Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.1
TASK 3.2.1: APPLICATION — NETS AND CHANGING HEIGHT
Solutions
•
•
Read through the directions before you begin.
Make sure your group has all the materials for this exploration.
o Ruler
o Grid paper
o Tape or Glue
o Color pencils
o Scissors
o Markers
o Chart paper
o Graphing calculators
Each group will be working with the original rectangular prism that has:
height of 1 unit,
depth of 2 units,
width of 3 units.
1 unit
2 units
3 units
Group 1 –
o Create a net for the original figure. Label this net A.
o Increase the height of the original prism by 2 units and create a net for the figure.
Label this net B.
o Increase the height of the original prism by 4 units and create a net for the figure.
Label this net C.
o Increase the height of the original prism by 6 units and create a net for the figure.
Label this net D.
o Increase the height of the original prism by 8 units and create a net for the figure.
Label this net E.
Create a table that shows:
o The dimensions of each prism
o The process for determining the surface area of each prism
o The total surface area of each prism
Height
1
3
5
7
9
11
Process
2[(1*2) + (1*3) + (2*3)]
2[(3*2) + (3*3) + (2*3)]
2[(5*2) + (5*3) +(2*3)]
2[(7*2) + (7*3) + (2*3)]
2[(9*2) + (9*3) + (2*3)]
2[(11*2) + (11*3) + (2*3)]
Surface Area (square units)
22
42
62
82
102
122
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
1
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.1
2
Create a graph of the surface area vs. height of the prism.
Surface Area
Changing Height
1. Describe the functional relationship.
The surface area depends on the height of the prism.
s = 2[(h + 2)d + (h + 2)w + (w*d)], where s is surface area, h is the height, w is
the width, and d is the depth of the rectangular prism.
2. Describe a reasonable range and domain for the situation and the range and
domain for the function that models this situation.
A reasonable domain for this situation is 1 unit to 11 units of height. This domain
matches our data and is about the limit for creating a net on one sheet of grid
paper. Since the range depends on the domain, the domain for this situation would
be 22 to 122 square units. The domain and range for the functional model of this
situation is negative infinity to positive infinity.
3. Explain how the surface area of the prism is changing. Use table values and the
graph to support your explanation. Use a color pencil to color and model the
growth on your nets.
The data table shows a constant change of 10 square units per 1 unit change in
height. On the graph, a horizontal movement of 2 requires a vertical movement of
20, resulting in a rate of change of 10 square units per each unit increase in the
height. Check participant’s nets to observe their recording of the changes in the
surface area.
4. Record all your information on chart paper.
Group 2 –
o Create a net for the original figure. Label this net A.
o Increase both the height and depth of the original prism by 2 units and create a net
for the figure. Label this net B.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.1
3
o Increase both the height and depth of the original prism by 4 units and create a net
for the figure. Label this net C.
o Increase both the height and depth of the original prism by 6 units and create a net
for the figure. Label this net D.
o Increase both the height and depth of the original prism by 8 units and create a net
for the figure. Label this net E.
Create a table that shows:
o The dimensions of each prism
o The process for determining the surface area of each prism
o The total surface area of each prism
Height
Process
Surface Area
(square units)
1
3
5
7
9
11
2[(1*2) + (1*3) + (2*3)]
2[(3*4) + (3*3) + (4*3)]
2[(5*6) + (5*3) +(6*3)]
2[(7*8) + (7*3) + (8*3)]
2[(9*10) + (9*3) + (10*3)]
2[(11*12) + (11*3) + (12*3)]
22
66
126
202
294
402
Create a graph of the surface area vs. height of the prism.
Surface Area
Changing Height and Depth
1. Describe the functional relationship.
The surface area depends on the height and depth.
2. Describe a reasonable range and domain for the situation and the range and domain
for the function that models this situation.
The domain for this situation is the same as the previous, 1 to 11 units of height,
however the range is now 22 to 402 square units. The domain for the function that
models this situation is negative to positive infinity; the range is 0 to positive infinity.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.1
4
3. Explain how the surface area of the prism is changing. Use the table values to
support your explanation. Use colored pencils to color and model the growth on your
nets.
Applying finite differences, the change in the table is not constant the first time;
however, it is constant the second time finite differences are checked, thus the data
represent a quadratic relationship. On the graph from the first coordinate point (1, 22)
to the next coordinate point (3, 66) there is a horizontal change of 2 and a vertical
change of 44. From point (3, 66) to point (5, 126) there is a horizontal change of 2
but a vertical change of 60. Since the change between the points is not constant, this
is not linear graph.
4. Record all your information on chart paper.
Group 3 –
o Create a net for the original figure. Label this net A.
o Increase the height, depth, and width of the original prism by 3 units and create a
net for the figure. Label this net B.
o Increase the height, depth, and width of the original prism by 5 units and create a
net for the figure. Label this net C.
o Increase the height, depth, and width of the original prism by 7 units and create a
net for the figure. Label this net D.
o Increase the height, depth, and width of the original prism by 9 units and create a
net for the figure. Label this net E.
Create a table that shows:
o The dimensions of each prism
o The process for determining the surface area of each prism
o The total surface area of each prism
Height
1
3
5
7
9
11
Process
2[(1*2) + (1*3) + (2*3)]
2[(3*4) + (3*5) + (4*5)]
2[(5*6) + (5*7) +(6*7)]
2[(7*8) + (7*9) + (8*9)]
2[(9*10) + (9*11) + (10*11)]
2[(11*12) + (11*13) + (12*13)]
Surface Area
(square units)
22
66
126
202
294
402
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.1
Create a graph of the surface area vs. height of the prism.
Surface Area
Changing Height and Depth
1. Describe the functional relationship.
The surface area depends on the height, depth, and width.
2. Describe a reasonable range and domain for the situation and a reasonable range
and domain for the function.
The domain for this data is the same as the original domain. The range is 22 to
402 square units. The domain for the function that models this data is negative
infinity to positive infinite. The range for this function is 0 to positive infinity.
3. Explain how the surface area of the prism is changing. Use table values and the
graph to support your explanation. Use colored pencils to color and model the
growth on your models.
Applying finite differences, the change in the table is not constant the first time;
however, it is constant the second time finite differences are checked, thus the
data represent a quadratic relationship. On the graph from the first coordinate
point (1, 22) to the next coordinate point (3, 94) there is a horizontal change of 2
and a vertical change of 72. From point (3, 94) to point (5, 214) there is a
horizontal change of 2 but a vertical change of 120. Since the between the points
on the graph are not constant, this is not a graph of linear data.
4. Record all your information on chart paper.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
5
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.1
6
TASK 3.2.1: APPLICATION — NETS AND CHANGING HEIGHT
•
•
Read through the directions before you begin.
Make sure your group has all the materials for this exploration
o Ruler
o Grid paper
o Tape or Glue
o Scissors
o Markers
o Chart paper
o Graphing calculators
Each group will be working with the original rectangular prism that has:
height of 1 unit,
depth of 2 units,
width of 3 units.
1 unit
2 units
3 units
Group 1 –
o Create a net for the original figure. Label this net A.
o Increase the height of the original prism by 2 units and create a net for the figure.
Label this net B.
o Increase the height of the original prism by 4 units and create a net for the figure.
Label this net C.
o Increase the height of the original prism by 6 units and create a net for the figure.
Label this net D.
o Increase the height of the original prism by 8 units and create a net for the figure.
Label this net E.
Create a table that shows:
o The dimensions of each prism
o The process for determining the surface area of each prism
o The total surface area of each prism
Create a graph of the surface area vs. height of the prism.
1. Describe the functional relationship.
2. Describe a reasonable range and domain for the situation and a range and domain
for the function that models this situation.
3. Explain how the surface area of the prism is changing. Use table values and the
graph to support your explanation. Use colored pencils to color and model the
growth on your nets.
4. Record all your information on chart paper.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.1
7
Group 2 –
o Create a net for the original figure. Label this net A.
o Increase both the height and depth of the original prism by 2 units and create a net
for the figure. Label this net B.
o Increase both the height and depth of the original prism by 4 units and create a net
for the figure. Label this net C.
o Increase both the height and depth of the original prism by 6 units and create a net
for the figure. Label this net D.
o Increase both the height and depth of the original prism by 8 units and create a net
for the figure. Label this net E.
Create a table that shows:
o The dimensions of each prism
o The process for determining the surface area of each prism
o The total surface area of each prism
Create a graph of the surface area vs. height of the prism.
1. Describe the functional relationship.
2. Describe a reasonable range and domain for the situation and a reasonable range
and domain for the function.
3. Explain how the surface area of the prism is changing. Use table values and the
graph to support your explanation. Use colored pencils to color and model the
growth on your models.
4. Record all your information on chart paper.
Group 3 –
o Create a net for the original figure. Label this net A.
o Increase the height, depth, and width of the original prism by 3 units and create a
net for the figure. Label this net B.
o Increase the height, depth, and width of the original prism by 5 units and create a
net for the figure. Label this net C.
o Increase the height, depth, and width of the original prism by 7 units and create a
net for the figure. Label this net D.
o Increase the height, depth, and width of the original prism by 9 units and create a
net for the figure. Label this net E.
Create a table that shows:
o The dimensions of each prism
o The process for determining the surface area of each prism
o The total surface area of each prism
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 2. Quadratics and Solutions;
Task 3.2.1
Create a graph of the surface area vs. height of the prism.
1. Describe the functional relationship.
2. Describe a reasonable range and domain for the situation and a reasonable range
and domain for the function.
3. Explain how the surface area of the prism is changing. Use table values and the
graph to support your explanation. Use colored pencils to color and model the
growth on your models.
4. Record all your information on chart paper.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
8