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# Quarters, Dimes and Linear Combinations

Lesson 7 of 14

## Objective: SWBAT model a combination of quarters and dimes as a system of linear equations and solve by elimination.

#### Warm up

*10 min*

I plan for this Warm up to take about 5 minutes for the students to complete and about 5 minutes to discuss the solutions found. Most of the students draw coins on their paper or make tables to work this problem. This is a concrete example that I allow students to solve using any method. Money is a concept that students understand, and a good concrete example for students. After students complete the warm up, I question them about the more abstract idea of writing algebraic equations to represent the situation. I use this lesson as an introduction to solve a system of equations using linear combinations, which is I also refer to as elimination.

I question students after the warm up:

- How did you find the number of quarters and dimes needed to represent $1.45?
- What 2 math operations do you use to find the money total and number of quarters and dimes?
- How many possible solutions are there to this problem?
- What are the 2 constraints of this problem?
- What 2 equations could you write to represent this problem as a system of equations based off of those constraints?
- How can you solve this system of linear combinations by Elimination?

I model showing students how to write the algebraic equations in the video below.

#### Resources

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#### Cooperative Activity

*30 min*

In this think-pair-share activity, I provide the students time to work in pairs to solve the system of equation word problems using elimination. After going over the warm up, I want students to be able to recognize the difference between these Linear Combination problems and the word problems we previously solved involving increasing and decreasing situations. I have focused several previous lessons on distinguishing between these two types of problems. In the increasing and decreasing problems, students should recognize an initial condition, and a constant rate of change. In linear combinations, there are two linear situations that can be defined with variables.

There are 4 word problems for students to solve. I purposely plan for all of the problems to be similar, so students can learn these type of problems from repeated reasoning. I use mathematical practice 7 and 8, to look at the repeated reasoning and the same structure.

I do specify students to write the equations of these problems to introduce solving a system of equations using elimination. After I present all of the different methods to solve a system of equations with practice, I give the students a summative assessment at the end of the unit. I allow students to use any method of their choice on the summative assessment. I teach my students that if a method is not specified, they may use any method of their choice as long as they support it with the math and reasoning.

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#### Exit Slip

*10 min*

I want students to be able to recognize the difference between a word problems that describe an increasing or decreasing situation, and, a word problems that describe a linear combination. So, I use this Exit ticket to summarize the differences, and to answer what is found by linear combinations. Why are linear combinations useful?

#### Resources

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- UNIT 1: Introduction to Functions
- UNIT 2: Expressions, Equations, and Inequalities
- UNIT 3: Linear Functions
- UNIT 4: Systems of Equations
- UNIT 5: Radical Expressions, Equations, and Rational Exponents
- UNIT 6: Exponential Functions
- UNIT 7: Polynomial Operations and Applications
- UNIT 8: Quadratic Functions
- UNIT 9: Statistics

- LESSON 1: Introduction to a System of Linear Equations
- LESSON 2: The Best of 2 Cell Phone Plans
- LESSON 3: Who is 1st in the Father Daughter Race?
- LESSON 4: Can You Save The Diver in 7 Minutes?
- LESSON 5: Make a Substitution
- LESSON 6: Alternate Method to Solve a System of Equations by Substitution
- LESSON 7: Quarters, Dimes and Linear Combinations
- LESSON 8: Define, Set, Go!
- LESSON 9: Khan Your Way Into Solving a System of Equations Using Elimination
- LESSON 10: Elimination with 2 Column Notes
- LESSON 11: Assessment of a System of Linear Equations
- LESSON 12: Use The TI-Nspire CX To Solve a System of Equations
- LESSON 13: Solving a System of Inequalities
- LESSON 14: Solve the System of Inequalities to Find The Treasure!