94 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Advantages Disadvantages Written Notes You know more detail about events, so you don’t have to guess what happened. It can be difficult to quickly see the whole pattern. Table Gives exact distances for given times. You have to do calculations to see increases and decreases in the patterns between the variables. Graph Shows the “picture” of all the data for the day. You can see when the change in distance is great and when it is small and when the distance does not change. You need to estimate some values from a graph, rather than knowing exact values. Many students will choose the table or graph instead of the written description because it is too vague and doesn’t give you specific data to work with. Possible answers: 1.4 Answers CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Mathematical Reflection 95 At a Glance What are the advantages and disadvantages of using different representations to show the relationship between two variables? The Mathematical Reflection provides an opportunity to discuss the goals of the investigation. Students can pull together their reasoning from the Now What Do You Know? questions to summarize their learning over the time. Students can record their responses to the Mathematical Reflection to create a record of their current understandings of the big ideas of the unit. The Mathematical Reflection can provide a self-assessment for students. Each student can have checkpoints of their understanding of the mathematics after each investigation. Students can use the reflection to consolidate their mathematical thinking, take notes, and provide evidence of what they know and can do. A teacher can gain an understanding of student thinking during a discussion of the reflection question. Then one can assess individual understanding based on each student’s written work. 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 Having students refer to their notes from the Now What Do You Know? in each problem in the investigation can help them to synthesize all their ideas around the advantages and disadvantages of using different representations to show the relationship between two variables. As a class, discuss the Mathematical Reflection. Use an idea like those on the next page to have students synthesize and record their thinking. Suggested Questions • After this investigation, what do we know about the advantages and disadvantages of using different representations to show the relationship between two variables? • What did we learn in each problem of this investigation? • How might we describe the “big mathematical idea(s)” of the investigation? Time Anchor Chart Portrayal Language MR Mathematical Reflection Arc of Learning™ Exploration Pacing __1 2 day CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
96 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs Student Responses At the beginning of the year, students will need more collaboration to outline and summarize the important ideas. They may need examples of writing, diagrams, and/or justifications from other students to help build their vision of what is expected when answering a Mathematical Reflection. Early in the year, you may want to start writing Mathematical Reflections as a whole group. Then as the year progresses, move to small groups, pairs, and finally individuals. Each investigation contributes to students’ conceptual understandings of the ideas in the unit. Students’ explanations at the beginning of a unit might be just forming. As you progress through the unit, students can use the contexts, representation, and connections to express a more solid understanding. By the end of the unit, students can create a complete picture of understanding. Example Strategies for Student Participation Here are a few creative strategies teachers use to encourage students’ ownership of their learning. Anchor Charts • After a discussion, chart the emerging understanding, and post it in the classroom. This can be done on poster paper or electronically. • Work with students throughout the unit to reference, add to, or refine their understandings. Note: For teachers who move classrooms or have multiple classes of the same grade level, create the chart in all classes, but keep just one to represent all of your classes. Post this one in the room, or bring it out when needed. Note Organization • Some teachers use the Mathematical Reflections as an organizer for note-taking during the investigation. • As part of the Summarize section of the problems, students record key ideas to the Now What Do You Know? reflection questions on a separate paper. • At the close of the investigation, students synthesize their notes into responses that summarize their emerging understandings of the ideas in the unit. Word Bank • As a class, create a word bank of terms from the Investigation. • Have groups of students write three or four statements using the words from the bank. • After formatively assessing their statements, you may choose to have a class discussion to refine the statements. Chalk Talk With a chalk talk, your writing does “the talking” instead of talking aloud. • Students post the question(s) on sheets of chart paper or on sections of your board. • Small groups record responses while collaborating in “chalk talk” format. • Students move to others’ work and add their thinking in the form of new ideas and connections. Final Reflection Presentation Teachers sometimes use the Mathematical Reflection after the last investigation as a summary of students’ learning. • Students consolidate their learning from the unit. • Teachers choose from various ways to present their ideas. Presentation choices might include creating a poster, written paper, presentation, or song/rap. Partner Write • Students create a written response to the reflection question with a partner. • Students discuss the reflection question with a partner. • Students create and write a response with a partner. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Mathematical Reflection 97 In this unit, we are studying variables and the relationship between two variables. We will use words, tables, graphs, and equations to represent these relationships and to solve problems. At the end of this investigation, ask yourself: What are the advantages and disadvantages of using different representations to show the relationship between two variables? Students should be just beginning to see key aspects of representing and making sense of variable relationships in tables, graphs, and words. They might notice that: • tables give explicit numerical values for the variables and this information from graphs requires reading scales and sometimes imprecise value estimates; • graphs show an overall trend in the data and make it easy to identify changes in a pattern; and • narratives/notes/descriptions/stories tell what is happening and can be vague or incomplete, requiring one to make interpretations. If you use different representations to show the relationship between two variables, then you can better understand and interpret the situation. You could also estimate the relationship between two variables even if you do not have exact data. When necessary, you could predict future patterns from the data you have. MR Answers Embedded in Student Edition Problems Mathematical Reflection CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
98 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs APPLICATIONS 1. The following table shows the number of bags of popcorn sold from 6:00 a.m. to 7:00 p.m. at the local outdoor market. Popcorn Sales Time Total Bags Sold 6:00 a.m. 0 7:00 a.m. 3 8:00 a.m. 15 9:00 a.m. 20 10:00 a.m. 26 11:00 a.m. 30 noon 45 1:00 p.m. 58 2:00 p.m. 58 3:00 p.m. 62 4:00 p.m. 74 5:00 p.m. 83 6:00 p.m. 88 7:00 p.m. 92 a. Describe the pattern of change in the number of bags of popcorn sold during the day. Answers will vary. This is assigned after the first problem of the year, so be lenient with what you accept. Sharing a quality description from a student the next day is helpful. Note: Problem options listed here are a collection from which to choose. It is always a good idea to do the problems yourself first so you know what is involved in each one. Especially with this unit, each problem may have several parts, and some require students to make tables and graphs or write descriptions. These types of questions take lots of time, so you will need to assign accordingly. See At a Glance for ACE that can support learning after each problem. Answers Embedded in APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) INVESTIGATION 1 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Applications—Connections—Extensions (ACE) 99 AACECE Example: Sales increase rather steadily throughout the day, with somewhat greater sales between 11 a.m. and 1 p.m. (the lunch hour) and somewhat slower sales for the following two hours. Students may mention that no popcorn was sold between 1 p.m. and 2 p.m. Early in the unit, some students will tend to describe every interval, and some will be too general, stating only that the number of bags of popcorn increased. b. During which hour did the market sell the most popcorn? During which hour did it sell the least popcorn? Greatest sales (15 bags) were in the sixth hour (11 a.m. to 12 p.m.); least sales (0 bags) were in the eighth hour (1 p.m. to 2 p.m.). 2. The following chart shows the water depth in a harbor during a typical 24-hour day. The water level rises and falls with the tides. Effect of the Tide on Water Depth Hours Since Midnight 0 1 2 3 4 5 6 7 8 Depth (meters) 10.1 10.6 11.5 13.2 14.5 15.5 16.2 15.4 14.6 Hours Since Midnight 9 10 11 12 13 14 15 16 Depth (meters) 12.9 11.4 10.3 10.0 10.4 11.4 13.1 14.5 Hours Since Midnight 17 18 19 20 21 22 23 24 Depth (meters) 15.4 16.0 15.6 14.3 13.0 11.6 10.7 10.2 a. At what time is the water the deepest? Find the depth at that time. Six hours after midnight, or 6:00 a.m.; 16.2 meters. b. At what time is the water the shallowest? Find the depth at that time. Twelve hours after midnight, or noon; 10.0 meters. c. During what time interval does the depth change most rapidly? Water depth changes most rapidly (by 1.7 meters per hour) between two hours and three hours after midnight, or 2 a.m. and 3 a.m.; between 8 hours and 9 hours after midnight, or 8 a.m. and 9 a.m.; and between 14 hours and 15 hours after midnight, or 2 p.m. and 3 p.m. (This pattern shows the physical property of tides that they move most swiftly at points halfway between high and low tides, which occur roughly every six hours.) CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
100 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs ACE 3. The Ocean Bike Tours partners went on a test ride. The (time, distance) data for their ride are shown in the following table. Ocean Bike Tours Test Ride Time (hours) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 5.0 5.5 6.0 Distance (miles) 0 10 19 27 34 39 36 43 53 62 66 72 a. Plot these data on a coordinate graph with scales and labels. Answer: 0 10 20 30 40 50 60 70 80 1.0 Time (hours) Distance (miles) 2.0 3.0 4.0 5.0 6.0 Ocean Bike Tours Test Ride y-axis x-axis b. At what time(s) in the ride were the four business partners riding fastest? How is that information shown in the table and on the graph? Fastest riding was in the first half hour and the half hour between 3.5 and 4.0 hours. In those time intervals, the riders covered 10 miles for an average speed of 20 miles per hour. By comparing adjacent entries in the data table, you can find the greatest increase in a half hour. The greatest increases are shown on the graph by the largest jumps upward from one point to the next. Students may also talk about the line between points being “steeper” where there is a greater increase. They may notice the distance between the two points being longer in these situations as well. c. At what time(s) in the ride were they riding slowest? How is that information shown in the table and on the graph? Slowest riding was in the half hour between 2.5 and 3.0 hours (when the riders were backtracking on their trip). In that time interval, the riders covered 3 miles for an average speed of 6 miles per hour. By comparing adjacent entries in the data table, you can find the least increase in a half hour. The least increases are shown on the graph by the smallest jumps upward from one point to the next. In this case, the jump is down, but the absolute value of the distance is 3. Students may talk about the line between these points being “less steep” or there being a shorter distance between points. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Applications—Connections—Extensions (ACE) 101 AACECE d. How would you describe the overall pattern in cyclist speed throughout the test run? The overall pattern in the data (excepting the backtrack interval) shows gradual slowing over the first half of the trip and again over the second half of the trip. This pattern is more apparent in the table than the graph. Note: There is a missing data point at time 4.5 hours. Students may say the graph shows a steady increase for the first half and then slows down in the middle of the trip. They may describe the last part of the trip as having a steady increase, as the dots appear to stay at the same “steepness.” e. What might explain the dip in the distance data between 2.5 and 3.5 hours? The dip might be caused by a wrong turn that required backtracking or by something dropped and only realized later down the road, requiring backtracking to pick up the lost item. 4. Three students made graphs of the population of a town called Greenville. The break in the y-axis in Graphs A and C indicates that there are values missing between 0 and 8. 2016 0 8 10 12 14 16 y-axis 2018 Year Graph A Population (in 1000s) 2020 2022 2024 x-axis 2016 0 2 4 6 8 10 12 14 2018 Year Graph B Population (in 1000s) 2020 2022 2024 y-axis x-axis 2016 0 8 12 16 20 24 28 2018 Year Graph C Population (in 1000s) 2020 2022 2024 y-axis x-axis a. Describe the relationship between time and population as shown in each of the graphs. Each graph shows a steady growth in the Huntsville population from about 10,000 in 2016 to about 14,000 in 2022. b. Is it possible that all three graphs correctly represent the population growth in Greenville? Explain. The graphs show the same data values. The different choices of scales cause the different appearances of the graphs. However, by making irregular y-axis scales, Graphs A and C give impressions that are misleading. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
