Problemson

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 < qc. p = qd.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 < qc. p = qd.None of the above

Fact: 1 km 0.62 mile; 1 mile = 5280 feet

HoM: Explore and generalize a pattern

p q1 3273.6

2 6547.2

10 32736

Procedure: 1 km 0.62 x 5280 feet = 3273.6 feet

Concept: Conservation (recognizing smaller units will produce larger counts)

p q1 3273.6

2 6547.2

10 32736

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 This misunderstanding appears to come from an incorrect over-come from an incorrect over-generalization of the very special generalization of the very special relationship that exists for a cube.”relationship that exists for a cube.”

(NCTM, 2000, p. 242)(NCTM, 2000, p. 242)

““[S]ome students may hold the [S]ome students may hold the misconception that if the volume of misconception that if the volume of a three-dimensional shape is a three-dimensional shape is known, then its surface area can be known, then its surface area can be determined. 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

HL

L

= ½L [L2 – (L/2)2]

0.5

= ½L (0.75L2)0.5

= ½L (0.75)0.5 L

0.433L2

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 m12 m

2 m16 m

1 m

20 m0.5 m

HoM: Reasoning with change and invariance

Concept:Perimeter = 2L + 2W ; Area = LW

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

(Driscoll, 2007, p. 83)(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 m4 m4 m4 m 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.

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 > 365b. x < 365c. x = 365d. 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 > 365b. x < 365c. x = 365d. 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 What HoM Have We Learned?Learned? Reasoning with Change and InvarianceReasoning with Change and Invariance

Reasoning with FormulasReasoning with Formulas

Reasoning with RelationshipsReasoning with Relationships

Seeking counter-exampleSeeking counter-example

Identifying quantities & relationshipsIdentifying quantities & relationships

Attending to meaningAttending to meaning

Assigning a value to an unknown and exploreAssigning a value to an unknown and explore