activity Book p. 2, Using Measurement Benchmarks and Measurement Tools. Read the Teacher Note, Introducing Benchmarks, p. 150 How accurate were your estimates? What might make them more accurate? How can this work with benchmarks support students to become more fluent and capable with measurement tools and skills? 3. Guess My Rule with Quadrilaterals (Session 2.4) In Investigation 2, students investigate the attributes of polygons. They focus particularly on quadrilaterals, categorizing them by side length, angle size, and parallel sides. In this session, students sort quadrilaterals according to a variety of criteria and they discuss the properties of quadrilaterals. Working in pairs, find all the quadrilaterals in the Shape Cards sets. When everyone is done, make certain you agree each pair has all the quadrilateral cards. Unit Guide for Grade 4, Unit 4 2

23 Read the Activity, Introducing Guess My Rule with Quadrilaterals, pp Play Guess My Rule with Quadrilaterals, p. 76. (For a full description of Guess My Rule refer to the directions, p. 70.) As you play, record your thinking about quadrilaterals on Student Activity Book p. 29, All or Some Quadrilaterals. Read the ion, All Quadrilaterals Some Quadrilaterals, pp What attributes did you pay attention to as you played Guess My Rule? How might this game support students to gain an understanding of the properties of quadrilaterals? How does the game help students develop geometric vocabulary? Read the Teacher Note, Classification of Quadrilaterals, p. 155, and the Dialogue Box, Are Squares Rectangles?, p How do students develop an understanding of the properties of quadrilaterals? What kind of reasoning do students use to explain the relationship between squares and rectangles? 4. Building Angles (Session 3.2) Students use Power Polygons to make right angles and then to determine the measure of some angles that are more or less than 90 degrees. These include angles that measure 60, 120, and 150 degrees. Read the Activity, Introducing Building Angles, pp With a partner, complete Student Activity Book pp , Building Angles. How did you use knowledge of 90 degrees to find angles that measure 30, 60, 120, etc.? What reasoning did you use to find the measure of the angles on the Student Activity Book pages? Read the Dialogue Box, Building Angles, p. 167 How do these students find the measures of angles? What knowledge are they building on? Unit Guide for Grade 4, Unit 4 3

24 5. Area of Polygons (Session 4.6) In this Investigation students measure the area of polygons in both nonsquare and square units of measure by decomposing the shapes and using symmetry. In this session, students find the area of different polygons by decomposing shapes in rectangles and triangles. Read and do Activity 2, Area of Polygons, pp Complete Student Activity Book pp , Area of Polygons. How did you find the area of each polygon? Did you use the same strategy as other members in your group? Did you decompose the shape? If so, what shapes did you find the most useful? How did you deal with the diagonal lines in the shapes? How could this activity support students to find the area of any polygon? 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 Size, Shape and Symmetry LogoPaths Software Teacher Note, Introducing and Managing LogoPaths, pp Teacher Note, About the Mathematics in the LogoPaths Software, p. 153 Ten-Minute Math in This Unit, p. 16 Today s Number: Broken Calculator Quick Images: 2-D Assessment: What is a Quadrilateral? (Session 2.5) Teacher Note, pp End-of-Unit Assessment (Session 4.7) Teacher Note, pp Unit Guide for Grade 4, Unit 4 4

25 Unit Guide for Grade 4, Unit 5: Landmarks and Large Numbers Addition, Subtraction, and the Number System Unit Guide for Grade 4, Unit 5

