The purpose of this Warm-up is to elicit the idea that organizing data is helpful for recognizing patterns, which will be useful when students work with ways or organizing data in a later activity. While students may notice and wonder many things about this table, working towards recognizing any patterns and associations are the important discussion points.
This Warm-up prompts students to familiarize themselves with the context and mathematics that might be involved by making sense of data before organizing it (MP1).
Arrange students in groups of 2. Display the table for all to see. Ask students to think of at least 1 thing they notice and at least 1 thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner.
Activity
None
Here is a table of data. Each row shows 2 measurements of a triangle.
What do you notice? What do you wonder?
length of short side (in)
length of perimeter (in)
0.25
1
2
7.5
6.5
22
3
9.5
0.5
2
1.25
3.5
3.5
12.5
1.5
5
4
14
1
2.5
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the table. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the pattern that both values increase together does not come up during the conversation, ask students to discuss this idea.
1.2
Activity
Standards Alignment
Building On
Addressing
8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
In this activity, students continue to look for patterns and associations for variables by sorting the values in a table. They use the table to make predictions about data that is not included in the table (MP2). In the discussion following the activity, students are asked to consider other methods of finding patterns within the data and see a graphical representation that more clearly shows the strong relationship between the variables. The graphical representation can also help improve predictions over those made with only the tables.
Launch
Keep students in the same groups of 2 from the Warm-up. Explain that the data in the Warm-up came from fourth grade students who were practicing drawing isosceles right triangles and measuring their perimeters. Display an example of an isosceles right triangle.
Ask students to recall what it means for a triangle to be isosceles and right as well as how to measure the perimeter of a triangle. Ask students if they think there should be a relationship between the length of the 2 short sides and the entire perimeter. Remind students that specifying 2 sides and the angle between them does determine a unique triangle, so we might expect that knowing the 2 side lengths and the right angle would be closely related to the length of the perimeter.
Activity
None
Here is the table of isosceles right triangle measurements from the Warm-up and an empty table.
length of short sides (in)
length of perimeter (in)
0.25
1
2
7.5
6.5
22
3
9.5
0.5
2
1.25
3.5
3.5
12.5
1.5
5
4
14
1
2.5
length of short sides (in)
length of perimeter (in)
How can you organize the measurements from the first table so that any patterns are easier to see? Write the organized measurements in the empty table.
For each of the following lengths, estimate the perimeter of an isosceles right triangle whose short sides have that length. Explain your reasoning for each triangle.
length of short sides is 0.75 in
length of short sides is 5 in
length of short sides is 10 in
Student Response
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Building on Student Thinking
Activity Synthesis
Select students to share their arrangements of the data, the patterns they noticed, and their predictions for the triangles with short side lengths. Display the predictions for all to see.
Ask students to share ideas for other ways to look at the data that might lead to better predictions.
Display a graph of the data for all to see.
To highlight features of the graph, ask:
"What patterns do you see in the data when it is graphed?" (It is close to linear.)
"How could you use the graph to estimate the perimeter for the isosceles right triangles with a short side length of 0.75 in, 5 in, or 10 in?" (It is okay for students to not be precise in their answer at this point. Future lessons will give more instruction in how to do this.)
Tell students that this graphical representation of data is called a scatter plot. A scatter plot is when two numerical variables are graphed by using one variable as the -coordinate and the other as the -coordinate. Data pairs are represented as plotted points.
Note that there is a difference between time series graphs and scatter plots. In time series graphs, a single variable is recorded at multiple time points and plotted on a graph. In a scatter plot, 2 variables are measured and plotted on a graph. For scatter plots, time may be one of the variables, but it should be possible to have more than one measurement for the second variable for the same time measurement. For example, when comparing the price of a car to its model year uses the year as one of the variables, but 2 cars made in the same year could have 2 different prices. The price for a single car throughout different years would be represented in a time series graph since it only has one price at any given time.
MLR8 Discussion Supports. Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement “The number is between the other values” as “Because the short side length is about halfway between these other 2 side lengths, the perimeter should also be about halfway between the other 2 perimeters.” Advances: Speaking, Representing
1.3
Activity
Standards Alignment
Building On
Addressing
8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
An essential part of creating and understanding scatter plots is interpreting the meaning of the points (MP2). In this activity, students match tables of data with scatter plots representing the same information (MP7). After matching appropriately, students are asked to include titles for the axes of the scatter plots. Following the activity, the importance of the axis labels is discussed.
Launch
Arrange students in groups of 2. Distribute 1 copy of the tables from the blackline master to each group and resolve any clarifying questions about the data in the tables. In particular, students may wish to know about particular terminology. For example, “kilowatt hours” are a unit of electrical energy that most electricity companies use to measure how much electricity a customer consumes to determine how much to charge them. Another example is “battery life,” which is how long a device can run before its battery dies.
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, check in with students within the first 2–3 minutes of work time. Supports accessibility for: Organization, Attention
Activity
None
Here are four scatter plots. Your teacher will give you four tables of data.
A
Scatterplot, x, 0 to 150,000 by 30,000, y, 6,000 to 21,000 by 3,000. Points begin around 2,000 comma 16,000 and trend down and right toward 120,000 comma 7,000.
B
Scatterplot, x, 72 to 102 by 6, y, 15 to 45 by 5. Points begin around 76 comma 18 and trend up and right toward 100 comma 40.
C
Scatterplot, x, 10 to 50 by fives, y 10 to 45 by fives. Points begin around 13 comma 12 and trend up and right toward 47 comma 41.
D
Scatterplot, x, 1,000 to 2,500 by 250, y, 14 to 32 by 2. Points begin around 1,100 comma 29 and trend down and right toward 2,300 comma 16.
Match each table with one of the scatter plots, then write a title and label the axes for each.
Why do you think some of the scatter plots do not include the origin?
Why do you think the values on the grid lines differ by numbers other than 1?
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to develop strategies for relating different representations of data.
Select 1–2 groups to share their responses to the questions about axis numbering with the class.
Tell students that they should think about the maximum and minimum values as well as the range (the distance between the maximum and minimum values) when setting the scale for the different axes. For the graphs in this unit, it is not usually essential to include the point in the graph, so that makes the axis labels even more important.
To conclude the discussion, consider asking some of these questions:
"How did you match the tables to the scatter plots?" (I looked at the scale of the values and tried to match them to the data in the tables. For example, the used car data had both variables in the thousands, which matches the scatter plot on the top left.)
"Why is it important to include labels for the axes on scatter plots?" (so that any patterns that are found can be recognized from the scatter plot without having to go back to the table)
"Why is it important to include units in the axis labels?" (so that the patterns found can be understood easily)
"The same data is presented in the tables as in the scatter plots. Which is easier to understand? Explain your reasoning." (I think the scatter plots are easier to understand. I can more easily see how the data are related to one another.)
Lesson Synthesis
To highlight the progression of representations seen today (unorganized table, ordered table, scatter plot) that help to highlight any patterns that may be present in data, ask:
“Why would sorting the information in the table be helpful?” (It makes it easier to see the minimum and maximum values.)
“When looking for relationships between 2 variables, what are some graphic representations that might be helpful to use?” (An organized table helps show trends of one variable while another goes from smallest to largest. A scatter plot also shows how one variable changes in relation to the other.)
“What is a scatter plot and how is it different from plotting points for a function?” (For a function, every input has a single output, but a scatter plot uses data that does not have an input or output, and a single value for one variable could have multiple values in the second variable.)
Remind students that, when reorganizing data, it is important to continue to label what the information represents. A table without titles or a scatter plot without labels may show some relationship between numbers, but is meaningless outside of the context.
Student Lesson Summary
Consider the data collected from pulling back a toy car and then letting it go forward. In the first table, the data may not seem to have an obvious pattern. The second table has the same data and shows that both values are increasing together.
Unorganized table:
distance pulled back (in)
distance traveled (in)
6
23.57
4
18.48
10
38.66
8
31.12
2
13.86
1
8.95
Organized table:
distance pulled back (in)
distance traveled (in)
1
8.95
2
13.86
4
18.48
6
23.57
8
31.12
10
38.66
A scatter plot of the data makes the pattern clear enough that we can estimate how far the car will travel when it is pulled back 5 in.
Patterns in data can sometimes become more obvious when reorganized in a table or when represented in scatter plots or other diagrams. If a pattern is observed, it can sometimes be used to make predictions. This is a scatter plot for this scenario:
A scatterplot. Horizontal, from 0 to 12, by 2’s, labeled distance pulled back, inches. Vertical, from 0 to 40, by 10’s, labeled distance traveled, inches. 6 data points. Trend linearly upward and right.
Standards Alignment
Building On
Addressing
Building Toward
8.SP.A
Investigate patterns of association in bivariate data.
