Lesson 3

What a Point in a Scatter Plot Means

3.1: The Giant Panda (5 minutes)

Warm-up

The purpose of this warm-up is for students to interpret the meaning of a single point in a scatter plot by looking at the point's coordinates and the graph's axis labels (MP2). 

Launch

Display the graph for all to see. Give students 1 minute of quiet think time followed by a whole-class discussion. 

Student Facing

A giant panda lives in a zoo. What does the point on the graph tell you about the panda?

Photograph. Giant Panda.
Point graphed in coordinate plane. 

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their responses. Record and display their responses for all to see. If all of the students responds, “At 36 months of age, the panda weighed 82 kilograms,” ask them the following questions to clarify where they found that information in the graph:

  • "What did the first coordinate of the point tell you?" (The first coordinate shows how far to go on the horizontal axis.)
  • "What did the second coordinate of the point tell you?" (The second coordinate shows how far to go on the vertical axis.)
  • "How did you know the meaning of each of the numbers?" (You can read the meaning of the numbers by looking at the labels on the axes.)
  • "What specific information did each axis give you?" (The labels on the axes indicate attribute, age or weight, and the units in which each attribute is measured, months or kilograms.)

3.2: Weight and Fuel Efficiency (15 minutes)

Activity

The analysis of scatter plots continues with data about the mass of automobiles and their fuel efficiency. Again, students connect points in the scatter plot with rows of data in a table, but now there is a third column that gives a name to each pair.

After picking out information about points in the scatter plot and table, students make their first foray into thinking about how scatter plots can help them make predictions. Specifically, when looking at two particular points in the scatter plot, students are asked if the results are surprising given the overall trend of the data. In later lessons, students develop more sophisticated tools for answering questions like this (specifically, by fitting lines to the data), so a heuristic discussion is all that is expected at this point.

Launch

Explain a bit about fuel efficiency so that students understand this measurement. “Fuel efficiency” is a measure of the average distance a car will travel using a certain amount of gas. Commonly this is measured in "miles per gallon." For example, a car that has a fuel efficiency of 25 miles per gallon (mpg) should be able to drive approximately 25 miles while using up a gallon of gas. Many factors can influence fuel efficiency, including the way an engine is engineered (to produce more power, an engine may use more gasoline), driving conditions (more frequent stopping and starting or hills will use fuel less efficiently), and what accessories are being used (air conditioning requires energy that is not used for actually driving the car). 

Before beginning the activity, ask “What would a scatter plot including these two variables look like?” to encourage students to predict how a car's mass might influence fuel efficiency.

If using the digital version of the activity, after completing the launch discussion, students will use an applet to analyze a scatter plot, identify points, and draw conclusions about the data. The digital activity mirrors the analog activity. 

Writing, Conversing: MLR5 Co-Craft Questions. Display only the table without revealing the graph or questions that follow. Ask students to work with a partner to write possible questions that could be answered by the data in the table. Invite 2–3 groups to share their questions with the class. Look to see whether/how the weights and fuel efficiencies of the vehicles shows up in students’ questions. Next, reveal the questions of the activity. This routine helps students consider the context of this problem and to increase awareness about language used to talk about weights and fuel efficiencies.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

The table shows the weight and fuel efficiency of 18 different cars.

car weight (kg) fuel efficiency (mpg)
A \(1,\!549\) 25
B \(1,\!610\) 20
C \(1,\!737\) 21
D \(1,\!777\) 20
E \(1,\!486\) 23
F \(1,\!962\) 16
G \(2,\!384\) 16
H \(1,\!957\) 19
I \(2,\!212\) 16
J \(1,\!115\) 29
K \(2,\!068\) 18
L \(1,\!663\) 19
M \(2,\!216\) 18
N \(1,\!432\) 25
O \(1,\!987\) 18
P \(1,\!580\) 26
Q \(1,\!234\) 30
R \(1,\!656\) 23

The data points in the table are shown in this scatter plot.

You can hide the expressions list using the double arrow. You can click on a point to see its coordinates.

 
  1. Which point in the scatter plot represents Car L’s measurements? Drag the circle around the correct point.

  2. What is the fuel efficiency of the car with the greatest weight?

  3. What is the weight of the car with the greatest fuel efficiency?

  4. Car S weighs 1,912 kilograms and gets 16 miles per gallon. On the scatter plot, plot a point that represents Car S’s measurements.

  5. Cars N and O, shown in the scatter plot, are made by the same company. Compare their weights and fuel efficiencies. Does anything surprise you about these cars?

  6. A different company makes Cars F and G. Compare their weights and fuel efficiencies. Does anything surprise you about these cars?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Explain a bit about fuel efficiency so that students understand this measurement. “Fuel efficiency” is a measure of the average distance a car will travel using a certain amount of gas. Commonly this is measured in "miles per gallon." For example, a car that has a fuel efficiency of 25 miles per gallon (mpg) should be able to drive approximately 25 miles while using up a gallon of gas. Many factors can influence fuel efficiency, including the way an engine is engineered (to produce more power, an engine may use more gasoline), driving conditions (more frequent stopping and starting or hills will use fuel less efficiently), and what accessories are being used (air conditioning requires energy that is not used for actually driving the car). 

Before beginning the activity, ask “What would a scatter plot including these two variables look like?” to encourage students to predict how a car's mass might influence fuel efficiency.

If using the digital version of the activity, after completing the launch discussion, students will use an applet to analyze a scatter plot, identify points, and draw conclusions about the data. The digital activity mirrors the analog activity. 

Writing, Conversing: MLR5 Co-Craft Questions. Display only the table without revealing the graph or questions that follow. Ask students to work with a partner to write possible questions that could be answered by the data in the table. Invite 2–3 groups to share their questions with the class. Look to see whether/how the weights and fuel efficiencies of the vehicles shows up in students’ questions. Next, reveal the questions of the activity. This routine helps students consider the context of this problem and to increase awareness about language used to talk about weights and fuel efficiencies.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

The table and scatter plot show weights and fuel efficiencies of 18 cars.

car weight
(kg)
fuel efficiency
A 1,549 25
B 1,610 20
C 1,737 21
D 1,777 20
E 1,486 23
F 1,962 16
G 2,384 16
H 1,957 19
I 2,212 16
J 1,115 29
K 2,068 18
L 1,663 19
M 2,216 18
N 1,432 25
O 1,987 18
P 1,580 26
Q 1,234 30
R 1,656 23
A scatterplot.
  1. Which point in the scatter plot represents Car L’s measurements?

  2. What is the fuel efficiency of the car with the greatest weight?

  3. What is the weight of the car with the greatest fuel efficiency?

  4. Car S weighs 1,912 kilograms and gets 16 miles per gallon. On the scatter plot, plot a point that represents Car S’s measurements.

  5. Cars N and O, shown in the scatter plot, are made by the same company. Compare their weights and fuel efficiencies. Does anything surprise you about these cars?

  6. A different company makes Cars F and G. Compare their weights and fuel efficiencies. Does anything surprise you about these cars?
     

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

After a board game competition, the tournament director collects 50 dice from the games played and rolls each one until he gets bored and tries a different one. The scatter plot shows the number of times he rolled each die and the number of 6s that resulted during those rolls.

Scatterplot, number of times die is rolled, 0 to 220 by 20, number of sixes, 0 to 40 by 10. Points include 50 comma 7, 100 comma 13, 140 comma 19, and 162 comma 28,

Select a point in the scatter plot and give its approximate coordinates, then tell the story of that point in the context of the problem.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of the discussion is for students to compare pairs of data in both representations and to begin thinking about trends in the data. Ask students to share their thinking about the last two questions. Draw out the fact that cars N and O seem to fit the overall trend that an increase in mass typically corresponds to a decrease in fuel efficiency but cars F and G don't fit that trend when compared with each other. Mention that we will see many examples of trends that don't necessarily apply to individuals in a population.

Consider asking some of the following questions to conclude the discussion:

  • “How did you find the automobile with the greatest weight?” (I looked for the point furthest to the left.)
  • “How did you find the automobile with the greatest fuel efficiency?” (I looked for the point that is highest.)
  • “The data in this table is from automobiles that run solely on gasoline. Where do you think a hybrid car would appear on the graph?” (A hybrid car would use less gas for the same weight, so it's point would be below the other points with a similar weight.)
  • “Did the scatter plot match your prediction?”

