Students work independently on the Think About It problem to find the area of the given triangle. Students then work to create an expression that can be used to find the area of a triangle.
Some students derived the formula for the area of a triangle by the end of the previous lesson. Other students may think that the expression they should use is A = b x h. This misunderstanding comes from the previous lesson, when we used A = b x h to find the area of the composed parallelogram (and then cut in half).
The goal for this part of the lesson is to have students derive the formula for finding the area of a triangle. Both the work on the Think About It problem and our follow-up conversation will target this objective.
During Intro to New Material section, I introduce students to two ways to express the formula for finding the area of a triangle:
A = (b x h)/2
A = (1/2)bh
Aside from exposing all of my students to a formula for calculating the area of a triangle, another goal of this section of today's lesson is to give students the opportunity to explain why the formula works. (Students are asked to articulate this understanding at multiple points throughout this lesson. )
The first version of the formula is the version that students are most likely to derive on their own because it captures the steps that we followed when finding the area using created parallelograms. The second version of the formula tends to be more common in texts. The existence of different formulas allows us to have a conversation around the commutative property and being really thoughtful about the order in which we multiply numbers.
Example 2 on the INM worksheet provides a great opportunity to talk about the commutative property of multiplication. I first have students solve for the area of the triangle. Then, I ask for a response. I expect some of my students to reply 60 sq. inches. Then, I will ask students how they derived the answer.
Students tend to multiply 10 in. and 12 in. first, and then cut the product in half. I then share with students that I might decide to first cut 10 in in half, and then multiply 5 in. by 12 in. I still get 60 sq. in. I ask students to try multiplying 1/2 by 12 in, and then by 10 in. I want students to realize that they can multiply the factors in any order to get to the area.
Students work in pairs on the Partner Practice problem set. As they work, I circulate around the room and check in with every pair. I am looking for:
I am asking:
After 10 minutes of work time, I will ask the class to discuss Problem 4. I expect that students will quickly determine that Sebastian did not divide by 2 when finding the area of the triangle. The reason why I ask students to discuss this problem is because of the explanations they're required to provide. Once one student shares out, I will ask the class to work together to make the explanation stronger. I will project the initial response on the board, using the document camera. I'll use markers on the board to add and change the response, so that the class feels we have a clear and thorough explanation.
Students work on the Independent Practice problem set.
There are very few problems in this set that ask students to simply find the area of a triangle. The problems ask students to apply what they know about the area of a triangle, and do it in a variety of ways. I design the progression of tasks to help students build on their previous conceptual understanding. My goal is to push my students to make the jump to more and more rigorous problems.
After independent work time, we discuss Problem 9, which requires students to find the value of the base, given the area of the triangle. I ask students if they considered using 4 mm as the value of the base. Many will admit that they considered this, because 8 x 4 is 32, which is the area of the given formula. I ask, then, what made them change their minds. I want students to articulate that they need to halve the value the get when multiplying the base and the height, because a triangle is half of a parallelogram, and the area of a parallelogram is b x h. I also want students to talk about testing their answer by using the formula, to be sure that their value for the base does indeed result in an area of 32 sq. millimeters.
Before completing the Exit Ticket, I have students turn and share their work on Problem 10 with their partners. I like using Problem 10 for student discussion because each student created his/her own triangle. Yet, every student eventually went through the same process when answering the questions. I expect that by this point in the year, my students can have rich mathematical conversation with a peer, including asking one another meaningful questions. That is what I will be listening for. When I am satisfied that I have given students ample opportunity to share and discuss, I will ask them to work independently on the Exit Ticket to close the lesson. An exit ticket sample has been provided.