Area and Perimeter: The Mysterious Connection
Student Worksheet
In these problems you will be working on understanding the relationship
between area and perimeter. Pay special attention to any patterns that arise in
your exploration.
Part 1
The question we are trying to answer in this lesson is what connection if any exists
between area and perimeter?
I.
Figure A and figure B below have different areas. Determine if the
perimeters are the same or different.
Figure A
Figure B
Area of Figure A ________square units
Area of Figure B ________ square units
Perimeter of Figure A _______ units
Perimeter of Figure B _______ units
Explain how you arrived at your conclusion. What was the process that you used to
find the perimeters? Show an example of your process with labels included.
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II.
Is there a square unit you can remove from figure A, changing the area,
but not changing its perimeter? If so, which one?
• Draw the resulting figure below.
Figure A
Use pictures and words to explain how you know the perimeters are the same.
III.
Is this the only square unit you can remove that would give you the same
perimeter? Discuss your answer with your partner and record it below.
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IV.
Can you keep reducing the area of figure A by removing square units, but
continue to leave the perimeter unchanged? If so, how many total square
units can you remove and continue to have the same perimeter? Show
your thinking below with words and pictures.
V.
What surprises you about the relationship between area and perimeter in
this exploration? Discuss with your partner and summarize your thoughts
below.
VI.
We want to know if this is true for other rectangles or just for Figure A.
Choose two more rectangles with your partner and record their
dimensions below.
Rectangle 1: ____________
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Rectangle 2 ______________
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VII.
Use square tiles (or centimeter grid paper) to explore your rectangles.
Each of you will explore one of the rectangles. Use the same process you
used for figure A. Remove one tile at a time until you can’t remove
anymore tiles without changing the perimeter.
• Can you keep the perimeters the same as you change the area of
each original rectangle by removing tiles?
• How many square units or tiles can be removed?
•
Does there seem to be any pattern in determining how many tiles can
be removed?
•
Explain what you observe.
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VIII.
•
•
Share what you discovered with your partner.
What conjectures can you and your partner make?
How can you explain them to someone else?
IX.
•
•
•
Make a small poster or use a small white board to show what you’ve
figured out so far.
Use words and diagrams to communicate your thinking about the
relationship between area and perimeter.
Be prepared to share your poster with the class.
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Part 2
I. Below are two rectangles that have the area of 24 square units.
II.
•
•
•
•
Can you draw any other rectangles that have the same area?
If so draw as many as you can on a sheet of grid paper.
Compare your rectangles with your partner.
How did you know that you have found them all? Explain why you think
you’ve found all the rectangles below.
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III.
IV.
Determine the perimeters of each of your rectangles and record your
results in the table below.
Area
Length
Width Perimeter
LXW
L
W
2L+2W
Square
units
units
units
units
24
•
Which rectangle has the largest perimeter?
•
Which has the smallest perimeter?
Draw all of the rectangles that have the area of 36 square units on another
sheet of grid paper. Complete the table below for this set of rectangles.
Area
Length
Width
Perimeter
LxW
L
W
2L+2W
Square
units
units
units
units
36
•
Which rectangle has the largest perimeter?
•
Which has the smallest perimeter?
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V.
Repeat III with a set of rectangles having the area 16.
Area
LxW
Square
units
16
Length
L
units
Width
W
units
Perimeter
2L+2W
units
•
Which rectangle has the largest perimeter?
•
Which has the smallest perimeter?
•
What generalizations about area and perimeter can you make
looking at sets of rectangles with the same area? What patterns to
you see?
•
•
Discuss this with your partner.
Make a complete list below.
VI.
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VII.
On a separate sheet of paper, apply the relationships that you discovered
in this exploration on the following problems:
•
Describe how you would construct a rectangle with the largest possible
perimeter given an area of 9 square units.
•
Mrs. Hill asked you to construct a pen for the class rat. You can use
100 square inches of space on the table in the back of the room, but
she wants you to use as little material as possible to make the sides of
the pen. How much material will you need? How do you know that
this is the least amount of material needed? Explain your answer
using ideas about area and perimeter that you have learned.
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Part 3
I.
II.
Consider a set of rectangles that has a perimeter of 12 units. Draw this
set of rectangles on a sheet of grid paper. Find the area of each rectangle
and complete a chart below.
Perimeter Length
Width
Area
2L+2W
L
W
LxW
units
units
units
Square
units
12
•
Which rectangle has the largest area?
•
Which has the smallest area?
Repeat number I for a family of rectangles that has a perimeter of 18 units
and then 24 units.
Perimeter
2L+2W
units
Length
L
units
Width
W
units
Area
LxW
Square
units
18
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Perimeter
2L+2W
units
Length
L
units
Width
W
units
Area
LxW
Square
units
24
III.
•
Which rectangle has the largest area?
•
Which has the smallest area?
Discuss with you partner how you know you drew all the possible
rectangles for the sets of rectangles you have drawn.
• What process did you use?
IV.
•
What observations do you make about these sets of rectangles that
have the same perimeter? What patterns do you see?
•
•
Discuss your ideas with your partner.
Make a complete list below.
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On a separate sheet of paper, apply the relationships that you discovered in this
exploration on the following problems:
•
Describe how you would construct a rectangle with the largest possible area
given a perimeter of 20 units.
•
You are making a card with a ribbon boarder. You have 14 inches of ribbon.
You have a lot to write on your card. What size card should you cut out of
card stock paper? How much area will you have to write on?
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Conclusion:
Now you should be able to confidently answer the following questions.
Make sure you use clear mathematical thinking and diagrams to explain
your answers.
1. True or False
Rectangles with the same area must have the same perimeters. Explain and
give an example.
2. True or False
Rectangles with the same perimeters can have different areas. Explain and give
an example.
Fill in the blank.
3. For a fixed perimeter the rectangle with the largest area is always
________________________________________________.
4. For a fixed perimeter the rectangle with the smallest area is always
________________________________________________.
5. For a fixed area the rectangle with the largest perimeter is always
________________________________________________.
6. For a fixed area the rectangle with the smallest perimeter is always
________________________________________________.
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Perimeter and Area
Pre-Check
Answer the following questions in the space provided. Use words and diagrams to
explain your thinking.
1. What does “area” mean?
2. How is “area” different from “perimeter”?
3. Which rectangle has the bigger area?
4. What is the area of this rectangle?
7
3
3
7
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Smallest to Largest
Pre-Exploration
In this exploration you will be ordering a set of rectangles according to their
relative sizes from smallest to largest. Pay attention to which attribute you are
asked to measure. Work in groups of two or three.
1.
Carefully cut out the set of rectangles you have been given. Make sure
everyone in your group has a set.
Put the rectangles on the desk in front of you. Without doing any measuring,
order the rectangles from smallest perimeter to largest perimeter.
Discuss your order with your group and come to agreement. Record your
order below.
2.
3.
4.
Again, without doing any measuring, order the rectangles from smallest area
to largest.
Discuss your order with your group and come to agreement. Record your
order below.
5.
6.
7.
Now you can measure. Find the perimeter and area of each rectangle by
comparing the rectangles to one another or by using some other object to
help you (not a ruler).
•
What is the order for the perimeters of the rectangles?
•
What is the order for the areas of the rectangles?
How do these actual orders compare to your original orders?
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8.
What did you learn about perimeter and area in this activity? What ideas
about measurement do you have to pay attention to with area and perimeter?
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