Geometry, Teacher\'s Edition

2.7 Similarity

Ratio and Proportion

Keep it in Order –When writing a ratio, the order of the numbers is important. When the ratio is written in fraction
form the amount mentioned first goes in the numerator, and the second number goes in the denominator. Remind the
students it is important to keep the values straight, especially if they are looking at the male to female student ratio
at their top three college choices.

To Reduce or Not to Reduce –When a ratio is written in fraction form it can be reduced like any other fraction.
This will often make the arithmetic simpler and is frequently required by instructors for fractions in general. But
when reducing a ratio, useful information can be lost. If the ratio of girls to boys in a classroom is 16 to 14, it may
be best to use the fraction^1614 because it gives the total number of students in the class where the reduced ratio^87 does
not.

Consistent Proportions –A proportion can be correctly written in many ways. As long as the student sets up the
ratios in a consistent, orderly fashion, they will most likely have written a correct proportion. There should be a
common tie between the two numerators, the two denominators, the numbers in the first ratio, and the numbers in
the second ratio. They should think about what the numbers represent, and not just use them in the order given in
the exercise, although the numbers are usually given in the correct order.

Key Example:

1. Junior got a new hybrid. He went 525 miles on the first five gallons that came with the car. He just put 12 gallons
   in the tank. How far can he expect to go on that amount of gas?

Answer:

525

5

=

x

12

He can expect to go 1,260 miles.

x= 1260

Note: Students will be tempted to put the 12 in the numerator of the second ratio because it was the third number
given in the exercise, but it should go in the denominator with the other amount of gas.

The Fraction Bar is a Grouping Symbol –Students know that parenthesis are a grouping symbol and that they
need to distribute when multiplying a number with a sum or difference. A fraction bar is a more subtle grouping
symbol that students frequently overlook, causing them to forget to distribute. To help them remember have them
put parenthesis around sums and differences in proportions before they cross-multiply.

Example:x+ 53 =x− 78 becomes(x+ 53 )=(x− 78 )

Properties of Proportions

Everybody Loves to Cross-Multiply– There is something satisfying about cross-multiplying and students are prone
to overusing this method. Remind them that cross-multiplication can only be used in proportions, when two rations
areequal to each other. It is not appropriate to cross-multiply when two fractions are being added or subtracted.

Chapter 2. Geometry TE - Common Errors