Lesson 7

Ways to Find Unknown Length (Part 1)

Warm-up: Number Talk: Multiple Thirds (10 minutes)

Narrative

This Number Talk encourages students to use multiplicative reasoning and to rely on properties of operations to mentally find the value of products of a whole number and a fraction. The reasoning elicited here will be helpful later in the lesson when students find the perimeter of a figure with fractional side lengths.

Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

Activity

• Keep expressions and work displayed.
• Repeat with each expression.

Student Facing

Find the value of each expression mentally.

• $$6 \times \frac{1}{3}$$
• $$30 \times \frac{1}{3}$$
• $$60 \times \frac{2}{3}$$
• $$90 \times \frac{2}{3}$$

Activity Synthesis

• “What do these expressions have in common?” (The first number in each sequence is a multiple of 3 and a multiple of 6. The second number is a fraction with 3 in the denominator.)
• “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: All the Way Around (20 minutes)

Narrative

In this activity, students find the perimeter of several shapes and write expressions that show their reasoning. Each side of the shape is labeled with its length, prompting students to notice repetition in some of the numbers. The perimeter of all shapes can be found by addition, but students may notice that it is efficient to reason multiplicatively rather than additively (MP8). For example, they may write $$4 \times 6\frac{1}{3}$$ instead of $$6\frac{1}{3} + 6\frac{1}{3} + 6\frac{1}{3} + 6\frac{1}{3}$$.

MLR8 Discussion Supports. Synthesis: Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.

Launch

• Groups of 2–4

Activity

• 3–4 minutes: independent work time
• 2 minutes: group discussion

Student Facing

1. Find the perimeter of each shape. Write an expression that shows how you find the perimeter.

Activity Synthesis

• Invite students to share the expressions they wrote for the perimeter of each shape. Record the expressions.
• For each shape, ask: “What is the same about these expressions? What is different?” (They each use the side lengths in the shape. They each show the total length around the shape. Some use addition, some use multiplication, some use both multiplication and addition.)
• Highlight that when two or more sides have the same length, the perimeter can be found by adding, multiplying, or a combination of both.

Activity 2: Ponder Perimeter (15 minutes)

Narrative

Earlier, students found the perimeter of polygons when all the side lengths were given. In this activity, only some of the sides are labeled with their length, but students are given some information about the attributes of the shapes (presence or absence of parallel sides and symmetry). Students use that information to determine the length of the unlabeled sides and the perimeter. They may also conclude that the length of the perimeter cannot be determined.

When students interpret and analyze Mai's and Andre's reasoning about the quadrilateral perimeters they critique the reasoning of others (MP3).

This activity uses MLR3 Clarify, Critique, and Correct. Advances: reading, writing, representing

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most important for solving the problem. Display the sentence frame, “The next time I am finding the perimeter of a quadrilateral and I don’t know all the side lengths, I will look for . . . ”
Supports accessibility for: Conceptual Processing, Memory, Attention

Required Materials

Materials to Gather

Launch

• Groups of 2
• 2 minutes: quiet think time for the first question

Activity

• 2 minutes: partner discussion about Mai and Andre's reasoning
• Monitor for students who use symmetry and what they know about parallel sides in their response.
• 3–4 minutes: independent work time

Student Facing

Here are four shapes and what we know about them:

• A, B, and C have no lines of symmetry.
• A has no parallel sides.
• B has 1 pair of parallel sides.
• C has 2 pairs of parallel sides.
• D has 1 pair of parallel sides and 1 line of symmetry.

Mai says, “We can’t find the perimeter of any quadrilateral because each one is missing one or more side lengths.”

Andre disagrees. He says, “We can find the perimeters for C and D but not for A and B.”

1. Do you agree with either one of them? Explain or show your reasoning.

2. Find the perimeters that could be found, if any.

Activity Synthesis

MLR3 Clarify, Critique, Correct
• “What do you think Mai means? What is unclear? Are there any mistakes?”
• 1 minute: quiet think time
• 2 minute: partner discussion
• “With your partner, work together to write a revised explanation for Mai.”
• Display and review the following criteria:
• “We can’t find the perimeter of a quadrilateral if one or more side lengths are missing and _____.”
• 3–5 minute: partner work time
• Select 1–2 groups to share their revised explanation with the class.
• Record responses as students share.
• “How did you know that the perimeter of C can be found by combining twice 4 cm and twice $$7\frac{1}{2}$$ cm or $$(2 \times 4) + (2 \times 7\frac{1}{2})$$?” (We saw earlier that when quadrilaterals have two pairs of parallel sides, opposite sides have the same length. If the sides are parallel, they are the same distance apart.)
• “How did you know the unlabeled side in Shape D is also $$7\frac{1}{2}$$ cm long?” (The figure has a line of symmetry through the midpoint of the top and bottom segment.)
• “What expression can we write for the perimeter of D?” ($$4 + 5\frac{3}{4} + 7\frac{1}{2} + 7\frac{1}{2}$$ or $$4 + 5\frac{3}{4} + (2 \times 7\frac{1}{2})$$ or equivalent)

Activity 3: Perimeter Expressions [OPTIONAL] (10 minutes)

Narrative

This optional activity gives students an additional opportunity to use the attributes and their emerging understanding of the properties of some categories of shapes to find their perimeter, or to conclude that the perimeter cannot be determined.

Required Materials

Materials to Gather

• Groups of 2

Activity

• 3–4 minutes: quiet think time
• 2 minutes: partner discussion
• Monitor for students who use symmetry and the properties of parallelograms in their response.

Student Facing

Here are five shapes and what we know about them.

• Not all the sides are labeled.
• The lines of symmetry are shown.
• Only the triangle has no parallel sides.
1. For which shapes is it possible to find the perimeter? For which shapes is it not possible? Be prepared to explain how you know.

2. Here are four expressions. Each one represents the perimeter of one shape. The $$6\frac{1}{2}$$ and $$3\frac{1}{4}$$ in each represent side lengths. Can you tell which expression represents which shape?

1. $$(2 \times 6\frac{1}{2}) + 3\frac{1}{4}$$
2. $$4 \times 6 \frac{1}{2}$$
3. $$(2 \times 6\frac{1}{2}) + (4 \times 3\frac{1}{4})$$
4. $$(2 \times 6\frac{1}{2}) + (2 \times 3\frac{1}{4})$$

Student Response

If students begin to label all opposite sides with the same length, consider asking:

• “How do you know these two sides have the same length?”
• “What do you need to know about a shape to be sure that any unlabeled sides have the same length as the labeled sides?”

Activity Synthesis

• Invite students to share how they knew for which shapes it is possible to find the perimeter and for which it is not.
• Select other students to share how they matched the expressions and the shapes.

Lesson Synthesis

Lesson Synthesis

“Today we found the perimeter of various flat figures. Sometimes the sides were labeled with their lengths. Other times they were not.“

“Do you agree with the following statements? Find an image from today’s lesson that supports your answer.”

Display and read, one at a time:

• “If a figure has line symmetry, we can sometimes tell the lengths of the segments even when not all segments are labeled.”

(Agree. If the lengths of the segments on one side of the line of symmetry are known, we can tell the lengths on the other side. See figure D in Activity 2 and figures X, Y, and Z in Activity 3.)

• “If a figure shows no line symmetry, we can’t tell the lengths of unlabeled segments.”

(Disagree. Sometimes we can tell. For example, a parallelogram has no line symmetry, but we know their opposite sides are the same length. See parallelogram C in Activity 2 and parallelogram W in Activity 3.)