102 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs ACE 5. The graph shows the number of cans of juice purchased each hour from a school’s vending machines in one day. On the x-axis of the graph, 8 means the time from 7:01 a.m. to 8:00 a.m., and so on. Juice Vending Machine Sales Time of Day (starting with 7 a.m.) Cans Sold 20 40 60 80 100 120 140 160 180 200 0 87 9 10 11 12 1 2 3 4 5 6 7 y-axis x-axis a. What might explain the high and low sale time periods shown by the graph? High sales probably occur just before school starts. Between 8 a.m. and 9 a.m., 80 cans were sold. This probably occurred because students were arriving at school and having breakfast before school started. High sales also occurred from 11 a.m. to 1 p.m., probably because of lunch, and again between 4 p.m. and 5 p.m., right after school when students may stay after for practice or other activities. Low sales, like between 10 a.m. and 11 a.m. and 3 p.m. and 4 p.m., probably occur while most students are in class. b. Does it make sense to connect the points on this graph? Why or why not? It does make sense to connect the points because sales are likely occurring throughout the hour-long interval period. 6. Students have a contest to see how many sit-ups they can complete in 10 minutes. Andrea and Ken plot their results. Their graphs are shown here. Andrea’s Graph Time (minutes) Number of Sit-ups 50 100 150 0 1 2 3 4 5 6 7 8 9 10 y-axis x-axis CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Applications—Connections—Extensions (ACE) 103 AACECE Ken’s Graph Time (minutes) Number of Sit-ups 40 20 80 60 100 0 1 2 3 4 5 6 7 8 9 10 y-axis x-axis a. Ken claims that he did better because the points on his graph are higher than the points on Andrea’s graph. Is Ken correct? Explain. No, Ken is not correct. His graph only seems to go higher because he has chosen a different scale on the y-axis. Both Andrea and Ken completed 90 sit-ups. The last point on both graphs sits at (10, 90). b. In what ways do the results of the sit-up test show a pattern of endurance in physical activity that is similar to the results of the test ride by the Ocean Bike Tours partners? These data show a gradual slowing of the rate of sit-ups, like what you would see (and expect) in the ride. Students may describe both graphs as increasing at a certain “steepness” and that “steepness” decreases as we read the graph from left to right. From 7 to 10 minutes, that invisible line between points appears to get less and less “steep.” c. Which person had the greatest number of sit-ups per minute? Overall, Andrea and Ken had the same average number of sit-ups. Andrea’s pace = 90 sit-ups/10 minutes = 9 sit-ups per minute. Ken’s pace = 90 sit-ups/10 minutes = 9 sit-ups per minute. d. Compare Ken’s pace in the first 2 minutes to his pace in the last 2 minutes. Ken’s pace in the first 2 minutes (approximately) = 39 sit-ups/2 minutes = 19 situps per minute. Ken’s pace in the last 2 minutes (approximately) = 10 sit-ups/2 minutes = 5 sit-ups per minute. This is the first unit of the year, so some students may not find the unit rate. They will see that he did about 38 sit-ups in the first 2 minutes and 10 sit-ups in the last 2 minutes but may not scale to how many per 1 minute at this early stage of the game. 7. Celia uses (time, distance) data from one part of the bike tour test run to draw the following graph relating time and speed. Celia forgot to include scales on the axes of the graph. Time Celia’s Graph Speed x-axis y-axis a. What does this graph show? The graph shows something moving at a constant speed over a period of time. As time passes, the speed remains the same. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
104 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs b. Is the graph most likely a picture of speed for a cyclist, the tour van, or the wind over a part of one day’s trip? Explain your reasoning about each possibility. The graph is not reasonable for a cyclist or for the wind under normal conditions. A rider’s speed can be affected by fatigue or environmental factors such as temperature, wind speed or direction, and terrain. A van could travel close to a constant speed on a flat surface. The wind usually comes in gusts. It does not seem that it would remain constant over a long period of time. Note: Some students might answer that they do not know what the scale is, so if a small amount of space on the y-axis means millions and millions, then this graph is possible for the cyclist, the van, or the wind because their small amount of speeding up and slowing down would not show up on the graph. 8. Katrina’s parents kept a record of her growth in height from birth until her 18th birthday. Their data is shown in the following table. Katrina’s Height Age (years) Height (inches) birth 20 1 29 2 33.5 3 37 4 39.5 5 42 6 45.5 7 47 8 49 9 52 10 54 11 56.5 12 59 13 61 14 64 15 64 16 64 17 64.5 18 64.5 a. Make a coordinate graph of Katrina’s height data. Answer: Katrina’s Height Age (years) Height (inches) 20 40 60 0 2 4 6 8 10 12 14 16 18 y-axis x-axis ACE CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Applications—Connections—Extensions (ACE) 105 AACECE b. During which time interval(s) did Katrina have her greatest “growth spurt”? Between birth and age 1 (9 inches). c. During which time interval(s) did Katrina’s height change the least? From age 14 to 16 and from age 17 to 18, when Katrina did not grow at all. d. Would it make sense to connect the points on the graph? Why or why not? It makes sense to connect the points because growth occurs between birthdays. Note: The question of how these points should be connected, by line segments or a curve, is another point of discussion. e. Is it easier to use the table or graph to answer? Explain. Answers will vary. The exact change in height is easier to read from the table. However, students may argue that the graph provides a better overall picture. 9. Here is a graph of temperature data collected on the Ocean Bike Tours test trip from Atlantic City to Lewes. Time (h) Temperature (°F) 20 40 60 80 100 10 Temperatures for Day 1 2 3 4 5 y-axis x-axis a. Make a table of (time, temperature) data from this graph. Answer: Time (hours) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Temperature (°F) 60 50 55 60 70 80 70 65 70 80 85 b. What is the difference between the day’s lowest and highest temperatures? The difference between the highest and lowest temperatures is about 35ºF (from 50ºF to 85ºF). c. During which time interval(s) did the temperature rise the fastest? During which time interval did it fall the fastest? The temperature rose at a rate of about 20 degrees Fahrenheit per hour (10 degrees Fahrenheit per half hour) between 1.5 and 2.0 hours, between 2.0 and 2.5 hours, and again between 4.0 and 4.5 hours. The temperature fell at a rate of about 20 degrees Fahrenheit per hour (10 degrees Fahrenheit per half hour) in the first half hour and again between 2.5 and 3.0 hours. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
106 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs d. Do you prefer using the table or the graph to answer questions like those in ACE 9b and 9c? Explain your reasoning. Answers will vary. Some students will express some preference for having numerical data on a table to calculate answers to questions like those in parts (b) and (c). Other students will prefer to “see” the pattern of change on graphs, as they tend to give overall pictures that are easier to remember in a global sense. Accept all opinions from students, as the question asked what they “prefer.” e. What information is shown by the lines connecting the points? Connecting the points shows the temperature changing at a steady rate between half-hour marks. It makes sense to connect the points because time is a continuous variable, so there will be temperatures after 15 minutes, after 37 minutes, and so on. The information may not be completely accurate because the temperature may not have changed at a constant rate. However, it is useful for making estimates. 10. Make a table and graph of (time, temperature) data that fit the following information about a day on the road with the Ocean Bike Tours cyclists: Trip Journal May 27 Entry 1: We started riding at 8 a.m. The day was quite warm, with dark clouds in the sky. Entry 2: About midmorning, the temperature dropped quickly to 63°F, and there was a thunderstorm for about an hour. Entry 3: After the storm, the sky cleared, and there was a warm breeze. Entry 4: As the day went on, the sun steadily warmed the air. When we reached our campground at 4 p.m., it was 89°F. All Entries Answers will vary, but the table and graph should show that it was warm at 8 a.m. (at time = 0 hours). Then, the temperature decreased rapidly to 63ºF by midmorning and stayed constant for about an hour. After this, the temperature increased at a fairly steady rate until it reached 89°F at 4 p.m. Students may think that “steady” means the temperature increased by the exact same number of degrees Fahrenheit each hour. A conversation about accepting between 4 and 6 degrees Fahrenheit per hour as “steady” may need to occur. ACE CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Applications—Connections—Extensions (ACE) 107 ACACEE Possible answer: Time (hours) 0 1 2 3 4 5 6 7 8 Temperature (°F) 80 80 63 63 70 75 80 85 89 Time (hours) Temperature (°F) 60 70 80 90 100 20 A Day on the Road 4 6 8 10 y-axis x-axis CONNECTIONS For ACE 11–13, order the given numbers from least to greatest. Then, for each ordered list, describe a pattern relating each number to the next number. 11. 1.75, 0.25, 0.5, 1.5, 2.0, 0.75, 1.25, 1.00 0.25, 0.5, 0.75, 1.00, 1.25, 1.5, 1.75, 2.0; add 0.25 each time 12. __3 8, 1, __1 4, __7 8, __3 4, __1 2, __1 8, __5 8 __1 8, __1 4, __3 8, __1 2, __5 8, __3 4, __7 8, 1; add __1 8 each time 13. __ 2 3, __ 4 3, __1 3, __1 6, __ 4 6, __ 8 3, ___ 32 6 __1 6, __1 3, __ 4 6 = __ 2 3, __ 4 3, __ 8 3, ___ 32 6 = __16 3 ; multiply by 2 each time 14. Consider the pattern that follows. a. Draw the next shape in the geometric pattern. Answer: CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
