Lessons preceding this address solving problems involving ratios and percents—problems they have had in seventh grade, but now possess more experience with the algebraic tools for solving them. In seventh grade, students solved ratio problems with “tape diagrams” but were encouraged to use algebra. Now students are only allowed to use algebraic equations for solving.
In this lesson, students learn how to multiply and divide fractions. They have already had some experience with this in this previous post. This lesson extends the concept to more complex forms, in which factoring and cancelling is part of the process. Identification of different types of polynomials that are factorable is essential, as is mastery of factoring in general.
Problem 1 is a percent problem which they have had in seventh grade. Both methods are shown. Problem 2 requires students to identify that the numerator is the difference of two squares. Also students must know how to reverse the denominator by factoring out -1. Excluded value is determined from the original fraction. Problem 4 is one that students have had in seventh grade using both tape diagrams and algebra. Problem 5 involves multiplying fractions which students have done before, and serves as a segue to today’s lesson.
Multiplying Fractions. After doing Problem 5 of the Warm-Ups I point out that the problem required some simplification.
“In this problem, like most fraction multiplications, we can cancel before or after we perform the multiplication. And in this case, it doesn’t matter, because it’s relatively easy in either case.” I illustrate this on the board:
“For more complex fraction, however, it is best to do any cancelling before you multiply to avoid a complicated expression in the end that is harder to simplify.”
Simplifying Products by Factoring.
“When you simplify fractions as we did in an earlier lesson, you factored as much as you could and then cancelled. This same rule applies for multiplying fractions. For example, suppose I have a multiplication like this.”
“Let’s factor the first fraction. Is there anything that can be factored?”
In fact, there is: the numerator, which most students identify straight off. I ask the same thing for the second fraction, and looking at notebooks, I pick a student who has factored both fractions correctly and have him/her write it on the board:
Students then proceed to cancel, and again I look at notebooks for the correct answer and have a student write it on the board.
Either answer is correct. I then ask them to find excluded values. In this case there are two:
Some students will still have difficulty with these, so I continue to ask for them to show excluded values for some portion of homework problems.
It is also important to remind students that any expression divided by itself becomes 1/1, and not zero!
Dividing Fractions. Dividing fractions are an extension of multiplying fractions. For division, the second fraction is inverted and then multiplied just as is done for numerical fractions. Division can be combined with this lesson, or treated as a separate lesson depending on how the students are doing with it.
Homework. Problems should consist of multiplication (and division if covered in this lesson), simplifying fractions and a few word problems involving ratios and percents.