Activity III: Surface Area of a Leaf (Grades 7-9)
Objectives:
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Complete a table of values.
Graph the values given in a table.
Create an equation representing the information in a table or graph.
NCTM Standards
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Standard
Standard
Standard
Standard
Standard
Standard
Standard
Standard
1
2
3
4
5
6
8
9
Problem Solving
Communication
Reasoning
Connections
Number and Number Relationships
Functions
Patterns and Functions
Algebra
Note to teachers:
Middle level: This activity can be modified for upper level classes by having the
class do a study of the finite differences of a quadratic formula.
Quadratic Growth
Through the process called photosynthesis plants absorb light through their leaves
and use it to split water molecules into hydrogen and oxygen molecules. The
oxygen is released into the atmosphere and the hydrogen is combined with carbon
dioxide from the atmosphere to create sugar to feed the plant.
Reprinted with permission from Mathematics in Context program, 2000
Encyclopædia Britannica Educational Corporation.
It is clear that the plant's ability to create food is dependent on the surface area of
its leaves. To determine the surface area of a leaf shine a light vertically at a leaf
held horizontally, trace and measure the shadow by subdividing it into geometric
figures. To determine a geometric model that might be similar and enable one to
approximate the surface area draw a square around it or its shadow.
Reprinted with permission form Mathematics in Context program, 2000
Encyclopædia Britannica Educational Corporation.=
Notice that the kite shaped model covers about the same proportion of the square
as does the leaf.
1. Determine what portion of the square is covered by the leaf. Explain how you
made your determination.
2. If you know the height, h, of such a leaf you would be able to determine its
surface area (A). Explain how you would determine a formula that could be
used to find the surface area of a black poplar leaf.
3. In the figure the leaf is symmetric. Draw a picture of a non-symmetric leaf
for which the formula will continue to work.
4. Use the formula from problem 2 to create a table of values. In your table,
include the values for the heights and areas of the black poplar leaf. Plot the
values on a graph.
5. Using the graph, determine the surface area of leaves having heights 4.5 cm,
8.3 cm and 11.5 cm; then check you results using the formula.
.
Student Activity Page
Activity III Quadratic Growth
Quadratic Growth
Through the process called photosynthesis plants absorb light through their leaves and
use it to decompose water into hydrogen and oxygen. The oxygen is released into the
atmosphere and the hydrogen is combined with carbon dioxide from the atmosphere to
create sugar to feed the plant.
Reprinted with permission form Mathematics in Context program 2000 Encyclopædia Britannica
Educational Corporation.
It is clear that the plants ability to create food is dependent on the surface area of its
leaves. To determine the surface area of a leaf shine a light vertically at a leaf held
horizontally, trace and measure the shadow by subdividing it into geometric figures. To
determine a geometric model that might be similar and enable one to approximate the
surface area draw a square around it or its shadow.
1
Reprinted with permission form Mathematics in Context program 2000 Encyclopædia Britannica
Educational Corporation.
Notice that the kite shaped model covers about the same proportion of the square as does
the leaf.
1. Determine what portion of the square is covered by the leaf. Explain how you made
your determination.
2. If you know the height (h) of such a leaf you would be able to determine its surface
area (A). Explain how you would determine a formula that could be used to
determine the surface area of a black poplar leaf.
3. In the figure the leaf is symmetric. Draw a picture of a non-symmetric leaf for which
the formula will continue to work.
4. Use the formula created in problem 2 and make a table of the heights and areas of the
black poplar leaf then plot its graph.
5. Using the graph determine the surface area of a leaves having heights 4.5 cm, 8.3 cm
and 11.5 cm the check you results using the formula.
2
Activity III Quadratic Growth Solutions
1. The leaf covers about half of the square.
o In the kite shaped model, fold the white part of the figure onto the shaded part and it
will cover it completely.
2. The formula is A = 1/2 (h2)
o The length of the side of the square is h and area is h2 and since the area of the leaf is
half the square the area formula is 1/2 (h2).
3.
4.
Height
(in cm.)
1
2
3
4
5
6
7
8
Area
(in cm2)
0.5
2
4.5
8
12.5
18
24.5
32
Leaf Surface Area
80
70
60
50
Series1
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10
11
12
Height (cm)
5.
Height
4.5
8.3
11
Area from Graph
10.1
31
62
Area from Calculation
10.12
34.45
60.2