44 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs 4. How can you use a graph to find the time it takes to do 20 jumping jacks? In the graph, the total number of jumping jacks is shown on the y-axis. You can point out the number 20 on the y-axis and draw a horizontal line from the scale of 20. But there is no data point that matches 20 jumping jacks. Since 20 is right in the middle of two numbers, 16 and 24, you can assume that the point that you are looking for might be also in the middle of these two points (20, 16) and (30, 24), which is (25, 20). So you may say it would take 25 seconds to do 20 jumping jacks. 10 20 30 40 50 100 Jumping Jacks Over Time Number of Jumping Jacks 20 30 40 50 60 x-axis y-axis (30, 24) (20, 16) 25 seconds Time (seconds) 5. How many jumping jacks were done at the end of 45 seconds? Explain your strategy. About 36 jumping jacks were done at the end of 45 seconds. You have two data points, (40, 32) and (50, 40). To get the data point (45, ?), you can assume the pattern is linear since the total number constantly increases by 8 as time increase by 10 seconds. Since 45 (seconds) is in the middle of 40 and 50, you can find that the number of jumping jacks for 45 seconds is in the middle of 32 and 40, which is 36. 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 (50, 40) (40, 32) About 36 jumps 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 45 6. Do you agree with Lashawn? Is it possible to use the table and the graph to answer questions about the experiment? Yes. You can answer questions from the graph. The coordinate pair of a point on the graph tells the value of the variables, which answers the questions. For example, if we have (40, 32), we look at the variable on the x-axis for the 40. So time is 40 seconds. We look at the variable on the y-axis for the 32. So the number of jumping jacks is 32. 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. In a table, you can find the rate of jumping jacks by checking the difference between two adjacent entries. In a graph, on the other hand, you can see the rate of jumping jacks by checking the gaps between two points. The greater the difference in the rates, the steeper the changes between two points you would see in the graph. The graphs might look different depending on the assignment of numbers on the axis even if they show the same data set. If you set short intervals for the y-axis, your graph might show steeper increase compared to other graphs that have greater intervals for the y-axis. The following graphs show the difference. The scale used to label the axes makes a difference in the way the graph looks, in particular how fast the points look like they are moving up. Two graphs can have different scales on the y-axis, and the patterns of data points will look different. 10 20 30 40 50 100 Time (seconds) Jumping Jacks Over Time 20 30 40 50 60 x-axis y-axis Number of Jumping Jacks 20 10 40 50 30 60 70 90 80 100 0 10 Time (seconds) Jumping Jacks Over Time 20 30 40 50 60 x-axis y-axis Number of Jumping Jacks 1.2 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
46 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs 4-by-4 Game Grid 1.2A TEACHING AID 1 2 3 4 0 1 2 3 4 © 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 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 47 Making a Graph 1.2B TEACHING AID Step 1. Identify the variables. In Problem 1.1, the two variables are time and the total number of jumping jacks. Step 2. Select an axis to represent each variable. Label the axis with the name of the variable. If time is a variable, you usually put it on horizontal axis, called the x-axis. The number of jumping jacks go on the vertical axis, called the y-axis. Step 3. Select a scale for each axis. For each axis, determine the least and greatest values to show on the axes. Then decide how to space the scale marks. This is the same as you do for a number line. Step 4. Plot the data points. Suppose that at 60 seconds, you had done 56 jumping jacks. To plot this information, start at 60 on the x-axis (time), and follow a line straight up. On the y-axis (number of jumping jacks), start at 56, and follow a line straight across. Make a point where the two lines intersect. This point indicates that in 60 seconds, you did 56 jumping jacks. You can describe this point with the ordered pair (60, 56), which is also called a coordinate pair. The first number in a coordinate pair is the x-coordinate, and the second number is the y-coordinate. Step 2 Select an axis to represent each variable. Total Number of Jumping Jacks Time (seconds) y-axis x-axis Jumping Jacks Over Time Jumping Jacks Over Time 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 Step 3 Select a scale for each axis. Jumping Jacks Over Time 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 Step 4 Plots the data points. © 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
48 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Summary Discussion Graphs 1.2C TEACHING AID Jumping Jacks Over Time 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 Time (seconds) Total Number of Jumping Jacks 10 20 30 40 50 60 70 80 90 100 110 120 0 y-axis x-axis Li Wei’s data Paula’s data Jumping Jacks Over Time 10 20 30 40 50 60 70 80 90 100 110 120 Jumping Jacks Over Time Time (seconds) Total Number of Jumping Jacks 10 20 30 40 50 60 70 80 90 100 110 120 0 y-axis x-axis Li Wei’s data Lashawn’s data Paula’s data 10 20 30 40 50 60 70 80 90 100 110 120 © 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 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 49 Ms. Park’s Class Data 1.2D TEACHING AID 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 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. 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 © 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
50 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs 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. 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 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 © 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 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 51 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. 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 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 © 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
52 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs 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. 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 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 © 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 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 53 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. 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 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 © 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
54 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Jackson’s Group Jackson started with a consistent pace. Then, as time went on, his total number of jumps grew more and more. 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 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 © 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 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 55 1 Centimeter Graph Paper 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
