Changing Dimensions to Create Different Areas Teacher Materials Lesson 1 – Plan a Garden • Have students complete Review 1. • This could be done as a warm-up. • Go over answers and have students write the formulas for finding the area and perimeter of rectangles. Area = length x width Perimeter = (length + width) x 2 •
Ask students to consider the following questions: • I’m thinking of a rectangle that has a perimeter of 12 units. Does anyone know the dimensions of the rectangle I am thinking of? • Do you know for sure? Is there only one possibility? • Do you think all of the possible rectangles have the same area?
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As a class, complete a table for all rectangles that have a perimeter of 12 units. Length
Width
Perimeter
Area
1 units
5 units
12 units
5 sq. units
2 units
4 units
12 units
8 sq. units
3 units
3 units
12 units
9 sq. units
4 units
2 units
12 units
8 sq. units
5 units
1 units
12 units
5 sq. units
• Give students tiles or paper squares to manipulate if they need help
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finding possible rectangles with a perimeter of 12 units. • Discuss any patterns in the table. Possible patterns: • Length increases as width decreases. • The width and length always add to 6, half of the perimeter. • Return to the questions asked earlier. • Do all rectangles with a perimeter of 12 units have the same area? Don’t worry if students don’t notice every pattern. You will come back to these at the end of the lesson. Introduce the day’s situation. • Read the situation aloud or have a student read it aloud. • Have a student restate what the problem is asking.
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• Instruct students to work together in pairs to find all the possible rectangular garden dimensions.
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Have students work in pairs to begin Student Activity Sheet 1. • Allow the students to use tiles or paper squares if they are having trouble. Give students time to explore the questions. • As students are working ask questions that prompt them to discover understandings. • See teacher notes. Discuss activity as a whole class. • What garden dimensions gave the greatest area? • What did it look like? • What garden dimensions gave the least area? • What did it look like? • What did you notice about the maximum and minimum areas when the perimeter was 18 yards? • If students are picking up on how to minimize or maximize the area bring the table from the beginning of the lesson with the perimeter of 12 units. • To secure the understanding of how to maximize or minimize the area with constant perimeter use a piece of string with the two ends tied together to demonstrate. • Start by holding it in a square to show the large amount of area inside. Move it slowly to a longer and skinnier rectangle to demonstrate the “loss’ of area. • Place the string on grid paper and have students count squares to determine the area. Summarize new understandings. Ask the following questions to prompt students. • What happens to the width as the length increases? • How do you maximize the area of a rectangle with a constant perimeter? • How do you minimize the area of a rectangle with a constant perimeter?
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Name __________________________ Class __________________________ Date __________________________ Lesson 1 – Plan a Garden– ANSWER KEY You want to build a rectangular garden in your back yard. You need a fence to go around the garden to keep the deer from eating all your vegetables. The community you live in has a community code that only allows 24 yards of fencing to be used to fence in a garden. The local hardware store sells units of fencing in 1-yard lengths. 1.
Make a sketch of all the possible rectangular gardens you could build with 24 yards of fencing.
(TC-1)
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2.
Using your sketches, record all the possible dimensions your garden could have along with their perimeter and area.
(TC-2) Length (yards)
Width (yards)
Perimeter (yards)
Area (sq. yards)
1 yards
11 yards
24 yards
11 sq. yards
2 yards
10 yards
24 yards
20 sq. yards
3 yards
9 yards
24 yards
27 sq. yards
4 yards
8 yards
24 yards
32 sq. yards
5 yards
7 yards
24 yards
35 sq. yards
6 yards
6 yards
24 yards
36 sq. yards
7 yards
5 yards
24 yards
35 sq. yards
8 yards
4 yards
24 yards
32 sq. yards
9 yards
3 yards
24 yards
27 sq. yards
10 yards
2 yards
24 yards
20 sq. yards
11 yards
1 yards
24 yards
11 sq. yards
3. What rectangular garden gives you the least, or minimum, area to grow vegetables? 11 yards by 1 yard - or – 1 yard by 11 yards________________________________ Describe its shape. Long and skinny. _____________________________________________________ 4. What rectangular garden gives you the greatest, or maximum, area to grow vegetables? 6 yards by 6 yards______________________________________________________ Describe its shape. It’s a square.__________________________________________________________
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5.
