*This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational.*

Students already know the rules about how to multiply and divide fractions. They also have learned the rules for multiplying negative numbers. Now these two skills come together so that students are doing problems like 2/3 x -5/6 and -4/5 ÷ 5/12.

Because they are familiar with these rules and procedures, I use this lesson to review the basics of fraction multiplication and division, as well as clear up misconceptions that occur along the way.

1. 3/5 ×10/4 Answer: 3/2

2. -3 × -15 Answer: 45

3. 24/40 ÷11/7 (Reduce to lowest terms) Answer: 3/5 ×7/11 = 21/55

4. (-10 × 5) – 45 Answer: -50-45 = -95

5. -4/5 +5/9 Answer: (-36+25)/45 = -11/45

**Teaching the Procedure. ** The teaching of this procedure typically goes very quickly and students find it accessible and easy. Starting with problem 1 of the warm-up, it is then easy to turn the problem into -3/5 ×10/4, and -3/5 ×-10/4 so that students can now apply the rules for multiplying negative fractions. Similarly, fractional division is handled the same way. Thus, variations to problem 3 using negatives can be introduced: -24/40 ÷11/7, and -24/40 ÷ -11/7.

**Cancelling, Reducing to Lowest Terms, and Misconceptions. **What becomes clear with these examples is the extent to which students cross cancel and reduce (i.e., simplify) fractions prior to the multiplication step. Some students’ answer to problems like 3/5×5/17 is 15/85 which takes more time and effort to multiply and reduce (simplify) to lowest terms. It takes much less effort to cancel the 5’s before multiplying, obtaining the answer of 3/17.

Other students may remember to cross cancel, but neglect to simplify fractions such as 24/40 , or 10/4. There is a mistaken notion that when multiplying fractions, one can cancel diagonally, but not within the same fraction. I like to put that misconception to rest quickly. In algebra they will need to get used to doing both.

And although we call it “cancelling”, some mention should be made that what we are doing when reducing fractions, or cross cancelling, is dividing. The fraction 10/4 is equal to 5/2 because both the numerator and denominator are divided by 2 to obtain the equivalent fraction. Similarly the fraction 5/5 is equivalent to 1/1. Another misconception to put to rest is that a fraction like 5/5 does not equal zero. It seems very obvious that it is not, but I have seen students in algebra mistake the variable version of this —e.g., 2x/x — as zero. Also, there is the common mistake of trying to cancel (i.e., divide) things that cannot be divided such as in (5+x)/(5+y), where students will think they can divide the 5’s. It’s a continual struggle and not just a one-off solution, but good to start putting some mistakes to rest right now.

**Multiplication by Reciprocals. **Another habit to instill is recognizing that multiplication of reciprocals equals one. A problem like 4/5 x 5/4 obviously equals1. A problem like 4/5 ×35/28 is less obvious—at first, until one realizes 35/28 is the same as 5/4. Students who are not in the habit of reducing fractions before multiplying will be making an easy problem harder. -4/5 x 35/28 is -1.

More importantly, however, is that students see that multiplication by a reciprocal equals one. Conversely, they also should be familiar with one divided by any fraction equals its reciprocal; i.e., 1/(a/b) = b/a . These properties will be revisited in the next unit on simple equations, since equations in the form a/b x = c are solved by multiplying both sides of the equation by the reciprocal of a/b . This procedure is also a key to proving why the “invert and multiply” rule for fractional division works as it does for common fractions.

**Word Problems.** Word problems that rely on multiplication and division of fractions are an important application of this topic. This should be taught as a separate topic on a different day. Students will need to recognize when a problem calls for division or multiplication. To scaffold these, it is helpful to give equivalent problems using whole numbers. For example, “How many 2 ft boards can be sawed from a 10 ft board?” Students will see immediately that solving the problem requires division: 10÷2. A problem like how many 1 1/2 ft boards can be sawed from a 15 ft board is then solved using the same structure: 15÷1/12.

Also, some problems require interpretation of a mixed number answer when the answer has to be a whole number. For example: Nina can carry 16 lbs. How many 1 1/2 lb books can she carry? One can divide 16 by 1 1/2 to get the answer of 32/3 or 10 2/3. But the answer must be a whole number since we are not dealing with fractions of a book. So the answer is that she can only carry 10.

1. The temperature is currently 0 degrees F. The temperature increases 2 1/2 deg F each hour. What will the temperature by 3 hours from now? Three hours ago?

Answers: Three hours from now is solved by 3 x 2 1/2: 3 x 5/2 = 15/2 or 7 1/2 degrees. Three hours ago would be -3 x 2 1/2 = – 7 1/2 or 7 1/2 degree decrease.

2. The temperature is currently 0 deg F. The temperature *decreases* 1 1/2 deg F each hour. What was the temperature 3 hours ago?

Answer: -3 x -1 1/2 = -3 x -3/2 = 9/2 or 4 1/2 degree higher.

3. 4/5 of a lasagna are shared by 3 people. What fraction of the lasagna does each person eat?

Answer: 4/5 ÷ 3 = 4/5 x 1/3 = 4/15

4. Sam rides his bike at 5 1/2 miles per hour. How far does he ride in 6 hours?

Answer: 5 1/2 x 6 = 11/2 x 6 = 33 miles

5. Julie rides her bike at 4 1/2 miles per hour. What fraction of a mile does she ride in 6 *minutes*?

Answer: Students must express 6 minutes in fraction of an hour: 6/10 or 1/10. Equation is 4 1/2 x 1/10 = 9/2 x 1/10 =9/20 of a mile

6. For problem 5, what distance is that in feet?

Answer: Students must know that there are 5,280 feet/mile. Then the answer is given by 9/20 x 5280 = 2,376 feet.

7. How many 1/2 cup servings are in 3/4 of a cup of yogurt?

Answer: 3/4 ÷ 1/2 = 3/4 x 2 = 6/4 = 1 1/2 servings. (Fraction of a serving is permissible here!)

8. Jim has 3/5 lb of dry pasta. He needs 3/16 lb of dry pasta to feed each person. How many people can he feed?

Answer: 3/5 ÷ 3/16 = 3/5 x 16/3 =16/5 or 3 1/5, so he can feed 3 people.

Biggest pet peeve is using grouping with any fractions. It’s fine for estimating, not so good for actually simplifying.

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“More importantly, however, is that students see that multiplication by a reciprocal equals one.”

This. This is one of my biggest pet peeves about students coming to high school math from middle school algebra – their understanding of the word “reciprocal” is “you flip it.” Yes, but, why does that matter? Why are reciprocal pairs special? These understanding are key to understanding the properties that make algebra happen. Yet many students struggle with properties because they don’t understand that these are the “rules of the game.”

An additional observation – students who don’t understand that reciprocal pairs multiply to 1 or that opposite pairs add to zero, are the first ones in Geometry who, when asked what supplementary angles are, will say “they equal 180.” The humanity!

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