26 Unit Guide for Grade 4, Unit 5 Landmarks and Large Numbers Addition, Subtraction, and the Number System Unit Summary: Students extend their knowledge of the number system by examining the structure of 10,000 and practice and refine strategies for adding and subtracting whole numbers up to 10,000. They continue to study place value by adding and subtracting multiples of 10 and 100 to numbers in the thousands, and they consolidate their understanding of the operation of addition by studying a variety of addition strategies and algorithms, including the U.S. algorithm for addition. Students continue their study of subtraction by solving, representing, and discussing their strategies for a variety of subtraction problems. Materials: Landmarks and Large Numbers (1 copy per person) Resource Masters M10-M11, Change Cards (1 set per pair, see Materials to Prepare, p. 25) Resource Master M12, Changing Places Recording Sheet (1 per person) Resource Master M15, Changing Places (1 per person) Student Activity Book p. 25, Addition Starter Problems, p. 1 of 2 (1 per person) Digit Cards (1 deck per pair, use manufactured decks or Resource Masters M16-M18, see Materials to Prepare, p. 27) Resource Master M21, Close to 1,000 (1 per person) Resource Master M22, Close to 1,000 Recording Sheet (1 per person) Student Activity Book p. 51, Subtraction Story Problems, p. 1 of 2 (1 per person) Do the following activities from Landmarks and Large Numbers: 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 on the table on p. 15, Assessing the Benchmarks. Unit Guide for Grade 4, Unit 5 1

27 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. Changing Places (Session 1.3) In the first two sessions of this Investigation, students create their own 1,000 books out of ten 100 charts. They write some numbers on each chart so they can easily find any number. (See pp for a further description of the 1000s book.) In the game Changing Places, students practice adding and subtracting multiples of 10 and 100 to create new numbers to write on each page of their 1000 books. Read the Activities, Introducing Changing Places, pp , and Changing Places, pp Refer to Resource Master 15, Changing Places, for the rules of the game. Play at least five rounds of this game, creating a new number with a different digit in the hundreds place for each round. How does playing Changing Places help students develop computational fluency in addition and subtraction? How does this game help students further their understandings of the base 10 number system? Read the Math Note, Combining Positive and Negative Numbers, p. 40. What math is described in this teaching note that is important for the teacher to understand? Is it important students share the same understanding? Why or why not? 3. Starter Problems (Session 2.3) Making an Equivalent Problem (Session 2.3) In this Investigation, students identify and compare different addition strategies. In this session, students solve addition problems after considering several different first steps. They represent and discuss how to create equivalent addition expressions. Read the Activity, Introducing Starter Problems, p. 73. Solve problems 2 and 3 on Student Activity Book p. 25, Addition Starter Problems, p. 1 of 2, with this adaptation: Use each of the 3 starts to solve the final problem. Compare solutions with a neighbor. How do starter problems help students develop computational fluency, particularly efficiency and flexibility, in addition? Read the ion, Making an Equivalent Problem, pp Read the Algebra Notes, Writing Equations, p. 76, and Generalizing in Mathematics, p. 77. Unit Guide for Grade 4, Unit 5 2

28 Look at the student work examples, pp In the sample work, discuss how each student proved that = How are students using story contexts and drawings to help them think about the early algebraic ideas of creating equivalent problems? How do these contexts and representations help students understand the mathematics? How does this work help students develop computational fluency for addition? 4. Close to 1,000 (Session 2.5) Students continue work on understanding the base-ten number system and place value, and on adding and subtracting efficiently. In Close to 1,000, students make combinations of 3-digit numbers with sums equal to or close to, 1,000. Read the Activity, Introducing Close to 1,000, pp Refer to Resource Master M21, Close to 1,000 for the directions to the game. Use the cards shown on p. 87 to play a sample round. Play several rounds of the game with a partner. How does this game help students develop their understanding of the number system and place value? How does it help them develop efficiency in adding and subtracting 3- digit numbers? 5. Subtraction Story Problems (Session 4.1) Strategies for Subtraction (Session 4.2) In Session 4.1, students solve subtraction story problems involving different contexts/situations for subtraction: missing part, comparison, and removal. They discuss solutions and strategies to one of these problems in the discussion in Session 4.2. Solve problems 1-3 on Student Activity Book p. 51, Subtraction Story Problems. If possible, use a number line or picture to show your solution. your solution and representation for problem 2 with a partner. Read the ion, Strategies for Subtraction, pp in Session 4.2, the Math Notes on p. 139, Recognizing Different Subtraction Situations, and The U.S. Standard Algorithm. Read the Math Note, Where is the Answer on the Number Line? p. 140 and the Ongoing Assessment, Observing Students at Work, p Unit Guide for Grade 4, Unit 5 3

29 As students recognize various situations as subtraction and see a comparison problem solved using two different strategies (take away, adding up), how are they developing computational fluency in subtraction? What do they have to be able to understand about subtraction? How do number lines help students keep track of their solutions? What confusions might students encounter in these different problems? Read the Teacher Note, Subtraction Strategies, pp How does understanding the mathematics of each of these subtraction strategies help you with instruction? What questions do you have about this work? 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 Landmarks and Large Numbers Ten-Minute Math in This Unit, p. 20 Today s Number, Broken Calculator Practicing Place Value Assessment Numbers to 1,000 (Session 1.5) Resource Master M20, Assessment Checklist Solving an Addition Problem in Two Ways (Session 2.6) Teacher Note, pp Numbers to 10,000 (Session 4.3) Resource Master M28, Assessment Checklist) End-of-Unit Assessment (Session 4.7) Teacher Note, pp Unit Guide for Grade 4, Unit 5 4

30 Unit Guide for Grade 4, Unit 6: Fraction Cards and Decimal Squares Fractions and Decimals Unit Guide for Grade 4, Unit 6

31 Unit Guide for Grade 4, Unit 6 Fraction Cards and Decimal Squares Fractions and Decimals Unit Summary: Students develop ideas about fractions by identifying fractions of an area (3/4 of a rectangle), fractions of a group of objects (3/4 of 24), and decimal fractions (.75). They compare fractions of different wholes (1/3 of a 6 x 4 rectangle and 1/3 of a 10 x 10 rectangle), and combine fractions using models and reasoning. Students use 10 x 10 grids to represent, compare, and combine common decimals in the tenths and hundredths. Materials: Fraction Cards and Decimal Squares (1 copy per person) Resource Master M7, 4 x 6 Rectangles (2 per person) Resource Master M11, 5 x 12 Rectangles (1 per person) Student Activity Book p. 37, Fractions in Containers (1 per person) Resource Master M15, 10 x 10 Squares (2 per person) Do the following activities from Fraction Cards and Decimal Squares: 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 4, Unit 6 1

32 2. Finding Fourths and Eighths (Session 1.1) and Fractions of 24 (Session 1.3) Students identify halves, fourths, and eighths of a 4 x 6 rectangle, and discuss how they can identify the value of each fraction. In Session 1.3, students work to identify fractions of a set of 24 objects. Read the Activity, Introducing Halves and Fourths, pp Use Resource Master M7, 4 x 6 Rectangles, to shade in 1/4, 2/4, 3/4, and 1/8 on separate rectangles. Read the Activity, Finding Fourths and Eighths, pp How did you find 1/4 and 1/8 of the rectangle? How do you know the piece you shaded in is exactly 1/4, 2/4, 3/4, or 1/8? Did you use ¼ to find 1/8? What is the relationship between these two fractions? What is the fractional part that is not shaded? Read the Activity, Fractions of 24, pp and solve the problems shown on p. 38. Read the Math Note, Fractions of a Group, p. 38. In what ways is it different to find a fraction of an area than to find a fraction of a set? The same? What kind of pictures or models help to identify fractions in both activities? What knowledge of fractions do you need in order to find 1/2, 1/4, 1/8, etc? Read the Teacher Note, Why Are Fractions so Difficult? Developing Meaning for Fractions, pp What are some of the important ideas students need to understand about fractions? What are some misunderstandings students often bring to their work with fractions? 3. Combinations that Equal 1 (Session 1.5) In this session, students find combinations of fractions that equal one whole rectangle. Read the Activity, Combinations that Equal 1, pp , and find several equations that equal 1. Record these on Resource Master M11, 5 x 12 Rectangles. Compare the equations you found and recorded that equal 1. What understandings about fractions did you use to find combinations? How might this activity support students in adding fractions? Unit Guide for Grade 4, Unit 6 2

