Checkpoint
- Course 2, Unit 1
Matrix Models
In each unit, there is a final lesson and Checkpoint that helps students
summarize the key ideas in the unit. The final Checkpoint will generally
be discussed in class, with the teacher facilitating the summarizing,
and students making notes in their Math Toolkits (teachers
may just refer to this as "notes") of any points they need to remember,
adding illustrative examples as needed. If your student is having difficulty
with any investigation in this unit, this Checkpoint and the accompanying
answers may help you recall the concepts involved, and give you the big
picture of what the entire unit is about. If your student has completed
the unit, then a version of this should be in his or her notes or toolkit.
Students should also have Technology Tips in their toolkits, which may
be useful for this unit.
Possible
Responses to Unit Summary Checkpoint
| In this unit, students used matrix models to analyze
problem situations and they examined properties of matrices. The
bolded words are vocabulary and concepts your student should be familiar
with. |
| a. |
In order for information
to be useful it must be organized.
- Describe how matrices can be used to organize information. Matrices are
rectangular arrays with rows and columns of numbers that can
be used to organize information about two variables. For example,
consider average running time for boys and girls of different
ages. The values of one variable, like gender, are the labels
for the rows, and the values of the other variable, like age,
are the labels for the columns. Then the individual cells or
entries in the matrix give the average running time for a particular
gender at a particular age. Once the information has been entered
into a matrix, it may be appropriate to operate on the matrix,
by multiplying, for example, to elicit more information.
In general, any model has two purposes:
to elucidate the mathematics of the current situation; and to
permit operations that will produce new information. Students
will learn about statistical models, function models, parametric
models, graph models, and many others in CMIC. For example, if
a coordinate graph models the relationship between dependent
and independent variables, say these are height (h) and time
(t), then replacing t with t - 3 produces a new graph
with new information about values of height if the times are
delayed by 3 units. Or, if a matrix models sales of three models
of cars in a current year, then multiplying by 0.1 will give
a new matrix predicting sales if there is an increase of 10%
across the board.
- Can the same information be displayed in a matrix in different
ways? Explain. Yes, you might switch the rows and columns,
or list the rows and columns of a matrix in different orders.
This would not alter the information at all. The dimensions would
be reversed.
- What are some advantages of using matrices to organize and
display information? What are some disadvantages? Matrices
are an extension of tables, usable for two-variable data. They
are easily stored in and operated on by computers, allowing quick
and convenient analysis of the data. Possible disadvantages include
that only two variables can be included in a matrix model. In
addition, matrices are not as visual as graphs; this means that
numerical patterns in the data are not so easy to spot in a matrix
format, and also that when using matrices to solve systems of
linear equations care must be taken with solutions to interpret
the results. The matrix solution of a system only works when
a matrix has an inverse, but there are two occasions when
a matrix has no inverse; one indicates many solutions,
the other no solution. A graphical solution makes this distinction
obvious; a matrix solution does not.
In Course 3, students will extend matrix
models to solving linear programming problems; in this case,
all intersection points are not helpful in the solution, a problem
that is clear in a graphical solution, but not a matrix solution.
In general, it is important that students develop an awareness
of different solution strategies and are able to weigh the pros
and cons of each.
|
| b. |
Sometimes a situation involves
two variables that are linked by two or more conditions. These situations
can often be modeled by a system of two linear equations of the format ax + by = c. Describe
at least three different methods for solving such a system of linear
equations.
- In Lesson 3, students learned how to set up a matrix
equation of the form Ax = B, where A and B
are matrices, and then solve by multiplying each side by
the inverse of matrix A, probably using a calculator
to find the inverse matrix. For example, if 2x + 3y = 5
and 3x - 6y = 11, then the matrix equation
is
  =  or
AX = B. The solution is X = A-1B =  .
- Students could also rewrite the equations in the format "y = ax + b" so
that they can both be graphed on a calculator, to find the
intersection point. This method is one students used in Course
1. For the above example, this would mean graphing y = 5/3 - (2/3)x
and y = (1/2)x - 11/6.
