Problems

1
Problems on Measurement Concepts
2

• Suppose p kilometers is equal to q feet, where p
  and q are positive numbers.
• Which statement is correct?
• p gt q
• p lt q
• p q
• None of the above

Item 1
3

• Suppose p kilometers is equal to q feet, where p
  and q are positive numbers.
• Which statement is correct?
• p gt q
• p lt q
• p q
• None of the above

p q
1 3273.6




2 6547.2




10 32736
Fact 1 km ? 0.62 mile 1 mile 5280 feet
Procedure 1 km ? 0.62 x 5280 feet 3273.6 feet
HoM Explore and generalize a pattern
4
Concept Conservation (recognizing smaller units
will produce larger counts)
p q
1 3273.6




2 6547.2




10 32736
HoM Explore and generalize a pattern
5
Concept Conservation (recognizing smaller units
will produce larger counts)
6
Concept Conservation (recognizing smaller units
will produce larger counts)
Concept Measurement involves iterating a unit
7
Concept Conservation (recognizing smaller units
will produce larger counts)
Concept Measurement involves iterating a unit
Concept Units must be consistent
Concept Inverse relationship between the size
of a unit and the numerical count
8
True or False
If the volume of a rectangular prism is known,
then its surface area can be determined.
Item 2
9
True or False
If the volume of a rectangular prism is known,
then its surface area can be determined.
Concept Volume Length ? Width ? Height
HoM Reasoning with Change and Invariance
10
Some students may hold the misconception that
if the volume of a three-dimensional shape is
known, then its surface area can be determined.
This misunderstanding appears to come from an
incorrect over-generalization of the very special
relationship that exists for a cube. (NCTM,
2000, p. 242)
11
True or False
If the surface area of a sphere is known, then
its volume can be determined.
Item 3
12
True or False
If the surface area of a sphere is known, then
its volume can be determined.
Concept A 4 ?r 2 V 4/3 ?r 3
HoM Reasoning with Formulas
13
True or False
If the area of an equilateral triangle is known,
then its perimeter can be determined.
Item 4
14
True or False
If the area of an equilateral triangle is known,
then its perimeter can be determined.
CU Area ½LH
½L ? L2 (L/2)2 0.5 ½L ? (0.75L2)0.5 ½L
? (0.75)0.5 L ? 0.433L2
L
L
L
HoM Reasoning with Relationships
15
True or False
As we increase the perimeter of a rectangle, the
area increases.
Item 5
16
True or False
As we increase the perimeter of a rectangle, the
area increases.
HoM Seeking causality
17
True or False
As we increase the perimeter of a rectangle, the
area increases.
4 m
8 m
Concept Perimeter 2L 2W Area LW
HoM Seeking counter-example
18
True or False
As we increase the perimeter of a rectangle, the
area increases.
4 m
8 m
Concept Perimeter 2L 2W Area LW
HoM Reasoning with change and invariance
19
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)
20
Consider this two-dimensional figure
21
Consider this two-dimensional figure

• Which measurement can be determined?
• Area only
• Perimeter only
• Both area and perimeter
• Neither area nor perimeter

Item 6
22
HoM Reasoning with Change and Invariance
23
Consider this two-dimensional figure

• Which measurement can be determined?
• Area only
• Perimeter only
• Both area and perimeter
• Neither area nor perimeter

Item 7
24
Consider this two-dimensional figure
4 m
HoM Reasoning with Change and Invariance
25
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.
Item 8
26
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.
HoM Reasoning with Change and Invariance
27
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.
Can you prove it using diagrams?
28
Consider a triangle inside a rectangle where one
of the triangles 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.
29
Consider a triangle inside a rectangle where one
of the triangles 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.
Item 9
30
Consider a triangle inside a rectangle where one
of the triangles 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
31
It takes approximately 720 small cubes (1cm on
each edge) to fit a prism.
Approximately how many big cubes (2cm on each
edge) would fit the prism?
Big Cube
Small Cube
32
It takes approximately 720 small cubes (1cm on
each edge) to fit a prism.
Approximately how many big cubes (2cm on each
edge) would fit the prism?
Big Cube
Small Cube

1. 80
2. 90
3. 180
4. 360
5. 1440

Item 10
33
It takes approximately 720 small cubes (1cm on
each edge) to fit a prism.
Approximately how many big cubes (2cm on each
edge) would fit the prism?
Big Cube
Small Cube

1. 80
2. 90
3. 180
4. 360
5. 1440

HoM Identifying quantities relationships
34

• Suppose 365 raisins weighs x pounds.
• Which statement is correct?
• x gt 365
• x lt 365
• x 365
• None of the above because it depends on the
  weight of each raisin.

Item 11
35

• Suppose 365 raisins weighs x pounds.
• Which statement is correct?
• x gt 365
• x lt 365
• x 365
• 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)
36
What HoM Have We Learned?

• Reasoning with Change and Invariance
• Reasoning with Formulas
• Reasoning with Relationships
• Seeking counter-example
• Identifying quantities relationships
• Attending to meaning
• Assigning a value to an unknown and explore