# Lesson 4

How Many Groups? (Part 1)

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

### 4.1: Equal-sized Groups

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

1. Eight $5 bills are worth$40.
2. There are 9 thirds in 3 ones.

### 4.2: 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.)

1. 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
2. Here are Elena’s diagrams for $$2 \boldcdot \frac12 = 1$$ and $$6 \boldcdot \frac13 = 2$$. Do you think these diagrams represent the equations? Explain or show your reasoning.

3. Use pattern blocks to represent each multiplication equation. Remember that a hexagon represents 1 whole.
1. $$3 \boldcdot \frac 16=\frac12$$

2. $$2 \boldcdot \frac 32=3$$
4. Answer the questions. If you get stuck, consider using pattern blocks.
1. How many $$\frac 12$$s are in 4?

2. How many $$\frac23$$s are in 2?

3. How many $$\frac16$$s are in $$1\frac12$$?

### Summary

Some problems that involve equal-sized groups also involve fractions. Here is an example: “How many $$\frac16$$ are in 2?” We can express this question with multiplication and division equations. $$\displaystyle {?} \boldcdot \frac16 = 2$$ $$\displaystyle 2 \div \frac16 = {?}$$

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

If the hexagon represents 1 whole, then a triangle must represent $$\frac16$$, because 6 triangles make 1 hexagon. We can use the triangle to represent the $$\frac 16$$ in the problem.
Twelve triangles make 2 hexagons, which means there are 12 groups of $$\frac16$$ in 2.
If we write the 12 in the place of the “?” in the original equations, we have: $$\displaystyle 12 \boldcdot \frac16 = 2$$
$$\displaystyle 2 \div \frac16 = 12$$