Lesson 1

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.

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 “\(x=0\).”

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).

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

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.)

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.

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

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.

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.)