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Problems on Measurement Concepts

This guide explores essential concepts in measurement, focusing on the relationships between volume, area, and perimeter in geometric shapes. It discusses the correct interpretations of converting between kilometers and feet, the implications of knowing volume versus surface area, and general misconceptions students hold about these relationships. Through various true/false questions and illustrative examples, the guide encourages critical thinking and reasoning about how changing dimensions affect measurements. Ideal for educators and students alike, fostering a deeper understanding of measurement concepts.

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Problems on Measurement Concepts

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  1. Problems on Measurement Concepts

  2. Suppose p kilometers is equal to q feet, where p and q are positive numbers. • Which statement is correct? • p > q • p <q • p = q • None of the above Item 1

  3. Suppose p kilometers is equal to q feet, where p and q are positive numbers. • Which statement is correct? • p > q • p <q • p = q • None of the above Fact: 1 km  0.62 mile; 1 mile = 5280 feet Procedure: 1 km  0.62 x 5280 feet = 3273.6 feet HoM: Explore and generalize a pattern

  4. Concept: Conservation (recognizing smaller units will produce larger counts) HoM: Explore and generalize a pattern

  5. Concept: Conservation (recognizing smaller units will produce larger counts) 1 wav ? wavs ? 1 arro ? arros

  6. Concept: Conservation (recognizing smaller units will produce larger counts) 1 wav  3.7 wavs 1 arro  7 arros Concept: Measurement involves iterating a unit

  7. Concept: Conservation (recognizing smaller units will produce larger counts) 1 wav  3.7 wavs 1 arro  9.6 arros Concept: Measurement involves iterating a unit Concept: Units must be consistent Concept: Inverse relationship between the size of a unit and the numerical count

  8. True or False: If the volume of a rectangular prism is known, then its surface area can be determined. Item 2

  9. True or False: If the volume of a rectangular prism is known, then its surface area can be determined. Concept: Volume = Length  Width  Height HoM: Reasoning with Change and Invariance

  10. “[S]ome students may hold the misconception that if the volume of a three-dimensional shape is known, then its surface area can be determined. This misunderstanding appears to come from an incorrect over-generalization of the very special relationship that exists for a cube.” (NCTM, 2000, p. 242)

  11. True or False: If the surface area of a sphere is known, then its volume can be determined. Item 3

  12. True or False: If the surface area of a sphere is known, then its volume can be determined. Concept: A = 4 r2 V = 4/3 r3 HoM: Reasoning with Formulas

  13. True or False: If the area of an equilateral triangle is known, then its perimeter can be determined. Item 4

  14. True or False: If the area of an equilateral triangle is known, then its perimeter can be determined. CU: Area = ½LH = ½L [L2 – (L/2)2] 0.5 = ½L (0.75L2)0.5 = ½L (0.75)0.5 L  0.433L2 L L H L/2 L HoM: Reasoning with Relationships

  15. True or False: As we increase the perimeter of a rectangle, the area increases. Item 5

  16. True or False: As we increase the perimeter of a rectangle, the area increases. HoM: Seeking causality

  17. True or False: As we increase the perimeter of a rectangle, the area increases. 16 m 2 m 4 m 8 m Concept: Perimeter = 2L + 2W ;Area = LW HoM: Seeking counter-example

  18. True or False: As we increase the perimeter of a rectangle, the area increases. 20 m 0.5 m 1 m 2 m 16 m 4 m 12 m 8 m Concept: Perimeter = 2L + 2W ;Area = LW HoM: Reasoning with change and invariance

  19. “While mixing up the terms for area and perimeter does not necessarily indicate a deeper conceptual confusion, it is common for middle-grades students to believe there is a direct relationship between the area and the perimeter of shapes and this belief is more difficult to change. In fact, increasing the perimeter of a shape can lead to a shape with a larger area, smaller are, or the same area.” (Driscoll, 2007, p. 83)

  20. Consider this two-dimensional figure: 4 cm 7 cm 10 cm

  21. Consider this two-dimensional figure: • Which measurement can be determined? • Area only • Perimeter only • Both area and perimeter • Neither area nor perimeter 4 cm 7 cm 10 cm Item 6

  22. 4 cm 7 cm 10 cm HoM: Reasoning with Change and Invariance

  23. Consider this two-dimensional figure: 4 m 3 m 10 m • Which measurement can be determined? • Area only • Perimeter only • Both area and perimeter • Neither area nor perimeter Item 7

  24. Consider this two-dimensional figure: 4 m 4 m 4 m 4 m 4 m HoM: Reasoning with Change and Invariance

  25. True or False: The area of the triangle is always ½ times the area of the rectangle as long as they share the same base, and the third vertex of the triangle lies on the opposite side of the rectangle. Item 8

  26. True or False: The area of the triangle is always ½ times the area of the rectangle as long as they share the same base, and the third vertex of the triangle lies on the opposite side of the rectangle. Concept: Area of Tria. = ½LW = ½ Area of Rect. HoM: Reasoning with Change and Invariance

  27. True or False: The area of the triangle is always ½ times the area of the rectangle as long as they share the same base, and the third vertex of the triangle lies on the opposite side of the rectangle. Concept: Area of Tria. = ½LW = ½ Area of Rect. Can you prove it using diagrams?

  28. Consider a triangle inside a rectangle where one of the triangle’s vertices lie on a vertex of a rectangle and the other two vertices of the triangle lie on the other two sides of the rectangle.

  29. Consider a triangle inside a rectangle where one of the triangle’s vertices lie on a vertex of a rectangle and the other two vertices of the triangle lie on the other two sides of the rectangle. True or False: The area of the triangle is always ½ times the area of the rectangle. Item 9

  30. Consider a triangle inside a rectangle where one of the triangle’s vertices lie on a vertex of a rectangle and the other two vertices of the triangle lie on the other two sides of the rectangle. The answer is false. HoM: Reasoning with Change and Invariance

  31. It takes approximately 720 small cubes (1cm on each edge) to fit a prism. Approximately how many big cubes (2cm on each edge) would fit the prism? Big Cube Small Cube Prism

  32. It takes approximately 720 small cubes (1cm on each edge) to fit a prism. Approximately how many big cubes (2cm on each edge) would fit the prism? Big Cube Small Cube • 80 • 90 • 180 • 360 • 1440 Prism Item 10

  33. It takes approximately 720 small cubes (1cm on each edge) to fit a prism. Approximately how many big cubes (2cm on each edge) would fit the prism? Big Cube Small Cube • 80 • 90 • 180 • 360 • 1440 Prism HoM: Identifying quantities & relationships

  34. Suppose 365 raisins weighs x pounds. • Which statement is correct? • x > 365 • x < 365 • x = 365 • None of the above because it depends on the weight of each raisin. Item 11

  35. Suppose 365 raisins weighs x pounds. • Which statement is correct? • x > 365 • x < 365 • x = 365 • None of the above because it depends on the weight of each raisin. HoM: Attending to meaning (e.g., benchmark for 1 pound) HoM: Assigning a value to an unknown and explore(e.g., if x = 365 pounds, then 365 raisins = 365 pounds)

  36. What HoM Have We Learned? • Reasoning with Change and Invariance • Reasoning with Formulas • Reasoning with Relationships • Seeking counter-example • Identifying quantities & relationships • Attending to meaning • Assigning a value to an unknown and explore

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