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Area Review Warm-Up
Area: The space that a two-dimensional shape occupies measured in square units.
For a rectangle, the area is found by multiplying the length by the width.
DIRECTIONS: Find the area of each rectangle below. Show all of your work.
1.
7m
16 m
2.
8 cm
8 cm
3.
11mm
3 mm
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Introduction
Contortionists are able to bend and flex in order to squeeze themselves into small
spaces and often perform as part of a circus or theatrical act. Most contortionists have
unusual natural flexibility enhanced through gymnastic conditioning and training. The
practice of squeezing one’s body into a small box appearing to be much too small for a
person to fit in is called enterology.
Examine these pictures of several contortionists.
What do you think? Can you fold yourself into a small box?
Try rolling yourself into a ball. Was it easy? Was it difficult?
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The contortionist in the above pictures is named Emma Tunbridge. She was a national
gymnast from England who now lives in Australia, studies yoga and performs in various
festivals and shows alongside great performers such as David Blaine, a magician and
illusionist. She is most commonly known for her ability to fold herself into a 17 inch
cube in less than a minute. Remember that a cube is a _______ dimensional shape
which has ____ equal faces and three equal dimensions of length, width, and
_________. A cube is also an example of a rectangular prism.
VOCABULARY
Rectangular prism:
a three-dimensional shape with six rectangular faces and the
dimensions of length, width, and height.
Example:
Label the length, width, and height on the rectangular prism above.
Rectangular Prism Search
Find two examples of rectangular prisms in the classroom. Name and describe why
your examples fit the definition above.
1.
2.
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EXPLORATION - Volume
Focus Question: What is volume?
In your small group, you will explore how much space is inside the 17 inch cube Emma
fits into. This space is called the volume of the figure. Then you will calculate the
volume of Emma to see how she is able to fit into the space in the cube.
STEP 1: Use the sheets of large inch grid paper to construct a full-size net (pattern) for
a 17-inch cube. Remember that a cube will have 6 equal sides. An example of a net
for a one-inch cube is shown below. The net should include all six sides and be able to
be folded to form a cube or box.
STEP 2: Fold your net into the cube. Tape all sides except the lid! Not a very big box,
huh? So, how does Emma fit into that small space? Well, as a class, let’s see how
many inch cubes would fit by filling the bottom of the paper cube with one layer of inch
cubes.
A. How many cubes does it take to fill the bottom of the paper cube? ________
B. How many layers tall is the paper cube? ____________
C. Check your answer by stacking inch cubes the height of the cube in one corner of
the paper cube.
D. So, if _________ cubes fill the bottom layer and there are _______ layers, how
many total inch cubes would fit inside the paper cube? Show your work.
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VOCABULARY
You just found the volume of the paper cube. Based on your work in part D,
define volume in your own words:
E. What are the dimensions of the cube you built?
Length ___ Width ___ Height ___
F. Compare the dimension numbers with the number of inch cubes used in the layers in
part D. Can you determine a general rule for finding the volume of rectangular
prisms or cubes? Explain and support with numbers and/or pictures.
STEP 3: Look at the first picture of Emma on page 2. If Emma was to stand straight
and we drew her as a rectangular prism with the three dimensions of length, width, and
height, her dimensions would be: length 12 inches, width 6.5 inches, and height 63
inches.
A. Draw a rectangular prism below with these dimensions.
B. Now use your drawing and general rule from Step 2 to calculate the volume of
Emma. Show all work.
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C. Compare Emma’s volume to the volume of the 17-inch cube. How do you account
for the differences? Be specific! Think carefully. Are humans really shaped like
rectangular prisms?
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D. Did the amount of space or volume of Emma change from when she was standing to
when she was squeezed into the cube box? Explain your reasoning.
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STEP 4: Time to practice what you now know about volume. You have a choice
between two problems. Try problem #1 if you are beginning to understand volume but
could use a little more practice. Try problem #2 if you really understand volume and
need a challenge. Or, you can try them both!!
#1 Using the picture of the rectangular prism below, find the volume.
