area and perimeter: the mysterious connection and perimeter: the mysterious connection ... can you...

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 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. (TC-1)

The Mysterious Connection Teacher Materials of 21

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.

Use pictures and words to explain how you know the perimeters are the same.

Figure A

(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)


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: ____________ Rectangle 2 ______________


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.


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.


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)


II. Determine the perimeters of each of your rectangles and record your results in the table below.

Area LXW Square units

Length L units

Width W units

Perimeter2L+2W 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 LXW

Square units

Length L units

Width W units

Perimeter 2L+2W units

36 • Which rectangle has the largest perimeter?

• Which has the smallest perimeter? (TC-10)


IV. Repeat exercise III with a set of rectangles having the area 16.


Length L units

Width W units


16

• Which rectangle has the largest perimeter?

• Which has the smallest perimeter? (TC-11)

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)


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)


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. Perimeter2L+2W units

Length L units

Width W units

Area LXW

Square 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 2L+2W units

Length L units

Width W units

Area

• Which rectangle has the largest area?

• Which has the smallest area?

(TC-15)

LXW Square units

18


Length L units

Width W


units

24


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)


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)


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

________________________________________________.


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 pre-requisite 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.


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.


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.


• 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


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.


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

7

33

A=21. Students may or may not include the label, square units. Because all the dimensions are labeled, students must be sure which attribute is being measured. If they are not sure they may find the perimeter instead of multiplying length by width


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 non-standard. 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.


These need to be enlarged by two for easier handling.

A

EB

F

DC

G


area and perimeter: the mysterious connection and perimeter: the mysterious connection ... can you...

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