TILING AND FILLING “THE POOL
Teacher Copy
The Pool
Part A
General overview of lesson for the teacher:
This lesson takes students through an activity to engage them in building their
understanding of perimeter, area, surface area, and volume of rectangles and
rectangular prisms through the experience of constructing a model. Their depth of
knowledge will also be built through dealing with a real life situation/context which they
can relate to, a swimming pool.
Students will begin by figuring the dimensions that the pool must have to fit the given
criteria. They will then construct an “open box” out of tag board, which will become their
pool model. A variety of questions and activities follow using their pool as a means for
students to build their understanding of area and surface area through the “tiling” of the
pool. Using cubes that have the same dimensions as the tiles, students will simulate
the “filling” of the pool with cubes to help them experience the difference between
area/surface area and volume.
Students will have opportunities to count units, square units and cubic units, to
determine “answers”, and then eventually calculate answers based on their experiences
(move towards the use of formulas). Students will then report their findings about their
pool in the form of an assessment question that is submitted for a check of
understanding. From this point Part B and Part C of the lesson can be used to further
engage the students in their understanding of surface area and volume and their
applications in the real world.
General notes:
1) This lesson can be done individually or in groups of 2-4.
2) Teacher involvement in model building can vary dependent on the abilities of the
student group.
3) Pacing is dependent on length of class periods, but Part A could easily be broken
into two to four class periods. Natural breaks occur after #1 (figuring out the site
plan/dimensions of pool), after #3 dealing with surface area, and then before #5
the assessment of Part A.
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The Pool
Part A Teacher Copy
At last, your parents have finally agreed that it might be possible to put a small pool in
the back yard. The only catch is, your parents are very busy and don’t have time to
spend figuring out the details of how much space it will take up and all of the related
construction costs. You are not responsible for all of the details, but your parents have
told you that if you want a pool, you are going to need to do some work convincing them
that you are serious about having the pool. They have told you that you must figure out
the some of the pool information and present it to them in the near future.
1) Your parents know that they want the pool to be a consistent depth, four feet. They
want to use the fencing that they bought last year, and never used, to surround the
pool for safety purposes. They have 96 feet of fencing and a four foot gate. The
gate can be positioned anywhere you want. This fence needs to be five feet from
the edge of the pool on all sides. The pool needs to be rectangular, two times as
long as it is wide.
Use the grid paper to help you figure out the dimensions (length and width) of
the pool, and where the fence and the gate will be.
Your parents need to know
how much of the yard the fenced area will be so that they will know if it is
reasonable to fit the pool into the back yard with the existing landscaping.
This will be a “bird’s eye view” (a view from above) and will not show the depth of
the pool.
TC-1: Make sure that students understand “bird’s-eye-view” and that this is not
intended to be a 3-Dimensional drawing. This is a good time to discuss the real world
context of “site plans” and “footprints”. Make sure that everyone understands
“dimensions” (the length, width, or height of an object.) Students’ sketches on the grid
will all look very different. If they become confused, refer them back to the given criteria
(twice and long as wide, fence needs to be five feet from pool, etc.)
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Scale: 1 square unit=1 square foot
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What will the dimensions (length and width) of the pool be?
___________length = 20 feet
width = 10 feet
____________
Once you have drawn the 2-dimensional diagram of the pool and fenced area, figure out
what the total area inside the fence will be. Do this by first counting all of the square
units inside of your fence. Record your answer. ___600__ square units. Is there
another way, besides counting them, that you could figure out how many square units it
would be? If so, explain the method giving examples that refer back to your diagram.
TC-2: If weaker students are struggling with this concept, lead them through looking at
how many rows there are and how many square units there are in each row.
____Yes there is another way. Each row across the pool has 20 square units. Since
there are 30 rows of 20, you can multiply 30 x 20 to get 600 square units.
Does this method give you the same answer as when you counted the square units?
______yes______
TC-3: Take time to discuss with students that the surface area of a 3-dimensional
object is not only the exterior surfaces of that 3-dimensional object, but also can refer to
the interior surfaces. Give students examples of when they would refer to exterior
surfaces in the real world (wrapping a present, painting the outside of a house, etc.) and
examples of interior surfaces (painting the walls and ceiling of a bedroom, gluing
construction paper to the inside of a shoe box to make a diorama, etc). Ask students for
other examples of each to check for understanding. Also make sure that students
understand that when they are using surface area in a real-world context there will be
times when one or more faces of the 3-Dimensional object will be eliminated (i.e. floor
or roof). These two ideas need to be clear to students before proceeding on to #2.
