Complete a table of values. Graph the values given in a table. Create an equation representing the information in a table or graph.

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1 Activity III: Surface Area of a Leaf (Grades 7-9) Objectives: 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 Standard 1 Problem Solving Standard 2 Communication Standard 3 Reasoning Standard 4 Connections Standard 5 Number and Number Relationships Standard 6 Functions Standard 8 Patterns and Functions Standard 9 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.

2 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.

3 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..

4 Student Activity Page

5 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

6 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

7 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 (h 2 ) o The length of the side of the square is h and area is h 2 and since the area of the leaf is half the square the area formula is 1/2 (h 2 ) Height (in cm.) Area (in cm 2 )

8 Leaf Surface Area Series Height (cm) 5. Height Area from Graph Area from Calculation

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