Students will explore the properties of congruent polygons and analyze how to apply those properties to solve problems.
How do the properties of congruent polygons help us solve problems?
Students use the Not Like the Others strategy to compare similar and congruent triangles with the same area.
Students explore the definition of congruence with the challenge of dividing a polygon into two congruent polygons.
Students complete guided notes with the class and label corresponding parts of a polygon that has been rotated or reflected.
Students apply what they have learned to find the values of unknown side lengths and angle measures of congruent polygons within a tessellation.
Students solve for unknown values given two congruent quadrilaterals.
Lesson Slides (attached)
Finding Matches handout (attached; one per pair; printed front only)
Guided Notes handout (attached; one per student; printed front only)
Trying Tessellations handout (attached; one per pair; printed front only)
Corresponding Parts handout (attached; one half per student; printed front only)
Colored pencils (5 different colors per pair)
Geoboards (optional for Explore)
Student devices with Internet access (optional for Explore)
Introduce the lesson using the attached Lesson Slides. Slide 3 displays the lesson’s essential question. Slide 4 identifies the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.
Instruct students to find a partner or assign students partners. Go to slide 5. Give each pair of students a copy of the attached Finding Matches handout. Using the Not Like the Others instructional strategy, ask students to consider the triangles shown in part one of the handout: Not Like the Others. Which polygon do they think is not like the others?
Tell students that they have 2 minutes to select the triangle that they believe is not like the others and share their thoughts with their partner. Allow students approximately 3 minutes to share their selection and reasoning with the class.
Display slide 6. Students continue to part two of the Finding Matches handout: Equal Halves. Each polygon can be divided into two congruent polygons by drawing a line segment from one point to another. Give students time to test and determine where to draw this segment. Consider reminding students that this line segment will likely not be the line of symmetry; this could potentially be a misconception.
Display slides 7–9 and discuss strategies for determining that the two halves are congruent. Remind students that the examples on the slides are just that – examples of line segments that would divide the polygon into two congruent halves. As time allows, ask for volunteers to share alternative segments.
Display slide 10. Give each student a copy of the attached Guided Notes handout. Each pair of students needs five colored pencils to share. Complete the handout as a class. Have students add this to their math notebook if that is a classroom norm.
Show slide 11 and direct students’ attention to polygon ABCDE and use their colored pencils to trace each side of the polygon with a different color. Display slide 12 as a model.
Show slide 13. Using the same five colored pencils, instruct students to trace the next polygon such that the corresponding sides are the same color as polygon ABCDE. Transition to slide 14 to show students the corresponding sides. Have students also label the three given angles. Display slide 15 and explain to students how to write the congruence statement and the importance of the order of the letters – corresponding parts are written in the same order.
Display slide 16 and direct students’ attention to the horizontal reflection of polygon ABCDE. Have them repeat the same steps as before, tracing and labeling corresponding parts. Display slide 17. Now if the students take a moment to determine how many vertices are on this handout, they should notice that there are 30 vertices, which is more than there are letters in our alphabet. Display slide 18 and instruct students to label the corresponding vertices A’, B’, …, E’; this notation is used to allow for students to use the remaining letters to label the last three polygons. Display slide 19 to show the congruence statement.
Show slide 20 and instruct students to repeat this process for the remaining polygons.
Display slide 24 and give each pair of students the attached Trying Tessellations handout. Here students are given a tessellation of one repeated pentagon and are asked to solve for each unknown using their knowledge of congruence. Give students a moment to read the directions on the handout, then show slide 25.
Optional: Display slide 26 as a hint for how to get started. Using the colored pencils, label corresponding sides, at least one, can be helpful.
Optional: Show slides 27–33 to review the value of each variable with the class.
Use the Exit Ticket strategy to individually assess what students have learned from the lesson. Go to slide 34 and give each student the Corresponding Parts handout. Students are asked to find the values of x and y given two congruent quadrilaterals.
After students have submitted their work, unhide and show slides 35–37. Give students time to reflect on their thinking. Use student responses to see what misconceptions still exist.