CMP4_6_1_TE_Investigation 1

Variables and Patterns Introducing Algebraic Reasoning 4 CONNECTED MATHEMATICS® Teacher Edition CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

xiii Quick Start Guide QS-1 Unit Planning UP-1 Unit Planning Chart� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-1 Unit Description � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-3 Summary of Investigations � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-3 Mathematics Overview � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-6 Student Edition Mathematical Goals of the Unit � � � � � � � � � � � � � � � � � � � UP-12 Unit Arc of Learning™ (AoL) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-13 Contents of the Student Edition� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-14 Unit Alignments: Goals, Arc of Learning™, Standards, Now What Do You Know?, and Emerging Mathematical Ideas � � � � � � � � � � � � � � � � � UP-15 Family Letter� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-23 Investigation 1. Organizing a Bike Tour: Variables, Tables, and Graphs 1 Investigation 1 Planning Chart� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �1 Problem 1.1 Organizing a Bike Tour Experiment: Variables and Tables � � � 3 At a Glance 3 Extended Launch—Explore—Summarize 5 Answers Embedded in Student Edition Problems 13 Learning Aids Learning Aid Template: Blank Tables and Graph 20 Learning Aid 1.1A: Jumping Jack Experiment Tables 21 Learning Aid 1.1B: Matching Descriptions and Tables (Part 1, Descriptions) 22 Learning Aid 1.1B: Matching Descriptions and Tables (Part 2, Tables) 23 Problem 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs� � � � � 24 At a Glance 24 Extended Launch—Explore—Summarize 26 Answers Embedded in Student Edition Problems 39 CONTENTS CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

xiv Contents Teaching Aids Teaching Aid 1.2A: 4-by-4 Game Grid 46 Teaching Aid 1.2B: Making a Graph 47 Teaching Aid 1.2C: Summary Discussion Graphs 48 Teaching Aid 1.2D: Ms. Park’s Class Data 49 Learning Aids Learning Aid Template: 1 Centimeter Graph Paper 55 Learning Aid 1.2A: Graphing the Jumping Jack Experiment 56 Learning Aid 1.2B: Matching Graphs to Jumping Tables and Descriptions 57 Problem 1.3 Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 59 At a Glance 59 Extended Launch—Explore—Summarize 61 Answers Embedded in Student Edition Problems 68 Teaching Aids Teaching Aid 1.3A: Atlantic City to Cape May Table and Graph 75 Teaching Aid 1.3B: Connecting Graph Points Examples 76 Learning Aids Learning Aid 1.3A: Atlantic City to Lewes 77 Learning Aid 1.3B: Matching Cases to Paths (Part 1, Paths) 78 Learning Aid 1.3B: Matching Cases to Paths (Part 2, Cases) 79 Learning Aid 1.3C: Day 2 of the Trip 80 Problem 1.4 Chincoteague Island to Norfolk: Stories, Tables, and Graphs � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 81 At a Glance 81 Extended Launch—Explore—Summarize 83 Answers Embedded in Student Edition Problems 90 Mathematical Reflection� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 95 At a Glance 95 Answers Embedded in Student Edition 97 Answers Embedded in Applications—Connections—Extensions (ACE) � � � 98 Assessment: Checkup 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �116 Answers for Assessment: Checkup 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �118 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Contents xv Investigation 2. Determining Tour Needs: Analyzing Relationships Among Variables 121 Investigation 2 Planning Chart � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �121 Problem 2.1 Renting Bicycles: Independent and Dependent Variables � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 122 At a Glance 122 Extended Launch—Explore—Summarize 124 Answers Embedded in Student Edition Problems 133 Teaching Aid Teaching Aid 2.1: Comparing Costs from Rocky’s and Adrian’s 137 Learning Aid Learning Aid 2.1: Rocky’s and Adrian’s Bike Rental 138 Problem 2.2 Finding Customers: More Variables � � � � � � � � � � � � � � � � � � � � � 139 At a Glance 139 Extended Launch—Explore—Summarize 141 Answers Embedded in Student Edition Problems 146 Learning Aids Learning Aid Template: 1 Centimeter Graph Paper 150 Learning Aid 2.2: Finding Customers 151 Problem 2.3 What’s the Story?: Interpreting Graphs� � � � � � � � � � � � � � � � � � 152 At a Glance 152 Extended Launch—Explore—Summarize 154 Answers Embedded in Student Edition Problems 159 Learning Aids Learning Aid 2.3: Matching Descriptions to Graphs (Part 1, Graphs) 163 Learning Aid 2.3: Matching Descriptions to Graphs (Part 2, Descriptions) 164 Mathematical Reflection� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 165 At a Glance 165 Answers Embedded in Student Edition 167 Answers Embedded in Applications—Connections—Extensions (ACE) � � 168 Assessment: Partner Quiz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 185 Answers for Assessment: Partner Quiz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 188 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

xvi Contents Investigation 3. Returning Home: Relating Variables, Expressions, and Equations 192 Investigation 3 Planning Chart � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 192 Problem 3.1 Returning Home: Equations with One Operation � � � � � � � � � 193 At a Glance 193 Extended Launch—Explore—Summarize 195 Answers Embedded in Student Edition Problems 205 Learning Aids Learning Aid 3.1A: Malcolm’s Table 213 Learning Aid 3.1B: Returning Home Speeds 214 Problem 3.2 Planning the Next Tour: More Equations with One Operation � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 215 At a Glance 215 Extended Launch—Explore—Summarize 217 Answers Embedded in Student Edition Problems 222 Teaching Aid Teaching Aid 3.2: Table to Equation 226 Learning Aid Learning Aid Template: 1 Centimeter Graph Paper 228 Problem 3.3 Planning Ahead: Connecting Equations with Tables and Graphs � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 229 At a Glance 229 Extended Launch—Explore—Summarize 231 Answers Embedded in Student Edition Problems 236 Teaching Aid Teaching Aid 3.3: Theo’s Matches for Situation A 244 Learning Aids Learning Aid 3.3: Sorting Theo’s Cards (Part 1, Notes) 247 Learning Aid 3.3: Sorting Theo’s Cards (Part 2, Graphs) 248 Learning Aid 3.3: Sorting Theo’s Cards (Part 3, Tables) 249 Learning Aid 3.3: Sorting Theo’s Cards (Part 4, Equations) 250 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Contents xvii Mathematical Reflection� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 251 At a Glance 251 Answers Embedded in Student Edition 253 Answers Embedded in Applications—Connections—Extensions (ACE) � � 254 Assessment: Unit Test� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 267 Answers for Assessment: Unit Test � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 271 Correlations 277 Problem Correlations with Common Core State Standards of Mathematics and Mathematical Practices � � � � � � � � � � � � � � � � � � � � � � � � 277 Mathematical Practices and Habits of Mind Description� � � � � � � � � � � � � � 278 Mathematical Practices and Habits of Mind Examples � � � � � � � � � � � � � � �280 Assessment Correlation with Common Core State Standards of Mathematics (CCSSM) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 281 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

1 Implementation Key Terms Materials Resources Problem 1.1 Groups of 3–4 students Pacing 2 days variables relationships between variables algebra For each student • Learning Aid Template: Blank Tables and Graph (optional) • Learning Aid 1.1A: Jumping Jack Experiment Table For each group of 3–4 students • Learning Aid 1.1B: Matching Descriptions and Tables (Part 1, Descriptions) • Learning Aid 1.1B: Matching Descriptions and Tables (Part 2, Tables) • timer (1 per group) • tape or glue • clothespin or clip (optional accommodation) Problem 1.2 Think, Pair, Share Pacing 1 day coordinate graph axes horizontal axis x-axis vertical axis y-axis scale coordinate pair or ordered pair For each student • Learning Aid Template: Blank Tables and Graph (optional) • Learning Aid 1.2A: Graphing the Jumping Jack Experiment For each pair of students • Learning Aid Template: 1 Centimeter Graph Paper* • Learning Aid 1.2B: Matching Graphs to Jumping Tables and Descriptions • tape or glue For the class • Teaching Aid 1.2A: 4-by-4 Board • Teaching Aid 1.2B: Making a Graph • Teaching Aid 1.2C: Summary Discussion Graphs • Teaching Aid 1.2D: Ms. Park’s Final Matches INVESTIGATION 1 PLANNING CHART Organizing a Bike Tour: Variables, Tables, and Graphs INVESTIGATION 1 Materials for All Investigations: calculators; student notebooks; colored pens, pencils, or markers; scissors; transparent sticky grid paper; rulers (continued) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

