Lesson 5How Many Groups? (Part 2)

Learning Goal

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.

Warm Up: Reasoning with Fraction Strips

Problem 1

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 reasoning.

A diagram of equivalent fractions from a whole down to ninths.

  1. How many s are in 2?

  2. How many s are in 3?

  3. How many s are in ?

Activity 1: More Reasoning with Pattern Blocks

Use pattern blocks to answer the questions.

Problem 1

If the trapezoid represents 1 whole, what do each of the other shapes represent? Be prepared to show or explain your reasoning.

open applet in presentation mode

  1. 1 triangle

  2. 1 rhombus

  3. 1 hexagon

Print Version

If the trapezoid represents 1 whole, what do each of the other shapes represent? Be prepared to show or explain your reasoning.

Four pattern blocks: One large yellow hexagon, one blue rhombus, one red trapezoid, and one green triangle.

  1. 1 triangle

  2. 1 rhombus

  3. 1 hexagon

Problem 2

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

Problem 3

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.”

Do you agree with either of them? Explain or show your reasoning.

Problem 4

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

Activity 2: Drawing Diagrams to Show Equal-sized Groups

Problem 1

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.

  1. 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?

  2. You need yard of ribbon for one gift box. You have 3 yards of ribbon. How many gift boxes do you have ribbon for?

  3. The water hose fills a bucket at gallon per minute. How many minutes does it take to fill a 2-gallon bucket?

Are you ready for more?

Problem 1

How many heaping teaspoons are in a heaping tablespoon? How would the answer depend on the shape of the spoons?

Lesson 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.

4 blue hexagons divided into 3 segments each. Each 2 segments are labeled as a whole and are marked from 1 to 6.

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 equal-sized groups involving fractions. This example shows how we might reason about the same question from above: “How many -cups are in 4 cups?”

4 stacks of 3 rectangles with circles grouping two one-third pieces ending up with with 6 two-third groups.

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: