Matrix Lesson Plan

Matrix Algebra Lesson Alana Linde Co-Teaching Christa Kordik

Overview of the Lesson:

Students will apply algebraic concepts of matrices within a real-world context to determine how

brand switching impacts a particulars shoe stores inclination to purchase their inventory.

Students will also understand how markups are affected by consumer discounts. Lastly, students

will work through a packet to understand the properties and how they are impacted when using

matrices.

Learning Goal:

Using contextual references, students will understand how to manipulate data using properties

and fundamentals pertaining to matrix algebra.

Standards to be achieved through this lesson:

HSN-VM.C. Perform operations on matrices and use matrices in applications.

HSN-VM.C.6. Use matrices to represent and manipulate data, e.g., to represent payoffs

or incidence relationships in a network.

HSN-VM.C.7. Multiply matrices by scalars to produce new matrices, e.g., as when all of

the payoffs in a game are doubled.

HSN-VM.C.8. Add, subtract, and multiply matrices of appropriate dimensions.

HSN-VM.C.9. Understand that, unlike multiplication of numbers, matrix multiplication

for square matrices is not a commutative operation, but still satisfies the associative and

distributive properties. (Addressed in the homework packet)

Type of Task: Procedures with Connections (Stein & Smith, 16).

Before the Lesson: Students will need to know what a matrix is and how you can perform

computations with them. Students will have been introduced to the basic concepts of adding,

subtracting, and multiplying matrices. Students can either have knowledge of computing these by

hand or on their calculators because the focus of this lesson is interpreting the matrices in a real-

world context. To activate this knowledge, we can ask students to recall what they learned from

the previous lesson on matrices and to keep this in mind for this lesson.


Launch:

To launch this lesson, we will ask the class the following questions.

Americans are extremely brand loyal as consumers.

Can someone tell us what this statement means?

What does brand loyalty mean?

What are brands that you are loyal to?

o If there is trouble with student participation ask about clothing, technology,

athletic gear, boots, school supplies, etc.

Then we will bring it into the context of our lesson – brand loyalty with shoes.

Have you ever been loyal to a brand of athletic shoes?

If so, which ones?

Have you ever changed brands over time?

What made you switch brands?

Did you ever return to the brand you began with?

Each student will be provided with the work packet for the lesson. The first page includes an

article called “Run Forrest Run.” After students have finished reading the short article

individually, students will be asked to complete the prediction questions at the bottom of the

page. These are the questions we want them to focus on for the lesson.

What brand of athletic shoes do you think males and females are most loyal to?

What gender generates the most profit for athletic shoes sales?

Once they have had time to do this, we will come together as a class and share out what the

students have predicted. While the students are sharing their information, we will write their

answers on the board to hold them accountable for what they are presenting as well as being a

facilitator.

Students will be placed into groups of four for this lesson and can use calculators as needed.

There are going to be students with varying abilities and we will make sure to differentiate for

them. Some challenges that ELLs might face would be that they might not be comfortable with

some of the business models in America if they have just moved here as well as not necessarily

knowing what certain brands are. If this were to arise, we would have to adjust it to something

that would be in their prior knowledge to relate it to first and then work our way into the

problem. However, after observing the class, we do not foresee this being an issue.


Explore:

When students walk into the room, we will hand them a card that had one of the four store names

on them (Scheels, Famous Footwear, Finish line, and Foot Locker). They will then sit at the

desks with the matching store name and this will be their ‘business’ for the day. Once they have

read and we have discussed the article, the students will begin to work on the packets that we

handed out with the rest of their peers focusing on the particular business they have been

assigned to.

The actual task given to students is attached, but the following is a detailed write up of each part.

Part 1:

Students will have to add the manufacturer's price and the markup to conclude what the retail

will be. There are questions that follow and they include:

What observations can you make between male and female shoes?

If you were managing a store, what factors would you consider when purchasing shoes

for your inventory?

Of the variables that are included in the matrices above, what type of shoe are you more

inclined to purchase for your inventory?

The next question is as follows and asks them to solve when there is a scalar multiple. Students

will have to multiply the retail value matrix by 85% (100% - 15%) and then subtract the

manufacturers price matrix to find the new profit.

Suppose a store is offering a sale on athletic shoes. What would be the profit for each

brand of shoes for males and females if the store is offering a 15% discount for their

customers?

