Unit 4, Lesson 10
Dividing by Unit and Non-Unit Fractions
Grade 6
Preparation
Lesson Narrative
In this lesson, students begin by using tape diagrams to recall and visualize two ideas from grade 5:
- Dividing by a whole number has the same outcome as multiplying by the unit fraction .
- Dividing by a unit fraction has the same outcome as multiplying by the whole number .
They apply the second idea to find the value of other expressions of the form without drawing tape diagrams.
Next, students use the same diagrams to analyze the effects of dividing by non-unit fractions. Through repetition, they notice a pattern in the steps of their reasoning (MP8) and structure in the visual representation of these steps (MP7).
Students see that division of a number by a non-unit fraction can be thought of as having two steps:
- Finding how many unit fractions there are (dividing the number by the unit fraction).
- Putting the unit fractions into groups the size of the non-unit fraction to see how many groups there are (dividing the result by the numerator of the non-unit fraction).
For instance, to divide by is equivalent to dividing by , and then again by 2. Because dividing by a unit fraction is equivalent to multiplying by 5, to divide a number by , we can multiply it by 5 and divide the result by 2.
- Interpret and critique explanations (in spoken and written language, as well as in other representations) of how to divide by a fraction.
- Use a tape diagram to represent dividing by a non-unit fraction $\frac{a}{b}$ and explain (orally) why this produces the same result as multiplying the number by $b$ and dividing by $a$.
- Use a tape diagram to represent dividing by a unit fraction $\frac{1}{b}$ and explain (orally and in writing) why this is the same as multiplying by $b$.
Let’s look for patterns when we divide by a fraction.
Required Preparation
None