Triangle Construction Site
Lesson 5 of 15
Objective: SWBAT construct a triangle using a given set of sides and angles. Students will understand precise meanings of terms used to describe the positions of parts of a figure in relation to one another.
Using the Slideshow, I display the Warm-up prompt for the lesson as the bell rings. The warmup asks students to define the word polygon. Since the warm-up is a team activity, I expect my students to share what they remember from previous math classes. Reviewing the team answers, I look to see if any of the definitions use terms from the previous lesson, such as "plane figure" or "2-dimensional".
I tell the class that we can define polygons very precisely by describing their structure: the parts of which they are composed and the way those parts are arranged relative to one another. It is no accident that figures are defined this way in geometry, because the structure of a figure tells a lot about the properties of the figure. In some cases, knowing just a few key details will tell us everything else there is to know about the figure. The purpose of this lesson is to help students learn to recognize those details--what to look for when they are given a figure in a problem (MP7).
Displaying the Agenda and Learning Targets, I tell the class that we will start with the simplest polygon: a triangle.
The goal of this activity is for students to learn to analyze the structure of a polygon and to learn the language used to describe it. I also want students to walk away with the realization that the parts of a triangle--its sides and angles--determine its properties and that--sometimes--knowing just 3 of those parts is enough. Later in the year, students will learn why that is so. Until then, I will ensure that they have many opportunities for hands-on experience with the phenomenon.
I display the instructions and distribute the handout for the activity. I ask students to read the instructions as a team. As students are ready to begin, I distribute the Problems on half-sheets of paper, one at a time. This activity follows the Rally Coach format, because I want students to help one another with the constructions.
The Activity begins with a construction that students working in pairs will usually be able to perform: constructing a triangle using two sides and the included angle. To perform this construction, students can simply construct each part in order: segment, angle, segment. Students will still sometimes struggle: usually, they are concentrating on the details of the construction and having trouble seeing the whole picture. I suggest that students try Sketching a Triangle before trying the formal construction. As with many problems in math, it is often wise to get a rough idea of how the solution will look before trying for a precise, final product (MP1).
Students will usually find the next problem--constructing a triangle using two sides and a non-included angle--much harder. This is because it is not clear how to construct the second side. If many students are stuck, I am prepared to call the class together for demonstration, which I perform as a think aloud. This construction requires students to do some problem-solving, using a circle to model the possible locations of the third vertex (MP4, MP7). There is only one possible location for the vertex, which determines the position of the second side. I assign this problem second, so that students will see that a construction can often be a puzzle, and they are expected to apply what they have already learned in the course to solve it.
Armed with this insight, I let the class continue with the third and fourth problems.
At the end of the 25 minute time limit, I summarize with the class. Which sets of sides and angles determined the final properties of the triangle? Which sets allowed the possibility of constructing more than one triangle? The focus is on helping students to recognize that the structure of a triangle--just a few sides and angles--can determine its final properties (MP7). A side benefit is that by exposing students to this idea early, I can give them many opportunities to form conjectures on their own before I ask them to prove triangle congruence theorems later in the year.
I ask students to take out their Guided Notes on Structure, which we began in the previous lesson. We complete the sections on polygons: polygon, side, vertex, the interior of a polygon, adjacent angles and vertices of a polygon, and included sides and angles. We will return to the notes following the lesson on polyhedra to reinforce or extend the meaning of these terms. More on how I use Guided Notes can be found in my strategies folder.
The Lesson Close asks students to name one case where the properties of a triangle are determined by its sides and angles and one case where more than one triangle can be constructed from the same set of parts. I'm not really looking for students to identify conditions for triangle congruence so much as to see whether they can describe the cases in terms of structure, using the vocabulary that was introduced in the lesson. This activity follows our Team Size-Up routine.
I assign problems #18-20 from Homework Set 1. Problems #18 and #19 ask students to practice constructing triangles using given sets of sides and angles. Problem #19 offers a challenge for some students, because one angle must be constructed in a position different from that which is given. (It must be reflected.) Look for an opportunity to have a discussion about what it means to 'copy' a geometric figure. Is it still the same angle if it is flipped? Problem #20 asks students to imagine how a cylinder can be generating by rotating or translating plane figures. While the standards do not address 'solids of translation', I find the idea useful for helping students to visualize the cross-section of a cylinder. I return to this image during the unit on volume relationships, in which we examine the properties of cylinders.