Part 2:

This allows the students to analyze the brand switching matrix that is given and answer the

questions:

What is the percent of people who currently own a pair of Under Armour shoes and will

most likely purchase Nike next year?

According to this research, which show buyer is the most brand loyal? How do you know

this? Explain your reasoning.

Part 3:

Students will have to take the brand switching matrix and multiply that by the total number of

shoes that 16 year olds bought last year in one of the four stores that they were assigned to.

How many shoes of each brand should Scheels purchase for next year?

Based on the predicted totals of shoes to purchase, calculate the expected profit for

Scheels next year on shoes purchased by 16-year-olds for males and females.


Part 4:

Once each business has had time to complete their findings, they will then share out with the rest

of the class. The student groups that are not presenting at that time have a table where they can

fill out what their other classmates found in their investigation. The purpose of having the

students present their findings is to reach the goal of the task and to demonstrate how matrices

can be utilized in a real-world context. Because each group represents a different shoe retailer,

students will need to gather this information to compare it to their store to analyze the data and

notice trends in consumers.

Two questions will be asked afterwards

Compare your results to the other stores. What do you notice?

How does this relate to what we read in the article?

From speaking to Mr. Seeley prior to observing the class we will be teaching, we understand that

other than the mini lesson that he will be teaching prior to our lesson, the students will not have

much experience with matrices. We hope that providing a real-world context will allow the

concept to resonate with them more so than just manipulating a set of values inside

parentheticals

We anticipate students to have the following misconceptions/troubles:

The brand switching matrix for the first task needs to add up to 100%.

Putting the brand switching matrix in the calculator incorrectly (i.e. 20 % will be typed in

as 20 instead of .20)

Question 5 is a multi-step problem and may need some guidance to arrive at the answer

o Student might multiply the retail value by .15 instead of .85 (100% - 15%) to

solve for the consumer’s cost after the discount is applied.

Anticipating and Monitoring:

As we monitor the students working, we will visit the groups and ask them questions to probe

their thought processes. Some of the questions we may ask are:

For Part 1:

o What does the addition problem tell you about businesses and how they set

prices?

o Do you think that businesses have a particular markup for each type of shoe or is

it even across the board?

o Is the retail value you solved for practical?

o According to equation, what is the most enticing for stores to purchase if they are

seeking profit?


o Students may need help with question 5 because it is a multi-step problem. They

first need to just the scalar multiple

For Part 2:

o We want to be sure how to interpret the data in the matrix. Asking students to

verbalize their reasoning to question 7 will help us to determine if the students do

in fact understand if we do not have sufficient evidence from their writing.

For Part 3: (Students work in groups to solve for each store)

o We hope that having students work in groups for these questions that they will

catch each other if minor errors occur along the way. We want students to be able

to compare their results with the other groups to look for trends in the data.

o Asking students how they know they can multiply the ‘total shoes bought by 16-

year-olds’ matrix by the ‘brand switching’ matrix?

This may bring to light their knowledge of matrix multiplication.

o Posing a circumstance such as, is it fair if we only had the data for two brands of

shoes to multiply it by the brand switching matrix to predict for the next year?

For Part 4:

o When students are presenting we want to ask how they solved for their problem

set and to explain their findings to the class.

o Asking students to visit their thought processes for question 10 will be helpful

too.

o Although the students are solving for similar concepts, the values are different.

We want them to make connections across the different stores to determine how

the brand switching matrix impact future purchases for each store.

For students that are struggling grasping the concepts, we may need to clarify the terms of the

questions so that they clearly understand what the question wants from them. Asking students

who are completely lost to refer back to the content that they previously were taught may help

them as well.

While monitoring we will be keeping track of students’ progress and learning. We will have a

table similar to what is below and write in students’ responses as well as student’s names that

had a particularly insightful thought that we may want to address in the discussion. We will also

keep track of particular difficulties that student were having as a point of reference for the next

time this lesson is taught. In addition to these things we will be tracking, we will also be

monitoring the responses to the questions that we will ask individuals.


