The easiest (and the most common) way to teach division of fractions is to just give the rule: To divide 5/8 by 2/3, just multiply 5/8 by the reciprocal of the divisor, 2/3. That is, 5/8 ÷ 2/3 = 5/8 × 3/2 = 5×3/8×2 = 15/16. If this is all we want our students to learn why bother teaching? I’m sure our students will be old enough to buy/download a calculator before they find themselves working on a problem that involves division of fraction.  Will someone please tell me what mathematics will the students learn here?

The following solutions show how we can explain to students where the rule comes from. It involves the same idea used for finding equivalent fraction or proportions – multiplying the upper and lower number of a proportion by the same number preserves the proportion. This is something that they learned before. The task of division of fractions becomes a simple extension of finding an equivalent fraction.

##### Teaching Sequence on Division of Fractions
###### Task 1 – Find fractions equivalent to the fraction 5/8.

This should be easy for learners as all they needed to do is to multiply the same number in the numerator and denominator. This is more of a revision for them.

###### Task 2 – Find fractions equivalent to

Here they will apply the same idea used in task 1: If you multiply, the same number (except 0) to numerator and denominator, you produce equivalent fractions. You can ask the students to classify the fractions they made. One group I’m sure will have a fraction for numerator and whole number for denominator; another group will have fractions for both numerator and denominator; and, another will have whole numbers for numerator and denominator. The last group is what you want. This fraction is in simplest form. They should be ready for Task 3 after this.

###### Task 3 – Find the fraction in simplest form equivalent to

From here you can ask the students to express the fraction as a division (this is one of the meaning of fraction – an indicated division) and rework their solution. It should be something like this:

You can challenge your students to find the shortest possible solution of getting the correct answer. It will involve the same idea of multiplying the dividend and the divisor by the same number. I’m sure that after doing the tasks above, they will be able to figure out the following solution which now leads to the the procedure ‘when dividing fractions, just multiply it by the reciprocal of the divisor’:

Procedural fluency does not mean doing calculation with speed and accuracy even without understanding. Remember that procedure is only powerful and useful in problem solving when students understand what it means and why the procedure is such. I suggest you also read my post on what it means to understand fractions and math knowledge needed by teachers to teach fractions and decimals.

Note:

The above lesson is not just about division of fractions. I made it in such a way that weaved in the lesson are the ideas of equivalent fractions, proportion, the property that when you multiply same number to the numerator (or dividend) and to the denominator (divisor) it does not change the value of the quotient, division by 1, etc. Working with the tasks engages students to the same process/technique they will be applying when they work with rational algebraic expressions. The main point is to use the lesson on division of fractions as context to make connections and to teach important ideas in mathematics. I think this is how we should teach mathematics.

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### 4 Responses to “Why do we ‘invert’ the divisor in division of fractions”

1.

Good post. The question that (imo) still needs to be answers is this: Why when multiplying fractions a/b x c/d, the answer is the fraction ac / bd…you just “multiply straight across?” Why doesn’t division of fractions work the same way? Why can’t you just “divide straight across?”

In fact, you CAN divide straight across. For example, 3/4 ÷ 5/6 = 3÷5 / 4÷6 = 18÷30 / 20÷30 = 18/20 = 9/10.

Now, multiplying fractions and dividing fractions work the same way.

2.

How about a simple intuitive approach? If you divide a number by 2, you have ½ as much. If you divide a number by 3, you have ⅓ as much. If you divide a number by x, you have 1/x as much. \$10/2 = ½ of \$10 = \$5. \$10/\$½ = 10 x 2 = 20 (# of half-dollar coins in \$10 is 20. A \$10 roll of quarters = 40 quarters: \$10/\$¼ = \$10 x 4quarters/\$1 = 40. It’s intuitive. It shows a pattern. It isn’t merely memorizing a rule and bypassing understanding. And it isn’t the complex example above with extra steps that will overwhelm and impel struggling kids to give up after being thrown down a rabbit hole of ineffiency.

3.

My sixth graders suffer from the “Curse of Knowledge.” Because they learned how to solve this kind of problem in 5th grade (invert and multiply) they do not see the need to understand how or why it works. So I ban this method, just as I don’t allow them to move the decimal point when multiplying decimals.

I eventually lead students to the steps in the article, but even more meaningful is the more concrete understanding that the Connected Math Program offers by supplying stories that involve the amount of cheese needed to make a pizza. I also have students draw illustrations of these problems using number lines and area models (rectangular units are easiest):

(Set 1): For 6 ÷ 1/3: If one small pizza requires 1/3 cup of cheese, how much cheese is needed for 1 pizza? 3 pizzas? 6 pizzas?

(Set 2): For 6 ÷ 2/3: If one large pizza requires 2/3 cup of cheese, how much cheese is needed for 1 pizza? 3 pizzas? 6 pizzas?

How do the answers to the 2nd set of problems relate to those in the first set?

With enough examples, students see that the dividend gets multiplied by the divisor’s denominator (Set 1), and then divided by the divisor’s numerator. They can then generalize to division problems in which the dividend is also a fraction.

Do my students like this method? A few do, but most find it rather painful to have to think things through, when most of their school experience has taught them that finding the correct answer quickly – if magically – is what is rewarded.