Lesson 10

Composing Figures

10.1: Angles of an Isosceles Triangle (10 minutes)

Warm-up

Isosceles triangles are triangles with (at least) one pair of congruent sides. Isosceles triangles also have (at least) one pair of congruent angles. In this warm-up, students show why this is the case using rigid motions by exploiting the fact that rigid motions of the plane do not change angle measures (MP7).

Launch

Give students 3 minutes of quiet work time followed by whole-class discussion.

Student Facing

Here is a triangle.

  1. Reflect triangle \(ABC\) over line \(AB\). Label the image of \(C\) as \(C’\).
  2. Rotate triangle \(ABC’\) around \(A\) so that \(C’\) matches up with \(B\).
  3. What can you say about the measures of angles \(B\) and \(C\)?

A triangle labeled A B C with horizontal side B C labeled 2 and sides A B and A C are each labeled 3.

Student Response

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Activity Synthesis

Select students to share their images and conclusions about the measures of angles B and C.

Time permitting, mention that it is also true that when a triangle has two angles with the same measure then the sides opposite those angles have the same length (i.e., the triangle is isosceles). This can also be shown with rigid transformations. Reflect the triangle first and then line up the sides containing the pairs of angles with the same measure.

10.2: Triangle Plus One (10 minutes)

Activity

The purpose of this task is to use rigid transformations to describe an important picture that students have seen in grade 6 when they developed the formula for the area of a triangle. They first found the area of a parallelogram to be \(\text{base} \boldcdot \text{height}\) and then, to find \(\frac{1}{2}\text{base} \boldcdot \text{height}\) for the area of a triangle, they “composed” two copies of a triangle to make a parallelogram. The language “compose” is a grade 6 appropriate way of talking about a \(180^\circ\) rotation. The focus of this activity is on developing this precise language to describe a familiar geometric situation.

Students need to remember and use an important property of 180 degree rotations, namely that the image of a line after a 180 degree rotation is parallel to that line. This is what allows them to conclude that the shape they have built is a parallelogram (MP7).

Launch

Arrange students in groups of 2. Provide access to geometry toolkits. A few minutes of quiet work time, followed by sharing with a partner and a whole-class discussion.

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students who struggle with visualizing the 180 degree rotation using center \(M\). Consider pausing for a brief discussion to invite 1–2 pairs of students to demonstrate and explain how to do the rotation.
Supports accessibility for: Memory; Organization

Student Facing

Here is triangle \(ABC\).

  1. Draw midpoint \(M\) of side \(AC\).

  2. Rotate triangle \(ABC\) 180 degrees using center \(M\) to form triangle \(CDA\). Draw and label this triangle.

  3. What kind of quadrilateral is \(ABCD\)? Explain how you know.
Triangle A B C is scalene with side B C as base and longest side, and side AB as the shortest side.

Student Response

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Student Facing

Are you ready for more?

In the activity, we made a parallelogram by taking a triangle and its image under a 180-degree rotation around the midpoint of a side. This picture helps you justify a well-known formula for the area of a triangle. What is the formula and how does the figure help justify it?

Student Response

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Anticipated Misconceptions

Students may struggle to see the \(180^\circ\) rotation using center \(M\). This may be because they do not understand that \(M\) is the center of rotation or because they struggle with visualizing a \(180^\circ\) rotation. Offer these students patty paper, a transparency, or the rotation overlay from earlier in this unit to help them see the rotated triangle.

Activity Synthesis

Begin the discussion by asking, "What happens to points \(A\) and \(C\) under the rotation?" (They end up at \(C\) and \(A\), respectively.) This type of rotation and analysis will happen several times in upcoming lessons.

Next ask, "How do you know that the lines containing opposite sides of \(ABCD\) are parallel?" (They are taken to one another by a 180 degree rotation.) As seen in the previous lesson, the image of a \(180^\circ\) rotation of a line \(\ell\) is parallel to \(\ell\). Students also saw that when \(180^\circ\) rotations were applied to a pair of parallel lines it resulting in a (sometimes) new pair of parallel lines which are also parallel to the original lines. The logic here is the same, except that only one line is being rotated \(180^\circ\) rather than a pair of lines. This does not need to be mentioned unless it is brought up by students.

