# Lesson 23

Modeling Constraints

These materials, when encountered before Algebra 1, Unit 2, Lesson 23 support success in that lesson.

## 23.1: Notice and Wonder: The Wonderful World of Finance (5 minutes)

### Warm-up

Students will answer questions about similar situations in their Algebra 1 class, and later in this lesson. The contexts of bank accounts and insurance purchases may be unfamiliar to students, especially English learners and students whose communities are not well served by banks or whose families do not have cars or other insurance policies.

By engaging with this explicit prompt to take a step back and become familiar with a context and the mathematics that might be involved, students are making sense of problems (MP1).

### Launch

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the situations for all to see. Display each situation one at a time. For each situation, give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

- Jada received \$100 on her birthday. She has a savings account and a checking account that she can deposit the money in.
- Han’s uncle is an insurance agent. He sells customers two types of car insurance policies: a cheap one and an expensive one. The cheap car insurance has a value of \$7,000 and the expensive one has a value of \$18,000. His goal for the month is to sell policies valuing over \$400,000 total.

### Student Response

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the situation. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If new vocabulary or connections or lack of connections with the situation don’t come up, here are some possible questions for discussion:

- What is the difference between a checking account and a savings account?
- Can Jada put some money in each account or does she have to choose one?
- What do you know about car insurance? What does it mean to sell car insurance?
- Why might a car insurance company sell cheap and expensive policies?
- I was wondering what “valuing over $400,000 total” means. Does anyone have any ideas?

## 23.2: Insurance Policies (15 minutes)

### Activity

The mathematical purpose of this activity is to introduce a specific strategy that students can use to write inequalities that describe a given situation, using their skill at finding specific values that satisfy a real-life situation.

Using a mix of structuring students’ thinking using tables, and teacher demonstration, students encounter a strategy for recording their thinking in a way that facilitates spotting patterns in the calculations and using patterns to generalize an equation.

This strategy will be useful if students are struggling to write equations based on situations in the associated Algebra 1 lesson.

Monitor for students who record their work in a way that makes it easy to spot patterns.

### Launch

Display Jada’s bank account situation for all to see: “Jada received \$100 on her birthday. She has a savings account and a checking account that she can deposit the money in.” Ask students, “What are some amounts of money Jada could deposit into each account?” (Sample responses: \$50 in each account, \$20 in one account and \$60 in the other and keep \$20 in cash, \$100 into one account and \$0 in the other, or other valid distrubutions.)

Ask students:

- “What would we have to check to be sure those totals worked in the story?” (They have to add up to no more than \$100.)
- “What are the quantities in the problem that we are testing?” (Amount deposited in checking and amount deposited in savings.)

Create a table with 4 rows and 4 columns as shown here for all to see. Label the columns in the table based on students’ names for the quantities. Begin to complete the table with some of students’ suggestions. Ask students what calculations they are doing with their guesses in order to make sure they work. Then ask students what they are doing to check that their guess works in the story. If needed, check in with students about whether less than or greater than makes sense, and whether strictly less than or less than and equal to makes more sense in the story. Finally, ask students if Jada deposited \(x\) dollars in checking and \(y\)\(\) dollars in savings, how they could fill out the table. A possible completed table may look like this.

amount deposited in checking | amount deposited in savings | calculation | check |
---|---|---|---|

\$50 | \$50 | \$50 + \$50 | \$50 + \$50 \(\leq\) \$100 true |

\$20 | \$60 | \$20 + \$60 | \$20 + \$60 \(\leq\) \$100 true |

\(x\) | \(y\) | \(x + y\) | \(x + y \leq 100\) |

Ask students, “What are some ways we could check that \(x + y \leq 100\) matches the original situation?” (Read the situation again, check that the guesses that worked with the situation also worked with the story.)

### Student Facing

Han’s uncle is an insurance agent. He sells customers two types of car insurance policies: a cheap one and an expensive one. The cheap car insurance has a value of \$7,000 and the expensive one has a value of \$18,000. His goal for the month is to sell policies valuing over \$400,000 total.

- List some different amounts of each policy Han’s uncle could sell.
- What calculations could you do to check whether Han’s uncle reached his goal?
- What could you compare your answers to in order to see if he reached the goal?
- Complete the table using the values from the previous questions.
number of cheap policies sold number of expensive policies sold calculation check \(x\) \(y\) - Write an inequality using number of cheap policies, \(x \), and number of expensive policies, \(y\). The inequality should be true if Han's uncle meets his goal.

### Student Response

### Activity Synthesis

Ask previously identified students to share their tables. Ask students, “How did making the table help you spot patterns?” (By seeing the patterns in the calculation before the value is determined, we can notice what is the same and what is different in the situations.)

Explain that this strategy of trying some specific values and looking for patterns is a strategy used by mathematicians all the time.

## 23.3: Row Game: Writing Inequalities from Situations (20 minutes)

### Activity

In this activity, students get a chance to practice applying their skills in representing situations symbolically and finding and interpreting solutions.

The structure of a row game supports students to check their thinking because their partners should get the same answer to different questions. When student answers don't match, they must make sense of each other’s questions and reasoning to figure out a correct solution.