3.3: Coat Sales (15 minutes)

Activity

In this activity, students continue to identify points on a scatter plot as representatives of a single month when two variables are measured. Students are also asked to explain the meaning of some abstract points in the context of the problem and identify when it does not make sense to extrapolate information from the graph based on the context (MP2). 

Monitor for students who understand that 60 degrees Celsius is unreasonable for an average monthly temperature in most parts of the world, as well as those who identify the trend to predict negative sales which does not make sense in this context either.

Launch

Arrange students in groups of 2. Allow students 5 minutes quiet work time followed by 5 minutes partner discussion and whole-class discussion.

Ask students to predict what a scatter plot might look like when the temperature is along the \(x\)-axis and the sales of coats at a store is along the \(y\)-axis. 

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, ask students to use the different colors to represent and annotate temperature and coat sales in the graph and table.
Supports accessibility for: Visual-spatial processing

Student Facing

A clothing store keeps track of the average monthly temperature in degrees Celsius and coat sales in dollars.

temperature (degrees Celsius) coat sales (dollars)
-5 1,550
-3 1,340
3 1,060
8 1,070
15 680
21 490
23 410
21 510
17 600
11 740
6 940
-2 1,390
A scatterplot, x, temperature in degrees celsius, y, coat sales in dollars. Points given in the accompanying table.
  1. What does the point \((15, 680)\) represent?
  2. For the month with the lowest average temperature, estimate the total amount made from coat sales. Explain how you used the table to find this information.
  3. For the month with the smallest coat sales, estimate the average monthly temperature. Explain how you used the scatter plot to find this information.
  4. If there were a point at \((0, A)\) what would it represent? Use the scatter plot to estimate a value for \(A\).
  5. What would a point at \((B, 0)\) represent? Use the scatter plot to estimate a value for \(B\).
  6. Would it make sense to use this trend to estimate the value of sales when the average monthly temperature is 60 degrees Celsius? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

​The purpose of the discussion is for students to understand that each point in the graph represents a single month in which two measurements are made. Select students to share their responses, including those identified for the last problem. 

Consider asking some of the following questions:

  • “When discussing functions, what is a point of the form \((0,A)\) called? What about a point of the form \((B,0)\)?” (The \(y\)-intercept. The \(x\)-intercept.)
  • “In a scatter plot is there a \(y\)-intercept?” (It depends on the context. For some situations, it may not make sense for the variable represented along the \(x\)-axis to be zero. For example if the variable is the height of a building. On other situations, there may be more than one point along the \(y\)-axis. For example, in a situation like the one in the task, there may be multiple months where the average temperature is 0 degree Celsius.)
  • “When the average monthly temperature goes up, what seems to be happening to sales?” (They are decreasing.)
Writing, Speaking: MLR1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their response to the last question. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help students strengthen their ideas and clarify their language (e.g., “Can you give an example?”, “Why do you think…?”, “How do you know…?”, etc.). Students can borrow ideas and language from each partner to strengthen their final version.
Design Principle(s): Optimize output (for explanation)

Lesson Synthesis

Lesson Synthesis

Revisit the idea that a point in a scatter plot represents an individual in a population. Ask students:

  • "What kind of information does a point in a scatter plot represent?" (Two measurements for an individual in a population.)
  • "How do we know what information is captured by a point in a scatter plot?" (The axis labels tell us how to interpret the coordinates of the points.)
  • "Do trends in a scatter plot necessarily apply to individuals in the population?" (No.)

3.4: Cool-down - Quarterbacks (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Scatter plots show two measurements for each individual from a group. For example, this scatter plot shows the weight and height for each dog from a group of 25 dogs. 

Scatterplot. 

We can see that the tallest dogs are 27 inches, and that one of those tallest dogs weighs about 75 pounds while the other weighs about 110 pounds. This shows us that dog weight is not a function of dog height because there would be two different outputs for the same input. But we can see a general trend: Taller dogs tend to weigh more than shorter dogs. There are exceptions. For example, there is a dog that is 18 inches tall and weighs over 50 pounds, and there is another dog that is 21 inches tall but weighs less than 30 pounds.

When we collect data by measuring attributes like height, weight, area, or volume, we call the data numerical data (or measurement data), and we say that height, weight, area, or volume is a numerical variable. Upcoming lessons will discuss how to identify and describe trends in data that has been collected.