- Again relying on Course 1 methods, students could make tables
by hand of the equations in their current format, substituting
values for x and finding corresponding values for y, and then
graphing these by hand to find the intersection. In the above
example, the tables might look like:
|
x
|
y
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1
|
1
|
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2
|
1/3
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3
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-1/3
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4
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-1
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x
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y
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1
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-8/6
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2
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-5/6
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3
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-2/6
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4
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1/6
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- They might make tables by hand or calculator and use the
tables to find the solution.
- They might rewrite the equations in the format "y = ax + b" and
then substitute to make an equation with only one variable,
which can then be solved using algebraic rules for operating
on equations. In the above example, we would have 5/3 - (2/3)x = (1/2)x - 11/6,
which would then become 21/6 = (7/6)x, so x must
be 3. This is a skill students learned to use in Course 1.
|
| c. |
List all the different operations on matrices that you have investigated
in this unit.
Students learned, in Lesson 1, to use row or column sums to
give total sales of a particular brand of shoes or in a particular
month (see Investigation 1), to multiply a matrix by a number to
predict the auto production if there is a 10% increase (see Investigation
3), and to add or subtract two matrices to compare auto
production (see Investigation 3). In Lesson 2, students learned
to multiply two matrices to find the costs of ordering uniforms
from two different companies for three different teams (see Investigation
2). Still in Lesson 2, students extended multiplication of matrices
to finding powers, and used these results to analyze a food
web (see Investigation 3).
|
| d. |
For each operation listed
in Part c:
- Describe how to use the operation using paper and pencil. To
add (or subtract) matrices, you add (or subtract) corresponding
entries. To multiply by a number, you multiply every entry
of the matrix by that number. To multiply two matrices A and
B, you multiply rows of matrix A by columns of matrix B, as
follows: you multiply the first entry of row 1 matrix A by
the first entry of column 1 matrix B, then the second entry
of row 1 matrix A by the second entry of column 1 matrix B,
and so on until all row 1 and column 1 entries have been multiplied.
Then these are summed to make the (row 1, column 1) entry of
AB. Now multiply row 1 matrix A by column 2 matrix B in the
same way to get the (row 1, column 2) entry of AB. Proceed
doing this until row 1 matrix A has multiplied every column
of matrix B. Then multiply row 2 matrix A by each column of
matrix B until all the row 2 entries of AB are complete. And
so on. (See the Quick Summary for
this unit for examples of matrix operations.) Because this
is how matrix multiplication is defined, we can only multiply
matrices A and B if the number of entries in row 1 matrix A
matches the number of entries in column 1 matrix B, which is
to say that the number of columns in matrix A must match the
number of rows in matrix B. Thus, a matrix with dimension 3
by 4 (which means 3 rows and 4 columns) can multiply any matrix
with dimension 4 by Z (where Z can be any number of columns).
The resulting AB would have dimension 3 by Z. This definition
also has repercussions on the existence of inverse matrices.
Because the multiplications A(A-1) and (A-1)A
must result in the same identity matrix, only square matrices
have inverses. (If A were a 3 by 4 matrix then A-1 would
have to be a 4 by 3 matrix to give a 3 by 3 identity for the
result. But if we reverse the order, then (A-1)A
would require us to multiply a 4 by 3 matrix with a 3 by 4
matrix, for a 4 by 4 result, NOT the same identity in both
cases. A cannot be a non-square matrix if the inverse exists.)
- Describe how to perform the operation using a calculator. Different
calculators have different steps, but all require the user
to enter the cell values into a matrix with a particular name
and store that. You will have to specify the dimensions so
that there are enough spaces to be filled. Repeat with another
matrix with a different name. Then on the home screen call
up the names of the matrices with the operation required, paying
particular attention to multiplication since order matters.
- Give at least one example showing how the operation can be
used to help you analyze some situation. Students found the
difference in auto production of three car companies each quarter
for two years and the total production by subtracting or adding
matrices. They multiplied the number of houses roofed by three
crews in two time periods, by the time required per roof to do
two different processes, by these same three crews, to find the
total time required by all crews to complete the two different
processes.
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If you would like to see specific problems from Course 2, Unit 1, a
link is provided to Examples of Tasks from
Course 2, Unit 1. If you are interested in following up on the Algebra
or Discrete Mathematics strands in general, then the Scope
and Sequence will help you see where different concepts are introduced.
On the Algebra and Discrete
Mathematics pages, you can read an explanation of the main algebra
and discrete mathematics concepts as they are developed in all four courses.
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