Unit 4, Lesson 6
Modeling Constraints
Extra Support Materials for Algebra 1
6.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-CED.A.3
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Instructional Routines
Notice and WonderMaterials
NoneActivity 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
NoneWhat 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
Instructional Routines
NoneMaterials
NoneActivity 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.
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
Supports accessibility for: Language, Attention
6.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
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 as they 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 (MP2).
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 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 .”