Explorations –When students discover a property or relationship themselves it will be much more meaningful.
They will have an easier time remembering the fact because they remember the process that resulted in it. They will
also have a better understanding of why it is true now that they have experience with the situation. Unfortunately,
students sometimes become frustrated with explorations. They may not understand the instruction, or they may not
be carefully enough and the results are unclear. Some of the difficulties can be alleviated by have the students work
in groups. They can work together to understand the directions and interpret the results. Students strong in one area,
like construction, can take on that part of the task and help the others with their technique.
Some guidelines for successful group work.
- Groups of three work best.
- The instructor should choose the groups before class.
- Students should work with new groups as often as possible.
- Desks or tables should be arranged so that the members of the group are physically facing each other.
- The first task of the group is to assign jobs: person one reads the directions, person two performs the
construction, person three records the results. Students should regularly trade tasks.
Inequalities in Triangles
The Opposite Side/Angle –At first it may be difficult for students to recognize what side is opposite a given angle
or what angle is opposite a given side. If it is not obvious to them from the picture, obtuse, scalene triangles can be
confusing, they should use the names. For 4 ABC, the letters are divided up by the opposite relationship, the angle
with vertexAis opposite the side with endpointsBandC. Being able to determine these relationships without a
figure is important when studying trigonometry.
Small, Medium, and Large -When working with the relationship between the sides and angles of a triangle,
students will summarize the theorem to “largest side is opposite largest angle”. They sometimes forget that this
comparison only works within one triangle. There can be a small obtuse triangle in the same figure as a large acute
triangle. Just because the obtuse angle is the largest in the figure, does not mean the side opposite of it is the longest
among all the segments in the figure, just that it is the longest in that obtuse triangle. If the triangles are connected or
information is given about the sides of both triangles, a comparison between triangles could be made. See exercise
#9 is the text.
Add the Two Smallest –The triangle inequality says that the sum of the lengths of any two sides of a triangle is
greater than the length of the third side. In practice it is enough to check that the sum of the lengths of the smaller
two sides is larger than the length of the longest side. When given the three sides lengths for a triangle, students
who do not fully understand the theorem will add the first two numbers instead of the smallest two. When writing
exercises it is easy to always put the numbers in ascending order without thinking much about it. Have the students
try to draw a picture of the triangle. After making a few sketches they will understand what they are doing, instead
of just blindly following a pattern.
Indirect Proof –Students will not really understand the method of indirect proof the first time they see it. Let
them know that this is just the first introduction, and that in subsequent lessons they will be given more examples
and opportunities to learn this new method of proof. If students think they are supposed to understand something
perfectly the first time they see it, and they don’t, they will become frustrated with themselves and mathematics. Let
them know that the brain needs time, and multiple exposures to master these challenging concepts.
Inequalities in Two Triangles
Use Color –The figures is this section now have two triangles instead of just one and are therefore more complex.
2.5. Relationships Within Triangles