33 4. Fractions in Containers (Session 2.4) In Investigation 2, students compare fractions. They use a set of Fraction Cards that they make in Session 2.1, which include the name of the fraction and picture that represents the fraction. (See examples of Fraction Cards, p. 71.) In this session, students compare fractions to the landmarks 0, 1/2, 1, and 2. Read the Activity, Comparing Fractions to Landmarks, pp , to see how students use their Fraction Cards to compare the size of fractions. Complete Student Activity Book p. 37, Fractions in Containers. What reasoning did you use to place these fractions into the containers? Which fractions are easier or harder to place than others? Why? How might this activity support students in comparing the size of different fractions? Read the Dialogue Box, Comparing Fractions to Landmarks, pp Describe the reasoning that Jill uses to place some of the fractions. What does she seem to understand about fractions? 5. Representing Decimals (Session 3.1) In Investigation 3, students consider everyday uses of decimals and the relationship of decimals to fractions. They learn to read and write decimals in tenths and hundredths. Read the Activity, Decimals on the 10 x 10 Square, pp Using Resource Master M15, 10 x 10 Squares, shade in at least 3 of the decimals listed on p What did you consider as you shaded in each number on the 10 x 10 squares? What might students learn about place value as they work with the 10 x 10 squares? Read the Dialogue Box, Are These Equal?, p How can the 10 x 10 squares support students to understand the relationship between fractions and decimals? 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 4, Unit 6 3

34 Other Key Features of Fraction Cards and Decimal Squares Ten-Minute Math in This Unit, p. 16 Practicing Place Value Quick Survey Assessment Identifying and Comparing Fractions (Session 1.5) Teacher Note, pp Comparing Fractions (Session 2.6) Teacher Note, pp End-of-Unit Assessment (Session 3.7) Teacher Note, pp Unit Guide for Grade 4, Unit 6 4

35 Unit Guide for Grade 4, Unit 7: Moving Between Solids and Silhouettes 3-D Geometry and Measurement Unit Guide for Grade 4, Unit 7

36 Unit Guide for Grade 4, Unit 7 Moving Between Solids and Silhouettes 3-D Geometry and Measurement Unit Summary: Students examine the relationships between 3-D solids and their 2-D representations. They learn and use the mathematical terminology for these solids and their attributes. They translate between 3-D shapes and their 2-D representations as they build cube configurations from pictures and mental images and investigate silhouettes of solids from several different perspectives. Students build an understanding of measuring volume as they examine the structure of rectangular prisms and determine the number of cubes that fit inside given box patterns. Materials: Moving Between Solids and Silhouettes, (1 per person) Student Activity Book p. 5, Silhouettes of Geometric Solids (1 per pair) Student Activity Book pp , Front, Top, and Side Silhouettes (1 per pair) Student Activity Book p. 30, Building from Silhouettes (1 per person) Student Activity Book pp , Making Boxes From the Bottom Up (1 per person) Geometric Solids (1 set per group of 4) Overhead Projector Connecting cubes (30-40 per person) Do the following activities from Moving Between Solids and Silhouettes: 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 4. Unit 7 TERC,

