Sample Test Problems for Chapter 7

Transcription

1 Sample test problems for Mathematics for Elementary Teachers by Sybilla eckmann copyright c Addison-Wesley, 2003 Sample Test Problems for Chapter 7 1. The diagram in Figure 1 shows the Earth and Moon to scale, as might be seen from outer space. Approximately what fraction of the Moon s surface will people on Earth be able to see when the Sun s rays are hitting the Earth and Moon as indicated? Explain your answer in detail. (Consider only those people on that part of the Earth where it is possible to see the Moon at the time.) Sun rays Earth Moon Figure 1: The Earth and Moon Shown to Scale 2. The diagram in Figure 2 shows the Earth and Moon as seen from outer space above the north pole (not to scale!). (a) Approximately what time of day is it at point P? Explain how you can tell from the diagram and the fact that the Sun rises in the east and sets in the west. (b) How does the Moon appear to a person at point P? Explain your answer. 3. The diagram in Figure 3 shows the Earth and Moon as seen from outer space, looking down on the north pole (not to scale!). (a) For a person located at point P, approximately what time is it? Explain how you can tell from the diagram and the fact that the Sun rises in the east and sets in the west. (b) Is the Moon waxing or waning in Figure 3? Explain how you can tell from the diagram. 4. The diagram in Figure 4 shows the Earth as seen from outer space, looking down on the north pole (labeled N). (a) What time of day is it at point P? Explain how you can tell from the diagram and the fact that the Sun rises in the east and sets in the west. 1

2 sun rays P N Moon Figure 2: The Earth and Moon P Moon N sun rays Figure 3: The Earth and Moon 2

3 moon s orbit sun rays N P Figure 4: The Earth (b) If a person at point P in the picture above can see the Moon, and if the Moon is neither new nor full, then is the Moon waxing or is it waning? Explain how you can tell from the diagram. 5. The diagram in Figure 5 shows the Earth as seen from outer space, looking down on the north pole (labeled N). (a) What time of day is it at point P? Explain how you can tell from the diagram and the fact that the Sun rises in the east and sets in the west. moon s orbit N sun rays P Figure 5: The Earth (b) If a person at point P in the diagram can see the Moon, and if the Moon is neither new nor full, then is the Moon waxing or is it waning? Explain how you can tell from the diagram. 6. Use the fact that the Sun rises in the east to explain clearly why the time of day on the west coast of the U.S. is earlier than the time of day on the east coast of the U.S. Draw pictures to aid your explanation. 3

4 7. What are the two ways of defining (or thinking about) the concept of angle? riefly describe how these are related. Then give an example of a situation where both ways of thinking about angles is needed. 8. Discuss the two different ways of defining the concept of angle. Describe how to use an angle explorer, shown in Figure 6, to relate the two definitions. angle explorer Figure 6: An Angle Explorer 9. Explain clearly why Keisha s angle explorer in Figure 7 does not show a bigger angle than Aaron s. Aaron s Keisha s Figure 7: Two Angle Explorers 10. rad got in his car at point A and drove to point along the route indicated in Figure 8. (a) Show all of rad s angles of turning along his route. (b) What is the total amount of turning that rad did along his route? Describe how you can determine this without measuring individual angles and adding them up. 11. The diagram in Figure 9 shows Ashley in a room with two wall mirrors as seen from the point of view of a fly looking down from the ceiling. What will Ashley see when she looks at point P in the mirror? Explain. 12. The picture in Figure 10 is of a person, a full length wall mirror, and a chair, shown from the point of view of a fly on the ceiling. Can the person see the chair in the mirror? If so, show a place on the mirror where the person can see the chair. Either way, explain briefly why or why not. 13. (a) Informally, we might describe a circle as a perfectly round shape. What is the mathematical definition of a circle? 4