2) Next you will be making a three-dimensional model of the pool to determine its
interior surface area and its volume. Surface area is the sum of the areas of all
the faces of a three-dimensional object. You need to do this so that you can
determine how many one square foot tile it will take to tile the bottom and four sides
of the pool, and so that your can determine the volume of water that will be in the
pool. Your parents need to know the volume so they can make a decision on the
water filtration system that they will use with the pool.
TC-4: Review materials list to prepare for building the model.
To make the model and determine its surface area and volume, you will proceed
through the steps listed and record your findings along the way.
Step A
Collect your supplies. You will need tag board, grid paper (scale: one square unit = 1
square foot), a straight edge, scissors, tape, and cubes that match the dimensions of
your grid paper.
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TC-5: Print grid paper on 11 x 17 photocopy paper or have students tape smaller
pieces together if necessary (will depend on the size of cubes being used).
TC-6: Provide students with appropriate grid paper and have them match a cube on top
of one of the squares on the grid paper so they understand and experience early on that
the dimensions of the square and cubic units in their model will be the same.
Step B
Measure and cut out a piece of tag board that is 8 units longer and 8 units wider than
the dimensions that you previously determined that the pool would be.
Step C
Next measure and cut out a four unit by four unit square from each corner of the tag
board. Your tag board should look similar to this:
Step D
Fold your tag board figure as shown in the diagram (along the dotted lines).
Step E
Loosely tape the edges you have folded up together and you have created an open box
which is your scale model for your pool. It is a rectangular prism (in this case it does not
have a closed top since it is a swimming pool).
Why did we originally cut the paper eight units wider and longer than the pool? __To be
able to have the depth represented in the 3-dimensional model. Four units were folded
up in each direction. _______________________________________________
______________________________________________________________________
___________________________________________________________________
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TC-7: This would be a good time to discuss the term “net” (a representation of a
3-dimensional figure that is unfolded).
3) Now that the model is built, you are going to determine the interior surface area of
the pool. Keep in mind that the top of this rectangular prism will be open.
Cut pieces of grid paper out to match the size of the four walls and the bottom of the
pool (cut on the lines). You can take the tape off and flatten the model if it is helpful.
Of the five pieces you just cut out (not individual square units), how many pieces of
different sizes did you need to cut? Answer and explain.
__Five pieces altogether, three different sizes. One bottom, two ends the same size
(dimensions), and two sides the same size (dimensions).______________
a)
b)
c)
d)
e)
How many square units was the piece of paper that covered the bottom of the
pool? (count them) ___________200 square units
________________
How many square units was the piece that covered one of the ends? (count them)
_____40 square units_____ Was there another one exactly the same size?
(explain) ______Yes, the opposite end________________________
How many square units was the piece that covered one of the sides? (count them)
_____80 square units_______ Was there another one exactly the same size?
____________________
(explain) _______Yes, the opposite side
If you buy a pool cover, how many square units will it need to be to perfectly match
the top of the pool? _____200 square units ____How do you know this? _____It
is opposite of the bottom of the pool____________________
What is the surface area of the entire pool model (no top)? Make sure to think
________
about what the label should be. _________440 square units
TC-8: Filling in the table below gives the students another opportunity to see the pattern
involved in finding surface area. Continue to reinforce that the top of the pool is not
accounted for in this scenario.
Fill in the table below with the information that you have figured out.
SQUARE UNITS
NUMBER
OF TOTAL SQUARE
SURFACES WITH UNITS
THESE
DIMENSIONS
BOTTOM
200
1
200
END
40
2
80
SIDE
80
2
160
TOTAL SURFACE 440 square units
AREA OF POOL
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4) Now you are ready to figure out the volume of water that the pool will hold. In other
words, how many cubic units it will take to fill the space inside the threedimensional rectangular prism. How will this be different than figuring out the
surface area of the pool? ______You need to fill the space inside of the pool, 3Dimensional,
not
just
cover
the
walls
and
bottom,
2Dimensional.________________________________________________________
TC-9: For the next part of this activity you can lessen the need for “thousands” of
cubes by having a “filling station” or two someplace in the room (or have a couple of
sets available to go to student work areas).
TC-10: It is probably not necessary for all students to completely fill their models once
they can clearly explain what they are doing and why filling the pool determines the
volume of the pool, teacher discretion. You will need to make sure that you have 800
cubes available if students are completely filling their pools (reminder-sugar cubes are
cheap!)
