# Lesson 10

Fractional Side Lengths Less Than 1

## Warm-up: Estimation Exploration: What is the Area? (10 minutes)

### Narrative

The purpose of this Estimation Exploration is for students to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to estimate the area of a rectangle when one side is not a unit fraction.

### Launch

• Groups of 2
• Display the image.
• “What might be the area of the shaded region?”

### Activity

• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

### Student Facing

What is the area of the shaded region?

Record an estimate that is:

too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Is more than half or less than half of the rectangle shaded?” (More than half)
• “How can you use this to help make your estimate?” (Half of 7 is $$3 \frac{1}{2}$$ so it’s a little more than that.)
• “Based on this discussion does anyone want to revise their estimate?”

## Activity 1: Rectangle With a Fractional Side Length (20 minutes)

### Narrative

The purpose of this activity is for students to find the area of rectangles with a side length that is a non-unit fraction. Students may use a variety of strategies to find the areas of the shaded region. Monitor for students who are noticing and using the structure of the rectangle and expressions to determine the area (MP7) by:

• grouping the shaded pieces with a fractional area to make whole unit squares
• multiplying the whole number of units by the numerator of the fractional side length and dividing the result by the denominator
Representation: Internalize Comprehension. Begin by asking, “Does this problem/situation remind anyone of something we have seen/read/done before?”
Supports accessibility for: Memory, Conceptual Processing

### Launch

• Display image from warm-up.
• “If the height of the shaded region was $$\frac{5}{6}$$ of a square unit, what expression could you write to represent the area of the shaded region?” ($$\frac{5}{6} \times 7$$ or $$7 \times \frac{5}{6}$$)
• Groups of 2

### Activity

• 1–2 minutes: quiet think time
• 5–8 minutes: partner work time

### Student Facing

Write a multiplication expression to represent the area of each shaded region. Then find the area.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Select previously identified students to share their strategies.
• “How are the second and third shaded regions the same? How are they different?” (They are each 4 units long. There are 12 shaded pieces in each. The 12 pieces in the second example are $$\frac{1}{4}$$ of a unit square. The 12 shaded pieces in the third shaded region are $$\frac{1}{5}$$ of a unit square.)
• Display the expression: $$4 \times 3$$
• “How does this expression relate to the second and third shaded regions?” (In both of them, the number of shaded pieces is $$4 \times 3$$.)
• “Why are the areas of these two shaded regions different?” (The small pieces are not the same size. They are $$\frac{1}{4}$$ of a unit square in one and $$\frac{1}{5}$$ in the other.)
• Display the expressions: $$12 \times \frac{1}{4}$$ and $$12 \times \frac{1}{5}$$
• “The expressions for the area show that there are 12 shaded pieces in both but they are different sizes.”

## Activity 2: What Are the Side Lengths? (15 minutes)

### Narrative

The purpose of this activity is for students to find the area of rectangles with a fractional side length. The side lengths are not labeled, so students will have to determine them by considering the number and size of the shaded pieces.

In the second problem, students are given expressions and they determine whether or not each expression represents the shaded area in a given diagram. The numbers in the expressions are similar, so students need to consider the structure of the expressions and the shaded regions. While they may match the expressions with the diagrams based on their value, students are encouraged to look for and clearly express how the diagrams represent the different expressions (MP6, MP7).

MLR2 Collect and Display. Circulate, listen for and collect the language students use as they determine the area. On a visible display, record words and phrases such as: length, shaded, fraction, pieces, width, multiply, area, whole, part, expression. Invite students to borrow language from the display as needed, and update it throughout the lesson.

• Groups of 2

### Activity

• 2–3 minutes: quiet think time

• 5–6 minutes: partner work time

• Monitor for students who:

• correctly determine the side lengths of the shaded region.

• use the side lengths to determine a correct multiplication expression.

• refer to the number of shaded pieces and the size of the shaded pieces when determining the area.

### Student Facing

1. Write a multiplication expression to represent the area of the shaded region. What is the area?

2. Here are two diagrams. Consider each expression and decide whether it represents the shaded region in one of the diagrams. Be prepared to share your thinking.

1. $$\frac{3}{4} \times 5$$
2. $$3 \times \frac{3}{5}$$
3. $$3 \times 4 \times \frac{1}{5}$$
4. $$4 \times \frac{3}{4}$$
5. $$3 \times 3 \times \frac{1}{4}$$
3. For each diagram, what is the area?

### Student Response

For access, consult one of our IM Certified Partners.

If students do not write the correct expression, show them the correct expression and ask, “How does the expression represent the area of the corresponding diagram?”

### Activity Synthesis

• “How did you determine the side lengths for shaded region in diagram X?” (I looked at the side lengths of the squares and the partitions, then I counted how many are shaded.)
• Display the expression: $$3 \times 4 \times \frac{1}{5}$$
• “How does this expression represent the shaded region in diagram X?” (The vertical rectangle has 3 rows of 4 shaded pieces, so that’s $$3 \times 4$$ shaded pieces. Each one has an area of $$\frac{1}{5}$$ square unit.)

## Lesson Synthesis

### Lesson Synthesis

“Today we found the area of rectangles with a whole number side length and a fractional side length.”

Display image from first problem in the last activity.

“What are some different expressions that represent the shaded region?” ($$\frac{2}{3} \times 3$$, $$2 \times \frac{1}{3} \times 3$$, $$\frac{6}{3}$$)

“Pick one of the expressions and explain to your partner how it represents the shaded region.” (There are 3 unit squares and $$\frac{2}{3}$$ of them are shaded. There are two shaded rows and each one is $$\frac{1}{3}$$ of 3 squares. There are 6 shaded parts and each one is $$\frac{1}{3}$$.)

## Cool-down: A Fractional Side Length (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.