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Unit 3, Lesson 7

Confident Models

Extra Support Materials for Algebra 1

7.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Math Talk MLR8: Discussion Supports

Materials

None

Activity Narrative

This Math Talk focuses on ordering positive and negative decimals. It encourages students to think about how decimals compare to each other and to rely on what they know about ordering numbers to mentally solve problems. The understandings elicited here will be helpful in the associated Algebra 1 lesson when students interpret correlation coefficients.

In defending their answers, students need to be precise in their word choice and use of language (MP6).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Before moving to the next problem, use the questions in the Activity Synthesis to involve more students in the conversation.

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Activity

None

Mentally order the numbers from least to greatest.

20.2, 18.2, 19.2
-14.6, -16.7, -15.1
-0.43, -0.87, -0.66
0.50, -0.52, 0.05

Student Response

to access Student Response.

Activity Synthesis

The goal of this discussion is to review students’ strategies for ordering positive and negative decimals.

To involve more students in the conversation, consider asking:

“Who can restate ’s reasoning in a different way?”
“Did anyone have the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
"What connections to previous problems do you see?"

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:

“First, I because .”
“I noticed , so I .”

Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing

7.2

Activity

Instructional Routines

Aspects of Mathematical Modeling

Materials

None

Activity Narrative

The purpose of this activity is for students to look at different scatter plots and rank them in order from the linear model being a very bad fit to the data to the linear model being a very good fit to the data. As students evaluate how well a linear model fits the data, they are practicing choosing which type of function to use to model a set of data, which is an aspect of mathematical modeling (MP4).

Launch

Allow students to work individually or with a partner. Allow students about 10 minutes for the first part of the Task Statement.

Activity

None

7.3

Activity

Instructional Routines

MLR3: Critique, Correct, Clarify

Materials

None

Activity Narrative

The purpose of this activity is for students to synthesize data displayed in a scatter plot, using aspects of a linear model. Students practice using the linear models to interpret the data, reasoning abstractly and quantitatively (MP2) as they move between the graph of the linear model and what the graph means in context.

This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.

Launch

Arrange students in groups of 2.

Activity

None

Here are scatter plots that represent various situations. Order the scatter plots from “A linear model is not a good fit for the data” to “A linear model is an excellent fit for the data.”

A

B

C

Scatter plot on grid, origin O. Horizontal axis, men’s ages, from 0 to 40, by 10’s. Vertical axis, length of beard, centimeters, from 0 to 35, by 5’s. 39 data points at 15 comma 15, 18 comma 0, 19 comma 6, 19 comma 14, 20 comma 7, 20 comma 4, 22 comma 0, 22 comma 12, 23 comma 3, 25 comma 22, 25 comma 25, 26 comma 26, 26 comma 9, 27 comma 14, 27 comma 25, 27 comma 27, 28 comma 18, 28 comma 25, 29 comma 22, 29 comma 20, 30 comma 13, 32 comma 11, 32 comma 12, 34 comma 22, 32 comma 19, 35 comma 22, 25 comma 28, 36 comma 15, 36 comma 17, 36 comma 18, 36 comma 23, 37 comma 14, 37 comma 20, 38 comma 16, 38 comma 23, 38 comma 33, 39 comma 16, 39 comma 19 and 39 comma 32.

D

Scatter plot on grid, origin O. Horizontal axis, time eating lunch, minutes, from 0 to 45 by 5’s. Vertical axis, number of absences, from 0 to 20, by 5’s. 50 data points. 1 comma 3, 1 comma 14, 3 comma 14, 4 comma 5, 5 comma 17, 7 comma 1, 10 comma 2, 11 comma 10, 12 comma 3, 12 comma 19, 13 comma 11, 15 comma 8, 15 comma 12, 15 comma 14, 19 comma 5, 19 comma 15, 20 comma 4, 20 comma 13, 20 comma 17, 21 comma 5, 21 comma 10, 21 comma 10, 22 comma 7, 23 comma 1, 23 comma 6, 25 comma 10, 26 comma 7, 27 comma 0, 29 comma 6, 30 comma 1, 30 comma 3, 30 comma 8, 30 comma 17, 32 comma 3, 33 comma 0, 33 comma 8, 35 comma 11, 37 comma 11, 38 comma 2, 38 comma 15, 38 comma 19, 39 comma 8, 39 comma 18, 40 comma 0, 40 comma 12, 40 comma 18, 42 comma 12, 43 comma 7, 43 comma 7, 43 comma 15.

E

Scatter plot on grid, origin O. Horizontal axis, time sharpening the pencil, seconds, from 0 to 30 by 5’s. Vertical axis, length of pencil, millimeters, from 0 to 150 by 30’s. 50 data points. 1 comma 148, 2 comma 133, 3 comma 139, 4 comma 123, 4 comma 127, 5 comma 117, 7 comma 132, 8 comma 102, 8 comma 106, 9 comma 98, 9 comma 119, 9 comma 123, 10 comma 108, 10 comma 108, 11 comma 110, 12 comma 75, 12 comma 76, 12 comma 78, 15 comma 56, 15 comma 89, 16 comma 50, 16 comma 56, 17 comma 46, 17 comma 48, 17 comma 64, 19 comma 57, 19 comma 60, 19 comma 60, 19 comma 66, 20 comma 38, 22 comma 22, 22 comma 24, 22 comma 24, 22 comma 41, 22 comma 43, 23 comma 50, 24 comma 32, 24 comma 33, 25 comma 5, 25 comma 10, 25 comma 43, 26 comma 19, 26 comma 23, 26 comma 27, 27 comma 6, 27 comma 9, 27 comma 31, 28 comma 18, 29 comma 10, 29 comma 18.
Here are two scatter plots including a linear model. For each model, determine the when is 15. Which model prediction do you think is closer to the real data? Explain your reasoning.

<p>Scatter plot and line on grid.</p> — Graph F.

A first quadrant scatter plot comparing time for dog haircut (minutes) with dog hair length after grooming (mm) with the line of best fit. — Graph G.

Here are situations represented with graphs and linear models of the data. Use the information given to complete the missing fields for each situation.

The model predicts how much money, in dollars, the coach will make based on how many athletes sign up for one-on-one training. The model is represented with the equation .
Scatter plot and line on grid, origin O. Horizontal axis, number of athletes, from 0 to 35 by 5’s. Vertical axis, amount, dollars, from 0 to 1400 by 200’s. Line, moving upwards and to the right, passes through 5 comma 325, 10 comma 450, 15 comma 575, 20 comma 700, 25 comma 825, and 30 comma 950. 10 data points, located at 4 comma 300, 5 comma 375, 9 comma 410, 10 comma 420, 13 comma 525, 15 comma 60, 20 comma 700, 25 comma 820, 27 comma 830, and 30 comma 950.
- The slope of the model is (positive or negative).
- What does the model predict would be the amount the coach makes when there are 10 athletes present?
- Using the data points and the model as a reference, what is a reasonable range of money the coach will make when there are 10 athletes present?
- This model is a (great, good, okay, or bad) fit for the data.
- Using numbers between 0 and 1, rate your confidence in the model where 0 is no confidence and 1 is total confidence.
The model predicts the annual salary of a worker in a certain government position based on years of experience. The model is represented with the equation .
Scatter plot and line on grid, origin O. Horizontal axis, years of experience, from 0 to 20 by 5’s. Vertical axis, salary, thousands of dollars, from 0 to 70 by 10’s. Line moving upwards and to the right, passes through 3 comma 39 point 5, 10 comma 50, and ends at 15 comma 57 point 5. 9 data points at 3 comma 40, 5 comma 30, 7 comma 50, 9 comma 48, 10 comma 60, 11 comma 58, 12 comma 61 and 14 comma 59. 6 points fall above the line, 2 points fall on the line, 1 point falls well below the line.
- The slope of the model is (positive or negative).
- What does the model predict would be the employee’s salary when the employee has 10 years of experience?
- Using the data points and the model as a reference, what is a reasonable range for the salary of a worker based on 10 years of experience?
- This model is a (great, good, okay, or bad) fit for the data.
- Using numbers between 0 and 1, rate your confidence in the model where 0 is no confidence and 1 is total confidence.
The model predicts the number of absences a school will have based on the number of incentives given per month. The model is represented with the equation .
Scatter plot and line on grid, origin O. Horizontal axis, number of incentives, from 0 to 20 by 5’s. Vertical axis, number of absences, from 0 to 70 by 10’s. Line, moving downward and to the right, passes through 0 comma 54 point 78, 5 comma 65 point 68, 10 comma 76 point 58 and 15 comma 87 point 48. 30 data points at 0 comma 48, 1 comma 60, 2 comma 42, 3 comma 51, 5 comma 55, 6 comma 32, 6 comma 46, 6 comma 60, 7 comma 30, 7 comma 37, 7 comma 5, 8 comma 13, 8 comma 33, 9 comma 22, 9 comma 32, 9 comma 46, 10 comma 25, 10 comma 37, 10 comma 52, 11 comma 28, 11 comma 30, 11 comma 33, 12 comma 0, 13 comma 28, 13 comma 33, 13 comma 40, 13 comma 53, 14 comma 6, 14 comma 15 and 15 comma 28.
- The slope of the model is (positive or negative).
- What does the model predict would be the number of absences when 10 incentives are given for the month?
- Using the data points and the model as a reference, what is a reasonable number of absences when there are 10 incentives given?
- This model is a (great, good, okay, or bad) fit for the data.
- Using numbers between 0 and 1, rate your confidence in the model where 0 is no confidence and 1 is total confidence.

Activity Synthesis

The purpose of this discussion is for students to make a connection between scatter plots and how to use a linear model to interpret the relationship between the variables presented in the scatter plot.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response explaining how well a model fits the data, by correcting errors, clarifying meaning, and adding details.

Display this first draft:
The first graph’s model is not a good fit because, while it is close to the data, it does not touch every point. To be a good fit, every point should be on the line.
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Discuss how students made sense of the problems and ranked their confidence in the models. Here are sample questions to promote class discussion:

“How did you determine the ‘reasonable ranges' of values?” (I looked at how close the points are to the line and used that to determine a range on either side of the line for the given -value.)
“How did you determine what type of fit the linear model is to the data presented in the scatter plot?” (I looked at how close the points are to the line. If the points are very close to the line, it is a good fit. If the points are farther from the linear model, then the fit is not very good.)
“What things did you consider when determining how confident you are in the linear models?” (I considered if the linear model was a good fit to the data or not, how close the predicted -values were to the actual -values, and the strength of my confidence in the model.)

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Lesson 7

Confident Models

Warm-up

Standards Alignment

Building On

Addressing

5.NBT.A.3

Building Toward

HSS-ID.C.8

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Building on Student Thinking

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSS-ID.B

Building Toward

HSS-ID.C

Standards Alignment

Building On

Addressing

HSS-ID.C.7

Building Toward

HSS-ID.C.8