# Lesson 4

Symmetry in Figures (Part 1)

## Warm-up: Notice and Wonder: Seeing Double (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the idea of mirror images that match exactly, which will be useful when students generate a definition for symmetry in a later activity. Students may notice and wonder about possible real-life objects the block structure represents, but focus the discussion on describing how the two halves are reflections of each other.

Elicit the term “symmetry” or language that describes attributes of line-symmetric figures.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Highlight the language students used to describe the symmetry in the design such as:
• mirror images
• the same parts on both sides of the image
• “How might we check to see if the part on the left and that on the right are the same?” (We could place each half of the image on top of the other to see if they match.)

## Activity 1: Perfect Matches (20 minutes)

### Narrative

This activity uses the idea of folding to introduce students to line symmetry. Students analyze examples of figures that have a line of symmetry and those that don’t, and use their observations to formulate a definition of line of symmetry, which they then refine with their peers (MP6). Students later identify and draw lines of symmetry in various figures.

Some students may wish to use tools to help them define and find lines of symmetry. Provide access to patty paper, rulers, protractors, scissors, and copies of the figures (provided in the blackline master), if requested.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.

Representation: Develop Language and Symbols. Support student understanding of the first question by acting out Lin’s process with a class set of cutout figures. Invite students to examine and manipulate these figures as they work on this question.
Supports accessibility for: Visual Spatial Processing, Attention

### Required Materials

Materials to Gather

Materials to Copy

• Perfect Matches

### Required Preparation

• Make copies of the set of figures in the second question available for cutting and for demonstration during the lesson synthesis.

### Launch

• Groups of 2–4
• Give a ruler or straightedge to each student.
• Provide access to patty paper, protractors, scissors, and copies of the shapes in the second question.
• Display or sketch these parallelograms.

“How are the two figures alike? How are they different?”

• 1 minute: quiet think time
• Discuss responses. (Students may say:
• Both parallelograms show a dashed line through opposite corners.
• The dashed line creates two identical triangles.
• The triangles in the first shape would match up exactly if the shape is folded along the line, but not so with those in the second shape.)

### Activity

• 3 minutes: independent work time on the first question

MLR1 Stronger and Clearer Each Time

• “Share your response to ‘a line of symmetry is…’ with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 2 minutes: structured partner discussion
• Repeat with 2–3 different partners.
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time
• 3 minutes: independent work time on the remaining questions

### Student Facing

1. Lin has pieces of paper in different shapes. She folds each piece of paper once, creating two smaller parts.

She then sorts the pieces into two categories based on the folding lines.

Study the figures in each category. What do you think a line of symmetry means?

Complete this sentence:

A line of symmetry is . . .

2. Do the following figures have a line of symmetry? If so, draw the line. If not, explain how you know.

3. Are there any figures with more than one line of symmetry? If you think so, draw all the lines of symmetry.

### Student Response

For access, consult one of our IM Certified Partners.

If students identify a line of symmetry that would not result in two identical halves of the original figure, consider asking:

• “How did you decide where to draw your line? How do you know that it's a line of symmetry?”
• “How can you use a tool to prove that your line is a line of symmetry?”

### Activity Synthesis

• Ask groups to share their definitions of line of symmetry and record key phrases. Invite the class to comment.
• Discuss responses to the last two questions. Ask students to share the lines of symmetry they drew.
• “How might we check that the line we drew is a line of symmetry? How can we tell if the two halves that appear to be identical are indeed the same?” (Some ways:
• Cut out the figure and fold it along the line we drew.
• Trace one half of the figure on patty paper and turn it over to see if it matches the other half.
• Measure the segments and angles.)
• “If a figure can be folded along a line to create two parts that are a mirror reflection of one another and would match up exactly, we say that the figure has symmetry or has line symmetry.”
• “The line that splits a figure into two mirroring and matching parts is called a line of symmetry.”

## Activity 2: In Search of Symmetry (15 minutes)

### Narrative

In this activity, students practice identifying two-dimensional figures with line symmetry. They sort a set of figures based on the number of lines of symmetry that the figures have.

Continue to provide access to patty paper, rulers, and protractors. Students who use these tools to show that a shape has or does not have a line of symmetry use tools strategically (MP5). Consider allowing students to fold the cards, if needed.

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____ , so I matched . . .” Encourage students to challenge each other when they disagree.

### Required Materials

Materials to Gather

Materials to Copy

### Required Preparation

• Sort the shape cards from the previous lessons into three groups of 12 cards (A–L, M–X, and Y–JJ).

### Launch

• Groups of 2
• Give each group a set of 12 shape cards (A–L, M–X, or Y–JJ).

### Activity

• “Work together with a partner to sort the figures by the number of lines of symmetry.”
• 5 minutes: group work time

### Student Facing

1. Sort the figures on the cards by the number of lines of symmetry they have.
0 lines of symmetry 1 line of symmetry 2 lines of symmetry 3 lines of symmetry

2. Find another group that has the same set of cards. Compare how you sorted the figures. Did you agree with how their figures are sorted? If not, discuss any disagreement.

### Student Response

For access, consult one of our IM Certified Partners.

If students draw fewer or more lines of symmetry than the figure actually has, consider asking:

• “How did you decide where to draw your line(s)?”
• “How can you prove that each line is a line of symmetry?”
• “Have you found all the lines of symmetry for this figure? How do you know?”

### Activity Synthesis

• “How did you decide if a figure had a line of symmetry? What did you look for?” (Look for matching parts on opposite sides of a figure—parts that have the same size and mirror each other.)
• “Were there figures that you could immediately tell had no lines of symmetry? What was it about the figures that made it clear?” (The parts are different from one another. No folding lines could make two parts mirror each other.)

## Activity 3: Just Keep Folding [OPTIONAL] (10 minutes)

### Narrative

This optional activity gives students an opportunity to reason about lines of symmetry. Students see that some shapes can be folded in half again and again. Some students may notice that certain shapes, like a square, can be divided into two equal halves in perpetuity (MP8). Each new half created by a line of symmetry will continue to be line-symmetric.

### Required Materials

Materials to Gather

• Groups of 2

### Activity

• 4–5 minutes: independent work time
• 1–2 minutes: group discussion

### Student Facing

Priya is folding paper of different shapes along their lines of symmetry. She keeps folding each one until the folded shape has no more lines of symmetry.

1. How many times can she fold each shape before she can no longer continue?
2. What do you notice about each folded shape when it can no longer be folded?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “A, B, and C have more than one line of symmetry. Does the folding line we start with affect how many times the shape could be folded before there is no more line symmetry?” (In these shapes, no, the starting fold doesn’t determine how many folds can be made afterward.)
• 1 minute: quiet think time
• “How can you tell when a folded shape no longer has line symmetry?” (No segments are the same length and no angles are the same size.)
• “What new ideas about lines of symmetry did you learn?” (Sample responses:
• Folding a shape along a line of symmetry may create new shapes that also have lines of symmetry. These lines may create more shapes with symmetry.
• For some shapes, this process could go on even if we cannot physically fold the paper any longer.)

## Lesson Synthesis

### Lesson Synthesis

“Today we found lines of symmetry in flat figures.”

Display the two parallelograms from the first activity.

“In both figures, there’s a line that creates two identical triangles. Why does the first figure have line symmetry but the other doesn’t?“ (The triangles in the first parallelogram match each other exactly when folded along the line, which is not the case in the second parallelogram.)

“Not all lines that divide a figure into two identical halves are lines of symmetry.”

## Cool-down: One Line or More than One? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.