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

  • 5.NF.B.7

Criteria for Success

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  1. 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.
  2. 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.
  3. 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.

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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

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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

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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

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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

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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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