# Lesson 1

Rigid Transformations in the Plane

## 1.1: Traversing the Plane (5 minutes)

### Warm-up

In this activity students make connections between transformations and the coordinate grid. When defining transformations, they will notice and make use of the structure created by the grid (MP7). This task also presents an opportunity to refresh students’ memories of transformation language.

Monitor for students who draw a right triangle and use the Pythagorean Theorem to find the distance between points \(A\) and \(B\). Throughout this unit, distance will be viewed as an application of the Pythagorean Theorem. Distance calculations using the Pythagorean Theorem will lead into the development of equations for circles and parabolas. There will be no need to introduce a separate distance formula.

### Student Facing

- How far is point \(A\) from point \(B\)?
- What transformations will take point \(A\) to point \(B\)?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Invite a student who drew in a right triangle to share their method. If a student suggests the distance formula as an alternate method, ask the class how the formula connects to the Pythagorean Theorem. If no one uses the distance formula, there is no need to mention it.

Ask a few students to share their transformations. There are many possibilities. Transformations that take multiple steps are as valid as single step transformations. If students use descriptions such as “Move 3 units down and 4 units right,” connect this back to the language of translating and directed line segments. Remind students of this language by asking them to read the sentence frames for transformations from their reference chart:

- Translate (object) along the directed line segment from (point) to (point).
- Rotate (object) (clockwise or counterclockwise) using (point) as the center by angle (measure).
- Reflect (object) across line (name/equation).

Note that during this unit points could be named with letters (for example, point \(A\)) or with coordinates (for example, \((\text-3,5))\). Similarly, lines could be named in various ways, such as “\(y\)-axis” or “\(x=0\).”

## 1.2: Transforming with Coordinates (15 minutes)

### Activity

In this activity students practice transforming a figure on the coordinate plane. Students may choose to use tracing paper and perform these transformations as if there were no grid. Other students may notice the structure of gridlines and look for patterns in the coordinates. During the synthesis, students are reminded that rigid transformations produce congruent figures. This helps prepare students for the next activity in which they reason that given 2 congruent figures, there must be a sequence of transformations carrying one figure to the other.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

*Representation: Develop Language and Symbols.*Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide access to Figure \(H\) at scale or a large scale Figure \(H\) and a large scale grid to match.

*Supports accessibility for: Visual-spatial processing; Conceptual processing*

### Student Facing

First, predict where each transformation will land. Next, carry out the transformation.

- Rotate Figure \(H\) clockwise using center \((2, 0)\) by 90 degrees.

Translate the image by the directed line segment from \((2, 0)\) to \((3, \text-4)\).

Label the result \(R\). - Reflect Figure \(H\) across the \(y\)-axis.

Rotate the image counterclockwise using center \((0, 0)\) by 90 degrees.

Label the result \(L\).

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If students are stuck, suggest the use of tracing paper.

### Activity Synthesis

Invite students to share strategies such as, “Reflecting across the \(y\)-axis makes the \(x\)-values negative and keeps the \(y\)-values the same.” If students do not notice patterns like this one, there is no need to mention them. Students will investigate the effect of transformations on coordinates in a subsequent lesson.

Ask students what they notice about the 3 figures. (The figures are trapezoids. The figures have 3 right angles. All 3 figures are congruent.) Ask students how they know the figures are congruent. (They are congruent by definition of rigid transformations.)

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. As students share their strategies for transforming the figure, ask them to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped to clarify the original statement such as, “reflect across the \(y\)-axis” and “rotate clockwise or counterclockwise by 90 degrees.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.

*Design Principle(s): Support sense-making*

## 1.3: Congruent by Coordinates (15 minutes)

### Activity

In this activity students calculate side lengths and angle measures of triangles on the coordinate plane. In the process they demonstrate the two triangles are congruent. During the synthesis they discuss the minimum requirements for a proof of triangle congruence, since calculating *all* side lengths and angle measures goes above and beyond what is necessary. Finally, students specify a sequence of rigid transformations taking one triangle to the other.

### Launch

Tell students they can either leave answers as exact values or round sides to the nearest tenth and angles to the nearest degree.

*Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.*Use this routine to help students improve their written responses for the last question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the angle and center of rotation?”, and “What is the directed line segment of the translation?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students explain the sequence of transformations that takes triangle \(ABC\) to triangle \(DEF\).

*Design Principle(s): Optimize output (for explanation); Cultivate conversation*

### Student Facing

- Calculate the length of each side in triangles \(ABC\) and \(DEF\).
- Calculate the measure of each angle in triangles \(ABC\) and \(DEF\).
- The triangles are congruent. How do you know this is true?
- Because the triangles are congruent, there must be a sequence of rigid motions that takes one to the other. Find a sequence of rigid motions that takes triangle \(ABC\) to triangle \(DEF\).

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

What single transformation would take triangle \(ABC\) to triangle \(DEF\)?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If students are stuck on finding the measures of the angles, suggest they look at their reference chart for concepts from a prior unit that can help.

### Activity Synthesis

Invite a student to share their solution for calculating the measure of angle \(A\). Ask the class why they can use trigonometric ratios to find angle measures. (Triangle \(ABC\) is a right triangle.)

Now ask the students if they need to show that every pair of sides and every pair of angles is congruent to prove the triangles are congruent. (No.) Invite them to share strategies that can prove congruence with fewer pieces of information. (Theorems such as the Side-Angle-Side Triangle Congruence Theorem work. It is also possible to prove triangles congruent by finding a sequence of rigid motions that carry one triangle to the other.)

## Lesson Synthesis

### Lesson Synthesis

Arrange students in groups of 2. Display triangle \(END\) shown here. Instruct students to work with their partners to find a set of coordinates that forms a triangle congruent to \(END\). Then the students should write a proof showing the triangles are congruent.

Sample responses:

- We found the coordinates \(E'(\text-3,1), N'(3,\text-2),\) and \(D'(3,\text-5)\). Translate triangle \(END\) by the directed line segment from \((\text-2,4)\) to \((\text-3,1)\). Each vertex of triangle \(END\) will coincide with the corresponding vertices of triangle \(E'N'D'\), so the triangles are congruent.
- We found the coordinates \(E'(2,4), N'(\text-4,1),\) and \(D'(\text-4,\text-2)\). Then we calculated the lengths of the segments. The lengths of \(EN\) and \(E'N'\) are each \(\sqrt{45}\) units. The lengths of \(DE\) and \(D'E'\) are each \(\sqrt{72}\) units. The lengths of \(ND\) and \(N'D'\) are each 3 units. So, the triangles are congruent by the Side-Side-Side Triangle Congruence Theorem.

Invite a few pairs to present their triangles and proofs. If possible, select at least 1 pair who wrote a transformation proof and at least 1 pair who wrote a proof using calculations of side length or angle measure.

## 1.4: Cool-down - A Transformed Triangle (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

The triangles shown here look like they might be congruent. Since we know the coordinates of all the vertices, we can compare lengths using the Pythagorean Theorem. The length of segment \(AB\) is \(\sqrt{13}\) units because the segment is the hypotenuse of a right triangle with vertical side length 3 units and horizontal side length 2 units. The length of segment \(DE\) is \(\sqrt{13}\) units as well, because this segment is also the hypotenuse of a right triangle with leg lengths 3 and 2 units.

The other sides of the triangles are congruent as well: The lengths of segments \(BC\) and \(FE\) are 1 unit each, and the lengths of segments \(AC\) and \(DF\) are each \(\sqrt{10}\) units, because they are both hypotenuses of right triangles with leg lengths 1 and 3 units. So triangle \(ABC\) is congruent to triangle \(DEF\) by the Side-Side-Side Triangle Congruence Theorem.

Since triangle \(ABC\) is congruent to triangle \(DEF\), there is a sequence of rigid motions that takes triangle \(ABC\) to triangle \(DEF\). Here is one possible sequence: First, reflect triangle \(ABC\) across the \(y\)-axis. Then, translate the image by the directed line segment from \((\text-1,1)\) to \((\text-3,1)\).