Solutions to Check while Monitoring:

Retail Value: (sum of manufacturers price + markup)

Male Female

Adidas 119 109

UA 99 89

Nike 125 119

Question 5:

(.85 * Retail Value Matrix) - Manufacturer’s Price = New Profit Matrix

101.15 92.65

84.15 75.65

106.25 101.15

46.15 42.65

44.15 40.65

56.25 56.15

Total Shoes Sold to 16-year-olds

Predicted Total of Shoes Sold to 16-year-olds

Predicted Profit for Males and Females

Footlocker

[700 500 400] [420 610 571] [107,900 99,900]

Famous Footwear

[500 600 500 ] [370 600 630] [108,850 100,850]

Finish Line

[450 550 600] [350 575 675] [109,650 101,650]

Scheels

[450 400 750] [335 530 735] [110,775 102,775]


Close:

Once each group has had time to complete their findings, they will then share out with the rest of

the class. Prior to presenting, we will have asked tables to select a student to present or we will

select if we had a situation in which the student had an insightful discovery or approach. Staying

in the context of business, we want to be sure that as students are explaining their findings that

they are doing so in such a way that upholds the concept of a matrix, but also how that translated

for the retailers that they are representing.

We want students to make connections between the different retailers/student groups by

analyzing their data after it has been manipulated from the brand switching matrix as well as

calculated to find each store’s profit. We want students to see that for all cases but one, the brand

that the retailer will need to order the most of is Nike, which connects with the 70% brand

loyalty we found. For the profit calculations, we want students to see that male shoes are the

most profitable for each of the retailers. Between males and females, retailers will make $8,000

more on male shoes than female shoes.

It also might be valuable to ask them what operations they performed using matrices during this

investigation as well so that students can see that working through matrices isn’t just procedures

without connections, but rather when providing content it can flow nicely.

We will make it a point to check every students answer to question 11 ask a check for

understanding.

****

Homework/Extension to Next Lesson:

With the remaining time, we will move into our “homework/guided discovery” task.

This will guide students to working with matrix multiplication to determine if it is commutative,

associative, and/or distributive.

What is not finished will be considered homework to be discussed during the next time the class

meets.

Matrix Algebra Name: _________________________

Article Source: http://loyalty360.org/resources/article/run-forrest-run-men-go-farthest-in-footwear-loyalty-for-nike

Make a prediction:

What brand of athletic shoe do you think both males and females are most loyal to?

What gender generates the most profit for athletic shoe sales?

http://loyalty360.org/resources/article/run-forrest-run-men-go-farthest-in-footwear-loyalty-for-nike

Not only can matrices be used to simplify solving for systems of linear equations, but they can also be

used to help make predictions for businesses. The following data represents what a local store

computes based on high school students’ athletic shoe sales.

After purchasing a product, store must build in a markup to ensure that the store is receiving a profit on

their sales. Looking to stock inventory for men’s and women’s athletic shoes, stores use this markup

matrix to compute their retail value of athletic shoes for the following brands – Adidas, UnderArmour,

and Nike.

1. Add the manufacturer’s price matrix to the markup markup matrix to solve for the retail value of

tennis shoes.

Manufacturer’s Price Markup Retail Value

Male Female

Adidas 55 50

UA 40 35

Nike 50 45

Male Female

Adidas 64 59

UA 59 54

Nike 76 74

Male Female

Adidas

UA

Nike

2. What observations can you make between male and female shoes?

3. If you were managing a store, what factors would you consider when purchasing shoes for your

inventory?

4. Of the variables that are included in the matrices above, what type of shoe are you more

inclined to purchase for your inventory?

5. Suppose a store is offering a sale on athletic shoes. What would be the profit for each brand of

shoes for males and females if the store is offering a 15% discount for their customers?

Manufacturers of shoe brands carry out consumer research to gather information about brand loyalty.

Consumer research results from 2012 and 2013 are shown in the matrix below. This matrix represents

the percent of people who bought a particular brand, and their likelihood of switching brands the

following year. You can use this matrix to estimate how many customers will purchase each shoe brand

in the future.

Brand Switching Matrix:

Next Brand of Shoes

Adidas UA Nike

Current Brand

of Shoes

Adidas 40% 40% 20%

UA 20% 50% 30%

Nike 10% 20% 70%

6. What is the percent of people who currently own a pair of Under Armour shoes and will most

likely purchase Nike next year?

7. According to this research, which show buyer is the most brand loyal? How do you know this?

Explain your reasoning.

