is not isosceles unless at least two of the sides are marked congruent, no matter how much it looks like an isosceles
triangle. Maybe one side is a millimeter longer, but the picture is too small to show the difference. Congruent means
exactly the same. It is helpful to remind the students that they are learning a new, extremely precise language. In
geometry congruence must be communicated with the proper marks if it is known to exist.
Communicate with Figures –A good way to have the students practice communicating by drawing and marking
figures is with a small group activity. One person in a group of two or three draws and marks a figure, and then the
other members of the group tell the artist what if anything is congruent, perpendicular, parallel, intersecting, and so
on. They take turns drawing and interpreting. Have them use as much vocabulary as possible in their descriptions of
the figures.
Two-Column Proofs
Diagram and Plan –Students frequently want to skip over the diagramming and planning stage of writing a proof.
They think it is a waste of time because it is not part of the end result. Diagramming and marking the given
information enables the writer of the proof to think and plan. It is analogous to making an outline before writing an
essay. It is possible that the student will be able to muddle through without a diagram, but in the end it will probably
have taken longer, and the proof will not be written as clearly or beautifully as it could have been if a diagram
and some thinking time had been used. Inform students that as proofs get more complicated, mathematicians pride
themselves in writing simple, clear, and elegant proofs. They want to make an argument that undeniably true.
Teacher Encouragement –When talking about proofs and demonstrating the writing of proofs in class, take time
to make a well-drawn, well-marked diagram. After the diagram is complete, pause, pretend like you are considering
the situation, and ask students for ideas of how they would go about writing this proof.
Assign exercises where students only have to draw and mark a diagram. Use a proof that is beyond their ability at
this point in the class and just make the diagram the assignment.
When grading proofs, use a rubric that assigns a certain number of points to the diagram. The diagram should be
almost as important as the proof itself.
Start with “Given”, but Don’t End With “Prove” –After a student divides the statement to be proved into a given
and prove statements he or she will enjoy writing the givens into the proof. It is like a free start. Sometimes they get a
little carried away with this and when they get to the end of the proof write “prove” for the last reason. Remind them
that the last step has to have a definition, postulate, property, or theorem to show why it follows from the previous
steps.
Scaffolding –Proofs are challenging for many students. Many students have a hard time reading proofs. They are
just not used to this kind of writing; it is very specialized, like a poem. One strategy for making students accustom
to the form of the proof is to give them incomplete proofs and have them fill in the missing statements and reason.
There should be a progression where each proof has less already written in, and before they know it, they will be
writing proofs by themselves.
Segment and Angle Congruence Theorems
Number or Geometric Object –The difference between equality of numbers and congruence of geometric objects
was addressed earlier in the class. Before starting this lesson, a short review of this distinction to remind students
is worthwhile. If the difference between equality and congruence is not clear in students’ heads, the proofs in this
section will seem pointless to them.
Follow the Pattern –Congruence proofs are a good place for the new proof writer to begin because they are fairly
2.2. Reasoning and Proof