The practice will pay off when they model situations with inequalities in their Algebra 1 lessons.

### Launch

Students will work independently to complete their designated rows, and will work with their partners in the event that an answer is wrong or different from their partner’s answer.

Remind students that partner A completes only set A, and partner B completes only set B. Your answers in each question should match. Work on one question at a time and check whether your answer matches your partner’s before moving on. If you don’t get the same answer, work together to find your mistake.

### Student Facing

Your teacher will assign you a set. Work only on the problems in your set. Work on one question at a time and check whether your answer matches your partner’s before moving on.

Set A

- Clare has \$25.00 to spend on souvenirs during her class trip to Washington, D.C. She wants to buy some souvenirs from the Air & Space Museum and some from the National Museum of African American History and Culture. She might not spend all of her money. Let \(x\) represent the amount of money she spends at Air & Space and \(y\) represent the amount of money she spends at the African American museum.
- What is one ordered pair \((x, y)\) that will work in this situation?
- Write an inequality in terms of \(x\) and \(y\) that shows what Clare can spend on souvenirs.

- Dried apricots have 10 grams of sugar per ounce. Cashews have 2 grams of sugar per ounce. Diego wants to make bags of trail mix with no more than 50 grams of sugar per bag. Let \(x\) represent the number of ounces of apricots in a bag and \(y\) represent the number of ounces of cashews in each bag.
- What is one ordered pair \((x, y)\) that will work in this situation?
- Write an inequality in terms of \(x\) and \(y\) that shows how many ounces of dried apricots and cashews Diego can include in his trail mix bags.

- The band is raising money for their trip to Orlando. Each student needs to raise at least \$250. They are selling candles which earn \$7 each, and poinsettias which earn \$15 each. Let \(x\) represent the number of candles sold and \(y\) represent the number of poinsettias sold.
- What is one ordered pair \((x, y)\) that will work in this situation?
- Write an inequality in terms of \(x\) and \(y\) that shows how many candles and poinsettias each student needs to sell.

- Mai is trying to earn at least \$75 toward prom-related expenses. Her mom has offered to pay her \$3.00 every time she cleans the cat litter, and \$5.00 every time she walks the dog. Let \(x\) represent the number of times she cleans the cat litter and \(y\) represent the number of times she walks the dog.
- What is one ordered pair \((x, y)\) that will work in this situation?
- Write an inequality in terms of \(x\) and \(y\) that shows how many times Mai could walk the dog and clean the cat litter to meet her goal.

Set B

- Lin’s library sets a maximum of 25 items that can be checked out at one time. Lin likes to check out books and DVDs. Let \(x\) represent the number of books Lin checks out, and \(y\) represent the number of DVDs Lin checks out.
- What is one ordered pair \((x, y)\) that will work in this situation?
- Write an inequality in terms of \(x\) and \(y\) that shows how many books and DVDs Lin can check out.

- Noah is sending a care package to his cousin in the military. He has saved \$50 to spend. His cousin’s favorite items are movies, which Noah found on sale for \$10 each, and energy bars, which are \$2 each. Let \(x\) represent the number of movies Noah buys, and \(y\) represent the number of energy bars. Noah doesn’t have to spend all of the money on this care package.
- What is one ordered pair \((x, y)\) that will work in this situation?
- Write an inequality in terms of \(x\) and \(y\) that shows how many movies and energy bars Noah can send his cousin.

- A group of teachers is ordering school supplies online. They need pencils, which are \$7 a box, and paper, which is \$15 a box. They get free shipping on orders of \$250 or more. Let \(x\) represent the number of boxes of pencils they buy, and \(y\) represent the number of boxes of paper they buy.
- What is one ordered pair \((x, y)\) that will work in this situation?
- Write an inequality in terms of \(x\) and \(y\) that shows how many boxes of pencils and paper the teachers could buy to get free shipping.

- Priya is helping her cousins at their farm stand. Her aunt has asked them to try to sell at least 75 pounds of tomatoes by noon. They sell tomatoes in 3-pound and 5-pound bags. Let \(x\) represent the number of 3-pound bags of tomatoes they sell and \(y\) represent the number of 5-pound bags they sell.
- What is one ordered pair \((x, y)\) that will work in this situation?
- Write an inequality in terms of \(x\) and \(y\) that shows how many 3-pound bags and 5-pound bags Priya could sell to meet her goal.

### Student Response

### Activity Synthesis

The purpose of the discussion is to determine how students write inequalities from situations and find ordered pairs that work in the situations.

Consider asking:

- "How did you think of ordered pairs that work in your situation? Were you surprised that they worked for your partner's situation?" (I tried to come up with a reasonable value for the \(x\)-variable and then see what \(y\)-variables might fit the constraints. It was not too surprising since the context was different, but the values were very similar.)
- "How did you determine which direction the inequality should go?" (I thought about the constraint and whether they would be ok if they went over or under the target amount. If they could go over a certain amount, I used a greater than symbol. If they had to stay under a certain amount, I used a less than symbol.)
- "How can you determine whether an equation or inequality describes the situation better?" (An equation is best when a target value must be reached exactly. An inequality makes sense when there is a limiting value that someone must stay under or over.)