2 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Implementation Key Terms Materials Resources Problem 1.3 Groups of 3–4 students Pacing 1 day For each student • Learning Aid 1.3A: Atlantic City to Lewes • Learning Aid 1.3C: Day 2 of the Trip For each group of 3–4 students • Learning Aid 1.3B: Matching Cases to Paths (Part 1, Paths) • Learning Aid 1.3B: Matching Cases to Paths (Part 2, Cases) For the class • Teaching Aid 1.3A: Atlantic City to Cape May • Teaching Aid 1.3B: Connecting Graph Points Examples Problem 1.4 Groups of 3–4 students Pacing 1 day For each group of 3–4 students • large poster paper (1 per group)* • sticky notes (1 per student) Mathematical Reflection Whole Class Individual Notes Pacing __1 2 day For the class • large poster paper (optional)* Assessment Checkup 1 Individual Pacing __1 2 day For each student • Checkup 1 (continued from page 1) *not included in Classroom Materials Kit CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.1 Organizing a Bike Tour Experiment: Variables and Tables 3 At a Glance This problem asks students to think about the physical fitness challenge of riding a bicycle for long distances. It uses that context to develop basic ideas of data representation and interpretation using tables. In the Initial Challenge, students collect data in a table while doing jumping jacks for 2 minutes. The What If . . .? situation has students matching descriptions of others doing the jumping jack experiment with tables of data. PROBLEM 1.1 Organizing a Bike Tour Experiment: Variables and Tables Arc of Learning™ Introduction NOW WHAT DO YOU KNOW? How does a table help you make sense of the relationship between two variables in a situation? Key Terms Materials variables relationships between variables algebra For each student • Learning Aid 1.1A: Jumping Jack Experiment Table For each group of 3–4 students • Learning Aid 1.1B: Matching Descriptions and Tables (Part 1, Descriptions) • Learning Aid 1.1B: Matching Descriptions and Tables (Part 2, Tables) • timer (1 per group) • tape or glue • clothespin or clip (optional accommodation) Pacing 1 day Groups 3–4 students A 1–2 C 11–13 E 18 Note: If you have a Grade 6 Classroom Materials Kit, please refer to A Guide to Connected Mathematics® 4 for a detailed list of materials included or items you will need to prepare ahead of time. For more on the Teacher Moves listed here, refer to the General Pedagogical Strategies and the Attending to Individual Learning Needs Framework in A Guide to Connected Mathematics® 4. Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Tell the class about bicycles and the yearly bicycle tour across Iowa. Encourage students to share other facts about organized bicycle tours they might know. Then continue reading about the bicycle trip that the five college students are planning. PRESENTING THE CHALLENGE Connect the bike tour and the jumping jack experiment by pointing out that both activities involve physical exertion over a period of time. This experiment works nicely if students are divided into groups of four. Within the group, each student has a job: • performing jumping jacks • counting jumps out loud • calling time when 10 seconds have passed • recording the number of jumping jacks completed at the end of every 10 seconds for the 2-minute time period Since this is the first problem of the year, you may want to do a short summary after the Initial Challenge. (continued) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

4 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Facilitating Discourse Teacher Moves EXPLORE PROVIDING FOR INDIVIDUAL NEEDS Suggested Questions • What was the pattern of your group’s jumping jacks? • How did that pattern look when the person was jumping? • How do you see that pattern in the table? • Have students read the descriptions of the six jumpers in Ms. Park’s class. Compared to Ms. Park’s students, does your data have the same relationship between the variables? • Who seemed to be the most consistent jumper? How do you determine a consistent jumper? PLANNING FOR THE SUMMARY As you are circulating, listen for how students are describing the patterns in the data. How are they describing how the table shows when a jumper was jumping fast/slow, stopped, or jumped consistently? Students should be talking about the change in the number of jumping jacks for each interval of 10 seconds. For the What If . . . ? situation, you can assign one or two of the cards to each group of students. SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES The scientific issue in this problem is how performance can change over time. The mathematical issue is how that performance pattern between the variables is shown by data expressed in tables. You can give focus to the Summarize discussion by asking students what their experiment told them about each issue. Suggested Questions • What was the pattern of the relationship between the variables of your group’s jumping jacks? • How did that pattern look when the person was jumping? • How do you see that pattern in the table? MAKING THE MATHEMATICS EXPLICIT Help students visualize the patterns of change relationships by imagining someone jumping and how that would influence the data in the table. These questions can help students make sense of how the physical activity is represented in a table of data. Suggested Questions • Compared to Ms. Park’s students, does your data have the same relationship between the variables? • Did anyone in the class have steady or consistent jumping? How do you determine a consistent jumper? • What would a consistent jumper’s table of data look like? • What would you see if you watched them do the jumping jacks? • Can you be slow but still be consistent? • What if a jumper started off really fast then got slower over the jumping time? What would the table of data look like? What would you see if you watched them do the jumping jacks? As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Implementation Note: Have students keep their data from the jumping jacks to use in Problem 1.2. ProblemSolving Environment Portrayal 1.1 (continued from page 3) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 5 Problem Overview This problem asks students to think about the physical-fitness challenge of riding a bicycle for long distances. It uses that context to develop basic ideas of data representation and interpretation using tables. Comment: The main activity is a jumping jack experiment. Doing jumping jacks for 2 minutes is hardly the same as riding a bike for 6−8 hours. However, students can do the jumping jack experiment in the classroom, and experience has shown that it does give students a personal sense of the physical demands of a bike tour. This makes it easier for them to interpret bike-tour data that are given in subsequent problems. So as not to embarrass any student who might not be physically up to jumping jacks, use volunteers. Sometimes, all students want to try doing the jumping jacks. It is a nice way to encourage participation at the beginning of the year. For students who are not able to jump, an alternative would be to use a clothespin or clip. (Portrayal) Opening and closing the clip can count in place of one jumping jack. Implementation Note: Since this is the first problem of the year, you may want to do a short summary after the Initial Challenge. Launch (Getting Started) Connecting to Prior Knowledge Tell the class about bicycles and the yearly bicycle tour across Iowa. Encourage students to share other facts about organized bicycle tours they might know. Then continue reading about the bicycle trip that the five college students are planning. Have students share their ideas about the questions in the introduction. Students should justify their guesses about the distance they think they could ride in a day and consider ways in which their speed might vary throughout the day. Suggested Questions • How far do you think you could ride in a day? (Answers will vary.) • How do you think the speed of your ride would change during the course of the day? (Most students will indicate that their speed would slow down over the course of the day as they grew fatigued. Others might say that they could get surges of energy, especially toward the end.) EXTENDED LAUNCH—EXPLORE—SUMMARIZE CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

