# Lesson 8

Ways to Find Unknown Length (Part 2)

## Warm-up: True or False: Equations with Fractions (10 minutes)

### Narrative

The purpose of this warm-up is to elicit strategies and understandings students have for adding, subtracting, and multiplying fractions and mixed numbers. The series of equations prompt students to use properties of operations (associative and commutative properties in particular) in their reasoning, which will be helpful when students solve geometric problems involving fractional lengths (MP7).

### Launch

- Display one equation.
- “Give me a signal when you know whether the equation is true and can explain how you know.”
- 1 minute: quiet think time

### Activity

- Share and record answers and strategy.
- Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

- \(1\frac{1}{5} + 2\frac{2}{5} + 3\frac{3}{5} + 4\frac{4}{5} = 12\)
- \(10 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - \frac{4}{2} = 5\)
- \(1\frac{1}{6} + 2\frac{2}{6} + 3\frac{3}{6} + 4\frac{4}{6} + 5\frac{5}{6} = 15\frac{3}{6}\)
- \(\frac{1}{3} + \frac{2}{3} + \frac{3}{3} = 3 \times \frac{2}{3}\)

### Student Response

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### Activity Synthesis

- “What strategies did you find useful for adding or subtracting these numbers with fractions?” (Possible strategies:
- Adding whole numbers separately than fractions
- Noticing that \(1 + 2 + 3 + 4\) is 10 and using that fact to add or subtract fractions
- Combine fractions that add up to 1 (such as \(\frac{1}{5} + \frac{4}{5}\) and \(\frac{2}{5} + \frac{3}{5}\)).
- In the second equation, add up the fractions and subtract the sum from 10, instead of subtracting each fraction individually.)

- Consider asking:
- “Who can restate _____’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone approach the expression in a different way?”
- “Does anyone want to add on to _____’s strategy?”

## Activity 1: Unknown Lengths (20 minutes)

### Narrative

Previously, students reasoned about the perimeter of two-dimensional figures based on given side lengths and known attributes of the figures, including symmetry. In this activity, students find unknown side lengths given the perimeter, some side lengths, and information about the symmetry of the figures. Students have opportunities to practice adding, subtracting, and multiplying numbers with fractions, as not all of the given measurements are whole numbers.

*MLR8 Discussion Supports.*Synthesis: Create a visual display of the shapes. As students share their strategies, annotate the display to illustrate connections. For example, next to each shape, write expressions and draw the lines of symmetry.

*Advances: Speaking, Representing*

*Engagement: Provide Access by Recruiting Interest.*Leverage choice around perceived challenge. Invite students to select 2 of the 4 shapes to work with. Offer feedback that emphasizes effort and time on task, and invite them to try another shape if time allows.

*Supports accessibility for: Organization, Attention, Social Emotional Functioning*

### Required Materials

Materials to Gather

### Launch

- Groups of 2
- Give a ruler or a straightedge to each student.
- Provide access to patty paper.

### Activity

- 5 minutes: independent work time
- 2–3 minutes: partner discussion
- Monitor for students who:
- can clearly articulate how lines of symmetry help them determine unknown side lengths
- can explain how they know that all four sides of Q are equal
- write expressions to show their reasoning

### Student Facing

Here are four shapes.

- Each shape has a perimeter of 64 inches.
- P, R, and S each have 1 line of symmetry.
- Q has 4 lines of symmetry.

- Draw the lines of symmetry of each shape.
- Find the unknown side length in each shape. Show your reasoning.

### Student Response

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### Advancing Student Thinking

If students find the unknown sides lengths for shapes P and Q, but say they need more information to find the unknown length for shapes R and S, consider asking:

- “How can you use the line of symmetry to label more of the sides?”
- “Which sides are still unlabeled? How are these sides related?”
- “How could you use an expression or an equation to help you find the unknown side lengths?”

### Activity Synthesis

- Select students to share their responses and reasoning.
- “How do the lines of symmetry in P, R, and S help you find the unknown side lengths?” (The lines of symmetry tell us the lengths of unlabeled sides that mirror labeled sides, making it possible to find the length of the side with a question mark.)
- “What about the lines of symmetry in Q?” (The vertical line of symmetry tells us the left and right sides have the same length and the horizontal one tells us the top and bottom sides are of equal length, so all sides have the same length.)

