Lesson plan

At this point, students will have already learned how to divide fractions using models. If they haven’t, check out our Dividing fractions with models lesson! In this lesson, we’ll focus on dividing fractions without models by following the algorithm “dividing by a number is the same as multiplying by the reciprocal”.

Grade 6

Number System

6.NS.A.1

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Students will be able to divide fractions using the concept of multiplying by the reciprocal.

- Slideshow
- Online Practice

Start the lesson by sharing slide `1` of the slideshow with a notice and wonder. This will allow students to look at different types of fraction division problems and what they notice and wonder about their patterns.

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Once students have written down noticings and wonders, have students share them with the class. Write them down on the board for all students to see. After students have shared, you might want to ask some targeted questions that will help bring up these points if students have not already noticed these.

- when dividing by a whole number, the answer is smaller than the first number.
- when a whole number is divided by a fraction, the answer is more than the whole number
- when dividing by a whole number, it makes the denominator that many times larger.
- when a larger fraction is divided by a smaller fraction, the answer is more than a whole and the other way around
- that there are `\frac{1}{2}` fits into `4` eight times but why does `\frac{1}{4}` divided by `\frac{1}{2}` result in `\frac{1}{2}`.
- that when you divide by a fraction, it is like you are multiplying the first number by the denominator

At this point, you can introduce these two division equations that students would already have worked on in the lesson “Dividing fractions with models”.

`3\div \frac{3}{4} = 4` `4\div \frac{2}{3} = 6`

Ask the question, “What is common to these two equations?” Students are likely to bring up the following points:

- you can divide the first number by the numerator and then multiply by the denominator
- if you flip the second number and multiply it, you get the same answer

You might need to add these two equations if they do not notice a pattern:

`3\div \frac{3}{4} = 4` `4\div \frac{2}{3} = 6`

`3\times \frac{4}{3} = 4` `4\times \frac{3}{2} = 6`

If students recognize a pattern similar to the division equation, congratulate them for discovering the division algorithm. Tell them what they figured out is that “to divide by a number, we have to change division to multiplication and flip that number”. When you flip the number, it has a fancy name called “reciprocal”. It is used in many places but definitely in division! So, we will learn more about reciprocals and practice finding reciprocals.

Review the definition with students. You can remind students that reciprocals are multiplicative inverses. Through this example, you may want to show students that finding the reciprocal of a fraction is like “flipping” that fraction.

Allow students a few minutes to find the reciprocals on their own. They can check their answers with a classmate. Based on the previous example, students should be able to “flip” the fractions to find the reciprocals of `\frac{2}{3}` and `\frac{8}{5}`.

- When you get to `\frac{1}{9}`, students will likely write `\frac{9}{1}`. Remind them that any number written as a fraction over `1` can be simplified to a whole number. So the reciprocal of `\frac{1}{9}` is `9`.
- Then with `6`, you’ll want to encourage students to write that as a fraction over `1`, so `\frac{6}{1}`. From there, students can flip the fraction to find the reciprocal, `\frac{1}{9}`.

Tell students that we learned that we can divide fractions by multiplying - but we have to flip the second number first. That the flipped number is called a reciprocal.

Complete the rest of the example problems as a class, following the same steps. Remind students that whole numbers can be made into a fraction by putting the number `1` in the denominator.

- Students may divide the numerators and divide the denominators.
- Students might find the reciprocal of both the numbers or just the first.
- Students might change the division to multiplication but not find the reciprocal of the second.

After you’ve completed the examples with the whole class, it’s time for some independent practice! ByteLearn gives you access to tons of dividing fraction activities. Check out their online practice and assign to your students for classwork and/or homework!

Dividing Fractions Without Models Practice

Problem 1 of 4

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