108 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs b. Make a table of (number of squares in bottom row, total number of squares) data for the first ten shapes in the pattern. Answer: Number of Squares in Bottom Row 1 2 3 4 5 6 7 8 9 10 Total Number of Squares 1 3 6 10 15 21 28 36 45 55 c. Describe the pattern of increase in total number of squares as length of the bottom row increases. There are several ways to describe the pattern of increase in total number of squares. Students may describe the pattern they see as the total number of squares increasing by 2, then 3, then 4 as the number of squares in the bottom row increases by 1. Note: Students probably will not come in to sixth grade with many experiences writing formal rules or formulas. A formal rule for this would be as the number of squares in the bottom row increases from n to n + 1, the total number of squares increases by n + 1. One formula for the total number of squares on a base of length n is T = ______ n(n + 1) 2 or T = n2 __ 2 + __ n 2. It is not expected that students come up with this formal rule. This is the first unit of the year and the first formal algebra unit. This is for your information. 15. Make a table to show how the total number of cubes in these pyramids changes as the width of the base changes from 3 to 5 to 7. Then use the pattern in those numbers to predict the number of cubes for pyramids with base width of 9, 11, 13, and 15. Answer: Width of Base 3 5 7 9 11 13 15 Total Number of Cubes 10 35 84 165 286 455 680 As the width of the base increases from 3 to 5, the total number of cubes increases by 25 (5 × 5). From 5 to 7, the total number of cubes increases by 49 (7 × 7). If this pattern continues, the increase from 7 to 9 should be 81 (9 × 9) and so on. ACE CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Applications—Connections—Extensions (ACE) 109 AACCEE 16. The partners in Ocean Bike Tours want to compare their plans with those of other bicycle tour companies. The bike tour they are planning takes 3 days, and they wonder if this might be too short. Malcolm called 18 different companies and asked, “How many days is your most popular bike trip?” He recorded the answers on the clipboard. Bike Tour Data 3 5 4 3 5 4 6 10 2 5 7 3 7 7 3 14 12 6 a. Make a line plot of the data. Answer: Number of Days Length of Bike Tours 1 2 3 4 5 6 7 8 9 10 11 12 13 14 b. Based on ACE 16a, should Ocean Bike Tours change the length of the 3-day trip? Explain. Answers will vary. Some students may note that the 3-day tour is the most preferred length and surmise that a 3-day trip is the best option. However, other students may observe that half of the most popular tours are shorter than 5 days and half are longer than 5 days, so a 5-day tour is the average length and would be a popular option. Accept all reasonable answers from students. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
110 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs 17. The graph that follows shows the results of a survey of people over age 25 who had completed different levels of education. The graph shows the average salary for people with each level of education. Education and Salary Years of Education Completed Average Salary ($) 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 y-axis x-axis a. Make a table that shows the information in the graph. Answer: Years of Education Completed 8 9 10 11 12 13 14 15 16 Average Salary ($) 12,500 14,000 16,500 17,500 28,000 30,500 34,000 36,000 49,000 Data relating approximate annual salary to level of schooling completed will look like that in the table. b. After how many years of education do salaries take a big jump? Why do you think this happens? The greatest increases occur after 12 and 16 years of education. This is probably because a diploma qualifies a person for higher-paying jobs. (You may want to point out to students that these are not starting salaries. Some of these people have been in their field for years. The participants of this study are people over 25.) c. Do you find it easier to answer ACE 17b by looking at the graph or at your table? Explain. Answers will vary. It is often easier to see changes, or jumps, in a graph, but it is easier to use a table to find the exact amount of those changes. EXTENSIONS 18. Think of something in your life that varies with time. Make a graph to show the pattern of change. Answers will vary. Examples might be a student’s height or weight, number of friends, or amount of spending money. The key is getting students to explain how their tables and graphs tell a story about change over time of some quantity that interests them. Check students’ graphs. ACE CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Applications—Connections—Extensions (ACE) 111 ACACEE 19. Some students did a jumping jack experiment. They reported data on the student who could do the most jumping jacks in a certain amount of time. 10 20 30 40 50 0 10 Time (seconds) Our Jumping Jack Experiment Number of Jumping Jacks 20 30 40 50 60 x-axis y-axis a. According to the graph, how many jumping jacks did the jumper make by the end of 10 seconds? By the end of 20 seconds? By the end of 60 seconds? After 10 seconds, 10 jumping jacks; after 20 seconds, 20 jumping jacks; after 60 seconds, 40 jumping jacks. b. Give the elapsed time and number of jumping jacks for two other points on the graph. Other points show 5 jumping jacks after 5 seconds, 15 jumping jacks after 15 seconds, 25 jumping jacks after 25 seconds, 30 jumping jacks after 35 seconds, a d 35 jumping jacks after 45 seconds. c. What estimate would make sense for the number of jumping jacks in 30 seconds? The number in 40 seconds? The number in 50 seconds? Students may find the middle of the number of jumping jacks at 25 and 35 seconds, 25 and 30. They might