As students walk in the classroom, they are given a slip of paper on it. On the graph are several triangles. Some of the triangles are congruent to each other and others are not. Students are asked to identify the congruent triangles and explain why one pair of triangles is not congruent. This Do Now reviews concepts from the previous lesson, Rigid Motions and Congruence.
Students will use the graph from the Do Now during the lesson to connect the previous unit on performing transformations to the concept of congruence.
To begin the Mini-Lesson, we review the definition of congruence from the previous lesson. If an object can be transformed to occupy the same space as another object, the two objects are congruent. Then we go over the Do Now. I ask, “Which transformations can you use to prove the triangles are congruent?” We go over the specific transformations and their rules, which students investigated in the previous unit. Students use the coordinates of points on the grid to show that corresponding sides of the pre-image and image are the same length, and therefore prove the triangles are congruent (Congruence by Side Side Side will be investigated in a later lesson in the Proofs about Triangles unit).
Since all of the triangles on the grid are right triangles, this can be a good time to briefly review the Pythagorean theorem. At this point in the course, I don’t have the students use the distance formula to find the length of the sides of the triangles. I designed the activity so that the length of two sides of each triangle on the grid can be found by counting boxes and the third side can be found using the Pythagorean theorem (MP7).
I was pleased my students were applying the Pythagorean Theorem to prove a triangle and its reflection over the y-axis were congruent. In my class we hadn't explicitly used the Pythagorean Theorem which means they were able to connect what they had learned in previous classes to this lesson. Applying the Pythagorean Theorem helped students see that two triangles are congruent to each other if all three of the sides in the pre-image are congruent to the corresponding sides in the image.
Students work individually to identify pairs of congruent triangles on a grid. For each pair of congruent triangles, students describe the specific transformation that proves the triangles are congruent. They then label the congruent sides of the triangles and write statements of congruence for the triangle pairs and the congruent sides.
After about 12 minutes, I stop the students and we go over the transformations and the congruence statements. I ask for volunteers to show how they labeled the triangles using the document camera.
Exit Ticket: Explain why a triangle and its image after a reflection over the y-axis will be congruent.
Students often respond that the pre-image and the image have congruent sides. They can prove this using the Pythagorean theorem. Additionally, the pre-image and image are equidistant from the y-axis after a reflection.
During this lesson, all of the pairs of triangles were proved congruent using Side-Side-Side. In the next lesson, students will learn about other ways for proving triangles congruent.
Instead of collecting the Exit Ticket, we went over the question at the end of the lesson. My students were able to verbalize why a triangle and its image after a reflection over the y-axis are congruent; however, they had difficulty writing an explanation. Students often have difficulty applying mathematical vocabulary and writing a coherent explanation. I had them talk to each other first and then share out responses with the class. We came up with the explanation, "When a triangle is reflected over the y-axis, the pre-image and image will occupy the same space." Then we added, "The length of each of the corresponding sides will be the same." Although I may have answered the question differently myself, I thought it was important to use the students voice in answering the question. I did remind them of vocabulary and added the word "occupy" to clarify the description.