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Ratios and Proportions

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to reinforce the algebraic concept of ratios and proportions. Proportions are necessary when discussing similarity of geometric objects.

What’s the difference? Ratios and rates are both fractions. However, ratios compare same units, while rates compare different units. Ask students to brainstorm types of ratios and rates. Rates are typically much easier for students to identify – miles per hour, cost per pound, etc.

Look Out! Students easily get confused when we throw proportions into the mix. For some reason, students do not realize that the equal sign (=) in a proportion is different than a multiplication sign (*) when asking to find the product of two fractions. For example, students will attempt to solve these two statements the same way:

\frac{3}{4}* \frac{x}{7} && \frac{3}{4} = \frac{x}{7}

Encourage your students to understand the difference between finding the product of two fractions and using the means-extremes method of cross-multiplication

Look Out! Another pitfall is cross-multiplication versus cross-reducing. You may have to take some time to discuss the difference and allow your students to practice doing both.

Additional Example: A model train is built \frac{1}{64} scale. The stack of the model is 1.5”. How tall is the real smokestack?

Food For Thought! “Why are these the means?” The best answer I have heard was, “The extremes are called such because they are on the far ends of the equation, meaning ad = bc.” Before the fraction bar became commonplace, people would write fractions using the colon. 3:6 = 1:2. Therefore, 6 and 1 represent the means (middle values), while 3 and 2 represent the extremes.

Properties of Proportions

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to demonstrate to students that the order in which you write the proportion is irrelevant, the answer comes out identical.

Try This! Prior to reading through the lesson, and using the following example, ask students to create their proportion. Look for several different proportions. Spread these students around the room. Have the remainder of the class match their proportion to the “totem pole.” This allows students to see that there is no one correct way to write a proportion.

Question: A yardstick makes a shadow 6.5’ long. Raul is 6' \ 3". How long is his shadow? Be sure students convert a yardstick to 3’ before continuing with the proportion.

While there are many correct ways to write a proportion, encourage students to visualize what the proportion is stating. For example, the following are both correct (as are their reciprocals):

\frac{\text{length (stick)}} {\text{shadow (stick)}} = \frac{\text{length (person)}} {\text{shadow (person)}} && \frac{\text{length (stick)}} {\text{length (person)}} = \frac{\text{shadow (stick)}} {\text{shadow (person)}}

Nonetheless, the units are still the same in each ratio. The first proportion uses length and shadow as units while the second uses stick and person.  

Similar Polygons

Pacing: This lesson should take one class period

Goal: This lesson connects the properties of proportions to similar polygons. An introduction to scale factors is also presented within this lesson.

An alternative way of determining a scale factor is by using the fraction \frac{\mathrm{image}}{\mathrm{preimage.}} This relates to the notion that similar figures are formed by an applying an isometry, mapping an image onto its preimage. The first figure written in the similarity statement represents the preimage and the second figure represents its corresponding image. Therefore, the scale factor k, is a ratio of the length of the image to the preimage.

Extension! A golden rectangle is a rectangle in which the ratio of the length to the width is the Golden Ratio, approximately 1.6180339888. The Golden Ratio is denoted by the Greek letter Phi and can be found in nature, biology, art, and architecture. For example, golden rectangles can be found all over the Parthenon in Athens, Greece.

Using the diagram below, have your students measure distances and find how many golden rectangles can be found.


Similarity by AA

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to enable students to see the relationship between triangle similarity and proportions. While the angle-angle relationship does not necessarily lead to congruence, its properties are still imperative to similarity.

How Does it Work? Indirect measurement utilizes the Law of Reflection, stating that the angle at which a ray of light (ray of incidence) approaches a mirror will be the same angle in which the light bounces off (ray of reflection). This method is the basis of reflecting points in real world applications such as billiards and miniature golf.

Additional Example: Pere Noel is shopping for a Christmas tree. The tree can be no more than 4\;\mathrm{meters} tall. Mary finds a tree that casts a shadow of 2\;\mathrm{m}, whereas Mary (120\;\mathrm{cm} tall) casts a shadow of 0.8\;\mathrm{m}. Will the tree fit in Pere Noel’s room? Yes, the tree is 3\;\mathrm{meters} tall, therefore, it will fit in the room.

Similarity by SSS and SAS

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to extend the SSS and SAS Congruence Theorems to include similarity.

Visualization! Now may be a good time to discuss similarity and congruency by drawing a Venn diagram. Students may ask the question, “How can triangles be congruent and similar simultaneously?” The diagram below will help clear questions.

Similar figures are usually thought to be produced under dilations (size changes). However, congruent figures are a specific type of similarity transformation. Therefore, rotations, reflections, glide reflections, translations, and the identify transformation all yield similar figures.

Similarity Transformations

Pacing: This lesson should take one class period

Goal: Dilations produce similar figures. This lesson introduces the algorithm to produce similar figures using measurements and a scale factor, k

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 typical photograph is 4” \times 6”. An 8” \times 10” 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.


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 word frangere, 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.

  1. Hold your paper in landscape format.
  2. Draw a horizontal segment \overline{AB} in the center of the paper.
  3. Find C, the midpoint of \overline{AB}.
  4. Measure AC and take half of that distance. This is essentially \frac{1}{4}AB.
  5. Draw \overline{DC} (the length found in step 4) perpendicular to \overline{AB}.
  6. Draw lines \overline{AC} and \overline{BC}, forming \triangle ADC.
  7. Repeat steps 3 - 6 for \overline{AC}.
  8. Repeat steps 3 - 6 for \overline{BC}.

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