6 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs • What conditions would affect the speed and distance you could ride? (Answers might include the type of terrain [rocky or smooth]; how much of the ride is uphill, downhill, or flat; weather conditions and temperature; and how much gear you carry.) • How are the cyclists’ speed and distance likely to change throughout a day? (Answers will vary.) Presenting the Challenge After a short class discussion, move on to the jumping jacks stamina experiment. This problem begins the discussion of patterns that describe the relationship between two variables. Students will use the experiment data to describe the way the number of jumping jacks changes as time increases in 10-second intervals. The experiment provides a physical model to give students a sense of the relationship. The connection between the physical movement and the pattern in the table helps students interpret the data. (Portrayal) Connect the bike tour and the jumping jack experiment by pointing out that both activities involve physical exertion during a period of time. This experiment works nicely if students are divided into groups of four. Within the group, each student has a job: • performing jumping jacks • counting jumps out loud • calling time when 10 seconds have passed • recording the number of jumping jacks completed at the end of every 10 seconds for the 2-minute time period The directions suggest that students do jumping jacks for 2 minutes. Two minutes has worked well in many classes. We suggest that you tell students to talk to you if they are not physically able to do the experiment. Inform everyone that if they get tired, they should stop. Every student does not need to jump. Many students like to volunteer. This is a nice way to encourage participation in the classroom activities in the beginning of the year. Emphasize the following points: • The jumper performs a complete jumping jack when they complete these three steps: 1. Start with feet together and hands at sides. 2. Jump, landing with legs apart and hands touching above the head. 3. Jump again, returning to the starting position with feet together and hands at sides. LES CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 7 LES • The counter counts out loud, adding an additional jump to the total each time the jumper returns to the starting position. • The timer calls out “time” when each 10 seconds passes. • The recorder listens for the timer to call “time” and then writes the last number the counter called into the table. After the demonstration, give students copies of Learning Aid 1.1A: Jumping Jack Experiment Tables and a timer. Have students perform the experiment in groups of four and then complete Problem 1.1. Have as many students as possible take a turn at each task. Remind them that they need to count and record the total number of jumping jacks their teammates complete by the end of each time interval. After collecting and reflecting on their own jumping jack data, students match tables of data to descriptions of the relationships between variables from Learning Aid 1.1B: Matching Descriptions and Tables (Part 1, Descriptions) and Learning Aid 1.1B: Matching Descriptions and Tables (Part 2, Tables). You can assign one or two of the jumpers (Group cards) to each group of students in your class. (Problem-Solving Environment) Explore (Digging In) Providing for Individual Needs When students have collected their jumping jack data, have them examine their own data and compare it to the students in the What If . . . ? questions. Suggested Questions • What was the pattern of your group’s jumping jacks? (Answers will vary.) • How did that pattern look when the person was jumping? (Descriptions should include if the jumper sped up, slowed down, or kept a steady pace.) • How do you see that pattern in the table? (Descriptions should include changes in the way the number of total jumps changes. If the jumper sped up, there will be a larger increase in jumps. If the jumper slowed down, there will be less increase in jumps. If the jumper kept a steady pace, the increase in jumps is consistent.) • Have students read the descriptions of the six jumpers in Ms. Park’s class. Compared to Ms. Park’s students, does your data have the same relationship between the variables? (Answers will vary.) • What do you have to know to fill in the data for Lashawn’s group? (You need to consider the variables, how they are changing, and any patterns between the variables.) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

8 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs While this unit is focused on algebraic reasoning, the unit also informally begins proportional reasoning for grade 6. The vocabulary is first introduced to students in Problem 3.3. By looking at the pattern of relationships between two variables, some situations provide opportunities for students to intuitively develop understandings of ratio and unit rate. Students describe many of the patterns as “consistent” or “steady” when proportional reasoning is involved. You may hear phrases like “for every” or “each time.” The formal language and more explicit reasoning will be developed in later grades 6 and 7 units. It is important to encourage this reasoning. Suggested Questions • Who seemed to be the most consistent jumper? How do you determine a consistent jumper? (Lashawn was consistent. He did 8 jumps every 10 seconds. [Ratio Reasoning] Paula was consistent. She did 1 jump a second. [Unit Rate Reasoning]) Planning for the Summary Since this is the first problem of the school year, you may want to move to a class discussion to do the What If . . . ? questions. What evidence will you use in the summary to clarify and deepen understanding of the Now What Do You Know? NOW WHAT DO YOU KNOW? How does a table help you make sense of the relationship between two variables in a situation? (As you are circulating, listen for how students are describing the patterns in the data. How are they describing how the table shows when a jumper was jumping fast/slow, stopped, jumped consistently, and so on? Students should be talking about the change in the number of jumping jacks for each interval of 10 seconds.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies The scientific issue in this problem is how performance can change over time. The mathematical issue is how that performance pattern between the variables is shown by data expressed in tables. You can give focus to the Summarize discussion by asking students what their experiment told them about each issue. Suggested Questions • What was the pattern of the relationship between the variables of your group’s jumping jacks? (Answers will vary.) LES CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 9 LES • How did that pattern look when the person was jumping? (Descriptions should include if the jumper sped up, slowed down, or kept a steady pace.) • How do you see that pattern in the table? (Descriptions should include changes in the way the number of total jumps changes. If the jumper sped up, there will be a larger increase in jumps. If the jumper slowed down, there will be less increase in jumps. If the jumper kept a steady pace, the increase in jumps is consistent.) • The instructions told you to use 10-second intervals. Could you have chosen a different time interval for recording data in your table? (Yes.) • Would your choice have affected your table of data? If so, in what way? (For smaller intervals, the number of jumping jacks is lesser, and conversely, for larger intervals, the number of jumping jacks is greater. However, in either case, the number of jumps in a time interval tends to decrease over time.) • What does the jumping jack experiment suggest about bicycleriding speed over time? (Usually, the distance in a time interval decreases as time passes.) Making the Mathematics Explicit Help students visualize the patterns of change relationships by imagining someone jumping and how that would influence the data in the table. These questions can help students make sense of how the physical activity is represented in a table of data. (Portrayal) This can help students abstract information from a table without a physical representation in later problems. Suggested Questions • Compared to Ms. Park’s students, does your data have the same relationship between the variables? (Answers will vary. There may be some “consistent” jumpers in the classroom like Paula or Li Wei.) • Did anyone in the class have steady or consistent jumping? How do you determine a consistent jumper? (It may be someone from their group, someone from the class, or one of the characters in the What If . . . ? A consistent jumper has the same number of jumping jacks for every 10 seconds. Or some students might say the jumper did 1 jump for every 1 second.) • What would a consistent jumper’s table of data look like? What would you see if you watched them do the jumping jacks? (The table would increase by the same amount or almost the same amount for each 10-second interval. The jumper would move in a steady way.) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

10 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs • Can you be slow but still be consistent? (Yes.) What would the table of data look like? What would you see if you watched them do the jumping jacks? (The table still increases by the same amount or almost the same amount for each 10-second interval. However, the amount for each 10 seconds would be less. The jumper would move in a slower but still steady way.) • Were any of Ms. Park’s students consistent jumpers? How did their data compare? (Yes. Paula’s group and Li Wei’s groups were consistent. Paula did more jumps in a 10-second interval.) • Who was the least consistent? How do you know? What would the table of data look like? What would you see if you watched them do the jumping jacks? (It may be someone from their group, someone from the class, or one of the characters in the What If . . . ? The number of jumping jacks done in a 10-second interval will be very different each time. So how the table grows will vary. The number of jumps may increase a lot in 10 seconds, may increase only some, or may not increase at all. This jumper would be very inconsistent, sometimes going slow, going fast, maybe stopping, or doing any other varying jumping pattern. In Ms. Park’s class, Ana—Table 6 was the most inconsistent.) • What if someone stopped and started again? How would that change their table of data? What would you see if you watched them do the jumping jacks? (As time increases in the table, there would be no increase in the number of jumps when the jumper is stopped. When the jumper starts again, the number of jumps will begin to increase again as time continues. In Ms. Park’s class, Tori—Table 2 stopped, but she did not start again.) • What if a jumper started off really fast then got slower over the jumping time? What would the table of data look like? What would you see if you watched them do the jumping jacks? (For every 10 seconds, the table would increase quickly at the beginning, and the increase would be less and less as time continued. The student would jump quickly at first and slow down as time passes. In Ms. Park’s class, Sam—Table 3 started by her table increasing by many jumps in the beginning, about 15 jumps every 10 seconds. As she continued to jump, she did fewer jumps in an interval. At about 50 seconds, she was doing 5 jumps for every 10 seconds. Near the end of the 120 seconds, she was jumping about 2 jumps for every 10 seconds.) • What if a jumper started off slow then got really fast over the jumping time? What would the table of data look like? What would you see if you watched them do the jumping jacks? (For every 10 seconds, the table would increase slowly at the beginning, LES CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 11 LES and the increase would be more and more as time continued. The student would jump slowly at first and increase speed as time passes. In Ms. Park’s class, Jackson—Table 5 started a steady pace of about 5 jumps for every 10 seconds. By the end of the 120 seconds, he was doing about 12−13 jumps in 10 seconds.) Connecting the actions of the jumper to the change in data on the table will help students make sense of the patterns of change. (Portrayal) You may want to have student volunteers demonstrate the actions of a jumper and what the table of data will look like for those jumpers when you ask the previous questions. Briefly discuss filling in the data for Lashawn’s group in What If . . . ? Situation B. The pattern of change between the variables is consistent. This situation gives you the opportunity to informally discuss proportional relationships and ratios with your students. In Comparing Quantities, students will use “for every” statements to learn more formally about ratios. • What do you have to know to fill in the data for Lashawn’s group? (You need to consider the variables, how they are changing, and any patterns between the variables.) • What is the pattern of Lashawn’s data? (The jumper in the group did 8 jumping jacks for every 10 seconds.) • How can we use the pattern of 8 jumping jacks for every 10 seconds to find other amounts of jumping jacks that the group did? (We can keep counting 8 more jumps every time 10 more seconds pass.) Note that this informally builds on students’ understanding of multiples that will be further developed in the next unit, Number Connections: Expressing Factors and Multiples Algebraically. Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? questions. Since this is the first problem of the year, model thinking about the questions, reflecting back to the problem work, and answering the questions. You may want to have a class discussion, ask prompting questions, and write answers together. (Problem-Solving Environment) • How is this relationship between the two variables described in words? (At this time, many students will express this in 10-second intervals. For example, if their jumper slowed down over the 2 minutes, they might say: As we jumped a longer amount of time, the number of jumping jacks that we did in 10 seconds was less.) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

