Area and Perimeter: The Mysterious Connection
TEACHER EDITION
(TC-0)
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
units
Perimeter of Figure A ________ units
Area of Figure B __________ square
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.
(TC-1)
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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.
(TC-2)
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.
(TC-3)
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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.
(TC-4)
V.
What surprises you about the relationship between area and perimeter in this
exploration? Discuss with your partner and summarize your thoughts below.
(TC-5)
(TC-6)
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: ____________
The Mysterious Connection
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?
(TC-7)
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.
•
•
•
•
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.
(TC-8)
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II.
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?
(TC-9)
III.
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 units
W units
2L+2W
Square
units
units
36
•
Which rectangle has the largest perimeter?
•
Which has the smallest perimeter? (TC-10)
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IV.
Repeat exercise III with a set of rectangles having the area 16.
Area
LXW
Length
L
Width
W
Perimeter
2L+2W
Square
units
units
units
units
16
•
•
(TC-11)
Which rectangle has the largest perimeter?
Which has the smallest perimeter?
V.
•
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.
(TC-12)
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VI.
On a separate sheet of paper, apply the relationships that you discovered in
this exploration on the following problems:
a. Describe how you would construct a rectangle with the largest
possible perimeter given an area of 9 square units.
b. 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.
(TC-13)
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Part 3
I.
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
2L+2W
L
Width
W
Area
LXW
units
units
Square units
units
12
•
Which rectangle has the largest area?
•
Which has the smallest area?
(TC-14)
II.
Repeat number I for a family of rectangles that has a perimeter of 18 units
and then 24 units.
Perimeter Length Width
Area
Perimeter Length Width
Area
2L+2W
L
W
LXW
2L+2W
L
W
LXW
units
units
units
Square
units
18
units
units
units
24
•
Which rectangle has the largest area?
•
Which has the smallest area?
(TC-15)
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Square
units
III.
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?
(TC-16)
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.
(TC-17)
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V.
On a separate sheet of paper, apply the relationships that you discovered in
this exploration on the following problems:
a. Describe how you would construct a rectangle with the largest
possible area given a perimeter of 20 units.
b. 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?
(TC-18)
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Conclusion:
(TC-19)
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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TC-0
There are many misconceptions for students and adults in the complex relationship
between area and perimeter. It is challenging to keep track of which aspect of size is
being measured and what relationship, if any, exists between these aspects (area and
perimeter). Students need a variety of experiences with this relationship to develop a
strong grasp of the concepts. Students need to have experiences in which they are
manipulating the spaces that they are measuring to construct deep understanding.
Because of this, it is important to use a variety of manipulatives. In this lesson, grid
paper is essential, but I highly recommend students use square tiles as well to build the
different figures.
The Big Ideas:
• It is possible to change the area of a figure without changing its perimeter.
• It is possible for several rectangles to exist with the same area, but different
perimeters.
• It is possible for several rectangles to exist with the same perimeter, but different
areas.
• As the differences between the dimensions of a rectangle get smaller for a fixed
perimeter, the area of the rectangle increases. Maximizing as a square.
• As the differences between the dimensions of a rectangle get smaller for a fixed
area, the perimeter of the rectangle decreases. Minimizing when it is a square.
Before students grapple deeply with the relationship between perimeter and area it is
important they have had experiences isolating the aspect of a figure that they are trying
to measure, since any one figure has more than one aspect to be measured. In
particular, the perimeter of a shape and the area of that same shape. A good prerequisite activity to help students focus on these attributes would be a sorting activity
such as “Which way do they go?”. As students sort rectangles from smallest to largest
they are deciding which aspect they are measuring and how they will measure it.
TC-1
• This is the point where they are focusing on measuring the aspect of perimeter.
Watch to see that they are attending to perimeter and not area.
• Possible response: “I found the perimeters by counting the edges on the outside
or boundary of the figure.
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TC-2
• What you are looking for is that they identify that they are able to remove a
corner square unit and keep the perimeter the same.
• Their explanation may say something like, “I took two edges away and exposed
two edges, so the perimeter is the same.”
• They may have difficulty seeing that when they remove a square that is not a
corner that they are removing one edge and exposing 3.
• Ask them to point to the edges they are referring to as they explain their thinking.
TC-3
• Any corner square could be removed, but not any other one of the tiles. Again
two extra edges would be exposed.
TC-4
• Here they will find that they can remove two more square tiles, but the order in
which they remove the tiles matters.
• What you want to hear them discussing is that they have to pay close attention to
how many edges they are taking away and exposing. These must be equal.
TC-5
• You might see:
• “Figures that have the same perimeter can have different areas.”
