Lesson 4How Many Groups? (Part 1)

Learning Goal

Let’s play with blocks and diagrams to think about division with fractions.

Learning Targets

  • I can find how many groups there are when the amount in each group is not a whole number.

  • I can use diagrams and multiplication and division equations to represent “how many groups?” questions.

Warm Up: Equal-sized Groups

Problem 1

Write a multiplication equation and a division equation for each statement or diagram.

  1. Eight $5 bills are worth $40.

  2. There are 9 thirds in 3 ones.

  3. A tape diagram of 5 equal parts. Each part is labeled one fifth. Above the bar is a bracket, labeled 1, that spans the entire length of the bar.

Activity 1: Reasoning with Pattern Blocks

Use pattern blocks to answer the questions.

Problem 1

open applet in presentation mode

If a hexagon represents 1 whole, what fraction do each of the following shapes represent? Be prepared to show or explain your reasoning.

  1. 1 triangle

  2. 1 rhombus

  3. 1 trapezoid

  4. 4 triangles

  5. 3 rhombuses

  6. 2 hexagons

  7. 1 hexagon and 1 trapezoid

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

If a hexagon represents 1 whole, what fraction do each of the following shapes represent? Be prepared to show or explain your reasoning.

  1. 1 triangle

  2. 1 rhombus

  3. 1 trapezoid

  4. 4 triangles

  5. 3 rhombuses

  6. 2 hexagons

  7. 1 hexagon and 1 trapezoid

Problem 2

Here are Elena’s diagrams for and . Do you think these diagrams represent the equations? Explain or show your reasoning.

A red hexagon with a line going across the middle labeled "2 times one-half equals 1". Two blue hexagons divided into 3 pieces each labeled "6 times one-third equals 2".

Problem 3

Use pattern blocks to represent each multiplication equation. Remember that a hexagon represents 1 whole.

Problem 4

Answer the questions. If you get stuck, consider using pattern blocks.

  1. How many s are in 4?

  2. How many s are in 2?

  3. How many s are in ?

Lesson Summary

Some problems that involve equal-sized groups also involve fractions. Here is an example: “How many s are in 2?” We can express this question with multiplication and division equations.

Pattern-block diagrams can help us make sense of such problems. Here is a set of pattern blocks.

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

If the hexagon represents 1 whole, then a triangle must represent , because 6 triangles make 1 hexagon. We can use the triangle to represent the in the problem.

2 green hexagons divided into 6 parts.

Twelve triangles make 2 hexagons, which means there are 12 groups of in 2.

If we write the 12 in the place of the “?” in the original equations, we have: