To help students learn how to write two column geometric proofs, we begin by writing two column proofs of algebraic equations. In the Do Now, students solve two algebraic equations. This helps activate their prior knowledge and allows me to see what students remember from algebra.
The third question in the Do Now asks students to give an example of the Distributive Property. This property will be used during the lesson to write the algebraic proofs. I accept any example that accurately demonstrates the distributive property. We end the Do Now by writing a general statement of the distributive property in algebraic notation: a(b + c) = ab + ac.
I begin the Mini-Lesson by reviewing several postulates that are commonly used to write algebraic proofs. It is important that students know the names and descriptions of the postulates in order to use them for the proofs (MP6). First, I have three students come to the board and show their solutions to Questions 1, 2, and 4 from the Do Now. We then go over the Substitution Postulate, “A quantity may be substituted for its equal in any expression.” To illustrate this, I refer back to the equations from the Do Now. The correct value for the unknown variable can be substituted into the expression to make each statement true. My students are used to substituting values for variables, but today we are describing this process in a way that may be new to them.
Next, we review the addition, subtraction, multiplication and division postulates. I have written out the addition postulate on the presentation. After we go over the addition postulate, the students use it to write out the subtraction, multiplication and division postulates. We then go over examples to illustrate the postulates. For example, if 8 = 3 + 5, then 8 + 2 = 3 + 5 + 2. Similar examples can be used for the other postulates. Although in future lessons we use these postulates in a geometric context, I only use algebraic or numeric examples at this point.
We next move on to writing a two-column algebraic proof. We go over the components needed for the proof:
For our example proof we use the equation 3(4x +7) = 45. I have the students actually solve the equation to verify the solution x = 2. Then we write out the steps for the proof. For this proof, the students need the distributive property, the subtraction postulate, the division postulate and the substitution postulate. In the activity, students will write out algebraic proofs independently.
The main activity for today's lesson consists of three algebraic proofs. Before students write out the proofs, I have them brainstorm the process. They can solve the equation in this space and write out the steps needed in an informal way. Another way to describe this work might be a first, informal draft of the proof.
I want my students to work independently to solve each of the equations. As the students work, I check over their process. If students have difficulty solving the equations, I sometimes give them a simpler equation to work with.
Towards the end of the activity, I will ask students to work in pairs to write the proofs (MP3). I encourage students to:
After about 15 minutes, I call on students to show their work on the document camera. Students check their work using these examples.
In today's summary, we write the postulates using an algebraic notation. For example, the addition postulate can be written as: if a = b and c = d, then a + c = b + d. This leads us to applying the postulates to geometric concepts in future lessons.
Before we depart I briefly demonstrate to the students how their algebraic proofs connect to geometry by showing them an example of segment addition. They look at two segments made up of equal parts and can see how the sum of the equal parts are equal.