Finally, ask students "How is the area of parallelogram \(ABCD\) is related to the area of triangle \(ABC\)?" (The area of the parallelogram \(ABCD\) is twice the area of triangle \(ABC\) because it is made up of \(ABC\) and \(CDA\) which has the same area as \(ABC\).) Later in this unit, area of shapes and their images under rigid transformations will be studied further.

Writing, Speaking: Math Language Routine 1 Stronger and Clearer Each Time. This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and are asked to create a first draft response. Students meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means.
Design Principle(s): Optimize output (for justification)

How It Happens:

  1. Use this routine to provide students a structured opportunity to refine their justification for the question asking “What kind of quadrilateral is \(ABCD\)? Explain how you know.” Give students 2–3 minutes to individually create first draft responses in writing.

  2. Invite students to meet with 2–3 other partners for feedback.

    Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Listeners should press for details and clarity.

    Provide students with these prompts for feedback that will help individuals strengthen their ideas and clarify their language: “What do you mean when you say...?”, “Can you describe that another way?”, “How do you know the lines are parallel?”, and “What happens to lines under rotations?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss.

  3. Signal for students to move on to their next partner and repeat this structured meeting.

  4. Close the partner conversations and invite students to revise and refine their writing in a second draft. Students can borrow ideas and language from each partner to strengthen the final product.

    Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Quadrilateral \(ABCD\) is a ____ because.…” and “Another way to verify this is.…”.

    Here is an example of a second draft:

    \(ABCD\) is a parallelogram, I know this because a 180-degree rotation creates new lines that are parallel to the original lines. In this figure, the 180-degree rotation takes line \(AB\) to line \(CD\) and line \(BC\) to line \(AD\). I checked this by copying the triangle onto patty paper and rotating it 180 degrees. This means the new shape will have two pairs of parallel sides. Quadrilaterals that have two pairs of parallel sides are called parallelograms.”

  5. If time allows, have students compare their first and second drafts. If not, have the students move on by discussing other aspects of the activity.

10.3: Triangle Plus Two (15 minutes)

Activity

This activity continues the previous one, building a more complex shape this time by adding an additional copy of the original triangle. The three triangle picture in the task statement will be important later in this unit when students show that the sum of the three angles in a triangle is \(180^\circ\). To this end, encourage students to notice that the points \(E\), \(A\), and \(D\) all lie on a line.

As with many of the lessons applying transformations to build shapes, students are constantly using their structural properties (MP7) to make conclusions about their shapes. Specifically, that rigid transformations preserve angles and side lengths.

Launch

Keep students in the same groups. Provide access to geometry toolkits. Allow for a few minutes of quiet work time, followed by sharing with a partner and a whole-class discussion.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to highlight corresponding points in different triangles and consider redrawing the triangles in the same orientation to emphasize corresponding parts.
Supports accessibility for: Visual-spatial processing

Student Facing

The picture shows 3 triangles. Triangle 2 and Triangle 3 are images of Triangle 1 under rigid transformations.

Rotations of triangle ABC.
  1. Describe a rigid transformation that takes Triangle 1 to Triangle 2. What points in Triangle 2 correspond to points \(A\), \(B\), and \(C\) in the original triangle?

  2. Describe a rigid transformation that takes Triangle 1 to Triangle 3. What points in Triangle 3 correspond to points \(A\), \(B\), and \(C\) in the original triangle?

  3. Find two pairs of line segments in the diagram that are the same length, and explain how you know they are the same length.

  4. Find two pairs of angles in the diagram that have the same measure, and explain how you know they have the same measure.

Student Response

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Anticipated Misconceptions

Students may have trouble understanding which pairs of points correspond in the first two questions, particularly the fact that point \(A\) in one triangle may not correspond to point \(A\) in another. Use tracing paper to create a transparency of triangle \(ABC\), with its points labeled, and let students perform their rigid transformation. They should see \(A\), \(B\), and \(C\) on top of points in the new triangle.

