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DAY 1 - Hierarchy of Triangles, Triangle Inequality,
and Pythagorean Theorem
Title
Goals
Standard
Addressed
Materials for
Teacher
Materials for
Students
Description
Reflection
Looking Ahead
More about Triangles

Learn the relationships among various types of
triangles.
 The student will be able to recognize whether or not
three given lengths could be the measurements of
the sides of a triangle.
 Pythagorean Theorem
Current Standards: 3.MG 2.2; 4.MG 3.7; 5.MG2.1
Common Core Standards: 3.G.1; 5.G.4; 5.G.CA-1; 7.G.2
Uncooked Linguini noodles, rulers, pencils, and compasses.
Lesson handouts, uncooked Linguini noodles, rulers,
pencils, and compasses.
We use triangle definitions to create a Venn
diagram/hierarchy to show the relationships among them.
We also explore the side-length combinations, which can
result in a triangle using linguini noodles, as well as
through constrictions.
Triangles are the simplest polygons. Many properties of
polygons can be derived by “triangulating” them.
Link to text
Recognizing shapes and applying its properties are important components of geometric
thinking. These two skills provide a strong foundation for solving problems that appear
naturally in the real world. In this lesson we discuss triangles and apply their properties to
solve some problems. We start with the definition of a triangle. Next, we classify triangles
according to their sides, and according to their angles.
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Task 1: Classifying Triangles Continued
Construct a hierarchy (or a Venn diagram) showing the relationships among the following
figures: Polygon, Triangle, Scalene Triangle, Isosceles Triangle, and Equilateral Triangle.
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Task 2: Exploring the Side Lengths of Triangles
Option #1: Essential Question: What (if any) limitations are there on the measurements of
three sides of a triangle?
Materials:
Each group will need 7 full-length pieces of linguini and a ruler with centimeter markings.
Each individual will need an activity sheet and a note sheet.
Procedures:
 Place students into pairs or groups of three.
 Pass out the activity sheet.
 Preview activity and give instructions for obtaining required materials.
 Allow time for students to complete the Linguini activity (following sheet) while
teacher circulates to clarify, guide, and observe students.
 Conclude activity with a discussion of findings.
 Pass out note guide and put a copy on the overhead projector.
 Teacher calls on students to provide missing phrases, words, solutions, and
summary of what was learned through the activity. Teacher records given
responses, but then allows the class to modify or correct as necessary.
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Note Guide
1. This question uses 3 pieces of linguini.
Break one piece so that you have an 18cm length and a 4 cm length (there will be a little
leftover)
Break the second piece so that you have a 14cm length and an 8 cm length (there will be
leftover)
Break the third piece to create an 11 cm length (there will be significant amount left
over)
Below are listed all possibilities for triples of the lengths. If the three lengths will form a
triangle, put yes in the blank. If the three lengths will not form a triangle, put no in the
blank.
4,8,11
4,8,14
4,8,18
4,11,14
4,11,18
4,14,18
8,11,14
8,11,18
8,14,18
11,14,18
2. Use 2 pieces of linguini for this question.
Break one piece so that you have two 13cm lengths.
Leave the second piece unbroken.
Use the broken (13cm) pieces for two sides of a triangle. Use the unbroken piece to
represent the third side.
a. Make a single triangle (the unbroken piece may have extra past the triangle)
What is the length of the third side in the triangle?
b. What is the longest possible length for the third side?
c. What is the shortest possible length for the third side?
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3. Use 2 pieces of linguini for this question.
Break one piece so that you have a 20 cm piece and a 3 cm piece.
Leave the second piece unbroken.
Use the broken pieces (20cm & 3cm) for two sides of a triangle. Use the unbroken piece
to represent the third side.
a. Make a single triangle (the unbroken piece may have extra past the triangle)
What is the length of the third side in the triangle?
b. What is the longest possible length for the third side?
c. What is the shortest possible length for the third side?
Discussion: Triangle Inequality Theorem
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Problem Solving Using Triangle Inequality
Group Problem: TRIANGULAR ARRANGEMENTS
The three sides of a triangle have lengths a, b, and c. Also, all three lengths are whole numbers
and a  b  c.
Suppose c = 9. Find the number of different triangles that are possible.
Follow Up Questions:
1. In the problem your group just solved, would c = 9, b = 5, a = 4 be possible values for c, b,
and a? Explain why or why not.
2. If c = 4, how many triangles would be possible? Explain.
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Task 3: Pythagorean Theorem
1. Using the provided grid paper cut out three squares of sides 3 cm, 4 cm, and 5cm. On
each of your squares write the area of the respective areas.
2. What do you notice about the areas?
3. Can the sides of these three squares be used as the sides of a triangle?
4. If so, arrange the squares in such a ways that the form a triangle. What kind of a triangle
is it? How do you know?
Discussion: Pythagorean theorem —
Proof: Drawing on Algebra and Geometry to prove the Pythagorean theorem: Using the
diagram below to prove the Pythagorean theorem.
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Applications of the Pythagorean Theorem
Sample CST Question:
Sample CAHSEE Questions: