Mr. Javarra asked his students to make up some math games involving facts about triangles. Katie and Allie suggested the following game.
Triangle Trivia RulesProperties of Triangles—Perimeter Variation
Players: Th ree to four studentsMaterials: Th ree number cubes and a “segment pieces” set
of three each of the following lengths: 1 inch, 2 inches, 3 inches, 4 inches, 5 inches, and 6 inches.
Directions: Take turns. Roll the three number cubes. Find a segment piece to match each number rolled. See whether a triangle can be formed from those segment pieces. Th e value of the perimeter of any triangle that can be formed is added to that player’s score. Th e fi rst player to reach 50 points wins.
Amir said he thought that Katie and Allie’s game had nothing to do with triangles because all they did was fi nd the sum of the numbers rolled and then add that to their score. Katie and Allie told Amir that there was more to their game than he thought.
1. Play Katie and Allie’s game to see if what they told Amir is true. Follow the rules above. Record your results in the table.
LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell, Look for Pattern, Use Manipulatives
Player 1 Player 2 Player 3 Player 4Numbers Score Numbers Score Numbers Score Numbers Score
274 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle Trivia
LEARNING STRATEGIES: Quickwrite, Think/Pair/Share, Look for a Pattern
2. Is there more to the game than adding the three numbers on the cubes and then adding that total to your score? Explain below.
Amir found that he did not actually need to use the segment pieces to tell whether a triangle could be formed.
3. Explain how Amir could determine whether a triangle can be formed from three given lengths.
4. Katie and Allie’s game illustrates what is known as the Triangle Inequality Property. Use this property to determine whether a triangle can be formed with the given side lengths listed in inches. Show your work or explain.
a. a = 8, b = 6, c = 4
b. a = 3, b = 4, c = 7
c. a = 5, b = 5, c = 5
d. a = 3, b = 3, c = 7
e. a = 7, b = 4, c = 4
f. a = 8, b = 4, c = 5
g. a = 1, b = 2, c = 8
Amir had an idea that he thought would make Katie and Allie’s game more interesting. He calls his idea the Name My Sides variation. See the rules at the top of the next page.
Properties of TrianglesTriangle TriviaTriangle Trivia
LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell
Katie said she was not sure if she knew what scalene, isosceles, and equilateral meant. Amir showed her the following examples of each type of triangle.
The sets of matching tick marks, such as || and ||, show that the marked sides are congruent.
Triangle Trivia Rules—Name My Sides VariationNumber of Players: Th ree to four studentsEquipment needed: Th ree number cubes.Directions: • Take turns rolling three number cubes.
• If you can, form a scalene triangle .........................add 5 points an isoceles triangle ....................add 10 points an equilateral triangle ..............add 15 points no triangle ....................................add 0 points
• If you are caught making a mistake, deduct 10 points from your last correct score.
276 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle Trivia
LEARNING STRATEGIES: Quickwrite
5. Based on the examples that Amir showed Katie, describe each type of triangle.
a. Scalene triangle
b. Isosceles triangle
c. Equilateral triangle
6. When playing Amir’s Name My Sides variation of Triangle Trivia, suppose that your cubes landed on the following numbers. Tell how many points you would add to your score and why.
a. 5, 5, 5
b. 1, 6, 4
c. 3, 2, 4
d. 6, 6, 4
e. 1, 4, 1
7. Play the Name My Sides variation of Triangle Trivia. Use the table at the top of the next page to record your results.
Properties of TrianglesTriangle TriviaTriangle Trivia
Another way to classify triangles is by their angles. To do this, you need to know whether an angle is acute, obtuse, or right. A right angle has an angle measure of 90°. Th e angle measure of an acute angle is less than 90° and the angle measure of an obtuse angle is greater than 90°.
8. Identify each of these angles as right, acute, or obtuse.
LEARNING STRATEGIES: Create Representations, Use Manipulatives
Player 1 Player 2 Player 3 Player 4Numbers Score Numbers Score Numbers Score Numbers Score
278 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle Trivia
Now that you can identify angles as acute, right, and obtuse, you can classify triangles by their angles. Look at these examples and think about how each kind of triangle is related to its angles.
9. Describe each type of triangle.
a. Acute triangle
b. Obtuse triangle.
c. Right triangle.
10. A triangle that has been labeled as acute, obtuse, or right can also be labeled as scalene, isosceles, or equilateral.
a. Label each triangle at the top of this page as scalene, isosceles, or equilateral.
b. Choose a triangle from the table and explain how the two labels provide a better description of the triangle than either one alone.
LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Create Representations
Properties of TrianglesTriangle TriviaTriangle Trivia
11. Sketch a triangle described by each pair of words below or state that it is not possible. Use tick marks and right angle symbols where appropriate. If it is not possible to sketch a triangle, explain why not.
LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Create Representations
Scalene, right Isosceles, right Equilateral, right
280 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle Trivia
Amir wondered if he could design a variation of Triangle Trivia based on the measures of the angles of a triangle. He decided he would fi rst investigate the sum of the measures of the angles of a triangle. He measured the angles of some scalene, isosceles, and equilateral triangles and recorded his results.
12. Amir made some conjectures about triangles. Determine whether each conjecture below is always true, sometimes true, or never true. Explain why you chose each answer.
a. Th e acute angles of an isosceles triangle are complementary.
b. Th e three angles of any triangle have a sum of 180 degrees.
c. An isosceles triangle can have three equal sides.
d. An equilateral triangle can have a right angle.
e. Th e largest angle of a scalene triangle can be opposite the smallest side.
LEARNING STRATEGIES: Look for a Pattern, Quickwrite
A conjecture is a statement that seems to be true but has not been proven to be either true or false
Properties of TrianglesTriangle TriviaTriangle Trivia
Amir used what he learned about angle relationships in triangles to write a variation of Triangle Trivia. He called it the Th ird Angle Variation. Use his directions to answer Question 13.
Triangle Trivia Properties of Triangles—Th ird Angle Variation
Directions
Shuffl e the cards and place them facedown. Draw two cards. Th e number on each card is the measure of an angle of a triangle.
Find the third angle measure. If it is equal to:
• each of the other two, add 3 times the third angle measure to your score.
• one of the other two angles, add 2 times the third angle measure to your score.
• a right angle, subtract 90 from your score.• neither of the other two and is not a right angle, add the third
angle measure to your score.
Th e fi rst player to reach 300 points wins.
13. When playing Amir’s Th ird Angle variation of Triangle Trivia, suppose you drew cards with the following numbers on diff erent turns. Tell how many points you would add to your score each time and why.
a. 43, 94
b. 38, 52
c. 57, 39
d. 140, 12
e. 60, 60
LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell, Quickwrite
282 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle Trivia
Another student in Mr. Javarra’s class invented this variation. Play this game in groups of three or four.
Triangle Trivia RulesProperties of Triangles—Triangle Trio Game
Players: Th ree to four students
Materials: One set of Triangle Trio cards (24 cards on pages 285 and 287). All sides of equal length and all right angles are marked on the cards.
Goal: Be the fi rst player to make two sets. A set is three cards whose triangles have the same classifi cation either by sides or angles. For example, three acute triangles form a set or three equilateral triangles form a set. A card may be used only once to form a set.
Directions: • Deal all the cards face down so that each player has an equal number of cards.
• Players pick up their cards. If any player can make two sets of three cards, that player wins the round.
• If not, each player chooses one of their cards to pass to the player on their left . Th e players continue to try to make two sets of three cards to win the round.
• Play continues in this manner until someone wins the round.
• Use the answer sheet to verify that the winner has two correct sets.
14. Draw at least two examples of possible winning sets.
15. Explain what strategy you used to try to win the game.
LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell, Quickwrite
Properties of TrianglesTriangle TriviaTriangle Trivia
Write your answers on notebook paper. Show your work.
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show your work.
1. Use the Triangle Inequality Property to determine whether a triangle can be formed with the given length sides in centimeters. Show your work or explain.a. a = 4 b = 5 c = 9b. a = 2 b = 2 c = 5c. a = 6 b = 3 c = 8d. a = 3 b = 5 c = 5
2. Draw a triangle described by each pair of words below or state that it is not possible. If it is not possible, explain why not.a. Scalene, obtuseb. Acute, isoscelesc. Obtuse, equilateral
3. You are given two of the angles of a triangle. Find the third angle and use as many of the following words as possible to describe the triangle.(scalene, isosceles, equilateral, acute, obtuse, right)a. 32°; 58°
b. 162°; 9°
c. 60°; 60°
4. Read the following conjecture and determine whether it is always true, sometimes true, or never true. Explain your reasoning.Th e side of an isosceles triangle between two equal sides is longer than the other two sides.
5. MATHEMATICAL R E F L E C T I O N
Th e games in this activity were designed to help
you better understand the relationship between the sides and angles of diff erent kinds of triangles. Explain how triangles are classifi ed using angle measures and side lengths, and give two examples.
