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 \(1\frac 14\)?
- \(1 \div \frac {2}{6} = {?}\)
- \(2 \div \frac 29 = {?}\)
- \(4 \div \frac {2}{10} = {?}\)
![Fraction strips depicting 2 in 8 different ways.](https://staging-cms-im.s3.amazonaws.com/CxKRHHYovfQmSmvEbX3kLYRV?response-content-disposition=inline%3B%20filename%3D%226-6.4.B2.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%276-6.4.B2.Image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T092456Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=deae9cc8b614a67350c8bb541c1d239f7fd3787c501ab96d2f219409f1278bd0)
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.
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1 triangle
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1 rhombus
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1 hexagon
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Use pattern blocks to represent each multiplication equation. Use the trapezoid to represent 1 whole.
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\(3 \boldcdot \frac 13=1\)
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\(3 \boldcdot \frac 23=2\)
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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.
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Select all the equations that can be used to answer the question: “How many rhombuses are in a trapezoid?”
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\(\frac 23 \div {?} = 1\)
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\({?} \boldcdot \frac 23 = 1\)
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\(1 \div \frac 23 = {?}\)
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\(1 \boldcdot \frac 23 = {?}\)
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\({?} \div \frac 23 = 1\)
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5.3: Drawing Diagrams to Show Equal-sized 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 2-gallon 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.
![Diagram of 4 hexagons. Each hexagon is made up of 3 rhombuses using pattern blocks.](https://staging-cms-im.s3.amazonaws.com/mGCN6gAVYebSMTmYwXoAF3HB?response-content-disposition=inline%3B%20filename%3D%226-6.4.B1.Image.16b.png%22%3B%20filename%2A%3DUTF-8%27%276-6.4.B1.Image.16b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T092456Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a1c55dfb18c99d098ba4e8430027dc5703ebcfc07277ad7bb57fbc3654db4e7b)
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?”
![Diagram with 4 rectangles, partitioned into thirds.](https://staging-cms-im.s3.amazonaws.com/TEHizzc4j9PX93oqv9nEbFhq?response-content-disposition=inline%3B%20filename%3D%226-6.4.B1.Image.08a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.4.B1.Image.08a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T092456Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=877c4ba4cf2ed56b97714ea99aa6b36130831e854c686cc7a1ac4c8397b72e27)
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:
\(\displaystyle 6 \boldcdot \frac23 = 4\) \(\displaystyle 4\div \frac23 = 6\)