Triangle Inequality
MM1G2. Students will understand and use the language of mathematical
argument and justification.
a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples,
and indirect proof as appropriate.
MM1G3. Students will discover, prove, and apply properties of triangles,
quadrilaterals, and other polygons.
b. Understand and use the triangle inequality
MM1P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
MM1P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjecture.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
MM1P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through
communication.
b. Communicate their mathematical thinking coherently and clearly to peers,
teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
MM1P5. Students will represent mathematics in multiple ways.
b. Select, apply, and translate among mathematical representations to solve
problems.
Triangle Inequality
Questions:
1. How do we know if a triangle is truly a triangle?
2. What is the relationship between the angles and sides of a triangle?
Launch:
1.
List everything you know about triangles.
2.
If you are given 2 sides of a right triangle, how do you find the third side?
3.
What are the different ways you can classify triangles? Draw and label at least 3
different types.
Investigation:
Spaghetti Triangles
The students will be given 3-4 uncooked spaghetti noodles. In pairs they are to create 5
different lengths of spaghetti. 1 inch, 2 inch, 5 inch, 6 inch and 7 inch.
1. Using the 1, 2 and 5 inch pieces try and create a triangle. ( the spaghetti must
meet end to end). Draw your results below. Label your sides with the lengths.
What is the sum of the two smallest sides?
2. Using the 2, 5, and 7 inch pieces try and create a triangle. Draw and label your
results.
What is the sum of the two smallest sides?
3. Using the 2, 6, and 7 inch pieces try and create a triangle. Draw and label your
results.
What is the sum of the two smallest sides?
4. Using the 5, 6, and 7 inch pieces try and create a triangle. Draw and label your
results.
What is the sum of the two smallest sides?
5. Could you make a triangle the first 2 times you tried? Why do you think this
happened?
6. Could you make a triangle the last 2 times you tried? Why do you think this
happened?
7. What do you think made the difference in the last 2 tries?
8. Using a protractor, measure the angles of the triangles in #3 and #4. Label each
angle with its measure. What do you notice about the smallest angle and smallest
side? What do you notice about the largest angle and largest side?
Conclusion:
Come up with a statement or rule about the lengths of the sides of triangles that you got
out of this experiment that you might see in a geometry book.
Write a rule that is always true about the relationship between angles and sides.
In class problems:
1.
Find three side lengths that would make a right triangle. Does the angle to side
relationship still apply?
2.
Will you have a triangle if you have side lengths 20, 23, and 43? Why or why
not?
3.
Does the side lengths of 20, 23, and 44 make a triangle?
Closure:
1. How do we know if a triangle is truly a triangle?
2. What is the relationship between the angles and sides of a triangle?
Homework:
1.
Come up with 3 different sets of side lengths that will make a triangle. Label the
angles 1, 2, 3, from smallest to largest with 1 being the smallest. Explain or
show me with algebra how you know these side length form a triangle. Also
explain how you know which angle is the smallest and which is the biggest.
2.
A cell phone tower is placed in Hiram Ga. It can reach customers that are 45
miles away. If you live 60 miles south of the tower you will not get reception but
you have a friend that lives 10 miles northeast of you. Will your friend get
reception? Draw a picture and explain your reasoning.