56 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Name Date Class Graphing the Jumping Jack Experiment 1.2A 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 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs 57 Name Date Class Matching Graphs to Jumping Tables and Descriptions LEARNING AID 1.2B 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 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 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 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 © 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
58 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Name Date Class 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 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 © 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 1.3 Atlantic City to Lewes to Chincoteague: Time, Rate,and Distance 59 At a Glance This problem asks students to analyze data presented in a table and graph. Students move between the table and graph representations to relationships in the data. In the Initial Challenge, students look at data from Day 1 of the bicycle trip in a table, create a graph for it, and analyze the ride. In the What If . . . ? Situations, students will analyze how connecting points on a coordinate graph in different ways will tell a different story. Arc of Learning™ Introduction Exploration NOW WHAT DO YOU KNOW? Describe how distance changes over time. How is this pattern of change shown in tables and graphs? Key Terms Materials 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 Case 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 Pacing 1 day Groups 3–4 students A 5–7 C 15–16 E 20–21 Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance 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 Remind students of the five college students who wanted to operate bicycle tours as a summer business. Also, remind them that the jumping jack experiment was done to think about how far the touring group might be able to ride bikes in one day. PRESENTING THE CHALLENGE Tell the class how Sidney, Liz, Celia, Malcolm, and Theo plan to test their tour idea. Use the map inset showing where the cyclists are traveling. Day 1 is from Atlantic City, NJ, to Lewes, DE. Day 2 is from Lewes, DE, to Chincoteague Island, VA. Have students look at the table of (time, distance) data collected on Day 1. Suggested Question • Looking at this table, what do you notice and wonder about the first day of the trip? Notice Wonder Agency, Identity, Ownership PROBLEM 1.3 (continued) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
60 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Facilitating Discourse Teacher Moves EXPLORE PROVIDING FOR INDIVIDUAL NEEDS It will be important to monitor that students are connecting how the table and graph are giving them the same information. Point to a specific point on the graph and on the table. Ask what the coordinates mean and what story it tells. PLANNING FOR THE SUMMARY Watch and listen for how students are describing the patterns of change in the table and graph. It is important that students begin to relate how both representations can “tell the same story.” SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Use copies of student work to focus student observations about patterns in the (time, distance) data. MAKING THE MATHEMATICS EXPLICIT Focus on the relationship between the variables to explain the pattern of change. Suggested Questions • How would you describe the patterns in their trip from Atlantic City to Cape May? When did they travel the fastest? Slowest? How do you know? • What are some reasons for the patterns of change in the trip? • How does the data from Day 2 compare to the data from Day 1? As students give reasons why they selected each graph in What If . . . ? Situation A, make sure they are attending to such issues as why there is a horizontal segment in (3) and (4) and what it means. As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). 1.3 (continued from page 59) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Extended Launch—Explore—Summarize 61 Problem Overview This problem asks students to analyze data presented in a table and graph. Students move between the table and graph representations to relationships in the data. Students think about how these changes show up in a table and in a graph. They also assess the information that is represented by coordinates on a graph and pairs of values on a table. As students gain experience in looking for patterns of change in data, they will develop the ability to ask relevant questions that guide them in their analysis. Launch (Getting Started) Connecting to Prior Knowledge Remind students of the five college students who wanted to operate bicycle tours as a summer business. Also, remind them that the jumping jack experiment was done to think about how far the touring group might be able to ride bikes in one day. Suggested Questions • How were the number of jumping jacks and time related? What does this tell us about how far a bike tour group might be able to ride in one day? (Answers will vary. Students might discuss endurance over time. Students might also discuss other variables that could impact the distance traveled, such as wind, temperature, hills, age of the riders, etc.) • How did we represent the jumping jack data in a table? Graph? (Answers will vary. See examples in the previous questions.) Presenting the Challenge Tell the class how Sidney, Liz, Celia, Malcolm, and Theo plan to test their tour idea. Use the map inset showing where the cyclists are traveling (a story that you might embellish with other pictures and information about the places that the cyclists will be traveling). Day 1 is from Atlantic City, NJ, to Lewes, DE. Day 2 is Lewes, DE, to Chincoteague Island, VA. Have students look at the table of (time, distance) data collected on Day 1. EXTENDED LAUNCH—EXPLORE—SUMMARIZE CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
62 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Suggested Questions • Looking at this table, what do you notice and wonder about the first day of the trip? (Agency, Identity, Ownership) (Answers will vary. Get a few general noticings and wonderings from students to help them access the questions.) Have students work in groups of three to four on this problem. Distribute Learning Aid 1.3A: Atlantic City to Lewes Graph, Learning Aid 1.3B: Matching Cases to Paths (Part 1, Paths), Learning Aid 1.3B: Matching Cases to Paths (Part 2, Cases), and Learning Aid 1.3C: Day 2 of the Trip for students to use on this problem to organize and record their thinking. Explore (Digging In) Providing for Individual Needs There are three tasks for students in this problem: 1. make sense of the patterns shown by a table and graph of the same data; 2. identify the information given by a pair of coordinates or table values; and 3. consider the information that is not shown by looking between points or a set of points. Connecting points on a graph can help us see patterns more clearly. It also helps us consider what is happening in the interval between two points. Different ways of connecting the given data points tell different stories about that happens between the points. It will be important to monitor that students are connecting how the table and graph are giving them the same information. Point to a specific point on the graph. Ask what the coordinates mean and what story it tells. For example, the point (3.5, 31) tells us that at 3 __1 2 hours, the travelers were at 31 miles. This would suggest that the cyclists lost distances, because they had been at 34 miles the half hour before. This could happen if they went back to retrieve a lost item. Point to a specific point on the table. Ask what the coordinates mean and what story it tells. For example, the point (3.5, 31) tells us that at 3 __1 2 hours, the travelers were at 31 miles. This would suggest that the cyclists lost distances, because they had been at 34 miles the half hour before. This could happen if they went back to retrieve a lost item. To help students relate the stories to the graphs, you can try two strategies. (Portrayal) LES CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Extended Launch—Explore—Summarize 63 LES 1. Create a mental table of values. Pick points on the graph, and ask students to explain how the time and distance relate. This helps students to analyze the data as specific values to make sense of what is happening. Pick one of the curves or line segments, point to a series of points, and ask: What are coordinates at this point? What are coordinates at this point? What are coordinates at this point? Etc. So how would you describe the pattern? Which student’s story in the situations seems to match? 2. Relate back to doing jumping jacks. How would you do jumping jacks to create a graph that looked like Exercise 1? Exercise 2? (Students can talk about fast jumper data rising quickly or when someone stopped jumping a flat line occurred on the graph.) When matching the paths with the statements in What If . . . ? Situation A, students may question why the first statement (Situation Celia) does not correspond to path (3) and the second statement (Situation Theo) does correspond to path (4). Emphasize that the horizontal segment in these paths means the distance is not increasing and therefore the cyclist is not moving at all. Students may ask, “Why does the line go on if the cyclist is not moving?” Point out that, even though the cyclist is not moving, time continues to pass. Therefore, the segment gives us information about how long the cyclist was stopped. You may want to ask students to describe another situation that matches one of the graphs. Planning for the Summary Implementation Note: Depending on the time, you may want to stop the Explore before students have looked at Day 2 of the trip in the What If . . . ? Situation B. You can discuss this as a class during the Summarize. What evidence will you use in the summary to clarify and deepen understanding of the Now What Do You Know? question? What will you do if you do not have evidence? NOW WHAT DO YOU KNOW? Describe how the distance changes over time. How is this pattern of change shown in tables and graphs? (Watch and listen for how students are describing the patterns of change in the table and graph. It is important that students begin to relate how both representations can “tell the same story.” In tables, you can see the change by finding the difference of distance for two CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
64 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs LES adjacent numbers. In graphs, you can see the change by finding the y-coordinates of two data points. In graphs, you can also see which data point jumps the steepest from the previous one.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies Use copies of student work to focus student observations about patterns in the (time, distance) data. You can use Teaching Aid 1.3A: Atlantic City to Cape May Table and Graph at the end as a final wrap-up of the problem. Invite students to share their observations of interesting patterns in the graph and what they mean about the cyclists’ progress. Suggested Questions • Which variable is displayed on the x-axis? Why? (Time; it is usually put on the x-axis so that the graph tells a story of how a variable changes over time as you read from left to right. Also, it is the independent variable and thus should go on the x-axis.) • What is the greatest value needed to show on the x-axis? (The greatest value needed is 5.0.) • What is the least value needed to show on the x-axis? (The least value needed is 0. So the x-axis needs to count from 0 to at least 5.) • Why is it reasonable to have the size of the interval for the x-axis be 0.5 hours? (The data were collected on half-hour intervals.) • Which variable is displayed on the y-axis? (The variable displayed on the y-axis is distance.) • What is the least value needed to show on the y-axis? (The least value needed is 0.) • What is the greatest value needed to show on the y-axis? (The greatest value needed is 45. So the y-axis needs to count from 0 to at least 45.) • What could be the size of the interval for the y-axis on this graph? (Answers will vary. Some students may say 5 miles.) • What are the coordinates of the third point on the graph? What information do the coordinates represent? (The coordinates are (1, 15); it means the cyclists traveled 15 miles in one hour.) • How far did the group travel over the length of the trip? (They traveled 45 miles in 5 hours.) Making the Mathematics Explicit Focus on the relationship between the variables to explain the pattern of change. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Extended Launch—Explore—Summarize 65 LES Suggested Questions • How would you describe the patterns in their trip from Atlantic City to Cape May? When did they travel the fastest? Slowest? How do you know? (Answers will vary. Students should mention when the bikers were progressing the fastest [like in the first __1 2 hour or from 3.5 to 4 hours] or slowest [like between 2 and 2.5 hours or between 4 and 4.5 hours] or when the progress decreased or went backward from 3 to 3.5 hours.) • What are some reasons for the patterns of change in the trip? (Students should mention variables such as terrain, wind speed, food and rest breaks, and temperature as possible explanations for the data changes.) If students do not use the word variable to describe the things that might have affected the cyclists’ speed and distance traveled, make sure you add this word to the conversation. Also, let the class ask questions of each presenter and decide whether the presenter’s story fits or goes beyond the data given. Compare the different descriptions given by presenters. Connect the progress of the tour group on Day 1 to Day 2: • How does the data from Day 2 compare to the data from Day 1? (Answers will vary. Students might mention: the variables are the same, the distance goes backward on both days, Day 2 was longer, and the bikers traveled a longer distance.) As students give reasons why they selected each graph in What If . . . ? Situation A, make sure they are attending to such issues as why there is a horizontal segment in (3) and (4) and what it means. When matching the paths with the statements, students may question why the first statement (Situation Celia) does not correspond to path (3) and the second statement (Situation Theo) could correspond to path (4). Emphasize that the horizontal segment in these paths means the distance is not increasing and therefore the cyclist is not moving at all. Students may ask, “Why does the line go on if the cyclist is not moving?” Point out that, even though the cyclist is not moving, time continues to pass. Therefore, the segment gives us information about how long the cyclist was stopped. You may want to ask students to describe another situation that matches one of the graphs or have them act out the situation. Labeling the variables or seeing connections on a graph can help some students. Teaching Aid 1.3B: Connecting Graph Points Examples can help with this discussion. Pick specific points on a connection, and discuss what that coordinate says about time and distance. (Language) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
