Upload
alana
View
189
Download
3
Tags:
Embed Size (px)
DESCRIPTION
Applying Matrices in a Business Setting
Citation preview
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.
Matrix Algebra Lesson Alana Linde Co-Teaching Christa Kordik
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.
Matrix Algebra Lesson Alana Linde Co-Teaching Christa Kordik
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.
Matrix Algebra Lesson Alana Linde Co-Teaching Christa Kordik
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?
Matrix Algebra Lesson Alana Linde Co-Teaching Christa Kordik
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.
Matrix Algebra Lesson Alana Linde Co-Teaching Christa Kordik
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]
Matrix Algebra Lesson Alana Linde Co-Teaching Christa Kordik
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?
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
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 Finish Line at Jordan Creek 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
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
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 Footlocker 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
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
next year on shoes purchased by 16-year-olds for males and females.
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.
The following matrix represents the total number of shoes for each brand bought by people that are 16
years old at Scheels at Jordan Creek 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
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
next year on shoes purchased by 16-year-olds for males and females.
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?