Lesson 5

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.

1. How many $\frac 12$s are in 2?
2. How many $\frac 15$s are in 3?
3. How many $\frac {1}{8}$s are in $1\frac 14$?
4. $1 \div \frac {2}{6} = {?}$
5. $2 \div \frac 29 = {?}$
6. $4 \div \frac {2}{10} = {?}$

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

   1. 1 triangle

   2. 1 rhombus

   3. 1 hexagon

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

   1. $3 \boldcdot \frac 13=1$

   2. $3 \boldcdot \frac 23=2$

3. 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.

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

   • $\frac 23  \div {?} = 1$

   • ${?} \boldcdot \frac 23 = 1$

   • $1 \div \frac 23 = {?}$

   • $1 \boldcdot \frac 23 = {?}$

   • ${?}  \div \frac 23 = 1$

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 $\frac32$ 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 $\frac34$ 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 $\frac13$ gallon per minute. How many minutes does it take to fill a 2-gallon bucket?

Suppose one batch of cookies requires $\frac23$ cup flour. How many batches can be made with 4 cups of flour?

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 equal-sized 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?”