An easy way to remember expansions versus contractions is a rhyme. Have your students repeat the rhyme until it
sticks! “A contraction is a proper fraction!” Improper fractions are mixed numbers, thus creating expansions.
Dilations can also be clarified using a photograph. School pictures are great examples of dilations. Suppose a typi-
cal photograph is 4[U+0080][U+009D]× 6 [U+0080][U+009D].An 8[U+0080][U+009D]× 10 [U+0080][U+009D]
enlargement (expansion) does not alter the appearance. This is also true for shrinking photos for wallets. Using a
base picture, bring in several enlargements and contractions to further illustrate this concept.
The above visualization can be used when discussing notation. The first dilation is denoted using the apostrophe (‘)
symbol. Sub sequent transformations add an additional apostrophe (“, ‘”, and so on). Labeling each picture you’ve
made with this notation will allow your students to visualize how to use image notation.
In-Class Activity!Reproduce the smiley face drawn below for each student. Using the algorithm presented in this
lesson, instruct students to enlarge the smiley face by a scale factor of 3.
http://www.clker.com/clipart-4263.html
Self-Similarity (Fractals)
Pacing:This lesson should take one class period
Goal:Fractals, a term coined only approximately twenty years ago, are a newly discovered genre of mathematics.
This lesson introduces students to popular fractals. Fractals possess self-similarity, thus maintaining properties of
similarity.
History Connection! Mathematician Benoit Mandelbrot derived the term “fractal” from the Latin wordfrangere,
meaning to fragment. A fractal is a geometric figure in which its branches are smaller versions of the “parent”
figure. Most fractals are explained using higher level mathematics, however, students can create their own fractal
patterns easily.
Additional Example:Make your own fractals! Follow these instructions to create a cauliflower fractal.
a. Hold your paper in landscape format.
b. Draw a horizontal segmentABin the center of the paper.
c. FindC, the midpoint ofAB.
d. MeasureACand take half of that distance. This is essentially^14 AB.
e. DrawDC(the length found in step 4) perpendicular toAB.
f. Draw linesACandBC, forming 4 ADC.1.7. Similarity