5 A N W E S Figure 8: rad s Route mirror 1 P mirror 2 Ashley Figure 9: Ashley in a Room with Two Wall Mirrors 5

6 mirror chair person Figure 10: A Person, a Chair, and a Mirror (b) Solve the following problem and explain how the mathematical definition of circle is involved in solving the problem: The towns of Kneebend and Anklescratch are 10 miles apart, as shown on the map in Figure 11. The ig Savings store is 6 miles from Kneebend and 8 miles from Anklescratch. Where is the ig Savings store? 14. Which of the following provide a correct definition of the term circle. Check all that apply. (a) A collection of points in a plane that are all one fixed distance away from a point. (b) All the points in a plane that are one fixed distance away from each other. (c) All the points that are one fixed distance away from a point. (d) All the points in a plane that are one fixed distance away from a point. 15. A new Giant Superstore is being planned somewhere in the vicinity of Kneebend and Anklescratch, towns which are 10 miles apart (as shown on the map in Figure 11). The developers will only say that all the locations they are considering are less than 7 miles from Kneebend and more than 5 miles from Anklescratch. Indicate all the places where the Giant Superstore could be located. Explain your answer. Anklescratch Kneebend Figure 11: Where Could the Giant Superstore be Located? 16. John says that his house is more than 5 miles from Walmart and more than 3 miles from KMart. Indicate all possible locations for John s house on the map in Figure 12. Explain your answer. 6

7 5 miles Walmart KMart Figure 12: Where Could John s House be Located? 17. A GPS unit receives information from two satellites. The GPS unit learns that it is 10,000 miles from one satellite and 15,000 from the other satellite. Without any further information, describe the nature of all possible locations of the GPS unit. Explain your answer. 18. Use a compass and straightedge to construct an equilateral triangle in a careful and precise fashion. Explain why your construction must produce an equilateral triangle. 19. Use a compass and ruler to help you draw an isosceles triangle that has two sides of length 3 inches and one side of length 2 inches. 20. (a) Use a compass and ruler to help you draw a triangle that has one side of length 4 inches, one side of length 2 inches, and one side of length 3.5 inches. (b) Explain why your method of construction must produce a triangle with the required side lengths. 21. (a) Describe how 4 children could use 4 pieces of string to show a variety of different rhombuses. (b) Describe how you could show a variety of different rhombuses by threading straws onto string or by fastening sturdy strips of cardboard with brass fasteners. (c) Describe how to fold and cut an ordinary piece of paper so that when you unfold the paper, the resulting shape is necessarily a rhombus. Explain why the resulting shape must be a rhombus. 22. People sometimes get confused about the exact relationship between rectangles and squares. Describe and explain this relationship as clearly, thoroughly, and precisely as you can by referring to the definitions of rectangles and squares. 7

8 23. Draw a detailed Venn diagram showing how the set of rectangles and the set of rhombuses are related. Explain clearly why these two sets are related the way they are. 24. A problem given to 5th graders is shown in Figure 13. Criticize the problem on mathematical grounds and rewrite it to make a correct problem. A problem given to 5th graders: Put the terms squares, rhombuses, and quadrilaterals into the Venn diagram below. rectangles Figure 13: A Problem for Fifth Graders 25. (a) Draw a Venn diagram showing the relationships between the sets of squares, rectangles, and parallelograms. (b) Discuss whether the relationships in part (a) can be determined immediately from the definitions of these shapes or whether additional information that derives from the definitions but is not stated directly in the definitions is necessary. In other words, if a blind person knew only the definitions of the shapes, which relationships would they be able to deduce immediately and which would they not? 26. Suppose that builders pound four stakes into the ground to mark the four corners of a foundation for a house. The builders then measure the distances between opposite stakes and find that these distances are not the same. If the builders proceed to lay the house s foundation without moving the stakes, what can you conclude about this foundation? Relate this question to properties of shapes. 27. The line segments A and AC in Figure 14 have been constructed so that they could be two sides of a rhombus. (a) Use a compass and straightedge to finish constructing a rhombus that has A and AC as two of its sides. (b) y referring to the definition of rhombus, explain why your construction in part (a) must produce a rhombus. 8

9 C A Figure 14: The eginning of a Construction of a Rhombus 28. Suppose you use Geometer s Sketchpad (GSP) to construct two circles with centers A and, in such a way that the circles will always have the same radius, no matter how you move them. Suppose that the two circles meet at points C and D, and suppose that you construct line segments to make a quadrilateral ACD (by connecting connecting A to C, C to, to D, and D to A), as shown in Figure 15. What kind of special quadrilateral must ACD be, no matter how you move the points in your construction (as long as the circles still meet at two points)? Explain your answer clearly and in detail, as if you were explaining to someone who was just learning about the geometric concepts involved. C A D Figure 15: A Construction 29. Assume that the line segments A and AC in Figure 16 have been constructed using Geometer s Sketchpad so that they could be two sides of a rhombus. Explain how to use Geometer s Sketchpad in order to finish the sketch in Figure 16 so that it will always be a rhombus, no matter how points are moved around. In doing so, use only the definition of rhombus (not other properties that rhombuses have). Refer to the following list of Geometer s Sketchpad commands from the construct menu: construct segment, construct perpendicular line, construct parallel line, construct circle by center and point, construct circle by center and radius. 9

10 C A Figure 16: Construct a Rhombus 30. Emily wants to use Geometer s Sketchpad to construct a square in such a way that no matter how she moves the points in her construction, it will always remain a square. So far, Emily has constructed a line segment A and two lines that are perpendicular to A and pass through A and, as shown in Figure 17. Which of the GPS commands listed below should Emily use next to finish her construction? Explain how and why she should use those commands. construct segment construct perpendicular line construct parallel line construct circle by center and point construct circle by center and radius A Figure 17: Emily s Construction 31. (a) Use a compass and straightedge to construct a line that is perpendicular to the line segment A in Figure 18 and that divides the line segment A in half. 10

11 A Figure 18: A Line Segment (b) Show a rhombus that arises naturally from your construction in part (a). Use the definition of rhombus to explain why your shape really is a rhombus. (c) Which properties of rhombuses are related to your construction in part (a)? Explain. 32. (a) Use a compass and straightedge to divide the angle AC in Figure 19 in half. C A Figure 19: isect Angle AC (b) Show a rhombus that arises naturally from your construction in part (a). Use the definition of rhombus to explain why your shape really is a rhombus. (c) Which properties of rhombuses are related to your construction in part (a)? Explain. 33. Using a straightedge and compass, construct a line that is perpendicular to the line segment A in Figure 20 and passes through the point A. Leave the marks showing your construction. A Figure 20: A Line Segment 34. Use a compass and straightedge to construct a 45 degree angle in a careful and precise fashion. 35. Use a compass and straightedge to construct a hexagon for which all six sides have the same length and all six angles are equal. Leave your construction marks to show how you accomplished your construction (don t just show the finished hexagon). 36. Draw two (fairly) precise patterns: one for a triangular prism and one for a pyramid with a triangle base. Indicate which is which. Indicate which sides would be joined together if you were to fold up the patterns to form the shapes. 11

12 a b c Figure 21: A Triangular ase 37. Draw a a pattern for a triangular prism that has the triangle in Figure 21 as a base. On your pattern, label the sides that have length a, the sides that have length b, and the sides that have length c. (You may copy the triangle to other locations on the page.) 38. For each of the patterns in Figure 22, name the shape it would make if it were cut out, folded, and taped to make a closed shape. (Do this without cutting and folding!) In each case, label the base(s) (if any). Determine whether the first shape is oblique or right. precise name of shape: label the base(s) oblique or not? precise name of shape: label the base(s) precise name of shape: label the base(s) Figure 22: Patterns 39. What shapes would the patterns in Figure 23 make if they were cut out along the heavy lines, 12

13 folded along the dotted lines, and taped together to make closed shapes? Name each shape as precisely as you can. precise name of shape: precise name of shape: Figure 23: Patterns precise name of shape: 40. A cube is one of the Platonic solids. Name one other Platonic solid and describe its characteristics. How many faces does it have? What shape are the faces? How many faces come together at a point? 41. A dodecahedron has 3 regular pentagon faces coming together at each corner, but there is no Platonic solid that has 4 or more regular pentagon faces coming together at each corner. Why not? (See Figure Fill in the blanks appropriately: Prism is to cylinder as is to. 43. What is a prism? 44. How are prisms, cylinders, pyramids, and cones related? 13

14 ± 108 Figure 24: A Pentagon 45. Draw a pattern that could be cut out and taped together without overlaps to make a cone without a base (like an ice cream cone). Indicate which sides should be taped together. 46. Draw a pattern for a triangle base pyramid. You may leave the base off your pattern. 47. Which of the patterns in Figure 25 could be cut out and folded up without overlaps to make a closed cube? Which cannot? (Circle does or does not.) Solve this problem by visualizing (do not cut out patterns!). does does not does does not Figure 25: Which Would Make a Cube? 48. Answer the following without using a model. (a) How many faces does a prism with a pentagon base have? What shapes are the faces? Explain briefly. (b) How many edges does a prism with a pentagon base have? Explain. (c) How many corners does a prism with a pentagon base have? Explain. 49. Answer the following without using a model. (a) How many faces does a pyramid with a pentagon base have? What shapes are the faces? Explain briefly. (b) How many edges does a pyramid with a pentagon base have? Explain. (c) How many corners does a pyramid with a pentagon base have? Explain. 14