# Lesson 15

Changing the Score

These materials, when encountered before Algebra 1, Unit 5, Lesson 15 support success in that lesson.

## 15.1: Math Talk: Rewriting Expressions (5 minutes)

### Warm-up

This warm-up is a review of previous work, and is intended to help students make calculations more efficiently in this lesson.

### Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

### Student Facing

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

\(x\) increased by 5%

\(y\) decreased by 10%

\(z\) increased by 25%

\(w\) decreased by 2.5%

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

- “Who can restate \(\underline{\hspace{.5in}}\)’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 \(\underline{\hspace{.5in}}\)’s strategy?”
- “Do you agree or disagree? Why?”

## 15.2: Your New Score (25 minutes)

### Activity

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 \(100 + 100 \boldcdot (1.12) \boldcdot 4\), rather than \(100 \boldcdot (1.12)^5\).

The purpose of this activity is to create a situation where 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.

### Launch

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.

### Student Facing

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

For access, consult one of our IM Certified Partners.

### 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, you want to turn this table 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 this. Point out that the outcome of the third roll is really \(50 \boldcdot (1.1) \boldcdot (1.1)\), and the outcome of the sixth roll is really \(50 \boldcdot (1.1) \boldcdot (1.1) \boldcdot (1.1)\), or \(50 \boldcdot (1.1)^3\). Tell students that in their Algebra 1 class, they’ll create exponential functions to model situations like these, where a starting amount is repeatedly increased (or decreased) by a percentage.

roll | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

calculation | \(50\boldcdot1.1\) | \(55\boldcdot1.1\) | \(60.5\boldcdot1.1\) | \(66.55\boldcdot1.1\) | |||||||

new score | 50 | 50 | 55 | 60.5 | 60.5 | 60.5 | 66.55 | 66.55 | 73.205 | 73.205 | 73.205 |

## 15.3: Bad Assumptions (15 minutes)

### Activity

The purpose of this activity is to focus on explaining why the common mistake that we are trying to avoid is a mistake. Students are critiquing the reasoning of others (MP3).

### Student Facing

- Mai started with 100 which increased by 10% for each successful roll. She had 2 successful rolls.
- Mai thinks her score is 120. Explain why this is incorrect.
- What is Mai’s score, really?

- Han started with 100 points and lost 10% for each successful roll. He had 2 successful rolls.
- Han thinks his score is 80. Explain why this is incorrect.
- What is Han’s score, really?

- Suppose you have 100 points. Would you rather be in a group that gets a 5% increase per successful roll and makes 4 successful rolls, or in a group that gets a 10% increase per successful roll but makes 2 successful rolls?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ensure that students can articulate that the fault with Mai 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.

- “Why does calculating the second percent increase or decrease from the original versus the new score result in the wrong answer?” (Additional percentages are calculated from 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?” (\(100 \boldcdot (1.15)^5\))