# 3.5 Fractions as Numbers

## Unit Goals

• Students develop an understanding of fractions as numbers and of fraction equivalence by representing fractions on diagrams and number lines, generating equivalent fractions, and comparing fractions.

### Section A Goals

• Understand that fractions are built from unit fractions such that a fraction $\frac{a}{b}$ is the quantity formed by $a$ parts of size $\frac{1}{b}$.
• Understand that unit fractions are formed by partitioning shapes into equal parts.

### Section B Goals

• Understand a fraction as a number and represent fractions on the number line.

### Section C Goals

• Explain equivalence of fractions in special cases and express whole numbers as fractions and fractions as whole numbers.

### Section D Goals

• Compare two fractions with the same numerator or denominator, record the results with the symbols >, =, or

### Problem 1

#### Pre-unit

Practicing Standards:  2.G.A.2

Partition the rectangle into 10 equal squares.

### Problem 2

#### Pre-unit

Practicing Standards:  2.G.A.3

Here are two equal-size squares. A part of each square is shaded.

Is the same amount of each square shaded? Explain or show your reasoning.

### Problem 3

#### Pre-unit

Practicing Standards:  2.MD.B.6

1. Label the tick marks on the number line.
2. Locate and label 45 and 62 on the number line.

### Problem 4

#### Pre-unit

Practicing Standards:  2.NBT.A.4

Fill in each blank with $$<$$ or $$>$$ to compare the numbers.

1. $$718\, \underline{\hspace{1cm}}\, 817$$

2. $$106\, \underline{\hspace{1cm}} \,89$$

3. $$806\, \underline{\hspace{1cm}} \,809$$

### Problem 5

Partition the rectangle into 6 equal parts.

### Problem 6

1. What fraction of the rectangle is shaded?

2. Partition the rectangle into 8 equal parts.

What fraction of the whole rectangle does each part represent?

### Problem 7

1. What fraction of the rectangle is shaded? Explain how you know.

2. Shade $$\frac{4}{6}$$ of the rectangle.

### Problem 8

Jada walks across the street at a stoplight $$\frac{5}{6}$$ of her way from home to school. Represent the situation on the fraction strip. Explain your reasoning.

### Problem 9

#### Exploration

Write a situation represented by the diagram. Explain why the diagram represents your situation.

### Problem 10

#### Exploration

Lin shaded part of some fraction strips. What fraction did she shade in each one? Explain how you know.

1.
2.
3.

### Problem 1

1. Locate and label $$\frac{1}{4}$$ on the number line. Explain your reasoning.

2. Locate and label $$\frac{1}{6}$$ on the number line. Explain your reasoning.

### Problem 2

1. Locate and label $$\frac{1}{8}$$ on the number line.

2. Locate and label $$\frac{1}{3}$$ on the number line.

### Problem 3

1. Locate and label $$\frac{4}{8}$$ on the number line.

2. Locate and label $$\frac{7}{6}$$ on the number line.

3. Diego marks and labels fourths on the number line like this:

Do you agree with Diego? Explain your reasoning.

### Problem 4

1. Label the tick marks on the number line.

2. Which numbers on the number line are whole numbers? Explain how you know.

### Problem 5

Locate and label 1 on the number line. Explain your reasoning.

### Problem 6

#### Exploration

How are the fraction strip and number line the same? How are they different?

### Problem 7

#### Exploration

Han says that he can find 1 on the number line without finding $$\frac{1}{8}$$. What might Han’s method be?

### Problem 1

Select all correct statements.

A:

$$\frac{1}{2}$$ is equivalent to $$\frac{3}{6}$$

B:

$$\frac{1}{2}$$ is equivalent to $$\frac{1}{3}$$

C:

$$\frac{2}{2}$$ is equivalent to $$\frac{4}{4}$$

D:

$$\frac{2}{2}$$ is equivalent to $$\frac{6}{6}$$

E:

$$\frac{2}{3}$$ is equivalent to $$\frac{4}{6}$$

F:

$$\frac{2}{3}$$ is equivalent to $$\frac{3}{4}$$

### Problem 2

Write as many fractions as you can that represent the shaded part of each diagram.

### Problem 3

1. Tyler draws this picture and says that $$\frac{3}{4}$$ is equivalent to $$\frac{2}{3}$$. Explain why Tyler is not correct.

2. Find a fraction equivalent to $$\frac{2}{3}$$.
3. Find a fraction equivalent to $$\frac{3}{4}$$.

### Problem 4

1. Write 10 as a fraction in 2 different ways.
2. Is $$\frac{88}{8}$$ equivalent to a whole number?

### Problem 5

#### Exploration

Decide if each fraction is a whole number. Explain or show your reasoning.

1. $$\frac{100}{2}$$
2. $$\frac{100}{3}$$
3. $$\frac{100}{4}$$
4. $$\frac{100}{6}$$
5. $$\frac{100}{8}$$

### Problem 6

#### Exploration

If you continue to fold fraction strips, how many parts can you fold them into? Can you fold them into 100 equal parts?

### Problem 1

1. Are $$\frac{2}{3}$$ and $$\frac{4}{6}$$ equivalent? Show your thinking using diagrams, symbols, or other representations.
2. Are $$\frac{6}{8}$$ and $$\frac{7}{8}$$ equivalent? Show your thinking using diagrams, symbols, or other representations.

### Problem 2

Han says there is no fraction with denominator 8 that's greater than $$\frac{8}{8}$$ because $$\frac{8}{8}$$ is a whole. Do you agree with Han? Explain your reasoning.

### Problem 3

Use the symbols $$>$$ or $$<$$ to make each statement true. Explain your reasoning.

1. $$\frac{5}{3} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{2}$$
2. $$\frac{3}{4} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{4}$$

### Problem 4

1. Jada threw the ball $$\frac{3}{4}$$ of the length of the gym. Clare threw the ball $$\frac{6}{8}$$ of the length of the gym. Clare says she threw the ball farther. Do you agree? Show your thinking.
2. Tyler kicked the ball $$\frac{7}{8}$$ the length of the playground. Andre kicked the ball $$\frac{7}{6}$$ the length of the playground. Andre says he kicked the ball farther. Do you agree? Show your thinking.

### Problem 5

#### Exploration

Clare walked $$\frac{3}{4}$$ of the way around a park. Tyler walked $$\frac{3}{6}$$ of the way around a different park. Who walked farther? Explain your reasoning.

### Solution

Choose a fraction that you can compare with both $$\frac{3}{8}$$ and $$\frac{5}{6}$$ by looking at the numerators and denominators.