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Unit 6, Lesson 15

Changing the Score

Extra Support Materials for Algebra 1

15.1

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 expressing percent change as a single expression. It encourages students to think about percent change and to rely on what they know about the distributive property to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students make calculations.

To express the percent change as a single expression, students need to look for and make use of structure (MP7).

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.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.

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

Express mentally each percent change using an expression that uses only multiplication.

increased by 5%
decreased by 10%
increased by 25%
decreased by 2.5%

Activity Synthesis

The goal of this discussion is to review students’ strategies for each problem. 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

15.2

Activity

Instructional Routines

None

Materials

To Gather

Number cubes Scientific calculators

Activity Narrative

A common mistake when learning about compounding, as students will in the associated Algebra 1 lesson, is to repeatedly apply the percent change to the original amount instead of the new amount. For example, let’s say a credit card balance is \$100, and 12% interest is charged each year. How much is owed after 5 years? Someone making the common mistake might compute , rather than .

The purpose of this activity is to create a situation in which it’s natural to compute the percent change on the most recent amount, so that students get a feel for the mechanics of this type of change. Monitor for students who are recording meaningful or helpful expressions in the “calculation” row. In this activity, students look for and express regularity in repeated reasoning (MP8) when calculating their scores using percent change.

15.3

Activity

Instructional Routines

MLR1: Stronger and Clearer Each Time

Materials

None

Activity Narrative

The purpose of this activity is to focus on explaining why multiplying the initial amount by a multiple of the growth factor is a mistake. Students construct viable arguments and critique the reasoning of others (MP3) when explaining why the reasoning is incorrect. When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.

Launch

Allow students to work in groups of 2 or individually. Tell students that they will be answering questions about situations similar to those in the previous activity.

Activity

None

Launch

Provide access to scientific calculators. Distribute 1 number cube to each student, and assign each student to one of three groups: A, B, or C. Ask students to read how Round 1 works. Before they get started, ask them to anticipate which group will end up with the greatest scores, and why.

Activity

None

Round 1: Your starting score is 50. Roll your number cube 10 times. If you are in group:

A, your score increases by 5% every time you roll a 4, 5, or 6 (and stays the same otherwise).
B, your score increases by 10% every time you roll a 5 or a 6 (and stays the same otherwise).
C, your score increases by 20% every time you roll a 6 (and stays the same otherwise).

Compute your new score after each roll.

roll	0	1	2	3	4	5	6	7	8	9	10
calculation
new score	50

Round 2: Your starting score is the result from Round 1. Roll your number cube 10 times. If you are in group:

A, your score decreases by 5% every time you roll a 6 (and stays the same otherwise).
B, your score decreases by 10% every time you roll a 5 or a 6 (and stays the same otherwise).
C, your score decreases by 20% every time you roll a 4, 5, or 6 (and stays the same otherwise).

Compute your new score after each roll.

roll	0	1	2	3	4	5	6	7	8	9	10
calculation
new score

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

The main goal of this activity is to understand how the new score was calculated after each roll. Invite a selected student to share their table, including the calculation row, with the class. Ideally, this table will be turned into a representation that shows repeatedly multiplying to calculate the new values. Emphasize that it’s the new score that is increased by the given percentage (not the original score). For example, a completed table for someone in group B, after discussion, might look like the completed table to follow. Point out that the outcome of the third roll is really , and the outcome of the sixth roll is really , or . Tell students that, in their Algebra 1 class, they’ll create exponential functions to model situations like these, in which a starting amount is repeatedly increased (or decreased) by a percentage.

roll	0	1	2	3	4	5	6	7	8	9	10
calculation
new score	50	50	55	60.5	60.5	60.5	66.55	66.55	73.205	73.205	73.205

Activity Synthesis

The goal is to clarify that a repeated percent change happens to the new value, not the original value.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to what was incorrect about Mai’s score. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback:

“ makes sense, but what do you mean when you say ?”
“Can you describe that another way?”
“The part I understood best was .”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, ensure that students can articulate that the fault with Mai's and Han’s reasoning was that they were adding (or subtracting) a percentage of their original score twice, rather than adding (or subtracting) a percentage of their new score for the second successful roll. Here are some questions for discussion:

“Why does calculating the second percentage increase or decrease from the original instead of the new score result in the wrong answer?” (Additional percentages are calculated with the new score. This changes the answer since a new score is more or less than the original.)
“Someone starts with 100 points and makes 5 successful rolls, increasing their score by 15% each time. What is the quickest way to compute their new score?” ()

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Lesson 15

Changing the Score

Warm-up

Standards Alignment

Building On

Addressing

7.RP.A.3

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSF-BF.A.1.a

Building Toward

HSF-LE.A.2

Standards Alignment

Building On

Addressing

HSF-BF.A.1.a

Building Toward

HSF-LE.A.2