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.