symmetry around they−axis, and odd functions have 180 degree rotational symmetry about the origin. Once a
function is classified as even or odd, properties and theorems can be applied to it.
Draw –Have students be creative and create their own logos or designs with specific types of symmetry. Using these
concepts in many ways will build a deeper understanding and the ability to apply the new knowledge in different
situations.
Dilations
Naming Conventions –Mathematics is a language, an extremely precise method of communication. While matrices
are named with uppercase letters, scalars are represented by lower case letters. Many times students do not know to
look for these types of patterns. Point out these conventions when an appropriate example arises and tell the students
to look for the subtle differences that have major significance when communicating with mathematics.
The Scale Factor and Area –Students frequently forget to square the scale factor when comparing the area of the
original figure to the image under a dilation. This relationship has been covered several times before when studying
area, similar figures, and when dilation was first introduced. Make a point of it again. This omission is quite common
and the concept is often used on the SAT and other standardized tests.
Additional Exercises:
- Does the multiplication of a scalar and a 2×2 matrix commute? If so, write a proof. If not give a counterexample.
Answer: Yes, multiplication of a scalar with a 2×2 matrix does commute.
Letkbe a scalar andA=
[
a b
c d]
.
ThenkA=k
[
a b
c d]
=
[
ka kb
kc kd]
=
[
ak bk
ck dk]
=
[
a b
c d]
k=AkHere a 2×2 matrix was used, but this same proof can be done with a matrix of any dimensions.
- What scalar could be multiplied by a matrix containing the vertices of a polygon to produce an image with half
the area as the original figure?
Answer: √^1
2
since(
√^1
2
) 2
=^12
- Will a dilation followed by a reflection produce the same image if the order of the transformations is reversed?
Why or why not?
Answer: The image will be the same regardless of the order in which the transformations are applied. This can be
justified with the commutative property of scalar multiplication.
Chapter 2. Geometry TE - Common Errors