Lesson 4 | Multi-Digit and Fraction Computation | 6th Grade Mathematics

Objective

Use visual models and patterns to develop a general rule to divide with fractions.

Common Core Standards

Core Standards

The core standards covered in this lesson

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?

The Number System

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

The foundational standards covered in this lesson

5.NF.B.6

Number and Operations—Fractions

5.NF.B.6 — Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5.NF.B.7

Number and Operations—Fractions

5.NF.B.7 — Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

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

Suggestions for teachers to help them teach this lesson

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.
There are several resources in the Teacher Tune Ups section of this SERP resource, including several short videos that demonstrate dividing with fractions using models and methods alternative to the general algorithm, as well as the post "Delaying 'Invert and Multiply'", a valuable resource for teachers to deepen their understanding of fraction division and to understand why the invert and multiply rule works prior to teaching the method to students.

Fishtank Plus

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

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

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

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

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

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

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Illustrative Mathematics How Many ___ Are in ... ?

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

A set of suggested resources or problem types that teachers can turn into a problem set

Fishtank Plus Content

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

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 $${6 \times \frac{3}{2}}$$.

Student Response

An example response to the Target Task at the level of detail expected of the students.

Create a free account or sign in to view Student Response

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

RDA Performance Task Bank Grade 6 Mathematics Sample SR Item C1 TB
Illustrative Mathematics Reciprocity
Illustrative Mathematics Making Hot Cocoa, Variation 2
Illustrative Mathematics Dan's Division Strategy — Common denominator strategy, goes well with EngageNY Lesson 3
EngageNY Mathematics Grade 6 Mathematics > Module 2 > Topic A > Lesson 3 — Exercises and Problem Set; These are fraction-divided-by-fraction-with-same-denominator problems, good for reinforcing visual models
Illustrative Mathematics How Many Containers in One Cup/Cups in One Container?

Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Lesson 3

Lesson 5

Lesson Map

Topic A: Dividing with Fractions

Interpret division problems as the number of items in each group or the number of groups of a given number of items. Write corresponding multiplication and division problems.

6.NS.A.1

Divide a fraction by a whole number using visual models and related multiplication problems.

6.NS.A.1

Divide a whole number by a fraction using visual models.

6.NS.A.1

Use visual models and patterns to develop a general rule to divide with fractions.

6.NS.A.1

Solve and write story problems involving division with fractions.

6.NS.A.1

Solve problems involving division with fractions.

6.NS.A.1

Topic B: Computing with Decimals

Add and subtract decimals using the standard algorithm.

6.NS.B.3

Multiply decimals using strategies, and develop an understanding of the standard algorithm.

6.NS.B.3

Multiply decimals using the standard algorithm.

6.NS.B.3

Divide multi-digit whole numbers using the standard algorithm.

6.NS.B.2

Divide numbers with decimal quotients. Divide decimals by whole numbers.

6.NS.B.2 6.NS.B.3

Divide decimals by decimals using the standard algorithm.

6.NS.B.3

Solve problems involving decimals using all four operations.

6.NS.B.3

Topic C: Applying the Greatest Common Factor and the Least Common Multiple

Use prime factorization to represent numbers as products of prime factors.

6.NS.B.4

Find the greatest common factor of two numbers. Solve application problems using the greatest common factor.

6.NS.B.4

Find the least common multiple of two numbers. Solve application problems using the least common multiple.

6.NS.B.4

Solve mathematical and real-world problems using the greatest common factor and least common multiple.

6.NS.B.4

Multi-Digit and Fraction Computation

Objective

Common Core Standards

Core Standards

The Number System

Foundational Standards

Number and Operations—Fractions

Number and Operations—Fractions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Problem Set

Target Task

Student Response

Additional Practice

Word Problems and Fluency Activities

Lesson Map

Request a Demo

Contact Information

School Information

What courses are you interested in?

ELA

Math

Are you interested in onboarding professional learning for your teachers and instructional leaders?

Any other information you would like to provide about your school?

Effective Instruction Made Easy