In the Do Now, students look at six equations, which look like proportions, and decide if they are true. There are various methods my students can use to show why or why not the equations are true. Some students will cross multiply and see if the products are equal. Others will divide to see if the quotients are equal or use common factors to check if the fractions are equivalent. I use this activity as a refresher for my students, as well as to gauge their fluency with proportions
To begin this part of the lesson, I show students a picture of two triangles with their angles and sides labeled. I tell students the triangles are similar and ask them to explain what that means using the image.
Usually, my students begin by explaining that each of the angles in Triangle ABC is congruent to each of the angles in Triangle DEF. It takes them slightly longer to see the relationship between the lengths of the sides. We then identify all of the corresponding parts and discuss ways we can write the proportions. For example, we can write side BC is to side EF or side AB is to side BC as side DE is to side EF. Then we verify that the sides are actually in proportion by using the lengths of the sides and write a statement describing similar triangles: If two triangles have three congruent angles, then they are similar and their sides will be in proportion.
Next, I hand out a small sheet with two triangles and some colored pencils or crayons. I ask students, “How can we find the measure of the missing sides? What should we do first?” Students look at the angles first and realize they are congruent. Then we identify and color corresponding sides with a different color for each pair. I give the student a few minutes to write and solve their proportions. Then we look at the solutions and discuss the different ways students used to find the lengths.
For today's independent work, my students work will try to find the missing measures of sides of triangles. The first three questions are straightforward. When they get to Question 4, they may need some help. At first, students may not think they have enough information. They will have to use the information they have to find the length of a different side and then use it to find the value of the variable. As I circulate, I guide students to figure out what to do.
I plan to give my students about 8 minutes and then go over questions 1 through 4. Before I have them work on Questions 5 and 6, we will briefly review the relationship between the angles formed by two parallel lines that are cut by a transversal. We will also practice identifying the corresponding parts of congruent or similar triangles. Students are then given a few more minutes to find the value of the variables. When students find the values of all of the variables, they can go back and use the information to find the lengths of any other missing sides.
For today's Exit Ticket, my students will use the properties of similar triangles to find the value of a variable and explain how they know the triangles are similar. I plan to use this task to determine how well the students understand the concept of similar triangles. I will look to see if the students have identified that two pairs of corresponding angles are congruent. In addition to identifying the congruent angles, I am hoping to see that my students correctly explain why the angles are congruent (e.g., "When two parallel lines are intersected by a transversal, corresponding angles are congruent.")