Lesson 5
How Many Groups? (Part 2)
Let’s use blocks and diagrams to understand more about division with fractions.
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 reasoning.
 How many \(\frac 12\)s are in 2?
 How many \(\frac 15\)s are in 3?
 How many \(\frac {1}{8}\)s are in \(1\frac 14\)?
 \(1 \div \frac {2}{6} = {?}\)
 \(2 \div \frac 29 = {?}\)
 \(4 \div \frac {2}{10} = {?}\)
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 these other shapes represent? Be prepared to explain or show your reasoning.

1 triangle

1 rhombus

1 hexagon


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

\(3 \boldcdot \frac 13=1\)

\(3 \boldcdot \frac 23=2\)


Diego and Jada were asked “How many rhombuses are in a trapezoid?”
 Diego says, “\(1\frac 13\). If I put 1 rhombus on a trapezoid, the leftover shape is a triangle, which is \(\frac 13\) of the trapezoid.”
 Jada says, “I think it’s \(1\frac12\). Since we want to find out ‘how many rhombuses,’ we should compare the leftover triangle to a rhombus. A triangle is \(\frac12\) of a rhombus.”
Do you agree with either of them? Explain or show your reasoning.

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

\(\frac 23 \div {?} = 1\)

\({?} \boldcdot \frac 23 = 1\)

\(1 \div \frac 23 = {?}\)

\(1 \boldcdot \frac 23 = {?}\)

\({?} \div \frac 23 = 1\)

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 \(\frac32\) 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 \(\frac34\) 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 \(\frac13\) gallon per minute. How many minutes does it take to fill a 2gallon bucket?
Summary
Suppose one batch of cookies requires \(\frac23\) cup flour. How many batches can be made with 4 cups of flour?
We can think of the question as being: “How many \(\frac23\) are in 4?” and represent it using multiplication and division equations.
\(\displaystyle {?} \boldcdot \frac23 = 4\) \(\displaystyle 4\div \frac23 = {?}\)
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 \(\frac13\) and 2 rhombuses represent \(\frac 23\). We can see that 6 pairs of rhombuses make 4 hexagons, so there are 6 groups of \(\frac 23\) 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 \(\frac 23\)cups are in 4 cups?”
We can see each “cup” partitioned into thirds, and that there are 6 groups of \(\frac23\)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:
\(\displaystyle 6 \boldcdot \frac23 = 4\) \(\displaystyle 4\div \frac23 = 6\)