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20.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR8: Discussion Supports

Materials

None

Activity Narrative

In this lesson, the meaning of the words “model” and “modeling” are explained in terms of a linear model. Students are not expected to define the words, but should be comfortable understanding and using them. A model is used to predict future battery life of a device not included in the data, as well as compare existing data points to the model.

In the digital version of the activity, students use an applet to analyze a scatter plot with a linear model. The applet allows students to interact with the linear model and data to answer questions.

If necessary, tell students that a “function” is another way to talk about a relationship between two variables.

Launch

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

To introduce the context, tell students that batteries in rechargeable devices will lose capacity over time and the device will not work for as long. In the experiment from this activity, when the devices were first bought, they were left on until they turned off automatically from not having enough battery to continue running. Then the devices were used regularly and the experiment was run again 1 year later.

Ask students,

“Do you think battery life 1 year later will be longer, about the same, or shorter than when the device is new?” (It will be shorter because the battery wears out through use.)
“What are some things that you think affect how the battery life changes from 1 year of use?” (the materials used in the battery, how much the device is used during the year, the type of device)

Tell students that they will look at some data for battery life in rechargeable devices. Previously, they have used mathematics to analyze real-world situations, identifying variables in a situation and describing their relationships mathematically. This process is called “modeling,” and the mathematical description is called a “model.” Sometimes students make assumptions about the situation or ignore some features so the model is simpler.

Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “It looks like . . . .” and “We are trying to . . . .”
Supports accessibility for: Language, Organization

MLR8 Discussion Supports. Display sentence frames to support students as they work with their partner and explain their reasoning: “I think because .” or “I (agree/disagree) because .”
Advances: Speaking, Conversing

Activity Synthesis

The goal of this discussion is to help students understand the relationship between the data and a linear model of the data.

To highlight the relationship, ask:

“What does a point in the scatter plot represent?” (the actual initial and later battery life of a device)
“What does the line represent?” (the predicted later battery life based on the initial battery life of a device)
“What does it mean when a point is close to the line? When it is far away from the line?” (When it is close, the model predicts the later battery life well. The farther away in the vertical direction, the worse the prediction.)
“How can you use the graph to predict the later battery life of a device that has an initial battery life of 20 hours? How can you use the equation?” (Find the point on the line that lines up with or substitute into the model equation.)

20.3

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR8: Discussion Supports Notice and Wonder

Materials

None

Activity Narrative

A scatter plot is shown and the points interpreted in context. Later, a linear model is graphed with the scatter plot to help students see an obvious outlier (MP7). In the discussion, the term "outlier" is introduced.

In the digital version of the activity, students use an applet to analyze a scatter plot with a linear model. The applet allows students to more easily examine the data and linear model. Use the digital version if available to allow students to focus on analyzing the scatter plot without worrying about construction.

Launch

Keep students in groups of 2. Display the image of the feet.

<p>Images of 3 different drawn feet.</p>

Ask students, “What do you notice? What do you wonder?” Ideally, they will notice that these are pictures of a foot of 3 different people. All 3 feet are approximately the same length, but different widths. Tell students that all of these feet are a size 8. However, they wouldn’t all necessarily find the same shoe equally comfortable, because of the varying widths. (Besides the numerical size, some shoes also come in different widths.) Students should understand that human feet can vary in both length and width.

MLR8 Discussion Supports. Use multimodal examples to show the meaning of width and length. Use verbal descriptions along with gestures, drawings, or concrete objects to show their meaning on the image.
Advances: Listening, Representing

Activity

None

Here is a scatter plot that shows lengths and widths of 20 different left feet.

Estimate the widths of the longest foot and the shortest foot.
Estimate the lengths of the widest foot and the narrowest foot.
Here is the same scatter plot together with the graph of a model for the relationship between foot length and width.

A scatterplot. Horizontal, from 20 to 32, by 2's, labeled foot length in centimeters. Vertical, from 7 to 12, by 1’s, labeled foot width in centimeters. 20 dots trend upward and to the right. Line drawn, trends linearly upward and right with 11 dots above lie and 9 below. No dots lie on the line. The line begins at about point 21 point 9 comma 9 and ends at about 31 point 25 comma 11 point 5.

Circle the data point that seems weird when compared to the model. What length and width does that point represent?

Activity Synthesis

Introduce the term outlier. An outlier is a point that is separated from the rest of the data. Sometimes data sets have outliers. There could be an outlier because there really is a data point that is very different than the others. Or, there could be an error in collecting or entering the data. When there are outliers, one has to make a judgment about whether to include it in the analysis or not.

In an experiment, a group gathers different rechargeable devices like phones, watches, electric toothbrushes, headphones, and cordless vacuums. They use each device until the battery runs out and record the initial battery life in hours. They use the devices for a year and then do the same experiment 1 year later, and record the battery life in hours again.

In this table, the first column shows battery life for 20 devices initially and the second column shows battery life 1 year later.

initial battery life (hours)	later battery life (hours)	predicted later battery life (hours)
12.8	12.5	10.78
14.1	12.6	11.885
17.5	14.7	14.775
19.1	16.1	16.135
22.7	17.8	19.195
25.8	25	21.83
26.6	21	22.51
26.8	21.5	22.68
27.5	22	23.275
29	25.7	24.55
31	27.6	26.25
31	24.3	26.25
33.1	26.8	28.035
33.7	29.5	28.545
34.7	29.3	29.395
35.1	29	29.735
35.6	31.7	30.16
42	34.8	35.6
42.9	37.1	36.365
46.8	41.1	39.68

The scatter plot shows the battery life measurements for 20 devices together with the graph of .

The function described by the equation is a model of the relationship between the initial and later battery life for a device.

This model predicts the later battery life from the device's initial battery life. These predicted battery lives are shown in the third column of the table.

Two devices that both have an initial battery life of 31 hours have different later battery life. What are their later battery lives? How can you see this in the table? How can you see this in the graph?
The model predicts that when the initial battery life is 31 hours, the later battery life will be 26.25. How can you see this in the graph? How can you see this using the equation?
One of the devices has an initial battery life of 29 hours. What does the model predict for its later battery life? How does that compare to the actual later battery life?
Find a device for which the model makes a very good prediction of the actual later battery life. How can you see this in the table? In the graph?
Find a device for which the model’s prediction is not very close to the actual later battery life. How can you see this in the table? In the graph?

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Lesson 20

Fitting a Line to Data

Warm-up

Instructional Routines

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Activity

Student Response

Building on Student Thinking

Activity Synthesis

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Addressing

8.SP.A.1

8.SP.A.2

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Activity

Standards Alignment

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Addressing

8.SP.A.1

8.SP.A.2

Building Toward

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Activity

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Standards Alignment

Building On

Addressing

8.SP.A.1

Building Toward

Activity

Student Response

Building on Student Thinking

Student Response

Building on Student Thinking