66 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Look at connection (A) on the graph, and relate to a table of values to represent the relationship shown by the connection. Time (hours) Distance (miles) 10 20 30 40 50 0 1.0 2.0 3.0 4.0 5.0 Atlantic City to Cape May y-axis x-axis If we mark a point on the connection, what does that point tell us about the time and distance? A (Estimate of the values: at 4.6 hours, the distance is 42.5 miles.) • If we look at time continuing and mark another point on the connection, what does that point tell us about the time and distance? A (Estimate of the values: at 4.7 hours, the distance is 43.5 miles.) • If we look at time continuing and mark another point on the connection, what does that point tell us about the time and distance? A (Estimate of the values: at 4.8 hours, the distance is 44 miles.) • If we look at time continuing and mark another point on the connection, what does that point tell us about the time and distance? A (Estimate of the values: at 4.9 hours the distance is 44.5 miles.) • Does this connection increase slowly or quickly? (It increases quickly. For an increase of __1 10 of an hour, the distance increases by 2.5 miles, then 1 mile, then __1 2 mile, then __1 2 mile. So the biker slows down as the time continues.) LES A. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Extended Launch—Explore—Summarize 67 LES Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? question. 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
68 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Answers Embedded in Student Edition Problems INITIAL CHALLENGE Every half hour, Sidney records the distances the cyclists have traveled from Atlantic City in a table. Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance PROBLEM 1.3 The business partners examine Sidney’s (time, distance) data. They look for patterns to help them improve the Ocean Bike Tours route and schedule. Malcolm writes a report of the trip and incudes a graph to illustrate the story of their journey. Use the table or graph to answer the following questions. • Make a graph of the data. Time (hours) Distance (miles) 10 20 30 40 50 0 1.0 2.0 3.0 4.0 5.0 Atlantic City to Cape May y-axis x-axis Atlantic City to Cape May Time (hours) Distance (miles) 0 0 0.5 8 1.0 15 1.5 19 2.0 25 2.5 27 3.0 34 3.5 31 4.0 38 4.5 40 5.0 45 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Problem 1.3 Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance 69 • How far did the group travel in the day? How much time did it take them? They traveled about 51 miles for Day 1, and it took 5 hours. From hour 0 to 3.0, they traveled 34 miles. From 3.0 to 3.5, they traveled 3 miles. From 3.5 to 5.0, they traveled 14 miles. • What patterns do you see in the (time, distance) data? Students will have a variety of observations about the pattern of travel shown in the table and the graph. They will probably notice that the cyclists went faster in some time periods than others. For example, they might point out that the speed from hour 0.5 to 1.0 is the same as that from hour 2.5 to 3.0, which is faster than the speed from hour 1.0 to 1.5 or from hour 2.0 to 2.5. They might notice that there is a “dip” in the graph between hour 3.0 and hour 4.0 and wonder what this could mean. One possible explanation would be that the cyclists made a wrong turn and had to back to get on course. Or one of the cyclists might have dropped something and had to turn back to retrieve it. • Pick a point on the graph. What are its coordinates? What information do the coordinates represent about the situation? Where is this information shown in the table? See the table that follows. • Pick an entry in the table. What are its coordinates on the graph? What information does it represent about the situation? Answer will vary depending on which entry a student picks. For all entries, from left to right, possible answers are as follows. Entry Coordinates Information In the Table 1 (0, 0) They were at the starting point in Atlantic City. In the 1st row of the table 2 (0.5, 8) After 0.5 hours, they were 8 miles away from the starting point. 2nd row 3 (1.0, 15) After 1.0 hours, they were 15 miles away from the starting point. 3rd row 4 (1.5, 19) After 1.5 hours, they were 19 miles away from the starting point. 4th row 5 (2.0, 25) After 2.0 hours, they were 25 miles away from the starting point. 5th row 6 (2.5, 27) After 2.5 hours, they were 27 miles away from the starting point. 6th row 1.3 Answers (continued) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
70 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs • During which interval(s) did the riders make the most progress? The least progress? They made the most progress for the first half hour because they went 8 miles from hour 0 to hour 0.5. For the time period from hour 0.5 to 1.0 and from 2.5 to 3.0, they went 7 miles, which was also good progress. The least progress was between hour 3.0 and 3.5 because the distance decreased during the period. Students may say the least progress was made between 2.0 and 2.5 and between 4.0 and 4.5 because they only rode 2 miles during those half-hour periods. WHAT IF . . . ? Situation A. Connecting the Dots Malcolm reads Sidney’s notes. He wonders what might have happened between the last two points (4.5, 40) and (5.0, 45) on the graph. He came up with five possibilities. • Match the connecting paths represented in Graphs 1, 2, 3, 4, and 5 to the travel possibilities of the cases for Celia, Theo, Sarah, Tony, and Liz. Entry Coordinates Information In the Table 7 (3.0, 34) After 3.0 hours, they were 34 miles away from the starting point. 7th row 8 (3.5, 31) After 3.5 hours, they were 31 miles away from the starting point. 8th row 9 (4.0, 38) After 4.0 hours, they were 38 miles away from the starting point. 9th row 10 (4.5, 40) After 4.5 hours, they were 40 miles away from the starting point. 10th row 11 (5.0, 45) After 5.0 hours, they were 45 miles away from the starting point. 11th row Graph 1 Graph 2 Graph 3 Graph 4 Graph 5 1.3 Answers (continued from page 69) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Problem 1.3 Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance 71 Celia Celia rode slowly at first. Then, she gradually increased her speed. Theo Theo rode quickly. Theo reached the Cape May ferry dock early. Sarah Sarah had to fix a flat tire, so she started after the others. She rode at a consistent speed. Tony Tony started off fast. He soon felt tired and slowed down. Liz Liz pedaled at a steady pace throughout this part of the trip. Matches Celia Theo Sarah Tony Liz Graph 5 Graph 4 Graph 3 Graph 1 Graph 2 Situation B. Day 2 of the Trip On Day 2, the tour rode from Lewes, DE, through Ocean City, MD. They stopped at Chincoteague Island, which is famous for its annual pony auction. Celia collected data along the way. Here is a graph of her data. 20 40 60 80 100 10 Time (hours) Day 2 Progress Distance (miles) 2 3 4 5 6 7 y-axis x-axis 1. What are the variables? Variables are time (hours) and distance (miles). The distance is not total distance because they seem to “go backward” from 2.5 to 3 hours. So the distance is probably the distance away from the start. 1.3 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
