Teacher Candidate: Jan Drabek
Date: February 4, 2011
Cooperating Teacher: Mike Comiskey
Grade: 9-12
School District: Seattle
School: Nathan Hale
University Supervisor: Andrea Escame-Hedger
Unit/Subject: Algebra 2
# of periods: 3
Lesson Title/Focus: Matrix multiplication
Script:
Learning Target
What: Learn matrix multiplication.
Standard supported by Learning Target:
A2.7.A Solve systems of three equations with three variables. (6.3-6.4) (While the lesson does
not offer a solution to systems of three equations with three variables we are working up to
applying matrices to solve these equations.)
A2.8.A Analyze a problem situation and represent it mathematically. (6.1, 6.6)
A2.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in
the context of the original problem. (6.5, 6.6)
A2.8.D Generalize a solution strategy for a single problem to a class of related problems, and
apply a strategy for a class of related problems to solve specific problems. (6.3)
A2.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and
conventions of mathematics. (6.1, 6.2)
A2.8.F Summarize mathematical ideas with precision and efficiency for a given audience and
purpose. (6.5, 6.6)
Chapter 6.2 introduces Matrix multiplication on pages 327-331
How: Consider a problem involving two shoppers buying three products at two stores
and have students calculate the total amount each shopper will spend at each store and then
create matrices which accomplish these computations through matrix multiplication.
Why: To enable students to build a mastery of matrices so that they can solve real world
problems involving systems of equations through the simplicity of matrices and technology.
Warm-up: Complete the following matrix arithmetic problems. If a problem is impossible,
explain why.
3
4
0
[6 −2 1] + [ 3 −3 −1]
12 0
8
[
−2 3
5 −5
]−2 [
]
7 −2
−4 4
Anticipatory Set: We have learned of the tedium and mistake prone nature of solving systems of
3 or more equations. Yesterday we learned about matrices, how to add matrices, and how to
multiply matrices by scalars. These are straight forward operations which generally make sense
with our understanding of these operations upon regular numbers. Today we are going to learn
about matrix multiplication which diverges from our natural sense of multiplication.
The Launch/Hook: A digital camera produces a matrix of numbers when it takes a picture. The
lens of the camera focuses an image on a flat rectangular surface covered with tiny lightsensitive electronic devices. The devices detect the color and brightness of the image in focus,
and this information is saved as a matrix of numbers in the camera's memory. When a picture is
downloaded from a camera to a computer and altered using image-editing software, it is
subjected to mathematical manipulations described by matrix algebra. The picture may also be
"compressed" so as to take up less computer memory or transmit over the Internet more quickly.
QR code matrix barcodes :QR is the abbreviation for Quick Response, as the creator intended the
code to allow its contents to be decoded at high speed.
Direct Instruction, Instructional Plan, and Strategies: After an anticipatory warm-up and a
review of last night’s homework and quiz questions. State the anticipatory set, state today’s
learning objective and how it is building us towards more complex manipulation of matrices.
Offer two real world examples of matrices. Hand out matrix multiplication worksheet and work
in pairs to complete sections with teacher check points at each major step. Take notes: matrix
multiplication requires that the number of columns of the first matrix equal the number of rows
of the second matrix. The number of rows of the first matrix and the number of columns of the
second matrix will be the same as the solution matrix. Matrix multiplication is not commutative.
Cover matrix multiplication on the calculator. Time permitting work on the homework. Closure.
Learning Experiences: Learning activity: Rather than just state the procedure for multiplying
two matrices I want to show students that is the logical approach that one would take for
organized tables of numbers. The worksheet will walk students through calculations they can
grasp and extend these calculations to matrix multiplication. The sheet will offer a reference for
the homework problems which if we have time may offer independent practice.
Direct instruction: How to multiply matrices on the calculator.
Guided Practice:
Formative assessment(s) strategy(s) & appropriate Modifications (Gear Up Gear Down
Strategies):
Teacher check point ask for final sum of first equation.
Gear up: Show the explicit equation, continue to next equations.
Gear down: Show step by step procedure for writing the explicit equation for finding sum then
offer teacher check point on next sum.
How feedback will be offered, opportunities for self assessment and reflection: While I want
students to work out their equations if my assessment shows a need to gear down I will walk
through the first equation step by step.
Administrative: Write learning target on board with outline of lesson and homework. White
boards, pens, collect quiz.
Instructional Materials, Resources, and technology
Florence and Bert fruit worksheet on matrix multiplication, Homework worksheet, Last night’s
homework. Quiz.
Homework review: Both the warm-up and homework are matrix addition problems. I will take
questions on both.
Student Grouping: Pairs or triplet’s.
Activity Timing: one period.
Assigned Homework (which relates to the learning target): Meadows or Malls Multiplying
Matrices.
Attachments: Florence and Bert fruit worksheet on matrix multiplication,
Reflections: This lesson was the first lesson where I was responsible entirely for the content. I
would typically cut and paste existing worksheets together to work on in class. I developed the
Florence and Bert worksheet myself. It felt good to create something from scratch and to receive
such high praise on the lesson from my CT and my clinical faculty. I can see how continually
creating worksheets from scratch could be unsustainable in terms of effort.
PPA Comments
9a
Observe norms associated with the white boards, compared to the initial introduction of
them we have progressed.
9b
If not observed in class CT will confirm a situation this week where a student threw a pen
at another student causing the other student to almost come to blows with the thrower. I asked
the victim to sit down and things settled down. I later thanked the victim for his not escalating
the situation and for showing good judgment. We referred the thrower to a vice principle.
9c
With regular reminding we settle into the warm-up and take the homework out to be
checked. I keep the hook short and then pass out worksheets or ask students to take notes.
Worksheets can simply lose the students; this worksheet is in two parts and by not handing out
the second half I want to keep the students together on it, not racing ahead and then checking out.
9d
The most likely suggestions you will observe are my calls to settle down and start the
warm-up, calls to attention, requests to change seats to break up chatter or to promote pairs or
triplet’s.
9e
To be determined, I will consult with CT.
3a
Post card to X’s family. X has been in this country since August, I think she is doing a
great job and succeeding even though her language skills are developing
3b
See included email exchange with Y , Z’s mother. Z is in first period and I make it a
habit to check in with him on his homework, and whether he is organized in his warm-ups.
3c
Check in on my contact with counselor concerning X.
3d
Y’s emails support this.
5a
The Florence and Bert Worksheet will satisfy all the learning standards listed as
supporting the Learning Target.
5b
This will be observed as the lesson will satisfy the LO and there will be an explicit
anticipatory set based on the previous day’s lecture introducing matrices.
5c
The plan will incorporate High Leverage teaching moves including:



A stated Learning Objective related to the standards.
Looking for Justification or Reasoning in the laying out of how matrix multiplication can
be applied to problem that can be understood without matrices.
Promoting Rich and Engaging Discourse in the form of having pairs and triplet’s
collaborate on the worksheet encouraging the sharing of ideas.


The use of Public Records to support student’s reconnection with skills previously
learned.
Ongoing Formative Assessment to keep in touch with student learning and consider
teaching moves in an informed way in the form of personal white boards.
5d
While the original draft of the worksheet included Ernie and Bert and Drabek and
Comiskeys’ the final draft has added Florence and Hudson instead of Ernie and Drabek to limit
the dominance of males in the story problem.
5e
The worksheet is designed to encourage critical thinking and collaborative learning rather
than just directly lecture the procedure of matrix multiplication. The plan incorporates a pencil
and paper approach to matrix multiplication stressing the basis of the procedure as well as
covering how to multiply matrices on a calculator for speed and accuracy.
5f
The worksheet represents more than a single gender.
5g
If not in this lesson instructions on how to use the ti-84 for matrix operations will be
introduced. Use of the internet to find images of the QR barcode.
5h
Students will be intentionally paired or tripled, if any students are sitting alone due to
absences of other students they will be asked to change or move seats to facilitate a role in a pair
or triplets that is one of equality.
5i
The launch will relate two everyday items students are familiar with which incorporate
matrices: digital cameras and QR barcodes.
Attachment 1 multiplication of matrices
Florence and Bert are planning to go shopping for fruit for the next week. They each want to
buy some kiwis, pineapples, and mangos, but in differing amounts. Table 1 lists what they
intend to buy. There are two fruit markets nearby- Comiskey’s and Hudson’s- and their prices
are given in Table 2. How much will it cost Florence and Bert to do their shopping at each of the
two markets?
Table 1:
Florence
Bert
Table 2:
Kiwis
6
4
Pineapples
3
8
Mangos
10
5
Comiskey’s
Hudson’s
Kiwis
$0.40
$0.55
Pineapples
$2.30
$3.35
Mangos
$0.90
$1.15
Write an equation showing how much Florence will spend at Comiskey’s:
Write an equation showing how much Florence will spend at Hudson’s:
Write an equation showing how much Bert will spend at Comiskey’s:
Write an equation showing how much Bert will spend at Hudson’s:
Create a table describing how much each will spend at each store:
Comiskey’s
Hudson’s
Florence
Bert
If we get rid of the headings in the tables our equations can be interpreted this way:
6
[
4
. 40
3 10
] ∗ [2.30
8 5
. 90
. 55
18.3 24.85
]
3.55] = [
24.5 34.75
1.15
If this is true what are the procedures for matrix multiplication?
What must be true of two matrices in order for them to be multiplied? Can you think of two
matrices that can’t be multiplied?
What is the product of the two matrices on the left hand side of the equal sign?
𝑥
1 −2 3
5
[−1 3 1] ∗ [𝑦] = [ 1 ]
𝑧
2 −1 4
14
[
5
]=[1]
14
How might you express this as a system of equations?
Attachment 2: multiplication homework
State whether the product AB is possible. Show work that supports your conclusion. If it is
possible, give the dimensions of AB.
1)
A: 2 x 2, B: 2 x 2
2)
A: 3 x 4, B: 4 x 2
3)
A: 2 x 1, B: 2 x 2
4)
A: 2 x 1, B: 1 x 5
Find the product. If the product is not possible, state the reason. (Show all calculations)
5)
3
5 
 1   =
7 
6)
7)
 1 0
 5 4 4  6 =


9  3 0 1 
0 2  4  2 =



8)
9)
8
0  4  2
2 5   3
0 


4 0   5  2
1
9
1 3 0  

2 12  4  4  3

  2 4 


Be the Teacher.
Describe and correct the error in finding the entry in the first row and first column of the matrix
product.
10)
Describe error:
3  1 7 0 
6 2   1  6  =



Correct multiplication steps:
3(7)  (1)(0)


 21
=
 



Using the given matrices, evaluate the expression
  1 2
2
3
  4 5
A   3 0 , B  
, C

 2  1
1 0


 4 1
11) A(B – C)
12) 2BC
13) Given the matrix equation, find the value of x and y.
  2 1 2 1   6 
3
2 4  x   19

 0  2 4 3  y 