Lesson 5How Many Groups? (Part 2)
Let’s use blocks and diagrams to understand more about division with fractions.
Learning Targets:
 I can find how many groups there are when the number of groups and the amount in each group are not whole numbers.
5.1 Reasoning with Fraction Strips
Write a fraction or whole number as an answer for each question. If you get stuck, use the fraction strips. Be prepared to share your strategy.
 How many s are in 2?
 How many s are in 3?
 How many s are in ?
5.2 More Reasoning with Pattern Blocks
Use the pattern blocks in the applet to answer the questions. (If you need help aligning the pieces, you can turn on the grid.)
 If the trapezoid represents 1 whole, what do each of the following shapes represent? Be prepared to show or explain your reasoning.

1 triangle

1 rhombus

1 hexagon


Use pattern blocks to represent each multiplication equation. Use the trapezoid to represent 1 whole.


Diego and Jada were asked “How many rhombuses are in a trapezoid?”
 Diego says, “. If I put 1 rhombus on a trapezoid, the leftover shape is a triangle, which is of the trapezoid.”
 Jada says, “I think it’s . Since we want to find out ‘how many rhombuses,’ we should compare the leftover triangle to a rhombus. A triangle is of a rhombus.”

Select all equations that can be used to answer the question: “How many rhombuses are in a trapezoid?”

5.3 Drawing Diagrams to Show Equalsized Groups
For each situation, draw a diagram for the relationship of the quantities to help you answer the question. Then write a multiplication equation or a division equation for the relationship. Be prepared to share your reasoning.
 The distance around a park is miles. Noah rode his bicycle around the park for a total of 3 miles. How many times around the park did he ride?
 You need yard of ribbon for one gift box. You have 3 yards of ribbon. How many gift boxes do you have ribbon for?
 The water hose fills a bucket at gallon per minute. How many minutes does it take to fill a 2gallon bucket?
Are you ready for more?
Lesson 5 Summary
Suppose one batch of cookies requires cup flour. How many batches can be made with 4 cups of flour?
We can think of the question as being: “How many s are in 4?” and represent it using multiplication and division equations.
Let’s use pattern blocks to visualize the situation and say that a hexagon is 1 whole.
Since 3 rhombuses make a hexagon, 1 rhombus represents and 2 rhombuses represent . We can see that 6 pairs of rhombuses make 4 hexagons, so there are 6 groups of in 4.
Other kinds of diagrams can also help us reason about equalsized groups involving fractions. This example shows how we might reason about the same question from above: “How many cups are in 4 cups?”
We can see each “cup” partitioned into thirds, and that there are 6 groups of cup in 4 cups. In both diagrams, we see that the unknown value (or the “?” in the equations) is 6. So we can now write:
Lesson 5 Practice Problems

Use the tape diagram to represent and find the value of .
Mark up and label the diagram as needed.

What is the value of ? Use pattern blocks to represent and find this value. The yellow hexagon represents 1 whole. Explain or show your reasoning.

Use a standard inch ruler to answer each question. Then, write a multiplication equation and a division equation that answer the question.

How many s are in 7?

How many s are in 6?

How many s are in ?


Use the tape diagram to represent and answer the question: How many s are in ?
Mark up and label the diagram as needed.

Write a multiplication equation and a division equation to represent each question, statement, or diagram.
 There are 12 fourths in 3.
 How many s are in 6?
 There are 12 fourths in 3.

At a farmer’s market, two vendors sell fresh milk. One vendor sells 2 liters for $3.80, and another vendor sells 1.5 liters for $2.70. Which is the better deal? Explain your reasoning.

A recipe uses 5 cups of flour for every 2 cups of sugar.
 How much sugar is used for 1 cup of flour?
 How much flour is used for 1 cup of sugar?
 How much flour is used with 7 cups of sugar?
 How much sugar is used with 6 cups of flour?