# Thread: Dividing by Fractions and Multiplicative Inverses

1. ## Dividing by Fractions and Multiplicative Inverses

Hello, I am trying to develop a lesson plan for middle school students on dividing fractions using reciprocals without explicitly giving the algorithm of diving by a fraction by multiplying by its inverse.
I try to do this by asking scaffolding questions.

1. First, I ask "Suppose you divide 3 by 4, how would you write that using math symbols?" Anticipated response: "3 ÷ 4"

2. Then I ask, "How would you write that as a fraction?" Anticipated response: "3/4"

3. To bridge the connection, I ask, "How can we write 3/4 as a product of a whole number and a fraction?" Anticipated response $\displaystyle 3 * \frac{1}{4} = 3 \div 4$
This should help students understand the connection between dividing by a fraction and multiplying by its inverse.

However, the difficulty arises when I want to help student understand why and how they use the multiplicative inverse. For example, if I wanted students to understand why, $\displaystyle \frac{3}{4} \div \frac{1}{4} = \frac{3}{4} \times 4$

I would give a problem like, "Suppose I have 3/4 of a cookie. One person eats one serving of 1/4 of a cookie. How many persons can I feed?"

Essentially, how would I go about explaining why $\displaystyle \frac{3}{4} \div \frac{1}{4} = \frac{3}{4} \times 4$?

I thought of drawing block patterns - dividing a bar into quarters and shading in 3 blocks. I show that 1/4 goes 3 times in 3/4.

Thus, $\displaystyle \frac{3}{4} cookies \div \frac{1}{4} = 3 servings$

There is a disconnect, however, if I try to explain 3/4 x 4 since the servings and cookie example becomes meaningless because the divisor and dividend represent a different example.

Thank you for reading. If you have suggestions, I would love to hear it.

2. ## Re: Dividing by Fractions and Multiplicative Inverses

For starters, you need to emphasise that division does NOT mean "splitting into equal smaller pieces". They will be expecting to get smaller numbers, so will be surprised when they get an answer for $\displaystyle \displaystyle 2 \div \frac{1}{4}$.

You need to emphasise that division mean "how much" or "how many".

So $\displaystyle \displaystyle 2 \div \frac{1}{4}$ means "How many quarters in 2?" In other words, $\displaystyle \displaystyle 8$.

$\displaystyle \displaystyle \frac{1}{4} \div \frac{1}{2}$ means "How much of a half is in a quarter?" In other words, $\displaystyle \displaystyle \frac{1}{2}$.

Then you can get them to estimate an answer for harder questions, like $\displaystyle \displaystyle \frac{1}{2} \div \frac{2}{3}$, before formalising with "stay, change, flip".

And when you want to formalise, here is a proof...

Suppose you knew that $\displaystyle \displaystyle \frac{a}{b} \times n = \frac{c}{d}$, how would you find $\displaystyle \displaystyle n$?

Method 1: Divide both sides by $\displaystyle \displaystyle \frac{a}{b}$, so $\displaystyle \displaystyle n = \frac{c}{d} \div \frac{a}{b}$.

Method 2: Multiply both sides by $\displaystyle \displaystyle \frac{b}{a}$, giving $\displaystyle \displaystyle \frac{b}{a}\times \frac{a}{b} \times n = \frac{c}{d} \times \frac{b}{a} \implies n = \frac{c}{d} \times \frac{b}{a}$.

Therefore $\displaystyle \displaystyle \frac{c}{d} \div \frac{a}{b} = \frac{c}{d} \times \frac{b}{a}$.

Therefore to divide fractions, keep the first fraction the same, change the divide to a times, and flip the second fraction. "Stay, Change, Flip".