8 feet
2 feet
4 feet
#2 If the volume of the rectangular prism below is 40 units cubed, find the missing
length.
4 feet
2 feet
?
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EXPLORATION – Surface Area
Focus Question: What is surface area?
STEP 1: Paper, Paper, Paper! So, how many square inches of paper did your group
use to build the cube like the one Emma fits into? Sure, to figure this out you could
count all of the inch squares on the paper. But, as mathematicians, we like to calculate
things as quickly as possible. So, find the area of each face (all 6 of them!) and then
add all six areas together. Show your work.
And there you have it! The surface area of the cube! Surface area is simply the total
area of the faces of a three-dimensional shape. Is there a faster way to get the surface
area other than counting the inch squares on each face? What if the cube wasn’t made
of grid paper? Explain and show work.
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STEP 2: Now use your ideas for calculating surface area to find the surface area of
Emma as a rectangular prism. Again the dimensions would be: length 12 inches,
width 6.5 inches, and height 63 inches. Show all work!
STEP 3: Time to practice what you now know about surface area. You have a choice
between two problems. Try problem #1 if you are beginning to understand surface area
but could use a little more practice. Try problem #2 if you really understand surface
area and need a challenge. Or, you can try them both!!
#1 Using the picture of the rectangular prism below, find the surface area.
8 cm
2 cm
4 cm
#2 If the surface area of the rectangular prism below is 108 square cm, find the missing
length.
4 cm
1 cm
?
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BOXED IN! REFLECTIONS – Volume and Surface Area
1. Compare the volumes calculated and the surface areas calculated for Emma. How
did they differ? How were they the same? Use specific examples to support your
comparisons.
2. Describe what volume is and how you find it. Use words, numbers, and/or pictures
to support your answer.
3. Describe what surface area is and how you find it for a rectangular prism. Use
words, numbers, and/or pictures to support your answer.
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Stretch Out and Curl Up!
The surface area of an animal impacts their ability to conserve heat or cool
themselves in different climates/temperatures. On a hot summer day, a dog or cat will
stretch out on their bellies to cool off in a shady spot, but on a cold winter day the same
dog or cat will curl into a ball to stay warm.
• radiate means to emit rays of light or heat
• conserve means to save something from loss – to preserve
Focus Question: How does surface area and/or volume relate to an animal’s ability to
radiate or conserve heat?
1. Your group was given 27 unit cubes. Place the cubes in a straight length (row) to
represent a dog or cat stretched out on a warm day.
a. Calculate the volume of the dog/cat. Show your work!
b. Calculate the surface area of the dog/cat. Show your work!
2. Now rearrange (stack) the unit cubes into a cube to represent the same dog or cat
curled up on a cold winter day.
a. Calculate the volume of the dog/cat. Show your work!
b. Calculate the surface area of the dog/cat. Show your work!
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3. Compare the volumes and surface areas of the stretched out and curled up dog/cats.
How are they they same? Different?
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4. Using your work on numbers 1-3, why do you think that surface area impacts an
animal’s ability to radiate heat on a warm day and to conserve heat (or avoid
radiating heat) on a cold day? Be specific!
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EXTENSION PROJECT – RESEARCH OPPORTUNITY!
1. Use the Internet and/or library resources to research other animals and their unique
ability to cool/stay warm based on their surface area and other characteristics.
A.
;
;
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In your research you must include:
The dimensions of the animal
The climate where the animal lives
Unique physical characteristics which also impact the animal’s heating/cooling
ability
B. Write a summary of your findings. Include
• calculations for surface area and any other math work
• your conclusions based on your research and mathematical work
C. Use the attached Notes and Summary Page
2. Additional challenge: Explore the relationship between an animal’s volume (mass)
and their surface area and how that relationship impacts their cooling/heating ability.
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Notes and Summary for Extension Project
NOTES/MATH WORK
Dimensions of Animal
Surface Area/Other Math Work
Climate
Unique Characteristics
SUMMARY OF FINDINGS
Write in complete sentences and give specific examples and support for your
conclusions.
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Net/Pattern for a Cube
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