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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.
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