72 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs 2. Make a table of the coordinates of the points on the graph. Note that only (0, 0) and (2.5, 30) appear to be on the grid lines. The other values for the coordinates will be approximations. Time (hours) 0 __1 2 1 1__1 2 2 2__1 2 3 3__1 2 4 4__1 2 5 5__1 2 6 6__1 2 7 7__1 2 Distance (miles) 0 8 12 21 21 30 21 31 38 49 49 57 62 71 73 80 3. During which interval(s) did the riders make the most progress? The least progress? Explain. The most progress: between hour 4.0 and 4.5. Two data points in the graph seem to have the largest distance among all adjacent two points. The coordinates are approximately (4.0, 36) and (4.5, 48). Then the progress would be 12 miles for the time period. Some students may notice that the two points: • have the greatest distance between them; • create the steepest line when you connect them with a line segment; and • make an angle that is that largest (from horizontal line to the height of the second point). 20 40 60 80 100 0 1 Time (hours) Day 2 Progress Distance (miles) 2 3 4 5 6 7 Increase of hour in the x variable 1 2 Increase of 12 miles in the y variable y-axis x-axis The least progress: between hour 2.5 and 3.0. For some reason, the distance from the starting point decreased over the __1 2 hour. At the hour 2.5, they were 30 miles away from the starting point and then, at the hour 3.0, they were 22 miles away from the starting point. Students may argue that between 1.5 to 2 or 4.5 to 5 there was not progress in distance made as time passed. They may perceive this as the least progress. 1.3 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Problem 1.3 Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance 73 Note: Some students may want to represent the data as a table to answer the progress question. Possible approximations of the data from the graph: 20 40 60 80 100 10 Time (hours) Day 2 Progress Distance (miles) 2 3 4 5 6 7 Increase of hour in the x variable 1 2 Decrease of 8 miles in the y variable y-axis x-axis 20 40 60 80 100 10 Time (hours) Day 2 Progress Distance (miles) 2 3 4 5 6 7 Increase of hour in the x variable 1 2 Increase of hour in the x variable 1 2 Increase of 0 miles in the y variable Increase of 0 miles in the y variable y-axis x-axis Time (hours) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Distance (miles) 0 7 12 22 22 30 22 31 36 48 48 57 62 71 74 81 4. What might have happened between hours 2 and 4? Answers will vary. Possible explanations include: Between hour 2 and 4, the riders needed to make a detour that brought them closer to Lewes. They may have forgotten something and had to go back to get it. After the detour, they continued to Chincoteague Island. Between hours 1.5 and 2, the riders may have taken a rest break for sightseeing. 5. What was the total distance the riders travel on Day 2? Assuming they only backtracked to mile 22, the total distance traveled for the day is the distance the riders are from Lewes at the end of the day plus the distance they traveled when they had to backtrack: 81 + 8 + 8 = 97 miles. NOW WHAT DO YOU KNOW? Describe how the distance changes over time. How is this pattern of change shown in tables and graphs? Variables in this problem are time (hours) and distance (miles). As time goes on, the distance the cyclists ride their bicycles is generally increasing. But there can be a decreasing pattern if they have to go back on the route that they have already traveled on. +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +__1 2 +7 +5 +10 +0 +8 −8 +9 +5 +12 +0 +9 +5 +9 +3 +7 1.3 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
74 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs You can see the change in each time period how faster/slower it was. In tables, you can see the change by finding the difference of distance for two adjacent numbers. In graphs, you can see the change by finding the y-coordinates of two data points. In graphs, you can also see which data point jumps the steepest from the previous one. Note: Time, rate, and distance are in the title of the problem. This problem is a very informal introduction to the Distance = Rate • Time relationship. The relationship will be brought up again in the Comparing Quantities unit. It will be more formally studied in grade 7. Students are expected to use rates implicitly with the context of traveling distance and the time that it takes. 1.3 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Problem 1.3 Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance 75 Atlantic City to Cape May Time (hours) Distance (miles) 0 0 0.5 8 1.0 15 1.5 19 2.0 25 2.5 27 3.0 34 3.5 31 4.0 38 4.5 40 5.0 45 Time (hours) Distance (miles) 10 20 30 40 50 0 1.0 2.0 3.0 4.0 5.0 Atlantic City to Cape May y-axis x-axis Atlantic City to Cape May Table and Graph 1.3A TEACHING 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
76 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Time (hours) Distance (miles) 10 20 30 40 50 0 1.0 2.0 3.0 4.0 5.0 Atlantic City to Cape May y-axis x-axis A Time (hours) Distance (miles) 10 20 30 40 50 0 1.0 2.0 3.0 4.0 5.0 Atlantic City to Cape May y-axis x-axis B Time (hours) Distance (miles) 10 20 30 40 50 0 1.0 2.0 3.0 4.0 5.0 Atlantic City to Cape May y-axis x-axis C Time (hours) Distance (miles) 10 20 30 40 50 0 1.0 2.0 3.0 4.0 5.0 Atlantic City to Cape May y-axis x-axis D Time (hours) Distance (miles) 10 20 30 40 50 0 1.0 2.0 3.0 4.0 5.0 Atlantic City to Cape May y-axis x-axis E Connecting Graph Points Examples 1.3B TEACHING 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 1.3 Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance 77 Atlantic City to Cape May Time (hours) Distance (miles) 0 0 0.5 8 1.0 15 1.5 19 2.0 25 2.5 27 3.0 34 3.5 31 4.0 38 4.5 40 5.0 45 Name Date Class Atlantic City to Lewes LEARNING AID 1.3A © 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
78 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Graph 1 Graph 2 Graph 3 Graph 4 Graph 5 Name Date Class Matching Cases to Paths (Part 1, Paths) 1.3B 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 1.3 Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance 79 Name Date Class Celia Celia rode slowly at first. Then, she gradually increased her speed. Tony Tony started off fast. He soon felt tired and slowed down. Theo Theo rode quickly. Theo reached the Cape May ferry dock early. Liz Liz pedaled at a steady pace throughout this part of the trip. Sarah Sarah had to fix a flat tire, so she started after the others. She rode at a consistent speed. Matching Cases to Paths (Part 2, Cases) LEARNING AID 1.3B © 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