12 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs • How is this relationship between the two variables shown in a table? (Answer will vary depending how they jumped. For example, if their jumper slowed down over the 2 minutes, they might say: As the time increases, the number of jumping jacks increases, but it increases more slowly.) Implementation Note: Have students keep their data from the jumping jacks to use in Problem 1.2. REFLECTING ON STUDENT LEARNING Use the following questions to assess student understanding at the end of the lesson. • What evidence do I have that students understand the Now What Do You Know? question? • Where did my students get stuck? • What strategies did they use? • What breakthroughs did my students have today? • How will I use this to plan for tomorrow? For the next time I teach this lesson? • Where will I have the opportunity to reinforce these ideas as I continue through this unit? The next unit? LES CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.1 Organizing a Bike Tour Experiment: Variables and Tables 13 Answers Embedded in Student Edition Problems Organizing a Bike Tour Experiment: Variables and Tables Equipment › timer, such as a clock or smartphone › paper to record the results in a table Jumping Jack Experiment Time (seconds) 0 10 20 30 40 50 60 70 . . . Total Number of Jumping Jacks Directions There are four roles: › A jumper to do the jumping jacks › A timer to keep track of time in seconds › A counter to count the jumping jacks › A recorder to write down the number of jumping jacks in a table Collecting the Data › The timer says “go,” and the jumper begins jumping. › The jumper continues jumping for 2 minutes. › The counter counts the jumping jacks out loud. › Every 10 seconds, the timer says “time,” and the recorder records the number of jumping jacks the jumper has done. INITIAL CHALLENGE Suppose you and your classmates did jumping jacks as fast as possible for a 2-minute test period. Make a Prediction • How many jumping jacks do you think you could do in 2 minutes? Student predictions will vary. Conduct the Experiment PROBLEM Answers 1.1 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

14 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Analyze the Data • For the jumper in your group, how did the number of jumping jacks change as time passed? How is this shown in your table? Students’ experiment data will vary. One possible table: Jumping Jack Experiment Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 14 26 38 49 58 70 80 93 102 112 123 134 Student answers will vary. Many students make general statements like “The number of jumping jacks increased over time.” Some students may need help looking at how the number of jumping jacks increases from one cell on the table to the next one. For example, in the table shown here, we can see that the jumper did 12 jumping jacks between 10 and 20 seconds. You may need to model how to look specifically at the amount of change between adjacent cells. Students may discuss the experiment rather than the table. For example, students might say, “Our jumper was really fast” or “Our jumper got faster at the end.” Help them connect how that pattern shows up in the table. Make sure that students are discussing both variables (not just the number of jumping jacks) so they are looking at the pattern of change between the variables. • What does this pattern of jumping jacks per second suggest about how bike-riding speed would change over a day’s time on the bicycle tour? Student answers will vary. It seems likely that students will find that their rate of jumping jacks slows near the end of the experimental time period. Or their rate of jumping jacks might slow if the experiment lasted for a longer amount of time. This pattern of jumping jacks would suggest that the speed of riding a bicycle will also decrease over a day’s time. WHAT IF . . . ? Situation A. Matching Descriptions and Tables Ms. Park’s class collected some interesting jumping jack data. The following are descriptions and tables from several groups. These describe the story told by the pattern of change from their experiments. 1. Match each group’s description with the correct table. 2. Describe how you decided that the table matches the story about the variables. 1.1 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.1 Organizing a Bike Tour Experiment: Variables and Tables 15 Sam’s Group Sam started out really fast. She did lots of jumps in the first few seconds. As time went on, her number of jumps for every 10 seconds was less and less. She was almost not jumping at the end of the 120 seconds. Li Wei’s Group Li Wei kept a consistent pace. As time increased by 10 seconds, he did 6 more jumps for each time interval. Tori’s Group Tori started with consistent jumping. As time increased by 10 seconds, he did about 10 jumps. Near the end of the time, his shoe came untied. So he stopped jumping. Paula’s Group Paula’s jumping was very consistent. She did about 10 jumps in every 10 seconds. She was able to keep this pace for 2 minutes. Jackson’s Group Jackson started with a consistent pace. Then, as time went on, his total number of jumps grew more and more. Ana’s Group Ana had a consistent pace for the first 20 seconds. Then the pace slowed down, increased, slowed down, and finally in the last 30 seconds increased a lot. Answer: Table 1 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 10 20 30 40 50 59 69 80 90 100 110 120 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +9 +10 +11 +10 +10 +10 +10 Table 1—Paula’s Group The rates of jumping jacks for every 10 seconds: 10, 10, 10, 10, 10, 9, 10, 11, 10, 10, 10, 10. This rate change matches Paula because she kept the pace of 10 jumps per 10 seconds. Students may interpret that a steady pace means no variation in the number of jumping in an interval. They may need help in considering that 9−11 jumps every 10 seconds is considered a fairly steady pace. Paula’s Group Paula’s jumping was very consistent. She did about 10 jumps in every 10 seconds. She was able to keep this pace for 2 minutes. 1.1 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

16 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Table 2 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 10 20 30 40 50 59 69 80 80 80 80 80 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +10 +9 +10 +11 +0 +0 +0 +0 Table 2—Tori’s Group The rates of jumping jacks for every 10 seconds: 10, 10, 10, 10, 10, 9, 10, 11, 0, 0, 0, 0. This rate change matches Tori because he jumped 10 jumps per 10 seconds in the beginning then stopped at the end. Table 3 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 15 31 44 54 60 65 69 73 79 81 83 84 Table 3—Sam’s Group The rates of jumping jacks for every 10 seconds: 15, 16, 13, 10, 6, 5, 4, 4, 6, 2, 2, 1. This rate change matches Sam because her rate was constantly decreasing as time went on. Tori’s Group Tori started with consistent jumping. As time increased by 10 seconds, he did about 10 jumps. Near the end of the time, his shoe came untied. So he stopped jumping. Sam’s Group Sam started out really fast. She did lots of jumps in the first few seconds. As time went on, her number of jumps for every 10 seconds was less and less. She was almost not jumping at the end of the 120 seconds. 1.1 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.1 Organizing a Bike Tour Experiment: Variables and Tables 17 Table 4 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 7 13 19 25 31 37 43 49 55 61 67 73 Table 4—Li Wei’s Group The rates of jumping jacks for every 10 seconds: 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6. This rate change matches Li Wei because, except for the first 10 seconds, he did 6 jumps for each time interval. Li Wei’s Group Li Wei kept a consistent pace. As time increased by 10 seconds, he did 6 more jumps for each time interval. Table 5 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 5 10 15 20 26 32 39 48 58 68 80 93 Table 5—Jackson’s Group The rates of jumping jacks for every 10 seconds: 5, 5, 5, 5, 6, 6, 7, 9, 10, 10, 12, 13. This rate change matches Jackson because he started at a steady pace and then the number went up more and more. Jackson’s Group Jackson started with a consistent pace. Then, as time went on, his total number of jumps grew more and more. 1.1 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

