Quarters, Dimes and Linear Combinations

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:

1. How did you find the number of quarters and dimes needed to represent $1.45?
2. What 2 math operations do you use to find the money total and number of quarters and dimes?
3. How many possible solutions are there to this problem?
4. What are the 2 constraints of this problem?
5. What 2 equations could you write to represent this problem as a system of equations based off of those constraints?
6. 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.

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.   

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?