Multi-Digit and Fraction Computation
Lesson 4
Objective
Use visual models and patterns to develop a general rule to divide with fractions.
Common Core Standards
Core Standards
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6.NS.A.1 — Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Foundational Standards
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5.NF.B.6
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5.NF.B.7
Criteria for Success
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- Determine that dividing a number by a unit fraction is the same as multiplying by the denominator of the fraction; show this in a visual model.
- Determine that dividing a number by a fraction is the same as multiplying by the denominator and then dividing by the numerator of the fraction; show this in a visual model.
- Understand the reasoning behind the invert and multiply rule or multiplying by the reciprocal.
Tips for Teachers
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- In this lesson, students use visual models and patterns to develop the general rule for dividing by fractions (MP.8). The focus of this lesson is on the development of the rule rather than on the use of it. Students will have opportunities for a lot of practice in upcoming lessons.
- Making Sense of Division of Fractions from SERP includes three great videos that demonstrate dividing with fractions using models and methods alternative to the general algorithm. They may be useful for your own reference or to use with students in smaller group settings.
- Additionally, Delaying "Invert and Multiply", also from SERP, is a valuable resource for teachers to deepen their understanding of fraction division and why the invert and multiply rule works prior to teaching the method to students.
Remote Learning Guidance
If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Problems 1 and 2 (benefit from discussion). Find more guidance on adapting our math curriculum for remote learning here.
Anchor Problems
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Problem 1
The number $$3$$ is divided by unit fractions $${\frac{1}{2}}$$, $$\frac{1}{3}$$, $${\frac{1}{4}}$$, and $${\frac{1}{5}}$$. For each division problem, draw a visual model to represent the problem and to find the solution. Then complete the rest of the chart and answer the questions that follow.
| Division Problem | Visual Model | Quotient | Multiplication Problem |
| $$3\div \frac{1}{2}$$ | |||
| $$3\div \frac{1}{3}$$ | |||
| $$3\div \frac{1}{4}$$ | |||
| $$3\div \frac{1}{5}$$ |
What pattern do you notice? What generalization can you make? Explain your reasoning.
Guiding Questions
Problem 2
The number 3 is now divided by fractions $${\frac {1}{4}}$$, $${\frac{2}{4}}$$, $${\frac{3}{4}}$$, and $${\frac{4}{4}}$$. For each division problem, draw a visual model to represent the problem and to find the solution. Then complete the rest of the chart and answer the questions that follow.
| Division Problem | Visual Model | Quotient | Multiplication Problem |
| $$3\div \frac{1}{4}$$ | |||
| $$3\div \frac{2}{4}$$ | |||
| $$3\div \frac{3}{4}$$ | |||
| $$3\div \frac{4}{4}$$ |
What pattern do you notice? What generalization can you make?
Guiding Questions
Problem 3
For each problem, draw a diagram and write a division problem. Find the solution using both the diagram and by calculating, and check that your answers are the same by each method.
a. How many fives are in 15?
b. How many halves are in 3?
c. How many sixths are in 4?
d. How many two-thirds are in 2?
e. How many three-fourths are in 2?
f. How many $${{\frac{1}{6}}}$$s are in $${\frac{1}{3}}$$?
g. How many $${{\frac{1}{6}}}$$s are in $${{\frac{2}{3}}}$$?
h. How many $${\frac{1}{4}}$$s are in $${{\frac{2}{3}}}$$?
i. How many $${\frac{5}{12}}$$s are in $${\frac{1}{2}}$$?
Guiding Questions
References
How Many ___ Are in ... ?, accessed on Sept. 28, 2017, 12:55 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
Problem Set
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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.
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Illustrative Mathematics How Many Containers in One Cup/Cups in One Container?
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EngageNY Mathematics Grade 6 Mathematics > Module 2 > Topic A > Lesson 3
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Exercises and Problem Set; These are fraction-divided-by-fraction-with-same-denominator problems, good for reinforcing visual models
- RDA Performance Task Bank Grade 6 Mathematics Sample SR Item C1 TB
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Illustrative Mathematics Reciprocity
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Illustrative Mathematics Making Hot Cocoa, Variation 2
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Illustrative Mathematics Dan's Division Strategy
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Common denominator strategy, goes well with EngageNY Lesson 3
Target Task
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Draw a visual model to represent the solution to the division problem $${6 \div \frac{2}{3}}$$.
Then, use your model to explain why this can also be solved with the multiplication problem $${6 \times \frac{3}{2}}$$.
Mastery Response
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