18 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Table 6 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 12 24 30 35 48 59 62 65 68 80 93 107 Table 6—Ana’s Group The rates of jumping jacks for every 10 seconds: 12, 12, 6, 5, 13, 11, 3, 3, 3, 12, 13, 14. This rate change matches Ana because she started out at a steady pace for the first 20 seconds, then the number decreased, then increased, and decreased again, and then she finished strong the last 30 seconds. Situation B. Lashawn’s Group The jumper in Lashawn’s group did 8 jumping jacks for every 10 seconds. They used a table to represent the relationship between time and total number of jumping jacks. The lunch bell rang before they finished filling in the table. 1. Fill in the missing entries in the table for the first 60 seconds. How did you decide which numbers to use? 2. How does the relationship in this table compare to those in Situation A? Answers: Total (seconds) 0 10 20 30 40 50 60 Total Number of Jumping Jacks 0 8 16 24 32 40 48 +8 +8 +8 +8 +8 +8 +10 +10 +10 +10 +10 +10 Ana’s Group Ana had a consistent pace for the first 20 seconds. Then the pace slowed down, increased, slowed down, and finally in the last 30 seconds increased a lot. Since the jumper did 8 jumping jacks for every 10 seconds, the total number of jumping jacks for each 10-second interval in the table is 8 more than the previous interval. To get the missing entries, you can add 8 to the number in the previous interval. 1.1 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.1 Organizing a Bike Tour Experiment: Variables and Tables 19 This jumper was consistent in terms of the rate of jumping jacks. Every 10 seconds, he jumped 8 times. Because the total number of jumping jacks is consistently increasing, this is like Paula’s group, who did 10 jumps every 10 seconds, and like Li Wei’s group, who did about 6 jumps every 10 seconds. Note: This is an informal introduction to proportional relationships. Students will look at proportional relationships formally in a future unit, Comparing Quantities. NOW WHAT DO YOU KNOW? How does a table help you make sense of the relationship between two variables in a situation? Possible student answers at this time: Tables help you to quickly notice differences or changes in the number of jumping jacks for each 10 seconds (rates). The variables in the relationship are named in the first cells of the table. So you can quickly see what the variables are. You can see if the relationship is consistent or not by looking at change in the number of jumping jacks for each interval of 10 seconds. 1.1 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

20 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Blank Tables and Graph Name  Date  Class TEMPLATE LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.1 Organizing a Bike Tour Experiment: Variables and Tables 21 Experiment 1 Jumper: _____________________________ Jumping Jack Experiment Tables Jumping Jack Experiment Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks Experiment 2 Jumper: _____________________________ WHAT IF . . . ?, SITUATION B Experiment Jumper: _____________________________ Jumping Jack Experiment Time (seconds) 0 10 20 30 40 50 60 … Total Number of Jumping Jacks 0 8 16 Jumping Jack Experiment Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks Lashawn Name  Date  Class 1.1A LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

22 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Name  Date  Class Sam’s Group Sam started out really fast. She did lots of jumps in the first few seconds. As time went on, her number of jumps for every 10 seconds was less and less. She was almost not jumping at the end of the 120 seconds. Tori’s Group Tori started with consistent jumping. As time increased by 10 seconds, he did about 10 jumps. Near the end of the time, his shoe came untied. So he stopped jumping. Paula’s Group Paula’s jumping was very consistent. She did about 10 jumps in every 10 seconds. She was able to keep this pace for 2 minutes. Jackson’s Group Jackson started with a consistent pace. Then, as time went on, his total number of jumps grew more and more. Li Wei’s Group Li Wei kept a consistent pace. As time increased by 10 seconds, he did 6 more jumps for each time interval. Ana’s Group Ana had a consistent pace for the first 20 seconds. Then the pace slowed down, increased, slowed down, and finally in the last 30 seconds increased a lot. Matching Descriptions and Tables 1.1B (Part 1, Descriptions) LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.1 Organizing a Bike Tour Experiment: Variables and Tables 23 Name  Date  Class Table 1 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 10 20 30 40 50 59 69 80 90 100 110 120 Table 2 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 10 20 30 40 50 59 69 80 80 80 80 80 Table 3 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 15 31 44 54 60 65 69 73 79 81 83 84 Table 4 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 7 13 19 25 31 37 43 49 55 61 67 73 Table 5 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 5 10 15 20 26 32 39 48 58 68 80 93 Table 6 Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 12 24 30 35 48 59 62 65 68 80 93 107 Matching Descriptions and Tables (Part 2, Tables) LEARNING AID 1.1B © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

24 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs PROBLEM 1.2 At a Glance In this problem, students graph their real-world data to represent the relationships between time and number of jumping jacks from the experiment in Problem 1.1. This problem continues the discussion of patterns that describe the relationship between two variables. In the Initial Challenge, students will use the experiment data from Problem 1.1 to create a coordinate graph of their data. In the What If . . . ? situations, students will match graphs with the tables and descriptions from Problem 1.1 and look at patterns from another student’s table and graph of jumping jack data. Note: If you have a Grade 6 Classroom Materials Kit, please refer to A Guide to Connected Mathematics® 4 for a detailed list of materials included or items you will need to prepare ahead of time. *not included in Classroom Materials Kit For more on the Teacher Moves listed here, refer to the General Pedagogical Strategies and the Attending to Individual Learning Needs Framework in A Guide to Connected Mathematics® 4. Organizing a Bike Tour: Variables, Tables, and Graphs NOW WHAT DO YOU KNOW? Describe how the pattern of change relationship between the variables—time and number of jumping jacks—is represented in a table and in a graph. Key Terms Materials coordinate graph axes horizontal axis x-axis vertical axis y-axis scale coordinate pair or ordered pair For each student • Learning Aid 1.2A: Graphing the Jumping Jack Experiment For each pair of students • Learning Aid Template: 1 Centimeter Graph Paper* • Learning Aid 1.2B: Matching Graphs to Jumping Tables and Descriptions • tape or glue For the class • Teaching Aid 1.2A: 4-by-4 Board • Teaching Aid 1.2B: Making a Graph • Teaching Aid 1.2C: Summary Discussion Graphs • Teaching Aid 1.2D: Ms. Park’s Final Matches Pacing 1 day Groups Think, Pair, Share A 3–4 C 14 E 19 Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE If your students need to review graphing, you might introduce them to Four-in-Row, a tic-tac-toe type game on a 4-by-4 grid. The winner is the person who gets four in a row first (horizontally, vertically, or diagonally). PRESENTING THE CHALLENGE Discuss how a set of axes is created. Read and discuss the four steps in the student book that review how to make a coordinate graph. Have students begin the problem on their own and then work with a partner as needed. Think, Pair, Share Arc of Learning™ Introduction Exploration CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 25 Facilitating Discourse Teacher Moves EXPLORE PROVIDING FOR INDIVIDUAL NEEDS If students forget which axis represents time, you can show the variables with color. Have them use one color for the time and the x-axis. Have them use another color for the number of jumps and the y-axis. Suggested Questions • What was the pattern of your group’s jumping jacks? • How did that pattern look when the person was jumping? • How do you see that pattern in the graph? • How does your group’s jumping data compare to Ms. Park’s students? • How does the relationship between the variables compare for the jumpers represented in Graph 3 and Graph 4? • What would a graph of a consistent jumper’s data look like? PLANNING FOR THE SUMMARY As you circulate, look for examples of how students have created graphs to represent their tables of jumping jack data that you can use during the summary discussion. Listen for the way students are describing the patterns between the variables represented on the graphs. Are students noticing how the steepness of a graph represents when the jumping jacks were faster or slower? Take note of the informal language that students are using to describe the way the graphs look. Students should begin discussing that by checking the difference between two entries on a table, you can find the rate. In a graph, you can see the rate by checking the gaps between two points. Implementation Note: For What If . . . ? Situation B, assign groups a graph from What If . . . ? Situation A, and have them pick two or three points on it to use for the situation. Selecting and Sequencing SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Have students review the process of making a graph to represent data. Suggested Questions • What are important things to think about when you are making a coordinate graph? • Why is a graph a helpful representation? Pick a point on one of the graphs, and ask: • What are its coordinates? • What information do the coordinates provide? MAKING THE MATHEMATICS EXPLICIT Suggested Questions • What was the pattern of your group’s jumping jacks? • How did that pattern look when the person was jumping? • Who seemed to be the most consistent jumper? How do you determine a consistent jumper? • Can you be slow but still be consistent? What would the graph of data look like? What would you see if you watched them do the jumping jacks? • How do you see that pattern in the graph? As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question. Portrayal 1.2 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