Suppose the community code has changed. You can now only use 18 yards of fencing to fence in your garden! Use what you discovered when using 24 yards of fencing to determine the new maximum and minimum areas with 18 yards of fencing.
(TC-3) What would the dimensions of the garden with the maximum area be? Length: _______5 yards _______________
Width: ______4 yards _______
How did you determine those dimensions? I found the dimensions that would make the rectangle closest to a square._______ Make a sketch of that rectangle and determine the area.
Area = length x width = 5 yards x 4 yards = 20 square yards
4 yards
5 yards 6.
What would the dimensions of the garden with the minimum area be?
Length: ________8 yards ____________
Width: _____1 yard_______
How did you determine those dimensions? I found the dimensions that would make the rectangle as long and skinny as possible. Make a sketch of that rectangle and determine the area.
8 yards
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7.
New Understandings: •
When the perimeter is held constant:
• The width decreases as the length increases. OR The length decreases as the width increases.
• To maximize the area, the rectangle needs to be as close to a square as possible.
• To minimize the area, the rectangle needs to be as long and skinny as possible.
Teacher Notes TC-1 Some students may put both orientations of rectangles. It is okay if they do. If you see students doing this ask them if the rectangles are different. TC-2 There is enough room for students to include information for both orientations of the rectangles. Some students may be worried that they have not completely filled their table. Ask them if they have a strategy to make sure they have all the possibilities. If they do tell them not to worry! TC-3 With a constant perimeter of 18, a perfect square cannot be made will whole number sides. Some students will struggle with this. Either • Provide additional graph paper so students can sketch out the possibilities. Ask them if there is one more like a square than the others, or • Challenge students to find the non-whole number dimensions.
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Lesson 2 - Landscaping •
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Have students complete Review 2. • This could be done as a warm-up. • Go over answers and have students write the formulas for finding the area and perimeter of triangles. Area = (base x height) / 2 Perimeter = add all three sides Have students restate what they learned from the previous day’s lesson. • Constant perimeter does not mean constant area. • To maximize the area of a rectangle with a constant perimeter, make the rectangle most like a square. • To minimize the area of a rectangle with a constant perimeter, make the rectangle as long and skinny as possible. Introduce the day’s situation. • Read the situation aloud or have a student read it aloud. • Have a student restate what the problem is asking. • Instruct students to work together in pairs to adjust the estimates, make sketches and then find the area and perimeter. Have students begin Student Activity Sheet 2. • Make sure students are making sketches of each set. Give students time to explore the questions. • As students are working ask questions that prompt them to discover understandings. Discuss activity as a whole class. • Go over the answers for the flower beds. • Be sure to ask how did the area change when one of the dimensions was doubled? How about when one of the dimensions was halved? Did perimeter change in the same way? • Ask students to predict what would happen to the area if a dimension was tripled or quadrupled. Go through a few examples together as a class. • Ask students if the same prediction is true for perimeter. Be sure students understand that perimeter does not behave the same as area because not all the lengths change. Summarize new understandings. • What happens if I increase one of the dimensions of the rectangle or triangle? • What happens if I decrease one of the dimensions of the rectangle or triangle? Parts 6, 7, and 8 (the shed, patio and decorative mosaic) could be given as an extension for students needing a challenge.
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Name __________________________ Class __________________________ Date __________________________ Lesson 2 – Landscaping – ANSWER KEY Now that you have your garden figured out, you want to design some flower boxes and other landscaping projects around your yard. You made some estimates but after measuring some things around the yard you realized you need to make some changes. On the centimeter graph paper, draw your original plan in one color. Then draw the new plan in another. Use the pictures to help you answer the questions. (scale: 1 centimeter = 1 foot) 1.
Rectangular Flower Bed 1 – You need to double the width.
Original Measurements: Length: 8 feet Width: Area:
Length: _8 feet ___________ 3 feet
8 feet x 3 feet = 24 sq. feet
Perimeter: (8 feet + 3 feet) x 2 = 22 feet
Width: 6 feet____________ Area: 8 feet x 6 feet = 48 sq. feet Perimeter: (8 feet + 6 feet) x 2 = 28 feet
New Measurements: 2.
Rectangular Flower Bed 2 – You need to cut the length in half.
Original Measurements:
New Measurements:
Length: 10 feet
Length: ___5 feet_________
Width: 7 feet
Width:___7 feet ____
Area: 10 feet x 7 feet = 70 sq. feet
Area: 5 feet x 7 feet = 35 sq. feet
Perimeter: (10 feet + 7 feet) x 2 = 34 feet
Perimeter: (5 feet + 7 feet) x 2 = 24 feet
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3. Right Triangular Flower Bed 3 – You need to cut the height in half. (TC-4) Original Measurements: New Measurements: Base:
3 feet
Base:
3 feet____________
Height:
4 feet
Height: 2 feet ___________
Area: (3 feet x 4 feet) / 2 = 6 sq. feet
Area:
(3 feet x 2 feet)/2 = 3 sq. feet
Perimeter:3 feet +4 feet + 5 feet = 12 feet
Perimeter: 3 feet +2 feet + 3.6 feet ≈ 8.6 feet
4. Right Triangular Flower Bed 4 – You need to double the base. Original Measurements:
New Measurements:
Base:
8 feet
Base:
__16 feet__
Height:
5 feet
Height: __5 feet___
Area: (8 feet x 5 feet) / 2 = 20 sq. feet
Area: (16 feet x 5 feet) / 2 = 40 sq. feet
Perimeter: 8 feet+5 feet+9.4 feet ≈ 22.4 feet
Perimeter: 16 feet+5 feet+16.8 feet ≈ 37.8 feet
5. New Understandings: (TC-5)
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When I increase one dimension, the area ___ increases._____
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When I increase one dimension, the perimeter increases.
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When I decrease one dimension, the area _ decreases._____________
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When I decrease one dimension, the perimeter _decreases_____________
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Only a few more projects and your plans for your yard will be complete! Make sure to sketch the original and new plans on the centimeter graph paper. (Possible extension problem.) (TC-6) 6. Shed – The area of the floor can only be 90 square feet. Change only one dimension. Original Measurements: New Measurements: Length: 10 feet Length: __10 feet________ Width: 12 feet Width: ___9 feet_______ Area: 10 feet x 12 feet = 120 sq. feet Area: 90 square feet Perimeter: (10 feet + 12 feet) x 2 = 44 feet Perimeter: (10 feet+ 9 feet) x 2 = 38 feet 7.
Patio – The perimeter of the patio needs to be 28 feet. Change only one dimension.
Original Measurements:
New Measurements:
Length: 8 feet
Length: __8 feet or 6 feet ____
Width: 8 feet
Width: ___6 feet or 8 feet___
Area: 8 feet x 8 feet = 64 sq. feet
Area: 6 feet x 8 feet = 48 sq. feet
Perimeter: (8 feet + 8 feet) x 2 = 32 feet
Perimeter:
8.
28 feet
You want to put a decorative circle mosaic in the center of the patio. What is the diameter of the largest possible circle that could be placed on both the original plan and the new plan?
(TC-7) Diameter:
___8 feet_________
Diameter:
___6 feet__________
Find the area and circumference of each circle. (area = πr2; circumference = dπ) (TC-8) Area:3.14 x 3 feet x 3 feet = 28.26 sq. feet Area: 3.14 x 4 feet x 4 feet = 50.24 sq. feet Circumference: 6 feet x 3.14 = 18.84 feet Circumference: 8 feet x 3.14 = 25.12 feet
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Teacher Notes for Lesson 2 TC-4 Students do not need to use Pythagorean Theorem to find the length of the hypotenuse of the triangle. They can measure the hypotenuse of the drawn triangle using a ruler. TC-5 For some students the answers may be intuitive. Others may struggle with what to write. Help them by finding an example that fits the situation. Ask them what happened to the area/perimeter with a dimension was increased/decreased. TC-6 The shed and patio problems ask the students to “work backwards” compared to the flower bed problems. If students get stuck have them list all the possible dimensions for an area of 90 square feet or a perimeter of 28 feet. Then ask them to look for a possibility where they would only need to change one of the original dimensions. TC-7 Students may think that the size of circle wouldn’t change. Have them sketch the circle on their centimeter graph paper drawings of the patio to help them visualize how the circle would need to change. TC-8 This problem is assuming students have had experience with finding the area and circumference of circles. Because the concept and procedures were not reviewed, the formulas were included to aid students.
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