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Unit 4, Lesson 6

Modeling Constraints

Extra Support Materials for Algebra 1

6.1

Warm-up

5 mins

Notice and Wonder: The Wonderful World of Finance

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

The purpose of this Warm-up is to allow students to pose questions about mathematical situations, which will be useful when students answer questions about similar situations in a later activity. While students may notice and wonder many things about these situations, comparisons between the different accounts and policies are the important discussion points.

This Warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1). 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 to insure or do not have other kinds of insurance policies.

Launch

Arrange students in groups of 2. Display the situations for all to see. Display each situation one at a time. Ask students to think of at least one thing they notice and one thing they wonder about each situation. For each situation, give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things they notice and wonder.

Activity

None

Student Task Statement

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: one with minimum coverage and one with full coverage. The minimum coverage insurance costs $1,000 each year, and the full coverage one costs $4,000 each year. His goal for the year is to sell policies costing over $100,000 total.

Student Response

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Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the situation. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

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?” (Usually checking accounts are for daily use and savings accounts are for saving money.)
“Can Jada put some money in each account or does she have to choose one?“ (She could split the money between accounts or choose to put all the money in one account.)
“What do you know about car insurance? What does it mean to sell car insurance?“ (Car insurance is money that is paid to cover potential future costs associated with having a car, including big repairs and accidents. People who sell car insurance are the ones who decide how much money is given when the car owner needs it.)
“Why might a car insurance company sell different policies?“ (Different people want different coverage or can afford only minimum coverage.)
“I was wondering what ‘costing over $100,000 total’ means. Does anyone have any ideas?” (People pay money, called a “premium,” to an insurance company in order to have the insurance coverage. People who sell insurance have many clients, and each client pays a premium of a certain amount, depending on the type of policy, every year. Han’s uncle wants enough people to buy insurance policies so that the insurance company gets at least $100,000 in premiums for the year.)

6.2

Activity

15 mins

Insurance Policies

Instructional Routines

None

Materials

None

Activity Narrative

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 inequality. Students look for and express regularity in repeated reasoning when they try different combinations of numbers to then generalize an inequality to represent the situation (MP8).

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.

6.3

Activity

20 mins

Row Game: Writing Inequalities from Situations

Instructional Routines

MLR8: Discussion Supports

Materials

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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?” ($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 distributions.)

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 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 or equal to” makes more sense in the story. Finally, ask students if Jada deposited dollars in checking and 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	true
$20	$60	$20 + $60	true

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

Representation: Access for Perception. Read the statement aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language, Attention

Activity

None

Student Task Statement

Han’s uncle is an insurance agent. He sells customers two types of car insurance policies: one with minimum coverage and one with full coverage. The minimum coverage insurance costs $1,000 each year and the full coverage one costs $4,000 each year. His goal for the year is to sell policies costing over $100,000 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, to see if he reached the goal?

Complete the table using the values from the earlier questions.

number of minimum coverage policies sold	number of full coverage policies sold	calculation	check

Write an inequality using the number of minimum coverage policies, , and the number of full coverage policies, . The inequality should be true if Han's uncle meets his goal.

Student Response

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Building on Student Thinking

Are You Ready for More?

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 to 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.

MLR8 Discussion Supports. Display sentence frames to support partner discussions. Examples:

“First, I because .“
“Why did you ?”
“I agree/disagree because .”

Advances: Speaking, Representing

Activity

None

Student Task Statement

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 represent the amount of money she spends at Air & Space and represent the amount of money she spends at the African American museum.
1. What is one ordered pair that will work in this situation?
2. Write an inequality in terms of and that shows what Clare can spend on souvenirs.
Diego is moving and needs to pack his books and games. His boxes of books weigh 10 pounds each. His boxes of games weigh 2 pounds each. Diego can move no more than 50 pounds of boxes at a time. Let represent the number of boxes of books and represent the number of boxes of games.
1. What is one ordered pair that will work in this situation?
2. Write an inequality in terms of and that shows how boxes of books and games Diego can move at one time.
The band is raising money for their trip to Orlando. Each student needs to raise at least $250. They are selling candles that earn $7 each, and poinsettias that earn $15 each. Let represent the number of candles sold and represent the number of poinsettias sold.
1. What is one ordered pair that will work in this situation?
2. Write an inequality in terms of and 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 represent the number of times she cleans the cat litter and represent the number of times she walks the dog.
1. What is one ordered pair that will work in this situation?
2. Write an inequality in terms of and 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 represent the number of books Lin checks out, and represent the number of DVDs Lin checks out.
1. What is one ordered pair that will work in this situation?
2. Write an inequality in terms of and 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 comic books, which are $2 each. Let represent the number of movies Noah buys, and represent the number of comic books. Noah doesn’t have to spend all of the money on this care package.
1. What is one ordered pair that will work in this situation?
2. Write an inequality in terms of and that shows how many movies and comic books 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 represent the number of boxes of pencils they buy, and represent the number of boxes of paper they buy.
1. What is one ordered pair that will work in this situation?
2. Write an inequality in terms of and 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 represent the number of 3-pound bags of tomatoes they sell and represent the number of 5-pound bags they sell.
1. What is one ordered pair that will work in this situation?
2. Write an inequality in terms of and that shows how many 3-pound bags and 5-pound bags Priya and her cousins could sell to meet the goal.

Student Response

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Unit

Lesson

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Assessment Access

Lesson 6

Modeling Constraints

Warm-up

Notice and Wonder: The Wonderful World of Finance

Standards Alignment

HSA-CED.A.3

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Activity Synthesis

Activity

Insurance Policies

Instructional Routines

Materials

Activity Narrative

Activity

Row Game: Writing Inequalities from Situations

Instructional Routines

Materials

Have feedback on the curriculum?

Building on Student Thinking

Are You Ready for More?

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

HSA-CED.A.3

HSA-REI.D.12

Standards Alignment

HSA-CED.A.3