##
* *Reflection: Problem-based Approaches
Introduction to systems of equations - Section 2: Investigation

The students did a really nice job finding the various combinations of coins in questions #1-5. This year, I let students share out after question number 1 how they came up with their equations. I had various students in the class explain to the rest of the class how to construct the equations to represent that particular situation.

We also got into an interesting discussion when one student suggested using the equation 5x+y=11 rather than 0.05x+0.01y=.11. I asked the students if they could show if these two equations were really the same. After discussing it with their partners for a short amount of time, many students were able to see that one equation was a multiple of the other.

Students did well with moving into the more abstract scenarios in questions 6 and 7. They were able to use a similar line of reasoning to determine the correct combination of bills to give them the values. As predicted, most students struggled with question #8. Quite a few students were able to determine the missing values using trial and error but it did take some time. The fact that the cost of a bag of chips was a decimal value (0.75) also slowed students down. I noticed once again how uncomfortable many students are with decimals and how to reason about their quantities in calculations.

*Investigation-Reflection*

*Problem-based Approaches: Investigation-Reflection*

# Introduction to systems of equations

Lesson 1 of 13

## Objective: Students will be able to set up a system of equations that can model a real world situation. Students will understand how the constraints of a system of linear equations leads to one simultaneous solution.

#### Investigation

*10 min*

During this portion of the lesson students will work with their partner using the coins to determine the appropriate combination of coins that will make the constrain equations true. I let all students go through and answer questions 1-5 first but leave the "equations" column blank. If students complete the task early they can attempt problems 6-8. Once all students have completed 1-5 we go back to write our algebraic representations. It is very important that we write a good description of our variables: e.g. *n* = the number of nickels and *p* = the number of pennies. Students can then write the constraint equations for each problem: e.g. 5n+p = 11 and n + p = 3. This is a good time to show how substituting the values that students came up with intuitively will make both equations true.

*expand content*

#### Closure

*10 min*

The objective of this lesson was to move students from a concrete understanding of constraint equations to being able to represent these constraint situations algebraically. The closure of this lesson (which I have student put on index cards or half sheets of paper as a **ticket out**) requires students to write one algebraic expression and one algebraic equation to represent two scenarios. The understanding of how to write these representations will be crucial for student success through the remainder of the unit.

#### Resources

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I am excited to use this as an Introduction activity. Thanks!

| 2 years ago | Reply

The opening is activity is awesome! I will definitely use this. Thank you for your creativity....

| 2 years ago | Reply

This looks like it would work very well in my class. I'm going to try it soon. Thank you!

| 2 years ago | Reply*expand comments*

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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