Anthony Carruthers JOHN MUIR HIGH, PASADENA, CA
Geometry : Unit #10 - Areas of Plane Figures : Lesson #2

Proving Area Formulas for Parallelograms, Triangles and Rhombuses

Objective: SWBAT use composition and decomposition to prove the area formulas for parallelograms, triangles, and rhombuses.
Standards: HSG-GPE.B.7 MP2
Subject(s): Math
60 minutes
1 Activating Prior Knowledge - 15 minutes

One of the essential understandings for this lesson is that we can compose or decompose figures to suit our purposes. As we compose and decompose figures in this lesson, the Area Congruence Postulate and Area Addition Postulate will be important. 

In this section of the lesson, I introduce students to these postulates via APK_Area Formulas.

I handout the resource and give students 5-7 minutes to work on it. When students have had enough time to complete it, I'll call on students to share their responses under the document camera.

I will coach students in order to improve their responses, but the main goal is for students to come away from this section understanding these two important postulates.

2 Prove Parallelogram Area Formula - 15 minutes

In this section of the lesson, we'll be proving the formula for the area of a parallelogram. The proof planning and strategizing as well as the proof writing, itself, is highly scaffolded on Proving Area Formulas. We go through this resource together as a class.

 

My role is basically to direct traffic and make sure that students are understanding things as we work our way through the proof process. For example, all of the places on the resource where it asks students to think, discuss, sketch, fill in the blanks, etc., I direct students to stop and work independently at first. Then I'll orchestrate a pair share. And finally I may have students share and explain their responses.

The second reason I stop the action is to make sure that students are understanding what we are reading. For example at the place where it explains why we introduce the Roman numerals to the diagram, I'll stop to make sure students can re-voice the purpose of doing this.

3 Prove Triangle Area Formula - 15 minutes

For this section, we'll continue working through Proving Area Formulas. This proof is very much analogous to the proof for the parallelogram area formula. For this reason, I want to make sure to engineer a high quality think-pair-share  and discussion on the first page when students are asked to predict the strategy we'll be using. I want to make sure that students are making connections to what they have just learned in the parallelogram formula proof. For one thing, I want to make sure that they have actually learned...as opposed to just completed...the first proof. Secondly, I want to be gradually releasing control to them over the proof-writing process because having students independently write proofs is the real long-term goal of the course.

Other than that, we proceed through resource together as a class, stopping to think-pair-share when there are directions for students to write or do something, or when there is a weighty concept that I want to make sure students are getting.

4 Prove Rhombus Area Formula - 15 minutes

We'll still be using Proving Area Formulas for this section. The rhombus area proof is somewhat different in nature than the first two proofs we've done in this lesson, but it is also a lot more accessible to students than the first two. So I expect that students will take the initiative on this proof.

For that reason, I have them work through the proof in pairs. I walk around checking in with groups and making sure that things are on track. When students have had enough time to complete, I'll call on students to share responses at the document camera.

5 Closure - 15 minutes

Students have just finished writing proofs with help and scaffolding. Now I want to make sure that they start getting the proofs, and the concepts/strategies behind them, into long term memory. To do that, I'll have students working on Closure_Area Formulas

Before handing it out, I'll give students 5 -10 minutes to review the proofs they've written and ask any questions they need to ask. Then I ask them to put everything away and get to work on the closure activity.

My emphasis is not on students producing perfect papers, but more on giving them and me an opportunity to see what they've actually retained and what still needs reinforcing. I will collect these when students are finished and look at them to get an idea of how students have processed the lesson. 

 

 

Giving the experience of Geometry as an axiomatic system
Connection to Prior Knowledge

I really enjoy teaching this lesson because it gives me a chance to give students the experience of how Geometry operates as an axiomatic system. I love how in Geometry we solve problems and then immediately use the solutions to these problems to solve other problems. It really gives me a sense of growth and building. So I like to give students opportunities to experience Geometry in this way as well.

In this lesson we postulate the rectangle area formula. Then we use it to prove the parallelogram area formula, which is in turn used to prove the trapezoid area formula. Not only that, we incorporate things we learned from much earlier in the course like congruent triangles, properties of quadrilaterals, etc. 

Geometry is not a learn and forget kind of subject and this lesson really drives that point home.