Geometry, Teacher\'s Edition

1.7 Similarity

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:

3

4

∗

x

7

3

4

=

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 641 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, meaningad=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 [U+0080][U+0099]long. Raul is 6′3”. How long is his shadow?Be
sure students convert a yardstick to 3 [U+0080][U+0099]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):

Chapter 1. Geometry TE - Teaching Tips