The students may need some help sorting out the shapes. A good way for them to begin this process is to draw the
figure on their paper and use highlighters to color code the information.
Both of the theorems presented in this section require two pairs of congruent sides. The first step is for student to
highlight these four sides in a common color, let’s say yellow. Once they have identified the two pairs of congruent
sides, they know the hypothesis of the theorem has been filled and they can apply the conclusion.
The conclusions of these theorems involve the third side and the angle between the two congruent sides. These parts
of the triangles can be highlighted in a different color, let’s say pink.
Now the students need to determine if they need to use the SAS Inequality theorem or the SSS inequality theorem.
If they know one of the pink angles is bigger than the other, than they will use the SAS Inequality theorem and write
an inequality involving the pink sides. If they know that one of the pink sides is bigger than the other, they will apply
the SSS Inequality theorem, and write an inequality involving the two pink angles.
Having a step-by-step process is good scaffolding for students as they begin working with new types of problems.
After the students have gained some experience, they will no longer need to go through all the steps.
Solving Inequalities –Students learned to solve inequalities in algebra, but a short review may be in order. Solving
inequalities involves the same process as solving equations except the equal sign is replaced with an inequality, and
there is the added rule that if both sides of the inequality are multiplied or divided by a negative number the direction
of the inequality changes. Students frequently want to change the direction of the inequality when it is not required.
They might mistakenly change the inequality if they subtract from both sides, or if result of multiplication or division
is a negative even if the number used to change the inequality was not negative. In geometry it is most common to
be working with all positive numbers, but depending on how the students apply the Properties of Inequalities, they
may create some negative values.
Indirect Proof
Why Learn Indirect Proof –For a statement to be mathematically true it must always be true, no exceptions. This
frequently makes it easier to prove that a statement is false than to prove it is true. Indirect proof gives mathematician
the choice between proving a statement true or proving a statement false and can therefore greatly simplify some
proofs. Letting the students know that indirect proof can be a potential shortcut will motivate them to learn to use
this type of logic.
Review the Contrapositive -Proving a statement using indirect proof is equivalent to proving the contrapositive
of the statement. If students are having trouble setting up indirect proofs, or even if they are not, it is a good idea
to have them review conditional statements and the contrapositive. The second section of Chapter Two: Reasoning
and Proof is about conditional statements. Have the students reread this section before working on indirect proof in
class. The first step to writing an indirect proof, can be to have them write out the contrapositive of the statement
they want to prove. This will reduce confusion about what statement to start with, and what statement concludes the
proof.
Does This Really Prove Anything? –Even after students have become adept with the mechanics of indirect proof,
they may not be convinced that what they are doing really proves the original statement. This is the same as asking if
the contrapositive is equivalent to the original statement. Using examples outside the field of mathematics can help
students concentrate on the logic.
Start with the equivalence of the contrapositive. Does statement (1) have the same meaning as statement (2)?
a. If you attend St. Peter Academy, you must wear a blue uniform.
b. If you don’t wear a blue uniform, you don’t attend St. Peter Academy.Let the students discuss the logic, and have them create and share their own examples.
Chapter 2. Geometry TE - Common Errors