How dividing fractions works
Teaching students how to divide fractions is part of the Common Core State Standards for Mathematical Practice. One of the most valuable things to teach your students when dividing fractions is what the answer means. Take a look at the example below:
½ ÷ ⅙ = 3
Step 1: Flip the divisor into a reciprocal
A reciprocal is what you multiply a number by to get the value of one. If you want to change two into one through multiplication you need to multiply it by 0.5. In fraction form this looks like:
²⁄₁ × ½ = 1
To find the reciprocal of a fraction you simply flip the numbers. The denominator becomes the numerator and vice versa.
Take a look at the example equation again:
½ ÷ ⅙ = ?
The first step to solve the problem is to turn our divisor, ⅙, into a reciprocal.
⅙ → ⁶⁄₁
Step 2: Change the division sign to a multiplication symbol and multiply
Dividing and multiplying are opposites of each other. When you create a reciprocal of a number, you’re creating its opposite as well. In a division problem, when you turn the divisor into a reciprocal, you also need to change the equation from division to multiplication.
Now that you’ve found the reciprocal of your divisor, you can change the equation from division into multiplication.
½ ÷ ⅙ = ? → ½ × ⁶⁄₁ = ?
We’ve got an extensive guide on how to multiply fractions, but here’s a quick tutorial:
- Multiply your numerators to get your new numerator
- Multiply your denominators to get your new denominator
- Simplify the final fraction, if possible
For the example equation you have two problems to solve:
1 × 6 = 6 2 × 1 = 2 ½ × ⁶⁄₁ = ⁶⁄₂
Step 3: Simplify your answer if possible
Fractions symbolize a part of a whole. This means many fractions represent the same value, so why not make the fraction as simple as possible?
For example, you almost never say five-tenths or ⁵⁄₁₀. Instead, you simplify that to one-half or ½.
To get a fraction down to its simplest form, you divide the numerator and denominator by their greatest common factor. The greatest common factor in ⁵⁄₁₀ is five. Dividing both numbers down by five leaves you with ½.
In the example question, the greatest common factor of ⁶⁄₂ is two. This turns your solution from ⁶⁄₂ to ³⁄₁, which is the same as saying three.
½ ÷ ⅙ = ? → ½ × ⁶⁄₁ = ⁶⁄₂ → ³⁄₁ → 3
Creating a reciprocal and multiplying an equation rather than dividing lets you skip several steps in an equation. It’s a shortcut that will make your students’ lives a whole lot easier!