1.1 The Pythagorean Theorem

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1 1.1 The Pythagorean Theorem Strand Measurement and Geometry Overall Expectations MGV.02: solve problems involving the measurements of two-dimensional shapes and the volumes of three-dimensional figures; NAV.02: simplify numerical and polynomial expressions in one variable, and solve simple first-degree equations. Specific Expectations MG2.01: relate the geometric representation of the Pythagorean theorem to the algebraic representation a 2 + b 2 = c 2 MG2.02: solve problems using the Pythagorean theorem, as required in applications NA2.01: simplify numerical expressions involving integers and rational numbers, with and without the use of technology NA2.02: relate their understanding of inverse operations to squaring and taking the square root, and apply inverse operations to simplify expressions and solve equations NA2.04: substitute into and evaluate algebraic expressions involving exponents (i.e., evaluate expressions involving natural-number exponents with rationalnumber bases) Learning Goals The side lengths of a right angle triangle are related by a 2 + b 2 = c 2. Connect the geometric representation of the Pythagorean theorem to the algebraic representation, a 2 + b 2 = c 2, and communicate this understanding. Prerequisite Skills/Knowledge The geometric representation of the Pythagorean theorem Solving equations that have one variable with integer solutions by the inspection/balancing method Determining a square root (geometrically "What is the side length of a square when you know its area?") NEL 1.1: The Pythagorean Theorem 1

2 MATERIALS For Minds On: BLMs with cutout shapes, protractor For Action: ruler, glue, chart paper, markers KEY VOCABULARY legs hypotenuse vertex (vertices) right triangle Pythagorean theorem NEL 1.1: The Pythagorean Theorem 2

3 1. Minds On TIME 20 min MATERIALS BLM 1-1a: Minds On protractor GROUPING Activity: small groups Discussion/Sharing: whole class Purpose of the Activity Activate knowledge of the geometric representation of the Pythagorean theorem. Making the Activity Work At the IWB, one student group can form triangles by rotating and dragging squares from each set. Then students can use the protractor tool or SMART Math Tools to determine whether any angle measures 90. They may wish to experiment by forming and measuring triangles with squares from different sets. Groups at their desks can cut out and use the squares in BLM 1-1a. Pasting the squares on a grid may be helpful with measurement and presentation, but this will limit experimentation unless each group has several copies of the BLM. Solution The squares in Sets 1 and 2 form right triangles. The squares in Sets 3 and 4 do not. Set 1 Set 2 NEL 1.1: The Pythagorean Theorem 3

4 Set 3 Set 4 To determine whether the triangle formed by any three squares will be a right triangle, sum the areas of the two smaller squares. If the total equals the area of the largest square, the triangle is a right triangle. Questioning How are the triangles you have formed the same? How are they different? How do you know that the triangles formed from the squares in Sets 1 and 2 are right triangles? How do the areas of the squares in Sets 1 and 2 compare? Is this relationship also true of the areas of the squares in Sets 3 and 4? What are the lengths of the sides of each triangle? How are the lengths of the sides of each right triangle related? Opportunities for Assessment ASSESSMENT OF LEARNING: During the discussion after the activity, at the IWB, ask a student to pick three squares from any of the sets and have the class predict whether the triangle formed will include a right angle. Then have students check their predictions. If any students do not demonstrate an understanding of the Pythagorean relationship, you could provide another opportunity to explore the Pythagorean theorem, perhaps through a video on the Internet. (Try searching for Pythagorean theorem. ) NEL 1.1: The Pythagorean Theorem 4

5 2. Action (open) TIME 30 min MATERIALS chart paper ruler GROUPING small groups Purpose of the Activity Use the algebraic or geometric representation of the Pythagorean theorem to solve a problem. Making the Activity Work At the IWB, a student group can solve the problem. Students will probably draw a diagram of a right triangle (including the distance from the ground to the window), estimate to determine how far the base of the ladder needs to be from the wall, and use the Pythagorean theorem to solve for the length of the hypotenuse (ladder) of their triangle. They may decide to add some length to the 7 m height of the window, so that the painter can reach the top of the window frame. Other groups can follow a similar process at a table using chart paper. After the students at the IWB have determined a solution, you may want to suggest how they can verify their solution using SMART Math Tools. Students can use the shape tool or irregular polygon tool to create a right triangle. By going into the drop-down menu and choosing Show/Hide Vertices, and Show/Hide Side Lengths to maintain 7 m or whatever distance the students have decided on in their solution, students can then take a vertex and drag it to different positions and use the measurements provided to determine if their solution is valid. Alternatively, you could make The Geometer's Sketch Pad applet attached to the IWB file available to students to verify their solutions. NEL 1.1: The Pythagorean Theorem 5

6 Solution For example, if the base of the ladder is 2 m from the wall, and the top of the ladder rests 7 m above the ground, (length of ladder) 2 = (length of ladder) 2 = (length of ladder) 2 = 53 length of ladder = 7.3 m Questioning Which length should you determine first? Why? Now that you have determined one length, how will you determine the remaining length? Would the lengths change if you wanted to paint the top of the frame of the window? Is there a maximum distance that the base of the ladder can be from the wall? Why? NEL 1.1: The Pythagorean Theorem 6

7 3. Consolidation TIME 20 min GROUPING whole class Consolidation Strategy Ask the student group that worked at the IWB and one other group to share their solutions. You may want to use a document camera to present a solution developed on paper. You can use the Reflecting and Connecting questions to help students frame their presentation. Highlights and Summary You can use the Highlights and Summary slides with SMART Response clickers to develop a summary for the lesson. Anticipated Key Learnings The Pythagorean theorem can be represented two ways. Geometric representation: The sum of the areas of the squares on the legs of a right triangle equals the area of the square on the hypotenuse. Algebraic representation: a 2 + b 2 = c 2, where a and b represent the lengths of the legs of a right triangle, and c represents the length of the hypotenuse. When you know the lengths of two of the three sides of a right triangle, or the areas of two of the squares on the sides, you can determine the length of the third side using the Pythagorean theorem. Other concepts that students should be able to express: The hypotenuse of a right triangle is always the longest side, and it is always opposite the 90 angle. The length of the hypotenuse is represented by c in a 2 + b 2 = c 2. It does not matter which of the other two side lengths is designated a or b. A triangle in which the sum of the areas of the squares on the two shorter sides does not equal the area of the square on the longest side is not a right triangle (i.e., if the Pythagorean theorem does not apply to a triangle, the triangle is not a right triangle). NEL 1.1: The Pythagorean Theorem 7

8 Independent Practice: Your Turn SAMPLE SOLUTION TO YOUR TURN I drew a right triangle and labelled one leg 6 cm. I decided to make the other leg 5 cm. a 2 + b 2 = c = c = c 2 61 = c = c I used the Pythagorean theorem to determine the length of the hypotenuse. I rounded my answer and labelled my diagram with the length of the hypotenuse. Opportunities for Assessment ASSESSMENT OF LEARNING: To determine whether students are ready for independent practice, observe whether students Construct a right triangle to represent the situation, with one leg 6 cm long. Use mental math strategies to determine the lengths of the other two sides (for example, using a triangle they have already worked with, or sums of common squares), or, alternatively, choose one other side length and use the Pythagorean theorem to determine the third side length. Consider having students who struggle with the Your Turn problem join you at the IWB to work through the first few Practice problems as the rest of the students work on assigned problems at their desks. NEL 1.1: The Pythagorean Theorem 8