80 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Name Date Class 20 40 60 80 100 10 Time (hours) Day 2 Progress Distance (miles) 2 3 4 5 6 7 y-axis x-axis Day 2 of the Trip 1.3C 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 1.4 Chincoteague Island to Norfolk: Stories, Tables, and Graphs 81 At a Glance This problem gives students the challenge of constructing a table and graph that match some written information expressed in words. There will not be one single correct answer to the questions, but the summary of work should provide a valuable opportunity to push students to justify their answers by explaining how their constructed data and graph match the story expressed in Malcolm and Liz’s notes. Chincoteague Island to Norfolk: Stories, Tables, and Graphs Arc of Learning™ Introduction Exploration NOW WHAT DO YOU KNOW? Describe how the relationship between two variables is shown in written notes, tables, and graphs. What are the advantages and disadvantages of each representation? Key Terms Materials For each group of 3–4 students • large poster paper (1 per group)* • sticky notes (1 per student) Pacing 1 day Groups 3-4 students A 17 E 22–23 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. Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE In the previous problems, students have collected information from an experiment and recorded it in a table and graph. They studied the patterns of change between the variables in the experiment. PRESENTING THE CHALLENGE In this problem, students will take two cyclists’ written notes and translate the information into a table and a graph. It is useful to make clear to students what their challenge is. Students should compose a data set that reflects the story in Malcolm and Liz’s notes, making sure to point out that there are several possible correct answers, but they should be able to justify their responses. Since there is room for interpreting the notes, each group will have slightly different graphs and tables. This is a good opportunity to have each group put their graph and table on large poster board. Then you can start the Summarize with a gallery walk. Students can leave sticky notes with questions or comments on the graphs as they walk around observing the various group work. Three Reads ProblemSolving Environment PROBLEM 1.4 (continued) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
82 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Facilitating Discourse Teacher Moves EXPLORE PROVIDING FOR INDIVIDUAL NEEDS Suggested Questions • What are the two variables? • What will you use for your time intervals? • At 0 hours, what is the distance traveled? At 9 hours? • What other data points are easy to fill in based on the set of notes? • How would riding into a strong wind affect the speed of the cyclists? Riding with the wind? PLANNING FOR THE SUMMARY As you circulate, look for examples where students have interpreted the notes correctly and incorrectly. Use these to lead the discussion in the summary. Listen for how students make sense of finding information from the different representations: In written notes, you can get the time information during the day, the distance information in terms of names of locations, and information on how long it may take based on the circumstances. In tables and graphs, time information is shown as increments associated the other variable, distance. Distance information in tables and graphs is shown as numbers representing how far it is from the starting point. Nonpermanent work surfaces can help students take more risks as they are interpreting the notes in the Initial Challenge. Selecting and Sequencing SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES It will be important for each group to display the table and graph that they produce to match the travel story and to explain to others in the class how they believe those representations do match the story. The gallery walk is very effective for eliciting helpful comparisons. Ask the class to look over all the displays and check for similarities and differences among them. Ask if others’ data seems reasonable and why. MAKING THE MATHEMATICS EXPLICIT Suggested Questions • What two variables are represented in the graph/table? • What does [this point] on the graph tell you? What does the x-coordinate of the point represent? What does the y-coordinate represent? • How do you enter the information about the point we looked at into the table? • How do you see where cyclists stopped on the table? Graph? • How do you see when the strong wind affected the cyclists on the table? Graph? • How do you see when the wind was at their backs on the table? Graph? As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Gallery Walk Agency, Identity, and Ownership 1.4 (continued from page 81) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Extended Launch—Explore—Summarize 83 Problem Overview This problem gives students the challenge of constructing a table and graph that match some written information expressed in words. There will not be one single correct answer to the questions, but the summary of work should provide a valuable opportunity to push students to justify their answers by explaining how their constructed data and graph match the story expressed in Malcolm and Liz’s notes. Launch (Getting Started) Connecting to Prior Knowledge In the previous problems, students have collected information from an experiment and recorded it in a table and graph. They studied the patterns of change between the variables in the experiment. Students also looked at data in a table and represented it in a graph. Students used both tables and graphs to look at the patterns of change and other information. Start with a discussion of the Chincoteague pony swim. Students may have read about this in the children’s book Misty of Chincoteague by Marguerite Henry (Aladdin, 2006). Implementation Comment: Since there is room for interpreting the notes, each group will have slightly different graphs and tables. (Problem-Solving Environment) This is a good opportunity to have each group put their graph and table on large poster paper. Then you can start the Summarize with a gallery walk. Students can leave sticky notes with questions or comments on the graphs as they walk around observing the various group work. Presenting the Challenge In this problem, students will take two cyclists’ written notes and translate the information into a table and a graph. It is useful to make clear to students what their challenge is. Students should compose a data set that reflects the story in Malcolm and Liz’s notes, making sure to point out that there are several possible correct answers, but they should be able to justify their responses. Have students read the notes three times, the first time thinking about what the task is about, the second time thinking about the given quantities in the task, and the third time thinking about mathematical questions they see in the task. EXTENDED LAUNCH—EXPLORE—SUMMARIZE CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
84 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs The Initial Challenge has students start by making a table and a graph to match the notes. Leave it to students to decide which representation to use or begin with. Have students work in groups of three to four. Distribute large paper for students to use to create their tables and graphs. Explore (Digging In) Providing for Individual Needs Be sure students are using all the relevant information given to make their data table and graph. Allowing students to use whiteboards, windows, desks, or laminated chart paper to try their ideas before committing them to paper can help students take more risks as there are interpreting the notes in the Initial Challenge. You can have the students start the table at 0 (represented as 8:00 a.m.) to focus on elapsed time rather than time of day. This sort of rescaling data is quite common to make graphing simpler and more standard. Some students may find it helpful to include the time of day, in addition to the elapsed time, on these representations to assist them in answering the questions. (Language) Encourage students to make a general sketch of the graph based on the constraints given in Malcolm and Liz’s notes to help them complete the problem. Suggested Questions • What are the two variables you will represent in the table and the graph? (The two variables are time and distance.) • What will you use for your time intervals? (A common suggestion is a half hour because previous problems have given distances for half-hour intervals.) • At 0 hours, what is the distance traveled? How do you know? (The distance traveled is 0 miles, because at 0 hours you are just about to start the day’s travel and have not gone anywhere yet.) • At 9 hours, what is the distance traveled? How do you know? (The distance traveled is 100 miles, because the last note states that the total distance traveled for the day, at the end of the 9 hours, was 100 miles.) • What other data points in the table would be easy to fill in based on the set of notes? (At approximately noon, or 4 hours into the trip, cyclists had traveled about half the total distance, or 50 miles.) LES CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Extended Launch—Explore—Summarize 85 LES • How would riding into a strong wind affect the speed the cyclists could travel? (It would slow them down.) • How would riding with the wind at their backs affect the speed? (It would help the cyclists go faster.) • Which clues give you information you could use to locate approximate points for your table? (Answers will vary, but students might mention the first, third, fifth, and sixth clues.) What If . . .? Situation A offers an extra challenge for students who are quick to make sense of making the table and graph from the trip notes. 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 relationship between two variables is shown in written notes, tables, and graphs. What are the advantages and disadvantages of each representation? (As you circulate, look for examples where students have interpreted the notes correctly and incorrectly. Use these to lead the discussion in the summary. Listen for how students make sense of finding information from the different representations: In written notes, you can get the time information during the day, the distance information in terms of names of locations, and information on how long it may take based on the circumstances. In tables and graphs, time information is shown as increments associated the other variable, distance. Distance information in tables and graphs is shown as numbers representing how far it is from the starting point.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies It will be important for each group to display the table and graph that they produce to match the travel story and to explain to others in the class how they believe those representations do match the story. Thus, it is critical that you allow extra time for the Summarize phase of this problem. The gallery walk is very effective for eliciting helpful comparisons. Ask the class to look over all the displays and check for similarities and differences among them. (Agency, Identity, and Ownership) Ask if others’ data seems reasonable and why. Let that student respond, either defending the data or agreeing and changing the values CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
86 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs accordingly. The tables and graphs will be different but should share these common characteristics: • The total distance must be 100 miles. • The early morning’s progress should be slower than the rest of the day because of the wind. • The second part of the morning’s progress should be faster because of the wind. • There are three breaks: at midmorning, at lunchtime for about an hour, and at around 3:00 p.m. • The class might assume that when the cyclists load their bicycles in the van, they will cover a greater distance in a shorter time than when they were pedaling. Making the Mathematics Explicit Ask two groups with different but correct graphs to share how their graphs match the trip notes. Encourage other students to ask questions so they can understand what assumptions and interpretations the presenters made in constructing their graphs. This is a conversation from a classroom around these two pieces of student work: LES Table Graph Time (hours) Distance (miles) 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 1 2 3 4 5 6 7 8 9 10 y-axis x-axis Hours Distance (miles) 1st hour 8 miles 2nd hour 15 miles 3rd hour 25 miles 4th hour 40 miles 5th hour 40 miles 6th hour 55 miles 7th hour 65 miles 8th hour 75 miles 9th hour 100 miles Group 1 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Extended Launch—Explore—Summarize 87 LES Teacher: I noticed that as everyone was looking at the tables and graphs during the gallery walk, there were several questions/ comments for these groups. I’d like to start our class discussion by looking at these. Who would like to start us out by discussing something they noticed or wondered? Vince: When I was looking at Group 2, I liked that they wrote halfway on the table and graph to emphasize that part of the notes, but I didn’t see where they showed the breaks. Sylvia: That was our group. We didn’t know how to show breaks, so we just put the things on that we knew how to show. It still gives you the idea of what happened during the day. Joan: It does, Sylvia, but if we are to show the whole story of the day in a table and graph, then we should include those things too. In Group 1, when I look at the graph, I can see that “flat line” (makes a horizontal motion with her hand), and that must be when the cyclists at their lunch. Vince: Yeah! Then when you look at Group 1’s table, you see 40 miles listed at the 4- and 5-hour mark. This says that the cyclists were not moving, so that must be the lunchtime. Graph Time (hours) Distance (miles) 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 8 9 10 11 12 1 2 3 4 5 x-axis y-axis halfway done Time Distance 8:00 am 20 miles 9:00 am 25 miles 10:00 am 30 miles 11:00 am 35 miles 12:00 am 45 miles 1:00 pm 56 miles 2:00 pm 65 miles 3:00 pm 75 miles 4:00 pm 100 miles Table Group 2 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
