# Problems on Measurement Concepts. Suppose p kilometers is equal to q feet, where p and q are positive numbers. Which statement is correct? a. p > q b.

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Problems on Measurement Concepts

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

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

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

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

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

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)

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

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

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.

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

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.

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

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

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

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

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

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

“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)

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

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

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

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

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

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.

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

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.

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.

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.

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

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

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

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

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.

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)

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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