The students’ responsibilities regarding each proof depend on the proof, the ability level of the students, and where
in the course the proof occurs. Some options are (1) The student should understand the logical progression of the
steps in the proof. (2) The student should be able to reproduce the proof. (3) The student should be able to create
proofs using similar arguments.
Classifying Triangles
Vocabulary Overload –Students frequently interchange the words isosceles and scalene. This would be a good time
to make flashcards. Each flashcard should have the definition in words and a marked and labeled figure. Just making
the flashcards will help the students organize the material in their brains. The flashcards can also be arranged and
grouped physically to help students remember the words and how they are related. For example, have the students
separate out all the flashcards that describe angles. The cards could also be arranged in a tree diagram to show
subsets, for instance equilateral would go under isosceles, and all the triangle words would go under the triangle
card.
Angle or Triangle –Both angles and triangles can be named with three letters. The symbol in front of the letters
determines which object is being referred to. Remind the students that the language of Geometry is extremely precise
and little changes can make a big difference.
Acute Triangles need all Three –A student may see one acute angle in a triangle and immediately classify it as an
acute triangle. Remind the students that unlike the classifications of right and obtuse, for a triangle to be acute all
three angles must be acute.
Equilateral Subset of Isosceles –In many instances one term is a subset of another term. A Venn diagram is a
good way to illustrate this relationship. Having the students practice with this simple instance of subsets will make
it easier for the students to understand the more complex situation when classifying quadrilaterals.
Additional Exercises:
- Draw and mark an isosceles right and an isosceles obtuse triangle.
Answer: The congruent sides of the triangles must be the sides of the right or obtuse angle.
This exercise lays the groundwork for studying the relationship between the sides and angles of a triangle in later
chapters. It is important that students take the time to use a straightedge and mark the picture. Using and reading the
tick marks correctly helps the students think more clearly about the concepts.
Classifying Polygons
Vocab, Vocab, Vocab –If the students do not know the vocabulary well, they will have no chance at leaning the
concepts and doing the exercises. Remind them that the first step is to memorize the vocabulary. This will take
considerable effort and time. The student edition gives a good mnemonic device for remembering the word concave.
Ask the students to create tricks to memorize other words and have them share their ideas.
Side or Diagonal –A side of a polygon is formed by a segment connecting consecutive vertices, and a diagonal
connects nonconsecutive vertices. This distinction is important when student are working out the pattern between
the number of sides and the number of vertices of a polygon.
Squaring in the Distance Formula –After subtracting in the distance formula, students will often need to square
a negative number. Remind them that the square of a negative number is a positive number. After the squaring step
there should be no negatives or subtraction. If they have a negative in the square root, they have made a mistake.
Additional Exercises:
Chapter 2. Geometry TE - Common Errors