Using Clicker Items to

i. Deepen Understanding of Measurement Concepts

ii. Foster Desirable Habits of Mind

Logging In Procedure

1. Turn-on your clicker

2. Wait until it says “Enter Student ID”(Enter your 5-digit ID)

3. The screen should display “ANS”

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

Revote 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.

Revote 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:

If the surface area of a sphere is known, then its volume can be determined.

Revote 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

True or False:

If the area of an equilateral triangle is known, then its perimeter can be determined.

Revote 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.

Revote 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 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

Note: Each corner is a right angle.

Consider this two-dimensional figure:

Item 6

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

Note: Each corner is a right angle.

Revote 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

Note: Each corner is a right angle.

4 cm

10 cm

7 cm

HoM: Reasoning with Change and Invariance

Consider this two-dimensional figure:

Item 7

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

Note: The two horizontal lines are parallel.

Revote 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

Note: The two horizontal lines are parallel.

Consider this two-dimensional figure:

HoM: Reasoning with Change and Invariance

4 m4 m4 m4 m 4 m

Note: The two horizontal lines are parallel.