##
* *Reflection: Connection to Prior Knowledge
Manufacturing Cat Toys (Day 1 of 2) - Section 1: Launch

Today was interesting. I did not anticipate that so many students would struggle with this initial task. It was a straightforward systems problem, but I think that students were so set on matrices that they automatically went to that strategy. I really had to rely on the line of questioning that I wrote about in the Launch narrative to get students on track.

Here are some samples of student work:

#1: This group wanted to use catnip and stuffing as the variables. I had to ask them what question we were trying to answer to get them on track. I also reminded them that we know the amount of catnip and stuffing, and if it isn't unknown then it shouldn't be the variable. Then they realized that the number of toys should be the variables.

#2: This student was on the right track and used substitution to solve.

#3: This students set up some matrices initially, but was unable to do anything with them. Interestingly, these matrices are really close to what we use later on in the lesson when we set up the system as a matrix equation.

# Manufacturing Cat Toys (Day 1 of 2)

Lesson 8 of 16

## Objective: SWBAT write systems of equations as matrix equations.

*50 minutes*

#### Launch

*20 min*

I have two **goals** for today’s lesson:

- Students will be able to take a system of equations and set it up as a matrix equation
- Students will realize that you cannot divide both sides of an equation by a matrix in order to solve.

Matrix equations are complicated and we will get to inverses and determinants, but today I want to lay the groundwork for those things.

Before we can set up a matrix equation we need to have a system of equations. I give students this task worksheet and have them work on Question #1 with their table groups. This is a problem that students will likely solve by setting up a system of equations. If students get stuck, here are some **hints** that are not too leading:

- How many variables will you need to use to solve?
- How many equations will you need to solve?
- What will the two variables represent?
- What will each equation represent?
- How can we solve for one of the variables? Is it easier to solve for
*y*or for*c*?

My students have used systems in Algebra 1 and Algebra 2, so by the time they get to Precalculus they are usually good at setting them up and are fluent in using elimination or substitution to solve for the variables. After giving them about 10-15 minutes to work on the problem, I will usually choose a student who solved using elimination to share and a student who solved using substitution to share. Graphing is not usually a common solution pathway, but if I notice that someone did try that, I will have them share. If not I will usually bring it up as a solution pathway.

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#### Explore

*15 min*

The purpose of solving this system with elimination or substitution is so students can check their answer once we solve using matrices. We are not going to formally prove why matrices could be used to solve a system of equations, but I think checking their solution with a method they know well will be convincing.

I will go through Questions #2 - 4 on the worksheet in a whole-class discussion format. I want students to figure out the two matrices that are needed to multiply together to get the left side of the system. Sometimes I will give hints if needed; I might tell students the dimensions of the matrices or that one matrix contains only variables. Usually 5 – 7 minutes is good for students to come up with the correct matrices. I talk more about this part of the lesson in the video below.

For Question #4, most students usually think we can divide each side of the matrix equation by the coefficient matrix. I will have them try it out on their graphing calculator to show that it does not work. It is a good idea to remind students that we have defined and used addition, subtraction, and multiplication with matrices, but we have not divided them; maybe a student has even brought this up before today. Once they realize that it does not work, we can work on gaining some understanding about how to solve the matrix equation in the next lesson.

#### Resources

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#### Summarize

*15 min*

Today’s assignment serves as a refresher on how to solve systems of equations with two or three variables. It also gives students some practice with the big idea from today - setting up the matrix equation given the system of equations. My hope is that students will come to class tomorrow being able to easily set up the matrix equation given a system so that we can start working on finding the inverse of a matrix.

#### Resources

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: The Purrrrrfect Cat Toy Company
- LESSON 2: Forecasting Sales and Calculating Revenue - Day 1 of 2
- LESSON 3: Forecasting Sales and Calculating Revenue - Day 2 of 2
- LESSON 4: Digital Images
- LESSON 5: Jet-Setting to the Purrrrrfect Cat Toy Offices
- LESSON 6: Market Research with a Markov Chain
- LESSON 7: Formative Assessment: Matrices
- LESSON 8: Manufacturing Cat Toys (Day 1 of 2)
- LESSON 9: Manufacturing Cat Toys (Day 2 of 2)
- LESSON 10: Hop on the Carousel
- LESSON 11: Partial Fraction Decomposition
- LESSON 12: Building Cat Furniture: An Introduction to Linear Programming
- LESSON 13: More Linear Programming
- LESSON 14: Formative Assessment: Systems of Equations and Inequalities
- LESSON 15: Unit Review and Cryptography
- LESSON 16: Unit Assessment: Matrices and Systems