• “You have to pay attention to how many edges you are removing and how
many are exposed.”
• “You can’t continue reducing the area and keeping the same perimeter.
There is a limit.”
TC-6
• Here in sections VI and VII they are collecting more evidence that their patterns
are true.
• Make sure they don’t pick rectangles that have too small of dimensions. This is
also a place to push a student to change their pattern of thinking. Is there more
than one way to remove the tiles?
• Shapes with the same perimeter do not have to have the same area.
• All tiles can be removed that are not necessary to preserve the dimensions of the
original rectangle.
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TC-7
• Here in VIII and IX they will be sharing their conjectures and constructing a
poster to explain what they have figured out to the rest of the classroom.
• Have them use words, as well as, diagrams to explain their ideas.
• Make sure they are using diagrams or have constructed figures with tiles that
they can share with their classmates.
• Push to have them show the process that they used not just explaining with
words. Physical action in developing measurement ideas is very important.
• Have they addressed why they think this works on their poster?
• Figures that have the same perimeter may not have the same area. This is the
main idea that they need to walk away with from this exploration.
• Stop here and have a classroom discussion around the posters students came
up with in their pairs. Start by asking for volunteers to share. Then ask if another
pair has something to add to what’s been shared already. If you observed a pair
that had an interesting idea or way of thinking, ask them if they would be willing
to share their interesting work. Make sure to ask students if they have questions
or constructive comments on their peers’ work.
Part II
TC-8
• They should come up with rectangles with dimensions 1 X 24 and 2 X 12.
• Students might want to say that 1 X 24 is a different rectangle that 24 X 1. This
is not a central question to this activity. You can ask them what their thinking is
about these rectangles. For a more advanced student you might want to
introduce the idea of congruent figures, but otherwise let it go with what they, as
partners, agree on.
• Some students may want to go to half units or even smaller units. This is a great
observation, don’t discourage them. Instead ask, “So, how many rectangles
could there be?” “What might be the best way to limit our exploration since we
can’t draw ALL rectangles?”
TC-9
• The rectangle with dimensions 1 X 24 will have the largest perimeter.
• The rectangle with dimensions 4 X 6 will have the smallest perimeter.
• The smaller the difference between dimensions, the smaller the perimeter.
Some students may extrapolate that a square will have the smallest perimeter. If
they see this, then push them to find out what those dimensions would be.
TC-10
• They should come up with rectangles with dimensions 1 X3 6, 2x 18, 3 X 12,
4 X 9, and 6 X 6.
• The rectangle with dimensions 1 X 36 will have the largest perimeter.
• The rectangle with dimensions 6 X 6 will have the smallest perimeter.
TC-11
• Students should draw rectangles with dimensions 1 X 16, 2 X 8, and 4 X 4.
• The rectangle with dimensions 1 X 16 will have the largest perimeter.
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•
The rectangle with dimensions 4 X 4 will have the smallest perimeter.
TC-12
• This is a good point to stop and have a brief discussion of the ideas that students
discovered. Have a pair offer one idea that they wrote down and then move to
the next pair.
• It is possible to have many (infinite) rectangles with the same area, but different
perimeters.
• Rectangles with the same area have dimensions that are factors of the fixed
area.
• When the difference between the dimensions of a rectangle with a fixed area is
the smallest you will have the smallest perimeter.
• When the difference between the dimensions of a rectangle with a fixed area is
the largest you will have the largest perimeter.
TC-13
• I would construct a rectangle where there is the largest possible difference
between the dimensions. In this case the dimensions would be 1 X 9.
• Mrs. Hill needs to have a pen that has the smallest perimeter possible. The
smallest perimeter will allow the least amount of material to be used. In this case
the dimensions of the pen will make a 10 X 10 square. I know this is smallest
perimeter, because the difference between the dimensions is the smallest it
could possibly be, zero.
Part III
TC-14
• Students should draw the rectangles 1 X 5, 2 X 4, and 3 X 3. The areas will be
5, 8, and 9 respectively.
• The rectangle with dimensions 1 X 5 will have the smallest area.
• The rectangle with dimension 3 X 3 will have the largest area.
TC-15
• For the set of rectangles that has a perimeter of 18, students will draw rectangles
with dimensions 1 X 8, 2 X 7, 3 X 6, and 4 X 5.
• For the set of rectangles that has a perimeter of 24, students will draw rectangles
with dimensions 1 X 11, 2 X 10, 3 X 9, 4 X 8, 5X 7, and 6X 6.
• The rectangle with dimensions 1 X 8 will have the smallest area.
• The rectangle with dimension 4 X 5 will have the largest area.
• The rectangle with dimensions 1 X 1 will have the smallest area.