Activity Synthesis

Ask students to list as many different pairs of matching line segments as they can find. Then, do the same for angles. Record these for all to see. Students may wonder why there are fewer pairs of line segments: this is because of shared sides \(AB\) and \(AC\). If they don’t ask, there’s no reason to bring it up.

If you create a visual display of these pairs, hang on to the information about angles that have the same measure. The same diagram appears later in this unit and is used for a proof about the sum of the angle measures in a triangle.

After this activity, ask students to summarize their understanding about lengths and angle measures under rigid transformations. If students don’t say it outright, you should: "Under any rigid transformation, lengths and angle measures are preserved."

Writing: Math Language Routine 3 Clarify, Critique, Correct. This is the first time Math Language Routine 3 is suggested as a support in this course. In this routine, students are given an incorrect or incomplete piece of mathematical work. This may be in the form of a written statement, drawing, problem-solving steps, or another mathematical representation. Pairs of students analyze, reflect on, and improve the written work by correcting errors and clarifying meaning. Typical prompts are: “Is anything unclear?” or “Are there any reasoning errors?” The purpose of this routine is to engage students in analyzing mathematical thinking that is not their own and to solidify their knowledge through communicating about conceptual errors and ambiguities in language.
Design Principle(s): Maximize meta-awareness

How It Happens:

  1. In the class discussion for this activity, present this incomplete description of a rigid transformation that takes Triangle 1 to Triangle 2:

    \(CB\) and \(AD\) are the same because you turn \(ABC\).”

    Prompt students to identify the ambiguity of this response. Ask students, “What do you think this person is trying to say? What is unclear?”

  2. Give students 1 minute of individual time to respond to the questions in writing, and then 3 minutes to discuss with a partner.

    As pairs discuss, provide these sentence frames for scaffolding: “I think that what the author meant by ‘turn \(ABC\)’ was….”, “The part that is the most unclear to me is … because.…”, and “I think this person is trying to say.…”. Encourage the listener to press for detail by asking follow-up questions to clarify the intended meaning of the statement. Allow each partner to take a turn as the speaker and listener.

    Listen for students using appropriate geometry terms such as “transformation” and “rotation” in explaining why the two sides are equivalent.

  3. Then, ask students to write a more precise version and explain their reasoning in writing with their partner. Improved responses should include for each step an explanation, order/time transition words (first, next, then, etc.), and/or reasons for decisions made during steps.

    Here are two sample improved responses:

    “First, I used tracing paper to create a copy of triangle \(ABC\) because I wanted to transform it onto the new triangle. Next, I labeled the vertices on my tracing paper. Then, using the midpoint of side \(AC\) as my center, I rotated the tracing paper 180 degrees, because then it matched up on top of Triangle 2. So, sides \(CB\) and \(AD\) are equivalent.”

    or

    “First, I accessed my geometry technology tool to create the drawing of only only Triangle 1 and 2. Next, I tried different transformations of Triangle 1 to make the points \(A\), \(B\), and \(C\) fall on top of points in Triangle 2. The one that worked was rotating Triangle 1 180 degrees using the midpoint of side \(AC\). Finally, I know that sides \(CB\) and \(AD\) are equivalent because Triangle 1 is exactly on top of Triangle 2.”

  4. Ask each pair of students to contribute their improved response to a poster, the whiteboard, or digital projection. Call on 2–3 pairs of students to present their response to the whole class, and invite the class to make comparisons among the responses shared and their own responses.

    Listen for responses that identify the correct pair of equivalent sides and explain how they know. In this conversation, also allow students the opportunity to name other equivalent sides and angles.

  5. Close the conversation with the generalization that lengths and angle measures are preserved under any rigid transformation, and then move on to the next lesson activity.

10.4: Triangle ONE Plus (10 minutes)

Optional activity

This activity builds upon the ideas of the previous one. This time a pattern is built from a single triangle via reflections and the goal is to study the angles and side lengths in this pattern as it grows. If they build the pattern carefully, students may notice after putting together 6 triangles, like in the previous activity, that two of the sides of the triangles (one side of the original and one side of the 6th) lie on the same line. The 6 triangle pattern can be reflected over this line to make it “complete” with 12 copies of the original triangle. Alternatively, students may notice the right angle made by 3 triangles and reason that they can complete a circle with 4 right angles. Both of these arguments are good examples of MP8. Rather than repeating the reflecting procedure 12 times, it is possible to use the structure of what they have learned along the way to accurately predict how many copies make a circle.