26 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Problem Overview In this problem, students graph their real-world data to represent the relationships between time and number of jumping jacks from the experiment in Problem 1.1. This problem continues the discussion of patterns that describe the relationship between two variables. Students will use the experiment data from Problem 1.1 to describe the way the number of jumping jacks changes as time increases in 10-second intervals. The connection between the representations of physical movement, the data in the table, and the visuals of the graph helps students interpret the data. (Portrayal) Students will probably have limited experience with coordinate graphing. The focus in this unit is around students understanding and interpreting what a graph is communicating about the data. Over the next three years, they will have many opportunities to plot points and make graphs. Launch (Getting Started) Connecting to Prior Knowledge Optional Review of Graphing. (Language) If your students need to review graphing, you might introduce them to Four-in-Row, a tic-tactoe type game on a 4-by-4 grid. The winner is the person who gets four in a row first (horizontally, vertically, or diagonally). The players take turns telling you two numbers that designate the location of the intersection point for their X or their O. This is different from the traditional game where the Xs and Os are placed in the middle of each square. With little instruction, nearly every student will recall how to graph points in a hurry. Display Teaching Aid 1.2A: 4-by-4 Game Grid or draw five horizontal and five vertical lines, equally spaced. EXTENDED LAUNCH—EXPLORE—SUMMARIZE 1 2 3 4 0 1 2 3 4 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 27 LES If you choose not to do this activity and students have not done graphing it is well worth a brief mention that coordinates correspond to intersections of lines, not squares (regions). Many students believe that the location corresponds to the space instead of the intersection. (2, 3) is here. 1 2 3 4 0 1 2 3 4 (2, 3) is NOT here. 1 2 3 4 0 1 2 3 4 You will need to model this as you play the game. Suggested Questions • How many of you know how to play tic-tac-toe? (Answers will vary. Some students may have played related target games where you give two “directions,” such as B-6, to find a location in rows and columns.) • How many do you need in a row to win? (Three.) Explain that today we will play a different version of tic-tac-toe. You will need four in a row to win. We’ll play the left side of the class against the right side of the class. The left side can go first and tell me two numbers. Have the first player tell you two numbers. If the numbers are between 0 and 4, they will be on the board. Otherwise, they will be off the board. For example, if a student says (5, 1), start at 0 and count until you get to 4 and say, “Oops, they fell off the board!” Then let the other team have a turn. The students quickly learn to use the correct numbers, and they quickly recognize that the order of the two numbers is important. In future units, you can change the number scheme by moving the origin to a different place in the grid and by using negatives, fractions, and so on. 0 y x –2 –1 1 2 –2 –1 1 2 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

28 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs LES Presenting the Challenge Discuss how a set of axes is created. A horizontal number line and a vertical number line are superimposed on top of each other at (0, 0), or the origin. If possible, place a transparent vertical number line over a horizontal number line. (Portrayal) Understanding that the axes are number lines may help students remember to set up the scales on the axes counting in equal increments that increase as you move to the right on the horizontal axis or increase as you move up the vertical axis. Suggested Questions • Can you remember seeing number lines in your elementary classrooms? What did they look like? (They counted by ones. They were across the wall. Students usually make a horizontal motion with their hand.) Why are there arrows at the ends of the line? (That is to show that the number line goes on and on. It does not stop there.) 0 1 2 3 4 5 6 7 8 9 10 11 We are going to call that number line the x-axis. It is the horizontal axis of two that we are going to talk about. (Language) You may want to ask students to describe looking at the horizon to connect to the word horizontal. • Have you ever seen number lines that are not horizontal? (Students may mention thermometers or other vertical number lines that they may have seen. If they have not seen a vertical axis, introduce it as a vertical number line.) 11 10 9 8 7 6 5 4 3 2 1 0 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 29 LES We can overlay the vertical number line on the horizontal number line at the zeros (called the origin) and make a set of axes. We can use these axes to relate two variables. The axes are perpendicular, or meet at a 90°angle. The horizontal axis is called the x-axis. The vertical axis is called the y-axis. 0 1 2 3 4 5 6 7 8 9 10 11 11 10 9 8 7 6 5 4 3 2 1 x-axis y-axis Origin is at (0, 0), where the number lines connect to make axes. 1 2 3 4 5 6 7 8 9 10 11 11 10 9 8 7 6 5 4 3 2 1 x-axis y-axis 0 There is a z-axis, too. But you will not study that right now. The z-axis also goes through the origin and lets us look at three variables at the same time. It is also perpendicular to the other two axes. 0 1 2 3 4 5 6 7 8 9 10 11 11 10 9 8 7 6 5 4 3 2 1 x-axis y-axis z-axis The z-axis would stick-out of the board or paper to make 3-dimensional graphs. Today you are going to plot the data that you collected in the jumping jack experiment. You will look at how the patterns between the variables is represented in a graph. Read and discuss the four steps in the student book and use Teaching Aid 1.2B: Making a Graph, which reviews how to make CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

30 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs LES a coordinate graph. Have students begin the problem on their own and then work with a partner as needed. Distribute Learning Aid 1.2A: Graphing the Jumping Jack Experiment and Learning Aid 1.2B: Matching Graphs to Jumping Tables and Descriptions for students to use as they work through the problem. Explore (Digging In) Providing for Individual Needs If students forget which axis represents time, you can show the variables with color. (Portrayal) Have them use one color for the time and the x-axis. Have them use another color for the number of jumps and the y-axis. Time (seconds) Number of Jumping Jacks x-axis y-axis When student have put their jumping jack data on a graph, have them examine their own data and compare it to the students in the What If . . . ? questions. Suggested Questions • What was the pattern of your group’s jumping jacks? (Answers will vary.) • How did that pattern look when the person was jumping? (Descriptions should include if the jumper sped up, slowed down, or kept a steady pace.) • How do you see that pattern in the graph? (Descriptions should include changes in the way the total number of jumps looks different. If the jumper sped up, there will be a bigger increase Jumping Jack Experiment Time (seconds) 0 10 15 20 25 30 35 40 45 50 55 60 Total Number of Jumping Jacks 0 8 12 16 20 24 28 42 36 40 44 48 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 31 LES between the points. Some students will even say that the angle is greater or the distance is greater between the points. When the jumper slowed down, there will be less distance between the points. With a steady pace, the points will line up.) • How does your group’s jumping data compare to Ms. Park’s students? (Answers will vary.) • Does your data seem to have the same pattern as any of Ms. Park’s students? (Answers will vary.) • If we looked at your data and one of Ms. Park’s students on the same graph, how would they compare? (Answers will vary with their own data.) Look at the graphs made with different scales. Suggested Questions • How does the relationship between the variables compare for the jumpers represented in Graph 3 and Graph 4? (It looks like the jumper in Graph 3 may have done more jumps. But this is not true. The ways the y-axes count are different. If we look at one point, we can see that the jumper in Graph 4 did more jumps. At 120 seconds, the jumper in Graph 4 had 120 jumps. If we put the data from the two groups on one graph, we would see that the jumpers were both consistent but that one jumped faster than the other.) While this unit is focused on algebraic reasoning, the unit also informally begins proportional reasoning. By looking at the pattern of relationships between two variables, some situations provide opportunities for students to intuitively develop understandings of ratio and unit rate. Students describe many of the patterns as “consistent” or “steady” when proportional reasoning is involved. You may hear phrases like “for every” or “each time.” The formal language and more explicit reasoning will be developed in later grades 6 and 7 units. It is important to encourage this reasoning. • What would a graph of a consistent jumper data look like? (The graphs of Lashawn’s and Paula’s data make a straight line because they jumped a steady pace. [Ratio Reasoning]) • What if their data were not steady? (The points would not line up. Slowing down or speeding up makes the points not line up. [Ratio Reasoning]) Implementation Note: For What If . . . ? Situation B, assign groups a graph from What If . . . ? Situation A, and have them pick two or three points on it to use for the situation. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

32 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs LES Planning for the Summary What evidence will you use in the summary to clarify and deepen understanding of the Now What Do You Know? question? NOW WHAT DO YOU KNOW? Describe how the pattern of change relationship between the two variables—time and number of jumping jacks—is represented in a table and in a graph. As you circulate, look for examples of how students have created graphs to represent their tables of jumping jack data that you can use during the summary discussion. Listen for the way students are describing the patterns between the variables represented on the graphs. Are students noticing how the steepness of a graph represents when the jumping jacks were faster or slower? Are students noticing that a line represents when jumping jacks were consistent? Are students noticing that a horizontal line represents when a student stopped jumping? Take note of the informal language that students are using to describe the way the graphs look. Students should begin discussing that by checking the difference between two entries on a table, you can find the rate. In a graph, you can see the rate by checking the gaps between two points. Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies Have students review the process of making a graph to represent data. Suggested Questions • What are important things to think about when you are making a coordinate graph? (A few answers may include: “Identify the variables.” “The axes have to count like number lines.” “Pick a scale so your data will be on the graph.” “Label the axes with the variable name.” “Plot points in the correct order.”) • Why is a graph a helpful representation? (It gives a visual for the relationship between the variables.) Pick a point on one of the graphs, and ask: • What are its coordinates? (Answers will vary.) • What information do the coordinates provide? (The coordinates tell the number of jumping jacks for a given 10-second interval. For example, the last point on Ana’s graph is (120, 107). We can find this point on the graph and see the values in the table. It tells us that at 120 seconds, Ana had completed 107 jumping jacks.) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 33 LES • Where on the corresponding table is this information? (Answers will vary depending on the point chosen.) To elicit a conversation about how the scales impact the look of the data, Graphs 3 and 4 were intentionally graphed with different scales. It is important that students experience how the look of the graph can give information. However, to interpret the information correctly, we must pay attention to the values of the coordinate pairs. Changing the scales on a graph can impact the look of the data, but it will not change the relationship between the variables. Teaching Aid 1.2C: Summary Discussion Graphs can assist you with this conversation. • How does the relationship between the variables compare for the jumpers represented in Graph 3 and Graph 4? (It looks like the jumper in Graph 3 may have done more jumps. But this is not true. The ways the y-axes count are different. If we look at one point, we can see that the jumper in Graph 4 did more jumps. At 120 seconds, the jumper in Graph 4 had 120 jumps. If we put the data from the two groups on one graph, we would see that the jumpers were both consistent but that one jumped faster than the other.) Display the data on the same graph so students can see how the jumpers compare on the same coordinate graph. For the students who are really thinking about how the relationship compares for the consistent jumpers have them look at adding Lashawn’s data to the graph. (These are students who are thinking proportionally about rates.) Time (seconds) Total Number of Jumping Jacks 10 20 30 40 50 60 70 80 90 100 110 120 0 10 20 30 40 50 60 70 80 y-axis x-axis Li Wei’s data Paula’s data 90 100 110 120 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

34 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs LES Help students read the data points on Lashawn’s graph and relate them to the relationships with the variables. • Which point on the graph represents 32 jumping jacks after 40 seconds? (Students will have to point to the dot on Lashawn’s graph at (40, 32). A student can track a vertical path from the 40 on the x-axis, hit the point, and then track a horizontal path to hit the y-axis at about 32.) • How can you use a graph to find the time it takes to do 20 jumping jacks? (Find 20 on the vertical axis that represents the jumps, track horizontally until you hit the point on Lashawn’s graph or approximately where that point would be, then track down from there to the x-axis to see it would be at about 25 seconds.) • How many jumping jacks were done at the end of 45 seconds? (From 45 on the x-axis, track vertically until you hit the point on Lashawn’s graph or approximately where that point would be, then track horizontally to the y-axis to see that it is about 36 jumps.) When looking at the questions for Lashawn’s data in Situation B, have students focus on the advantage of using the table and graph to answer questions about the data. • Which representation, a table or a graph, was more useful to answer the questions? (Student answers will vary based on the question being answered and/or their preferred representation to use.) Time (seconds) Total Number of Jumping Jacks 10 20 30 40 50 60 70 80 90 100 110 120 0 10 20 30 40 50 60 70 80 y-axis x-axis Li Wei’s data Lashawn’s data Paula’s data 90 100110 120 • What would Lashawn’s data look like on this graph? (Lashawn’s graph would line up in between Paula’s and Le Wei’s data.) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 35 LES Note on the Mathematics: When you are using the graph to approximate data in between the data points, this is called interpolation. It is not important for students to know this mathematical vocabulary at this time. Making the Mathematics Explicit Help students visualize the patterns of change by analyzing their own data. Suggested Questions • What was the pattern of your group’s jumping jacks? (Answers will vary.) • How did that pattern look when the person was jumping? (Descriptions should include if the jumper sped up, slowed down, or kept a steady pace.) • How do you see that pattern in the graph? (Descriptions should include changes in the way the total number of jumps looks different. If the jumper sped up, there will be a bigger increase between the points. Some students will even say that the angle is greater between the points. When the jumper slowed down, there will be less distance between the points. With a steady pace, the points will line up.) Help students visualize the patterns of change by imagining someone jumping and how that would influence the data in the graph. These questions can help students make sense of how the physical activity is represented in a graph of data. This can help students abstract information from a graph without a physical representation in later problems. (Portrayal) Use Teaching Aid 1.2D: Ms. Park’s Class Data for the description, table, and graphs to display as needed to help with the summary conversation. • Do you have any what if questions? • Who seemed to be the most consistent jumper? How do you determine a consistent jumper? (It may be someone from their group, someone from the class, or one of the characters in the What If . . . ? Paula and Li Wei were consistent. Tori was consistent until 80 seconds. A consistent jumper has the same number of jumping jacks for every consistent number of seconds. Or some students might say the jumper did some number of jumps for every 1 second. Here students are thinking informally about unit rates, which they will explicitly learn about in Comparing Quantities.) • What would a graph of a consistent jumper’s data look like? What would you see if you watched them do the jumping jacks? CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

36 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs LES (The graph would increase in a steady or consistent way. Students may say that there are no bumps in the points or that the points line up. The jumper would move in a steady way.) • Can you be slow but still be consistent? (Yes, like Li Wei with Graph 3.) What would the graph of data look like? What would you see if you watched them do the jumping jacks? (The graph still increases in a steady pattern. However, because the amount for each 10 seconds is less, the points of the graph do not go as high. The jumper would move in a slower but still steady way.) • Who was the least consistent? How do you know? What would the graph of data look like? What would you see if you watched them do the jumping jacks? (It may be someone from their group, someone from the class, or Graph 1 of Ana’s data in the What If . . . ? The number of jumping jacks done in a 10-second interval will be very different each time. Students often describe the graph as “bumpy.” From one point to the next, the points may be close, far apart, or somewhere in between. This jumper would be very inconsistent, sometimes going slow, going fast, maybe stopping, or doing any other varying jumping pattern.) • What if someone stopped and started again? How would that change the graph of data? What would you see if you watched them do the jumping jacks? (As time increases on the x-axis, there would be no increase in the number of jumps when the jumper is stopped. This would create a flat line segment on the graph. When the jumper starts again, the number of jumps will begin to increase, showing a slant up on the graph as time continues.) Connecting the actions of the jumper to the change in data on the table will help students make sense of the patterns of change to students. (Portrayal) You may want to have student volunteers demonstrate the actions of a jumper and what the table of data will look like for those jumpers when you ask the previous questions. • What if a jumper started off really fast then got slower over the jumping time? What would the graph of data look like? What would you see if you watched them do the jumping jacks? (For every 10 seconds, the graph would increase quickly at the beginning, and the increase would be less and less as time continued. This would make the points closer together or not getting as high. The student would jump quickly at first and slow down as time passes.) • What if a jumper started off slow then got really fast over the jumping time? What would the graph of data look like? What would you see if you watched them do the jumping jacks? (For every 10 seconds, the graph would increase slowly at the CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Extended Launch—Explore—Summarize 37 LES beginning, and the increase would be more and more as time continued. This would create a curve look with a slow or flatter start, and then it would curve up. The student would jump slowly at first and increase speed as time passes.) Check how students are interpreting the information from a point on the coordinate graph. See if students are looking to make sense what the values mean in terms of the two variables. Use the graphic from Teaching Aid 1.2C: Summary Discussion Graphs. What does the point (60, 30) tell us? (This point does not make sense. At 50 seconds, the jumper had almost 50 jumps. Since we are looking at the total number of jumps, as time increases the jumps must stay the same or increase. We cannot take jumps away.) Time (seconds) Total Number of Jumping Jacks 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 90 y-axis x-axis In the next investigation, students will look at relationships between variables in which changes in the value of one quantity cause or are caused by changes in the value of another correlated quantity. The words independent and dependent are more meaningful to describe variables in those contexts. So independent and dependent are not introduced until that point. Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? questions. Since this is the first unit of the year, model thinking about the questions, reflecting back to the problem work, and answering the questions. (Problem-Solving Environment) You may want to have a class discussion, ask prompting questions, and write answers together. • How is the pattern of change relationship between the two variables, time and number of jumping jacks, shown in a table? (As time increased, the jumps increased or stayed the same. A table shows the numbers of seconds and jumps from the experiment.) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

38 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs LES Shown in a graph? (A graph has the same patterns. With a graph, you can see a picture of the change in jumps as time increased. Sometimes with the graph you have to estimate the exact value of jumps because the dots are big or in between grid lines.) • How does the assignment of numbers on the axis number line affect the patterns of change? (We still have the same pattern of change. But if you change the numbering of the axis, the pattern might look different. So you have to look closely at the coordinate pair values. An example was Graph 3 and Graph 4. Graph 3 makes it look like Li Wei did more jumps than Paula (Graph 4). In Graph 3, the y-axis counted by 5, which spread out the look of the data. In Graph 4, the y-axis counted by 10, which compressed the look of the data together. This is similar to zooming in or zooming out on a picture to see it closer or farther away.) REFLECTING ON STUDENT LEARNING Use the following questions to assess student understanding at the end of the lesson. • What evidence do I have that students understand the Now What Do You Know? question? • Where did my students get stuck? • What strategies did they use? • What breakthroughs did my students have today? • How will I use this to plan for tomorrow? For the next time I teach this lesson? • Where will I have the opportunity to reinforce these ideas as I continue through this unit? The next unit? CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 39 Answers Embedded in Student Edition Problems INITIAL CHALLENGE Ms. Park’s students were curious about what information they might get from a graph of their jumping jack data. They labeled a grid to graph their data. Organizing a Bike Tour: Variables, Tables, and Graphs Time (seconds) Total Number of Jumping Jacks y-axis x-axis Jumping Jacks Over Time 20 40 60 80 100 200 Time (seconds) Jumping Jack Data Number of Jumping Jacks 40 60 80 100 120 x-axis y-axis • Make a graph of the jumping jack data for your group’s jumper. Student data will vary. One possible graph might look like this: • Describe the relationship between the number of jumping jacks the jumper did over time that is represented in the graph. The graph shows that the total number of jumping jacks increases over time and that for most intervals, the number of jumps in a 10-second interval decreases over time. PROBLEM 1.2 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

40 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Be sure students understand the graph shows this because the vertical distance between adjacent points becomes smaller as time passes. This may be hard for students now, but they will see the same concept in future problems, when they look at how distance changes over time. Students may talk about the “steepness” or “how high up” it is from one point to another. When there is a greater increase, the line between consecutive points will “be steeper, be longer, and have a larger angle.” • How does this relationship compare to the relationship displayed in your table? The graph shows that the total number of jumping jacks tends to increase as times goes on. The rate of jumping jacks in each time interval is the distance of two data points. On a graph, you can see that the greater rate is shown by bigger steps upward from one data point to the next. Students might note that they can see the exact numbers in the table and therefore more easily see that the jumper is slowing down. Other students might prefer the visual image of the graph. The graph gives the whole picture of the data in a glance so that changes can easily be seen. • How does your graph compare to other graphs in your class? Graphs will vary depending on the experiment. Students might find that the patterns tend to increase as time passes, even if different groups have different rates of jumping jacks. (All or most graphs will have a general increasing pattern.) By looking at other groups’ graphs, students can imagine the stories of other groups’ experiments like they matched descriptions with tables in Problem 1.1, What If . . . ? WHAT IF . . . ? Situation A. Matching Stories, Tables, and Graphs Ms. Park’s students made the following graphs from their tables and descriptions of their data in Problem 1.1. • Match the graph to the relationship shown in the table and description for each group. Explain how you decided each match. You can determine the data point using pairs of time and number of jumping jacks in the table and match them to points on the graph. Some patterns might help. For example, Tori stopped to tie her shoe, so her graph stops increasing. Li Wei and Paula were consistent jumpers, so their points form something like a line. Sam slowed down, so the graph starts to level off or not increase as much. See Teaching Aid 1.2C for the Match of the Descriptions, Tables, and Graphs. 1.2 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 41 Graph 1 Time (seconds) 10 20 30 40 50 60 70 80 90 100 110 0 10 20 30 40 50 60 70 80 90 y-axis x-axis Jumping Jacks Over Time Total Number of Jumping Jacks 100 110 120 Graph 1: Table 6 and Ana description. Graph 2 Time (seconds) 10 20 30 40 50 60 70 80 90 100 110 120 0 10 20 30 40 50 60 70 80 90 y-axis x-axis Jumping Jacks Over Time Total Number of Jumping Jacks 100 110 120 Graph 2: Table 3 and Sam description. 1.2 Answers Graph 3 Time (seconds) 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 0 10 20 30 40 50 60 70 80 90 y-axis x-axis Jumping Jacks Over Time Total Number of Jumping Jacks 100 110 120 Graph 3: Table 4 and Li Wei description. Graph 4 Time (seconds) 10 20 30 40 50 60 70 80 90 100 110 120 0 10 20 30 40 50 60 70 80 90 y-axis x-axis Jumping Jacks Over Time Total Number of Jumping Jacks 100 110 120 Graph 4: Table 1 and Paula description. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

42 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Situation B. Making Sense of Coordinates on Tables and Graphs Pick a point on a graph from Situation A, and answer the following questions. 1. What information do the coordinates of the point represent? Answers will vary. Possible answer: In Graph 5, there is a point at (20, 10). This point represents that at 20 seconds (the variable on the x-axis), the jumper had done 10 jumping jacks (the variable on the y-axis). 2. Where on the corresponding table is this information? Answers will vary. Possible answer to match the coordinate pair (20, 10). If we look at the time of 20 seconds, we can see the jumps of 10. Time (seconds) 0 10 20 30 40 50 60 70 80 90 100 110 120 Total Number of Jumping Jacks 0 5 10 15 20 26 32 39 48 58 68 80 93 1.2 Answers Graph 5 Time (seconds) 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 0 10 20 30 40 50 60 70 80 90 y-axis x-axis Jumping Jacks Over Time Total Number of Jumping Jacks 100 110 120 Graph 5: Table 5 and Jackson description. Graph 6 Jumping Jacks Over Time Time (seconds) Total Number of Jumping Jacks 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 0 10 20 30 40 50 60 70 80 90100 110 120 y-axis x-axis Graph 6: Table 2 and Tori description. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE

Problem 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 43 Lashawn’s Suggestion I made a table for my data in Problem 1.1. I also think the graph could be used to answer questions about the jumping jack experiment. Lashawn’s Group Jumping Jack Experiment Time (seconds) 0 10 20 30 40 50 60 Total Number of Jumping Jacks 0 8 16 24 32 40 48 1. Make a graph of Lashawn’s data. Graphs can vary depending on the scales on x-axis and y-axis that students choose. One possible graph might include: 2. What patterns do you notice on the graph? How are they shown in the table? A plot of the points corresponding to (Time, Total Number of Jumping Jacks) numbers in the table would produce a consistent (linear) pattern with the points rising up as you move from left to right 8 jumps for every 10 seconds. With the consistent pattern, the points create a straight line. The points on the graph go up at the same “steepness” and are the same distance apart as you read the graph from left to right. 10 20 30 40 50 100 Time (seconds) Jumping Jacks Over Time Number of Jumping Jacks 20 30 40 50 60 x-axis y-axis 3. Which point on the graph represents 32 jumping jacks after 40 seconds? Marked as red circle on the graph. 10 20 30 40 50 100 Time (seconds) Jumping Jacks Over Time Number of Jumping Jacks 20 30 40 50 60 x-axis y-axis 1.2 Answers Situation C. Lashawn Uses a Graph CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE