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Problems on Measurement Concepts

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

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Suppose p kilometers is equal to q feet, where p and q are positive numbers. Which statement is correct? a. p > q b. p < q c. p = q d.None of the above Fact:1 km 0.62 mile; 1 mile = 5280 feet HoM:Explore and generalize a pattern pq 13273.6 26547.2 1032736 Procedure:1 km 0.62 x 5280 feet = 3273.6 feet

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Concept:Conservation (recognizing smaller units will produce larger counts) pq 13273.6 26547.2 1032736 HoM:Explore and generalize a pattern

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

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

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

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

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

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This misunderstanding appears to come from an incorrect over-generalization of the very special relationship that exists for a cube.” (NCTM, 2000, p. 242) “[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.

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

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True or False: HoM: Reasoning with Formulas Concept:A = 4 r 2 V = 4/3 r 3 If the surface area of a sphere is known, then its volume can be determined.

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

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

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

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

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

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

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“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)

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

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Item 6 Consider this two-dimensional figure: 4 cm 10 cm 7 cm Which measurement can be determined? (A) Area only (B) Perimeter only (C) Both area and perimeter (D) Neither area nor perimeter

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

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Item 7 Consider this two-dimensional figure: Which measurement can be determined? (A) Area only (B) Perimeter only (C) Both area and perimeter (D) Neither area nor perimeter 4 m 10 m 3 m

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

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Item 8 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.

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HoM: Reasoning with Change and Invariance Concept:Area of Tria. = ½LW = ½ Area of Rect.

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Can you prove it using diagrams? 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.

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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.

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Item 9 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.

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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

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

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

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

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

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Suppose 365 raisins weighs x pounds. Which statement is correct? a. x > 365 b. x < 365 c. x = 365 d.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)

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

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