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Spatial Sense and Geometry

Spatial Sense and Geometry. You Be the Judge – Analyzing Answers. Directions:.

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Spatial Sense and Geometry

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  1. Spatial Sense and Geometry You Be the Judge – Analyzing Answers

  2. Directions: • Work independently to solve the problem below. Then compare your answers with those of others in your group. Finally, make any changes needed so that everyone has a correct answer. You may write your answer in any space on this page or on page 4.

  3. Without using a ruler, find the length of each of the sides of hexagon A: • Tell whether Hexagon ABCDEF is similar to Hexagon B. • Tell how you know. Make sure to include mathematical reasons in your answer.

  4. Solution • Sides BC and EF are each 2 units. Sides AB, CD, DE, and FA are each . Draw triangle AHF so that point H is at (3, 7), and use the Pythagorean Theorem to determine the length of FA. • Hexagon A is not similar to Hexagon B because corresponding sides are not in the same ratio. (Hexagon B is a regular polygon, Hexagon A is not, therefore they cannot be similar.)

  5. You Be the Judge • Pages 12-14 contain six responses written by students throughout New Jersey to the question you just answered on page 11. Read each of these responses carefully. In the space below indicate which three are the best responses and which three are the worst. Use the Comments section to tell why each answer is one of the best or one of the worst. In addition, answer these two questions: Does this response show an approach or method that is different from yours? If so, tell how it is different. • Does this response contain errors? If so, tell how to correct the errors. Three best responses: __________ Three worst responses: ____________

  6. Response 1: Hexagon A is similar to Hexagon B. The sides are in a ratio of 1:2: Each side of hexagon A measures 2 units. This response is one of the worst because most of the answers are incorrect. It shows some understanding of similarity in the reference to ratio. However, the response shows all the sides are 2 units and indicates that hexagon A is a regular hexagon, but A is not a regular hexagon. The non-horizontal sides are greater than, not equal to, 2.

  7. Response 2: The length of side AB is because I used the Pytagoren Therom. Side BC and EF are each 2 units. Hexagon A can't be similar to Hexagon B because only some of the sides are proportional This is one of the best responses. The answers are correct and the explanation is adequate. Nevertheless, the response would be improved if more explanation about similarity were provided.

  8. Response 3: I drawed the right triangle and used a2 + b2 = C2. So the big side is about 2.23 so all the diagonal sides are that. Then the other sides are 2. So the hexagons are the same shape but they are not the same size. This is one of the best responses. Even though it is vague about the “big” side and why it measures 2.23, and it does not specifically say that the hexagons are similar, the response does reflect an understanding of similarity by implying that the shapes are similar.

  9. Response 4: This response is one of the worst. The student applies the Pythagorean Theorem correctly, but does not realize that sides EF and BC are not included as the . They each should be exactly 2 units. This error probably led to the student to believe incorrectly that the hexagons were similar. There is some understanding of similarity but its application to the problem was based on wrong answers. The sides are all . So there is a ratio of : 1 so the hexagons are similar.

  10. Response 5: Hexagon A and Hexagon B are NOT similar. They are not in a proportion. The sides of the Hexagon A are not equal but the sides of the other one are. This is the best of the worst. The answers are correct but the supporting reasons are vague and do not convince the reader that there is a real understanding of similarity. Nor is the writer convincing that the Pythagorean Theorem was used to compute the length of the side CB.

  11. Response 6: Use the pythagorean thing and get a2 + b2 = c2 . so 4 of the sides are the square root of 6. The other 2 sies are 2. They are not Similar to the other one cause all the sides in B is 1. This response is one of the best because it shows understanding of the concept of similarity and the application and need for the Pythagorean Theorem. The only error is in computation, not in conceptual understanding.

  12. 1, 4, 5 2, 3, 6 Three best responses: __________ Three worst responses: ____________ You Be the Judge

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