37 2. Matching Solids and Silhouettes (Session 1.2) In this Investigation, students describe the properties and attributes of geometric solids. In this session, they identify the shapes of silhouettes projected by these solids and visualize what objects look like from different perspectives. Note: You may want to review the names of these geometric solids on p. 23, Session 1.1. Read the Activities, Introducing Matching Solids and Silhouettes, pp , and Matching Solids and Silhouettes, pp With a partner, complete Student Activity Book p. 5, Silhouettes of Geometric Solids. Read the Teacher Note: Difficulties in Visualizing Silhouettes, pp What parts of the 3-D solid do you focus on to determine the silhouette it will create? What confusions might students encounter as they work on this activity? 3. Drawing All Three Views (Session 2.2) In this Investigation, students move between 2-D and 3-D representations of cube buildings. They draw silhouettes from different buildings and construct the building from given silhouettes. Read the Activity, Introducing Different Silhouettes, pp Use the overhead projector to identify the perspective use for the top view. Read the Activity, Drawing All Three Views, pp Work with a partner to complete Student Activity Book pp , Front, Top and Side Silhouettes. Use the overhead to help identify the front, top, and side silhouettes. How are you figuring out the silhouettes of each side? What aspects of the 3-D shape are you paying attention to as you draw the silhouettes? Read the Teacher Note, Interpreting 2-D Diagrams of 3-D Shapes, p. 100 and the Teacher Note, Staying Properly Oriented, p What mathematics are students working on as they draw these silhouettes? How might this activity support students to move between 2-dimensional drawings and 3-dimensional figures? Unit Guide for Grade 4. Unit 7 TERC,

38 4. Building from Silhouettes (Session 2.4) In this session, students are given the front, top and right-side silhouettes and construct the matching cube buildings. Read the Activity, Introducing Building from Silhouettes, p. 65. Complete Student Activity Book p. 30, Building from Silhouettes. Read the Teacher Note, Integrating Three Views: How Students Try to Do It, p How did you construct your building to match all the silhouettes? (e.g. Did you work on each side, or integrate more than 2 sides at once?) How might students approach this task? How might you support students who are having difficulty? 5. Designing Box Patterns (Session 3.2) In Investigation 3, students work on finding the volume of rectangular prisms. In this session, students are given the shape of the bottom of a box and its volume and asked to draw the design of the box. Read the Activity, Designing Box Patterns, pp Complete Student Activity Book pp , Making Boxes from the Bottom Up. Use connecting cubes to build at least one of the boxes to see how the pattern and the box fit together. What strategies are you using to complete the patterns? How do the layers in each box relate to the pattern of the box? How does this activity support students in understanding and finding volume? 6. Wrap Up Look back at the unit overview, pp How do the activities done during this unit study fit into the mathematics of the rest of the unit? Unit Guide for Grade 4. Unit 7 TERC,

39 Other Key Features of Moving Between Solids and Silhouettes Ten-Minute Math in This Unit, p. 16 Practicing Place Value Quick Images Assessment Match the Silhouettes (Session 1.4) Teacher Note, pp. 99 Drawing Silhouettes (Session 2.5) Resource Master M18, Assessment Checklist End-of-Unit Assessment (Session 3.5) Teacher Note, pp Unit Guide for Grade 4. Unit 7 TERC,

40 Unit Guide for Grade 4, Unit 8: How Many Packages? How many Groups? Multiplication and Division 3 Unit Guide for Grade 4. Unit 8

41 Unit Guide for Grade 4, Unit 8 How Many Packages? How Many Groups? Multiplication and Division 3 Unit Summary: Students continue to develop efficient strategies for solving multiplication problems by breaking problems into smaller parts or changing one or both numbers to create an easier problem. Students also focus on recording their work with clear and concise notation. Students develop strategies for solving division problems (three-digit divided by two-digit), which involve making groups of the divisor. These problems are presented both in story contexts and numerically. Materials: How Many Packages? How Many Groups? (1 per person) Student Activity Book pp. 5-6, Solving 2-Digit Multiplication Problems (1 per person) Student Activity Book pp , More Multiplication Cluster Problems (1 per person) Student Activity Book pp , Problems About Teams (1 per person) Student Activity Book pp , Solving Division Problems (1 per person) Do the following activities from How Many Packages? How Many Groups?: 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 4, Unit 8 TERC,