Launch

Provide access to geometry toolkits. Allow for 8 minutes of work time, then a brief whole-class discussion.

Student Facing

Here is isosceles triangle \(ONE\). Its sides \(ON\) and \(OE\) have equal lengths. Angle \(O\) is 30 degrees. The length of \(ON\) is 5 units.

Triangle O N E. Angle N O E is labeled 30 degrees. Side O N is labeled 5.
  1. Reflect triangle \(ONE\) across segment \(ON\). Label the new vertex \(M\).

  2. What is the measure of angle \(MON\)?

  3. What is the measure of angle \(MOE\)?

  4. Reflect triangle \(MON\) across segment \(OM\). Label the point that corresponds to \(N\) as \(T\).

  5. How long is \(\overline{OT}\)? How do you know?

  6. What is the measure of angle \(TOE\)?

  7. If you continue to reflect each new triangle this way to make a pattern, what will the pattern look like?

Student Response

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Anticipated Misconceptions

If students are stuck with the first reflection, suggest that they use tracing paper. If you need to, show them the first reflected triangle, then have them continue to answer the problems and do the next reflection on their own.

Some students may have difficulty with the length of \(OT\), since it uses the initial information that triangle \(ONE\) is isosceles (otherwise, \(OT\) is unlabeled). Ask students what other information is given and if they can use it to figure out the missing length.

Activity Synthesis

The main goal of this activity is to apply and reinforce students’ belief that rigid transformations preserve distances and angle measures. Watch and make sure students are doing well with this. If they are not, reinforce the concept. It is critical that this is understood by students before moving forward.

Identify a few students, each with a different response, to share their description of the pattern they saw in the last question. The way in which each student visualizes and explains this shape may give insight into the different strategies used to create the final pattern.

Note: It is not important nor required that students know or understand how to find the base angle measures of the isosceles triangles, or even that the base angles have the same measure (though students do study this in the warm-up). Later in this unit, students will learn and prove that the sum of the angle measures in a triangle is 180 degrees.

Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Attention; Social-emotional skills
Speaking: MLR7 Compare and Connect. Use this routine when students describe the pattern they saw when reflecting the triangle multiple times. Ask students to consider what is the same and what is different about each description. Draw students’ attention to the different ways the pattern is explained. Some students may benefit from the use of gestures to support their understanding of the descriptions. These exchanges can strengthen students’ mathematical language use and reasoning to make sense of strategies used to create composite shapes.
Design Principle(s): Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Briefly review the properties of rigid transformations (they preserve lengths and angle measures). Go over some examples of corresponding sides and angles in figures that share points. For example, talk about which sides and angles correspond if this image is found by reflecting \(ABCD\) across line \(AD\).

Reflection of ABCD over AD.

Point out to students that sides on a reflection line do not move, so they are their own image when we reflect across a side. Also, the center of rotation does not move, so it is its own image when we rotate around it. All points move with a translation.

10.5: Cool-down - Identifying Side Lengths and Angle Measures (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Earlier, we learned that if we apply a sequence of rigid transformations to a figure, then corresponding sides have equal length and corresponding angles have equal measure. These facts let us figure out things without having to measure them!

For example, here is triangle \(ABC\).

A triangle A, B, C where the interior angle at A has measure 36 degrees.

We can reflect triangle \(ABC\) across side \(AC\) to form a new triangle:

Triangle A, B, C, with angle with measure 36 degrees at A. It has been reflected on the side A, C.

Because points \(A\) and \(C\) are on the line of reflection, they do not move. So the image of triangle \(ABC\) is \(AB'C\). We also know that:

  • Angle \(B'AC\) measures \(36^\circ\) because it is the image of angle \(BAC\).
  • Segment \(AB'\) has the same length as segment \(AB\).

When we construct figures using copies of a figure made with rigid transformations, we know that the measures of the images of segments and angles will be equal to the measures of the original segments and angles.