estimate 27 or 28 jumping jacks at 30 seconds. Using the graph, they may locate a point that looks “in between” the two points and assign it an appropriate number of jacks. d. What does the overall pattern in the graph show about the rate at which the test jumper completed jumping jacks? The rate of jumping jacks decreases as time passes. In the last 15 seconds (45−60), the jumper jumped only 5 times. e. Suppose you connected the first and last data points with a straight line. Would this line show the overall pattern? Explain. Connecting the first and last points of the graph with a straight line segment would suggest a constant rate of jumping jacks, a pattern that is quite different from that shown by the actual data plot. The graph shows an increase over time but not at a steady rate as it would be to connect the first and last data points. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
112 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs 20. The number of hours of daylight in a day changes throughout the year. We say that the days are “shorter” in winter and “longer” in summer. The table shows the number of daylight hours in Chicago, IL, on a typical day during each month of the year (January is Month 1 and so on). Daylight Hours Month Number of Hours 1 10.0 2 10.2 3 11.7 4 13.1 5 14.3 6 15.0 7 14.5 8 13.8 9 12.5 10 11.0 11 10.5 12 10.0 a. Describe any relationship you see between the two variables. The hours of daylight increase from a minimum of 10 in January to a maximum of 15 in June (midsummer) and then symmetrically decreases to the winter minimum of 10 in December. Since the winter solstice occurs in the latter part of December and the summer solstice in the latter part of June, these patterns make sense. b. On a grid, sketch a coordinate graph of the data. Put months on the x-axis and daylight hours on the y-axis. What patterns do you see? A graph of the (month, daylight hours) data illustrates the pattern described in part (a). 4 8 12 16 20 Month Daylight in Chicago Hours of Daylight 4 6 8 10 12 x-axis y-axis ACE CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Applications—Connections—Extensions (ACE) 113 ACACEE 21. The seasons in the Southern Hemisphere are the opposite of the seasons in the Northern Hemisphere. When it is summer in North America, it is winter in Australia. Chicago is about the same distance north of the equator as Melbourne, Australia, is south of the equator. Use the Chicago data in ACE 20 for Ace 21a. a. Sketch a graph showing the relationship you would expect to find between the month and the hours of daylight in Melbourne. Answer: 4 8 12 16 20 Month Daylight in Melbourne Hours of Daylight 4 6 8 10 12 x-axis y-axis b. Put the (month, daylight) values from your graph in ACE 21a into a table. Answer: Month 1 2 3 4 5 6 7 8 9 10 11 12 Daylight Hours 14.5 13.8 12.5 11.0 10.5 10.0 10.0 10.2 11.7 13.1 14.3 15.0 22. a. A school club sells sweatshirts to raise money. Which, if any, of the following graphs describes the relationship you would expect between the price charged for each sweatshirt and the profit? Explain your choice, or draw a new graph that you think better describes this relationship. Price Profit Graph 1 Price Profit Graph 2 Price Profit Graph 3 Price Profit Graph 4 Answers will vary. Students might make a reasonable argument for any of the graphs. Yet some graphs seem to be better than others. The following arguments assume that the intersection of the x- and y-axes is point (0, 0) on all graphs. Unlike Graph 1, Graphs 2, 3, and 4 show that there is a price that results in the maximum profit. Graph 4 is a better representation because Graph 3 shows the unlikely event of making a profit at a very low price for each shirt. Students might draw a more detailed graph that shows a negative profit (loss) when the price is too low. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
114 Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs b. What variables might affect the club’s profits? Answers will vary. Possible answers: selling price, price the club must pay for the sweatshirts, the location and times the club chooses for selling sweatshirts, and customer demand (which might depend on other variables such as income and weather). 23. In ACE 23a–e, how does the value of one variable changes as the value of the other changes? Estimate pairs of values that show the pattern of change you would expect, and record your estimates in a table with at least five data points. Sample: Number of hours of television you watch in a week and your school grade point average. Answer: As television time increases, I expect my grade point average to decrease. See the table below. TV Time (hours per week) 0 5 10 15 20 Grade Point Average 3.5 3.25 3.0 2.75 2.5 a. Distance from school to your home and time it takes to walk home You would expect that the greater the distance from school to your home, the longer it will take to walk home from school. For example: Distance (miles) Time (minutes) __1 4 5 __1 2 10 __3 4 15 1 20 1 __1 2 30 ACE CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Applications—Connections—Extensions (ACE) 115 ACACEE b. Price of popcorn at a theater and number of bags sold You would expect the number of bags of popcorn sold at a theater to decrease as the price increases. For example: Price of Popcorn at Theater ($) Number of Bags Sold 2 50 4 40 6 30 8 20 10 10 c. Speed of an airplane and time it takes the plane to complete a 500-mile trip You would expect the time it takes a plane to complete a 500-mile trip to decrease as the speed increases. For example: Speed (miles per hour) Time (hours) 100 5.0 125 4.0 150 3.33 175 2.86 200 2.5 d. Streaming TV bill and number of movies rented You would expect a streaming TV bill to increase as the number of movies rented increases. For example: Number of Movies Cost of Bill ($) 1 6 2 12 3 18 4 24 5 30 e. Points earned for reaching a new level of a game and number of levels accomplished You would expect the points to increase as the success in levels increases. For example: Levels Points 1 30 5 150 10 300 15 450 20 600 CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
116 Checkup 1 Checkup 1 Stefan and Anita do an exercise called a lunge. They experiment doing lunges for 40 seconds. 1. What are the two variables? 2. Whose data does this table represent? Circle the answer. Stefan’s Description of Data I started out doing about 1 lunge every 1 second. But as time went on, I got slower and slower. Anita’s Description of Data I was consistent with my lunges. I did about 5 lunges in every 5 seconds. I was able to keep this pace for most of the time. Stefan Anita Time (seconds) Total Number of Lunges 0 0 5 5 10 10 15 15 20 18 25 22 30 23 35 24 40 25 Stefan Anita Time (seconds) Total Number of Lunges 0 0 5 5 10 11 15 16 20 20 25 24 30 29 35 34 40 38 3. Whose data does this table represent? Circle the answer. Name Date Class CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Checkup 1 117 Name Date Class 4. Whose data does this graph represent? Circle the answer. 5. Whose data does this table represent? Circle the answer. 6. Estimate how many lunges Stefan would do at 45 seconds. Explain how you got your estimate. 7. Estimate how many seconds it took Anita to reach a total of 25 lunges. Explain how you got your estimate. Time (seconds) Total Number of Lunges 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 y-axis x-axis Stefan Anita Time (seconds) Total Number of Lunges 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 y-axis x-axis Stefan Anita CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
118 Checkup 1: Answers Checkup 1: Answers Stefan and Anita do an exercise called a lunge. They experiment doing lunges for 40 seconds. 1. What are the two variables? Time (seconds) and Total Number of Lunges 2. Whose data does this table represent? Circle the answer. Stefan Anita Time (seconds) Total Number of Lunges 0 0 5 5 10 10 15 15 20 18 25 22 30 23 35 24 40 25 Stefan Anita Time (seconds) Total Number of Lunges 0 0 5 5 10 11 15 16 20 20 25 24 30 29 35 34 40 38 Stefan’s Description of Data I started out doing about 1 lunge every 1 second. But as time went on, I got slower and slower. Anita’s Description of Data I was consistent with my lunges. I did about 5 lunges in every 5 seconds. I was able to keep this pace for most of the time. 3. Whose data does this table represent? Circle the answer. CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
Checkup 1: Answers 119 4. Whose data does this graph represent? Circle the answer. 5. Whose data does this table represent? Circle the answer. Time (seconds) Total Number of Lunges 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 y-axis x-axis Stefan Anita Time (seconds) Total Number of Lunges 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 y-axis x-axis Stefan Anita 6. Estimate how many lunges Stefan would do at 45 seconds. Explain how you got your estimate. About 25 or 26, maybe 27 Stefan said that he got slower. So he may have stopped by 45 seconds with 25 lunges. Or he may have continued to do 1 or 2 lunges like he was doing at the end. Estimating the table extended Estimating the graph extended Time in seconds Total Number of Lunges 0 0 5 5 10 10 15 15 20 18 25 22 30 23 35 24 40 25 45 26 Time (seconds) Total Number of Lunges 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 45 y-axis x-axis CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE
120 Checkup 1: Answers 7. Estimate how many seconds it took Anita to reach a total of 25 lunges. Explain how you got your estimate. A little more than 25 seconds About 26 or 27 seconds Informally using rate reasoning from the description Anita is doing about 1 lunge every 1 second. So 25 lunges would take about 25 seconds. Estimating from the table by extending the table in the middle Estimating in between values on the graph Time in seconds Total Number of Lunges 0 0 5 5 10 11 15 16 20 20 25 24 26 25 27 26 28 27 29 28 30 29 35 34 40 38 Time (seconds) Total Number of Lunges 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 y-axis x-axis CMP4 Sample © 2025 by Michigan State University. Published by Lab-Aids, Inc. All rights reserved. SAMPLE