Famous Footwear

The manager of the Famous Footwear at Valley West Mall provided the year-to-date totals for shoes

purchased. Before the holiday season, the manager wants to determine what shoes she needs to

purchase on inventory for the following year.

The following matrix represents the total number of shoes for each brand bought by people that are 16

years old at Famous Footwear at Valley West Mall.

8. Calculate what the predicted total number of shoes will be next year for each brand using the

brand switching matrix values.

Total Shoes bought by 16-year-olds

Adidas UA Nike

500 600 500

9. How many shoes of each brand should Famous Footwear purchase for next year?

10. Based on the predicted totals of shoes to purchase, calculate the expected profit for Famous

Footwear next year on shoes purchased by 16-year-olds for males and females.

Finish Line

The manager of the Finish Line at Jordan Creek Mall provided the year-to-date totals for shoes

purchased. Before the holiday season, the manager wants to determine what shoes she needs to



years old at Finish Line at Jordan Creek Mall.




Adidas UA Nike

450 550 600

9. How many shoes of each brand should Finish Line purchase for next year?

10. Based on the predicted totals of shoes to purchase, calculate the expected profit for Finish Line

next year on shoes purchased by 16-year-olds for males and females.

Footlocker

The manager of the Footlocker at Valley West Mall provided the year-to-date totals for shoes

purchased. Before the holiday season, the manager wants to determine what shoes he needs to



years old at Footlocker at Valley West Mall.




Adidas UA Nike

700 500 400

9. How many shoes of each brand should Footlocker purchase for next year?

10. Based on the predicted totals of shoes to purchase, calculate the expected profit for Footlocker


Scheels

The manager of the Scheels at Jordan Creek Mall provided the year-to-date totals for shoes purchased.

Before the holiday season, the manager wants to determine what shoes she needs to purchase on

inventory for the following year.


years old at Scheels at Jordan Creek Mall.




Adidas UA Nike

450 400 750

9. How many shoes of each brand should Scheels purchase for next year?

10. Based on the predicted totals of shoes to purchase, calculate the expected profit for Scheels


11. Complete the table from your peers’ results to determine the predicted values of shoes sold to

16-year-olds.

Total Shoes Sold to 16-year-olds

Predicted Total of Shoes Sold to 16-year-olds

Predicted Profit for Males and Females

Footlocker

Famous Footwear

Finish Line

Scheels

12. Compare your results to the other stores. What do you notice?

Name: ____________________ Period: _____ Date: ______________

1. Name one new thing you learned today about matrix multiplication

properties (commutative, associative, and distributive).

2. Name one thing you would like to understand better or know more about.

3. How is matrix multiplication different from the normal number

multiplication?

---------------------------------------------------------------------------------------------------------------------------------------------------------------------

Name: ____________________ Period: _____ Date: _____________

1. Name one new thing you learned today about matrix multiplication

properties (commutative, associative, and distributive).

2. Name one thing you would like to understand better or know more about.

3. How is matrix multiplication different from the normal number

multiplication?

Matrix Multiplication Properties

Name: ____________________________ Period: _____ Date: _____________________

A = 1 −23 −1

B = −4 5 01 −3 4

C = 3 01 15 −6

D = 0 4−1 8

Commutative property: AB = BA

Multiply the matrices, if possible:

1. CB 2. AC

3. AD 4. DA

5. Did you get the same answer for numbers 3 (AD) and 4 (DA)?

6. a. To ensure that it is not just a result from these given matrices, create two new

matrices, and perform the same operation as number 3 and 4.

b. What is your conclusion? Why do you conclude that?

A = 1 −23 −1

B = −4 5 01 −3 4

C = 3 01 15 −6

D = 0 4−1 8

E = 2 63 1

Associative property: (AB) C = A (BC) Distributive property: A (B + C) = AB + AC

Use the given matrices to solve the following problems (Hint: Keep in mind the order of

operations: PEMDAS- Parenthesis, Exponents, Multiply, Divide, Add, and Subtract).

7. (AD)E 8. A(DE)

9. What’s your conclusion about the associative property with matrix multiplication?

Why or why not?

10. A (D + E)

11. AD + AE

12. Is matrix multiplication distributive? Why or why not?

Matrix Lesson Plan

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Transcript of Matrix Lesson Plan