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# Penny Problem

Lesson 9 of 13

## Objective: Students will be able to solve a real life system of equations and present the solution graphically and algebraically. Students will also determine how the precision of measurements can effect the solutions to a system of equations.

#### Opening

*10 min*

During this opening, pose the initial questions and just let students work. Hopefully, based on all the work you have done so far on systems of equations you should hear students come up with the two equations listed on the third slide. Then have the students discuss these two equations with their partners and determine if the equations are the same or different. Students will come to the conclusion that these are really the same equation. Then have students *prove* that the equations are the same either by explaining, by solving them algebraically and showing they have an infinite number of solutions or by graphing them and showing that they are really the same line.

This opening is helping to plant the seeds for the elimination method of solving systems of equations. It is also getting students to work with one of the constraint equations from the penny problem. The penny problem itself brings in a new constraint (the weight of the older and newer pennies).

#### Resources

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#### Investigation

*30 min*

Access to the computer lab is ideal for this lesson so that all students can use the Geogebra software to graph the system of equations. In this lesson the emphasis is taken off of "how" to graph the system and is put on "what" is the solution based on the graph and what does that solution actually represent. If computers are not available, students could certainly graph the equations on the graphing calculator or by hand.

**Things to watch for:** (1) The danger of the opening activity is that some students will think that they need to use an equation for the value of the coins (the point of the opening was also to show the students that using the value equation and the number of coins will actually produce the same equation). For some students who struggle you may want to pose a question to them after they have time to read and understand the problem: "Which equation from the opening activity will be used and which will not?"

(2) What does the solution represent? I find myself asking this question over and over to students when they write and answer of (22,28). Continue to push your students to constantly think about the solution in the context of the problem (22 older coins and 28 newer coins)

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#### Closure

*10 min*

Ask students to reflect on the work that they just did. Name two things that you understood well as a result of solving this problem. Second, write down a question that you still have or something that you are unsure about. You can use the student reflections to group students that next day around common understandings and misunderstandings. This can be done in two ways: (1) Group students together who all have a common misunderstanding and give them a task that will help to clear up that misunderstanding. (2) Group students who have an understanding with those that don't so that they can work with each other. A word of caution about option #2...good partnerships are crucial so that the student with a good understanding can help the student who is not understanding the concept not just give them answers.

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- LESSON 1: Introduction to systems of equations
- LESSON 2: What Does a System of Equations Really Look Like?
- LESSON 3: What is the "Point" of Solving a System?
- LESSON 4: Fitness Center Question
- LESSON 5: Cell Phone Plans
- LESSON 6: How are Systems of Equations related to Equations and Functions?
- LESSON 7: Solving Systems of Equations Without a Graph
- LESSON 8: Practice with the Substitution Method
- LESSON 9: Penny Problem
- LESSON 10: Practice Solving Systems Algebraically
- LESSON 11: Pulling the Systems Concepts All Together
- LESSON 12: An Interesting Lottery
- LESSON 13: Don't Sink The Boat