Born to Puzzle! From Informal to Formal Geometric Thinking 1 Warm Up Identify the vertices of a square in each grid. 1. 2. 3. Learning Goals Recall properties of geometric figures. Understand that the goal of geometric reasoning in high school is to formalize earlier experiences using definitions and geometric properties. Understand that the results from measuring tools can be useful in composing a conjecture; however, they are not an acceptable form of mathematical reasoning to validate a conjecture. Make a geometric conjecture and use mathematical reasoning to prove a conjecture to be true. Key Term conjecture You have reasoned about lines and shapes in earlier grades and courses. How can you apply formal geometric reasoning to what you know? LESSON 1: Born to Puzzle! M1- GEO_SE_M01_T01_L01.indd 1
GETTING STARTED Three Squares a Day The diagram is composed of three adjacent squares. A B C D In mathematics, Greek letters such as α, β, and γ are often use to refer to angles and angle measures. H G F E 1. Draw AG, AF, and ĀE. 2. Label AGH using the Greek letter alpha, α. Label AFG using the Greek letter beta, β. Label AEF using the Greek letter gamma, γ. 3. Consider α in the diagram. a. Use a protractor to measure α. Enter the angle measure in the diagram. b. Classify AGH by its sides and angles. c. Using only the classification of AGH, determine an exact measurement for α. Does it confirm your answer in part (a)? M1- TOPIC 1: Using a Rectangular Coordinate System GEO_SE_M01_T01_L01.indd 2
4. Consider and β and γ in the diagram. a. Use a protractor to measure β and γ. Enter each angle measure in the diagram. b. How do your angle measurements compare to your classmates angle measurements? Why didn t everyone get the same answers? c. Classify AFG and AEF by their sides and angles. d. Unlike the measure of α in AGH, the exact measures of β and γ are not confirmed by the classifications of AFG and AEF. Explain. 5. Use your measurements to complete the blank. α 1 β 1 γ 5 Jack says in an ideal world, the statement should be completed as follows: Explain Jack s reasoning. A conjecture is a mathematical statement that appears to be true, but has not been formally proved. Conclusions such as α 1 β 1 γ 5 90, α 1 β 1 γ 5 89, and α 1 β 1 γ 5 91 are conjectures until they are validated using mathematical reasoning. The application of the properties of an isosceles right triangle in Question 3, part (c) is an acceptable form of mathematical reasoning and can be used to validate a conjecture. The use of measuring tools is not an acceptable mathematical reason due to the nature of its inaccuracies. α + β + γ = 90 LESSON 1: Born to Puzzle! M1- GEO_SE_M01_T01_L01.indd 3
ACTIVITY 1.1 Demonstrating Angle Relationships Let s use Jack s conjecture that α 1 β 1 γ 5 90 from the Getting Started. A jigsaw puzzle approach is an alternative way of thinking about this conjecture. Consider a puzzle that contains the diagram composed of three squares from the Getting Started and a vertical translation of those squares. Two copies of this puzzle are located at the end of this lesson. Remove both copies of the puzzle from the book. For this activity, carefully cut out AGH, AFG and AEF from one puzzle, keeping in mind you will use the remainder of the puzzle. The second copy of the puzzle will be used in the next activity. 1. Using the cutout triangles and two grid lines on the remaining puzzle piece, align α, β, and γ to demonstrate the sum of the angle measures is 90. This method establishes a foundation for understanding the relationship among α, β, and γ. It also eliminates inaccurate measurements caused by the use of a protractor. It is not an acceptable form of mathematical reasoning to validate a conjecture; however, it does give you insight into the mathematical knowledge and strategies required to validate the conjecture. M1- TOPIC 1: Using a Rectangular Coordinate System GEO_SE_M01_T01_L01.indd 4
ACTIVITY 1.2 Vallidating a Conjecture In the previous activity, you cut out puzzle pieces and aligned them within the puzzle to show the sum of their measures is 90. Can this be accomplished using only mathematical properties and definitions? To complete this activity, use the second copy of the puzzle located at the end of this lesson. Let s use Jack s conjecture that α 1 β 1 γ 5 90 from the Getting Started. 1. Recall in the Getting Started that α 5 45. a. What properties did you use to make that conclusion? b. Use α 5 45 to rewrite Jack s conjecture. α 1 β 1 γ 5 90 β 1 γ 5 You are 1 3 of the way to validating the conjecture that α 1 β 1 γ 5 90! Let s think of β 1 γ 5 45 as your conjecture and prove it. 2. To validate this conjecture, you need to connect a few existing points. a. Connect points A and K to form ĀK. b. Connect points E and K to form ĒK. LESSON 1: Born to Puzzle! M1- GEO_SE_M01_T01_L01.indd 5
3. Compare AHF, AIK, and KLE. a. Why are they all right triangles? b. What segment is the hypotenuse of each triangle? c. How do you know the three hypotenuses in are equal in length? d. What reasoning can you use to conclude that the three triangles are the same shape and same size? e. Write a congruent statement that describes the relationship among the three triangles. 4. Use the location of β in AHF. a. Label the angle that corresponds to β in AIK. b. Label the angle that corresponds to β in KLE. 5. Now, focus on AKE in the puzzle. a. Classify AKE by the length of its sides. b. At this point, do you have enough information to conclude AKE is a right angle? M1- TOPIC 1: Using a Rectangular Coordinate System GEO_SE_M01_T01_L01.indd 6
6. Investigate AKI, AKE, and EKL formed at point K. a. Label EKL with the expression (90 β) in the puzzle. Use KLE to explain why this expression represents the angle measure. b. What is the sum of the measures of the three angles? Explain. c. Write and solve an equation to determine the measure of AKE. 7. Reclassify the triangle. a. Classify AKE by the length of its angle and sides. b. Determine the measure of AEK. Explain. c. In Question 1 part (b), you rewrote Jack s conjecture as β + γ = 45. Did you prove your conjecture? Explain. LESSON 1: Born to Puzzle! M1- GEO_SE_M01_T01_L01.indd 7
NOTES TALK the TALK Wrap It Up and Tie a Bow On It! It all started when Jack conjectured that α 1 β 1 γ 5 90. You conjectured and proved that β 1 γ 5 45. 1. Use substitution to help Jack prove her conjecture. 2. List five definitions and/or properties that were used to prove the conjecture. 3. What is the connection between the jigsaw puzzle in Activity 1 and the strategy you used in Activity 2? M1- TOPIC 1: Using a Rectangular Coordinate System GEO_SE_M01_T01_L01.indd 8
Puzzle for Activity 1.1 I J K L A B C D α β γ H G F E LESSON 1: Born to Puzzle! M1- GEO_SE_M01_T01_L01.indd 9
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Puzzle for Activity 1.2 I J K L A B C D α β γ H G F E LESSON 1: Born to Puzzle! M1-1 GEO_SE_M01_T01_L01.indd 11
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