## Activity 2: Lin’s Design (15 minutes)

### Narrative

In this activity, students practice completing a geometric drawing given half of the drawing and a line of symmetry, and reasoning about the perimeter of a line-symmetric figure.

While a precise drawing is not an expectation here, if no students considered using tools and techniques—such as using patty paper or by folding—to complete the drawing precisely, consider asking how it could be done (MP5).

To find the perimeter of the design, students have opportunities to look for and make use of structure (MP7) to expedite their calculation. For instance, instead of adding all 10 segments individually, they may:

- add 5 segments on one side of the line of symmetry and double it: \((19 + 6 + 25 + 12\frac{3}{4} + 12\frac{3}{4}) \times 2\)
- multiply each segment by 2 and find the sum of those products: \((2 \times 6) + (2 \times 19) + (2 \times 25) + (2 \times 12\frac{3}{4}) + (2 \times 12\frac{3}{4})\)
- add 19 and 6 to get 25, then multiply 25 and \(12\frac{3}{4}\) by 4: \((4 \times 25) + (4 \times 12\frac{3}{4})\)

### Required Materials

Materials to Gather

### Launch

- Groups of 2
- Give a ruler or a straightedge to each student.
- Provide access to patty paper.
- Display the image for all to see.
- “What do you notice? What do you wonder?”
- 1 minute: quiet think time
- 1 minute: discuss observations and questions.

### Activity

- 5–6 minutes: independent work time
- 2–3 minutes: partner discussion
- Monitor for the different strategies students use to complete the drawing and to find the perimeter of the design (as noted in the Activity Narrative).

### Student Facing

Lin is using 145 inches of fancy tape for the outline of a design with line symmetry.

Here is half of the design. The dashed line is the line of symmetry.

- Sketch Lin’s entire design.
- Does she have enough tape for the entire outline? Show your reasoning.

If you have time: Lin has a sheet of fancy paper that she can cut up to cover the inside of the design. The paper is a rectangle that is 30 inches by 18 inches. If the angles in the design are right angles, does Lin have enough paper to cover the inside of the design? Show your reasoning.

### Student Response

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### Activity Synthesis

- Select students to present their completed drawings and share their process of drawing.
- If no one used patty paper, folding, or measurements to draw, consider asking: “Suppose we need to draw the other half of Lin’s design precisely. What strategies could we use?”
- Select other students to present their response and reasoning for the last question. Sequence the presentation in the order listed in the Activity Narrative and record the expressions students used to find the perimeter.

## Lesson Synthesis

### Lesson Synthesis

“Today we used attributes of figures to reason about their side lengths and perimeter.“

Display:

“Here are two shapes. Suppose we know the perimeter of each shape is 48 units. Shape A has a line of symmetry and B has none.”

“How can knowing the line of symmetry in A help us find the unknown side lengths?” (The line of symmetry tells us that the longer unlabeled side is 14 and the two shorter sides are equal. We can subtract \(8 + 14 + 14\) from 48 and divide the result by 2 to get the shorter sides.)

“Shape B has no line of symmetry. Can we figure out the unknown lengths?” (No. There isn’t enough information. We'd need to know if some of the sides are the same length.)

If students argue that they can tell that one of the other sides must also be 15 units long, ask: “Without measuring, what would you need to know to be sure one of the labeled sides is also 15 units long?”

“Suppose we know that B is a parallelogram. Would that help us find those lengths? Why or why not?” (Yes. Opposite sides of a parallelogram have the same length, so we know the unlabeled sides are 15 and 9.)

## Cool-down: Stage Symmetry (5 minutes)

### Cool-Down

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## Student Section Summary

### Student Facing

In this section, we used the attributes such as side lengths, angles, lines of symmetry, and parallel sides to solve problems about perimeter of shapes.

We learned that, if a shape has certain attributes, we can use them to find its perimeter, even if we don't have all of its side lengths. Or, if we know the perimeter of a shape, we can find its side lengths if there is enough information about their attributes.

For example, here are two shapes:

Shape B doesn't have a line of symmetry, but if we were told that its opposite sides have equal lengths, then we can also reason about the three missing side lengths.