• The rectangle with dimension 6 X 6 will have the largest area.
TC-16
• A student may say, “I started with a skinny rectangle with a width of 1. I then
doubled this and subtracted it from the perimeter. Then I took the remaining
units and split them between the remaining two sides.”
• Some students may observe that there could be many more rectangles if you
used fractional sides, but if the sides remain integers then they may recognize
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that they increased the width by one each time, and then at some point, the
dimensions started to repeat themselves.
TC-17
• This is a good point to stop and have a brief discussion of the ideas that students
discovered. Have a pair offer one idea that they wrote down and then move to
the next pair.
• If the perimeter is the same in a set of rectangles, then the area of those
rectangles does not have to be the same.
• Rectangles with the same perimeter have dimensions, where as the length
increases incrementally, the width decreases incrementally until they are as
close to the dimensions of a square as they can be.
• Given a fixed perimeter, the rectangle with the largest area will be the one with
the dimensions that are closest together. (A square.)
• Given a fixed perimeter, the rectangle with the smallest area will be the one with
the dimensions farthest apart.
TC-18
• Given a perimeter of 20 units, I would construct a rectangle where the
dimensions are as close together as possible. In this case it would be a 5 X 5
rectangle.
• If you have a lot to write, then you want as much area as possible. Since the
perimeter is 14 inches, I would construct a rectangle that has its dimensions
close together. In this case it would be a 3 x 4 rectangle.
TC-19
• False. Rectangles with the same area can have many different perimeters. For
example, a 3 x 4 and a 2 x 6 rectangle have the area of 12 square units, but their
perimeters are 14 units and 16 units respectively.
• True. Rectangles with the same perimeter can have many different areas. For
example, a 3 x 4 and a 2 x 5 rectangle have the perimeter of 14 units, but their
areas are 12 square units and 10 square units respectively.
• For a fixed perimeter the rectangle with the largest area is always the rectangle
where the difference between the dimensions is the smallest.
• For a fixed perimeter the rectangle with the smallest area is always the rectangle
where the difference between the dimensions is the largest.
• For a fixed area the rectangle with the largest perimeter is always the rectangle
where the difference between the dimensions is the largest.
• For a fixed area the rectangle with the smallest perimeter is always the rectangle
where the difference between the dimensions is the smallest.
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Perimeter and Area: The Mysterious Connection
Teacher Materials
Perimeter and Area
Pre-Check
KEY
Answer the following questions in the space provided. Use words and diagrams to
explain your thinking.
1. What does area mean?
•
•
•
The area of an object refers to the amount of space that is covered.
For example, it measures how many squares fit on a space without gaps
or overlaps.
Area is measured in square units.
2. How is area different from perimeter?
•
•
•
Perimeter measures around the boundary of an object.
For example, it measures how many edges of a square fit around an
object.
Perimeter is measured in linear units.
3. Which rectangle has the bigger area?
In this case both rectangles are the same
area. They have different perimeters; so
many students are surprised that the areas
are the same. (A=18, but P=18 and P=22
i l )
4. What is the area of this rectangle?
7
A=21. Students may or may not include
the label, square units. Because all the
dimensions are labeled, students must be
3
3
sure which attribute is being measured. If
they are not sure they may find the
perimeter instead of multiplying length by
7
width
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Perimeter and Area: The Mysterious Connection
Teacher Materials
Smallest to Largest
Pre-Exploration
Teacher Notes
•
•
•
•
•
•
Students will often choose to use a linear dimension (length or width) to order
rectangles by perimeter.
Ordering rectangles by area students will often just use their best guess.
They may determine which rectangle seems to cover more.
In these first two sections students are focusing on each attribute and using
their instincts to come up with orders. Do not worry about their orders at this
point, but pay attention to what they are attending to and how they are coming
to agreement. These discussions between peers can be very enlightening as
to what students are thinking.
When they begin to measure they are measuring with something nonstandard. They can use the rectangles to compare or they may choose
another object. You will want to provide a variety of objects for each group to
select from. For example, string, square tiles, popsicle sticks, straw, grid
paper, etc…
Once they have their orders, you want to listen to them discussing how they
varied from their original guesses and why they think they are the same or
different. What did they have to be careful of as they measured?
As they discuss what they learned you may want to use this as a class
discussion. You may hear things such as:
• When measuring perimeter you want to pay attention to the edges.
• Make sure you include all four edges in the perimeter.
• Perimeter is measured by using edges of objects.
• When measuring area you want to pay attention to how much space is
being covered.
• Area is measured by using solid objects rather than edges.
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These need to be enlarged by two for easier handling.
B
E
C
D
F
G
The Mysterious Connection
A
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