88 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs LES Sylvia: Oh yeah, like when we did the jumping jacks and I got tired and paused for 20 seconds, the number of jumping jacks didn’t change in my table. I forgot about that. Teacher: How do we see the midmorning break and/or the stop for a brief swim in these displays? Vince: I’m thinking that Group 1 didn’t show it as a stop but made the amount of distance in an interval smaller than others to indicate a brief stop. Like, between hours 1 and 2 there were 7 miles, and then between hours 3 and 4 there were 10 miles. Richie: I am in Group 1. That is exactly what we did. Then if you look at our graph at those times, you see that it is just a little bit steeper when they didn’t stop at all and the wind was helping them move forward. Jeff: I was in Sylvia’s group, which was Group 2. We didn’t think about that and just did around 5 or 10 miles for each hour of time. We see now how to change that to include all the notes from Malcolm and Liz. Suggested Questions • What two variables are represented in the graph? (The two variables are distance and time.) • When you are making a table, what role do variables play? (If students do not know, have them look at the table given for Problem 1.2 and the graph they constructed for that problem.) • What does this tell you about the labels for your table? (The labels for the columns are also distance and time.) • What does [this point] on the graph tell you? What does the x-coordinate of the point represent? What does the y-coordinate represent? (The x-coordinate tells the time, and the y-coordinate gives the distance covered up to this time.) • How do you enter the information about the point we looked at into the table? (Put the x-coordinate in the Time column and the y-coordinate in the corresponding place in the Distance column.) Ask students about the representation (table, graph, or story) that they found most helpful when trying to answer questions or analyze the patterns in the data. Discuss the advantages and disadvantages of each representation (trip notes, graph, and table) for looking at patterns of change in distance over time. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Extended Launch—Explore—Summarize 89 LES Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? questions. 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
90 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Answers Embedded in Student Edition Problems INITIAL CHALLENGE On Day 3, the group travels from Chincoteague Island to Norfolk, VA. Malcolm and Liz drove the tour van on the way. They forgot to record the time and distance data. Fortunately, they wrote some notes about the trip. Chincoteague Island to Norfolk: Stories, Tables, and Graphs PROBLEM 1.4 We started at 8:00 a.m. We rode into a strong wind until our midmorning break. About the time of our break, the wind shifted to our backs. Now the wind was helping to move us forward. Around noon, we stopped for lunch at a BBQ food truck. We rested for about an hour. By this time, we had traveled about halfway Around 3:00 p.m., we stopped for a brief swim in the ocean. Around 4:00 p.m., we reached the Chesapeake Bay Bridge and Tunnel. We stopped for a few minutes to watch the ships passing. Because riding bikes on the bridge is not allowed, we put the bikes in the van and drove across. We finished the We took 9 hours to complete today’s 100-mile trip. to Norfolk. last 25 miles in the van. • What are the variables in this situation? Variables in this situation are time (hours) and distance (miles). Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Problem 1.4 Chincoteague Island to Norfolk: Stories, Tables, and Graphs 91 • Make a table and graph of the data. How does your table and graph show the pattern of change in distance over time? Since the story does not include all the data, students’ tables and graphs will vary. Some students begin with a table and move to a graph. Other students begin with a graph then move to a table. Possible Answers: Time of Day 8:00 a.m. Midmorning, about 10:00 a.m. 12 p.m. (spend some time eating—no distance added during lunch) 3 p.m. (maybe __1 2 hour for a swim— no distance added in) 4 p.m. (watch ships then get in van) 5 p.m. Time (hours) 0 2 4 7 8 9 Distance (miles) 0 20 50 65 75 100 Time of Day 8 a.m. 9 a.m 10 a.m 11 a.m 12 p.m 1 p.m. 2 p.m. 3 p.m. 4 p.m. 5 p.m. Time (hours) 0 1 2 3 4 5 6 7 8 9 Distance (miles) 0 5 18 33 50 50 60 70 75 100 wind against wind with 25 miles in van or Graphs will vary. Possible graphs: Students might sketch the pattern between the variables at various points of time and distance. 40 50 30 60 70 90 80 100 0 1 2 3 4 5 6 7 8 9 10 11 12 Time (hours) x-axis y-axis Distance (miles) eating lunch going faster in the van 20 10 strong wind 1.4 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
92 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs or Students might plot points that seem to represent the pattern between the variables at various points of time and distance. • Are your table and graph the same as those of your classmates? Explain. Possible answer: Some points are the same; some points are different. We have different scales for the time. We have different estimates of the progress in distance. A few points that students might agree should be close to the same: • For (0, 0), at 8:00 a.m., they started the trip. • For (9, 100), they completed the trip. • At noon, they had traveled about halfway, so about (4, 50). • The last hour from 4 p.m. to 5 p.m., they stopped to watch the ships passing and then drove 25 miles in the van, so about (8.5, 75). WHAT IF . . . ? Situation A. Celia Loses Her Backpack When the group stopped for a swim, Celia realized she left her backpack where they had lunch. The group had to travel back to get the backpack. How does this information affect your table? Your graph? Explain. Possible answers: In both the table and the graph, you might see the distance increase as time moves on (except at lunch, when time continues but the distance does not). If the group goes back, they will decrease their distance as time continues until they get back to the lunch spot. Then, their distance will begin increasing again. Depending on the intervals used in the table, we might not be able to see the change. See the graph that follows. 40 50 30 60 70 90 80 100 0 1 2 3 4 5 6 7 8 9 Time (hours) x-axis y-axis Distance (miles) 20 10 1.4 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Problem 1.4 Chincoteague Island to Norfolk: Stories, Tables, and Graphs 93 NOW WHAT DO YOU KNOW? Describe how the relationship between two variables is shown in written notes, tables, and graphs. What are the advantages and disadvantages of each representation? In written notes, you can get the time information during the day, such as 8:00 a.m., midmorning, noon, 3 p.m., and so on, which is how we communicate with others in real life. In tables and graphs, time information is shown as increments associated with the other variable, distance. When you are at the starting point at 8:00 a.m., you have 0 (hours) in your table and graph. In the same way, in written notes, you can get the distance information in terms of the names of locations, such as a barbeque place or the north end of the Chesapeake Bay Bridge and Tunnel. In tables and graphs, distance information is shown as numbers representing how far it is from the starting point. Using written notes, you can understand where they were at certain times of the day, whereas in tables or graphs, you can understand the relationship between time and distance, showing how far they rode from the starting point as time went on. (See next page for possible answers.) 40 50 30 60 70 90 80 100 0 1 2 3 4 5 6 7 8 9 Time (hours) x-axis y-axis Distance (miles) 20 10 lunch swim possible change in the graph for going back to get the backpack, then catching back up to the others 1.4 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE