of the first step. Remind them of the composition of two functions if they have already learned about this topic.
Associative Property of Matrices –Students have heard about the commutative and associate properties many
times during their education in mathematics. These are important properties in the study of matrices and become
more meaningful for the students when applied to this new set. The fact that the commutative property does not
hold for matrix multiplication is surprising at first, and is a concept that needs to be revisited. Although it has been
discussed in recent lessons, it would be beneficial to go over it again here before discussing the property that is
really of interest in this section, the associate property of matrix multiplication. So far the students have seen that
the image of points can be found under a rotation or reflection by multiplying a matrix made up of the coordinates of
the points by a matrix specific to the chosen transformation. In this section they should discover that because matrix
multiplication is associative, they can multiply two or more transformation matrices together to get a matrix for the
composition.
Key Exercises:
Consider the triangle with matrix representationA=
2 1
5 3
4 4
.
Matrix Multiplication is NOT Commutative
- Use matrix multiplication to rotate the triangle 90 degrees, then take the image and reflect it in thex−axis.
- Use matrix multiplication to reflect the original triangle in thex−axis, then take the image and rotate it 90 degrees.
- Are the results of #1 and #2 the same?
Matrix Multiplication is Associate
- Multiply the matrix used to rotate the triangle 90 degrees by the matrix used to reflect the triangle in the liney=x.
- Use the matrix found in #4 to transform the triangle. Is the result the same as that in #1?
Answers:
1.
2 1
5 3
4 4
[
0 1
−1 0
]
=
−1 2
−3 5
−4 4
[
1 0
0 − 1
]
=
− 1 − 2
− 3 − 5
− 4 − 4
2.
2 1
5 3
4 4
[
1 0
0 − 1
]
=
2 − 1
5 − 3
4 − 4
[
0 1
−1 0
]
=
1 2
3 5
4 4
no
[
0 1
−1 0
][
1 0
0 − 1
]
=
[
0 − 1
− 1 0
]
5.
2 1
5 3
4 4
[
0 − 1
− 1 0
]
=
− 1 − 2
− 3 − 5
− 4 − 4
, yesTessellations
Review Interior Angles Measures for Polygons –Earlier in the course students learned how to calculate the sum of
the measures of interior angles of a convex polygon, and how to divide by the number of angles to find the measures
of the interior angle of regular polygons. Now would be a good time to review this lesson. The students will need
this knowledge to see which regular polygons will tessellate and the final is fast approaching.
Move Them Around –When learning about regular and semi-regular tessellation it is helpful for students to have
Chapter 2. Geometry TE - Common Errors