Using cubes that match the dimensions of the grid paper you used in #2 and #3, fill your
model with cubes to see how many cubic units it takes to fill your pool. Count them as
you put them in. How many cubes did you stack on top of each other to get to the top of
the pool? ___4__ How many cubes did it take next to each other to go from one side to
the other? __10__ How many cubes did it take to go from one end of the pool to the
other? __20___
a) What else could you do to figure out how many cubes it would take to fill the pool
besides counting them? Explain. _____Multiply 20 x 10 x 4. There are 20 rows of
ten cubes which equals 200 and there are four layers of 200 which equals 800
cubes._____________________________________________________________
________________________________________________________________
b)
What is the volume the pool? ________800 cubic units _____________
c) Was your label square units or cubic units? ______cubic units ____ Explain why.
_____Because
you
are
filling
a
3-Dimensional
space.___________
__________________________________________________________________
______________________________________________________________
TC-11: #5 is intended to be a check for understanding of the work done throughout
Part A and in the context of the beginning prompt.
5)
Now you are ready to present a summary to parents about what you have figured
out so that they can make further decisions about the pool. Keep in mind that you
won’t necessarily be there to answer questions for them, so make sure that your
information is clear. Show your work using words, numbers and/or diagrams (grid
paper is available for you to use if needed). Make sure that you include the
following information in your report to them:
• The dimensions of the pool (include a drawing)
• Total area that will be taken up in the back yard including the fenced area
(include a drawing)
• The surface area of the pool
• The volume of the pool
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______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
_______dimensions of the pool area: length = 20 feet
width = 10 feet___________
____total area taken in back yard including the fenced area = 600 square feet_______
________the surface area of the pool = 440 square feet_________________________
__________the volume of the pool = 800 cubic feet_____________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
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Scale: 1 square unit=1 square foot
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Scoring Rubric –The Pool Part A #5
ME01 (Attributes and Dimensions) Demonstrates understanding of the concepts of
volume and surface area of rectangular prisms. (GLE 1.2.1)
ME02 (Units and Systems) Demonstrates understanding of the differences between
square and cubic units. (GLE 1.2.2)
Process Strands: (GLE 4.2.3 and 5.1.1)
Item Characteristics:
• Items may ask students to use surface area and volume to describe a
rectangular prism.
• Items may ask the student to identify cubic units as a measure of volume.
• Items may ask students to clearly explain or describe mathematical ideas, facts,
properties, procedures, or strategies in a way that is appropriate for a given
purpose and/or audience using mathematical language and notation.
4-point response demonstrates mathematical content and communication by doing the
following:
•
•
•
•
Includes all four (4) components: total area in yard, dimensions of pool, surface
area of the pool, and volume of the pool.
Presentation has a layout that is clear, organized, has work that supports
answers and includes a drawing.
Labels for square feet and cubic feet are used sufficiently to demonstrate
understanding and appropriate use.
There are no errors in calculations.
3-point response includes three of the four listed above.
2-point response includes two of the four listed above.
1-point response includes one of the four listed above.
0-point response shows little or no mathematical understanding of task.
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The Pool
Part B
Teacher Copy
TC-1: Part B can be done as a part of the continuation of the lesson scenario.
Your parents have reviewed your summary and have gotten some additional
information about prices for different aspects of the pool’s construction and
maintenance. Based on the work that you have already done, they would like you to
figure out the cost of the items below and explain it to them. Create an organized list for
the different items’ costs and a total cost for all of these items combined.
TC-2: Make sure that students understand what all of the items below are. Also direct
students to look back at their plan from Part A where they determined the perimeter,
surface area and volume of the pool.
• Edging for around the perimeter of the pool (called bullnose) that is sold by the
foot and costs $1.25 per foot
• Tiles cost $4.00 per square foot (and will need to cover the entire surface area
of the pool)
• The material that adheres (glues) the tiles to the cement wall and bottom is
$0.90 per square foot
• The chemicals for the filtration system cost $0.25 per cubic foot (enough for a
year, which is what your parents want to know)
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Show your work using words, numbers and/or diagrams.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
__________________edging/bullnose $1.25 x 60 feet = $75______________________
__________________cost of tile $4 x 440 square feet = $1760____________________
_________________cost of adhesive $0.90 x 440 square feet = $396______________
________________chemical 800 cubic feet x $0.25 = $200______________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Total cost _______$2431________
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Scoring Rubric –The Pool Part B
MC01 (Connect within Mathematics) Use concepts and procedures from two or more of
the content strands in a given problem or situation. (GLE 5.1.1)
ME03 (Procedures) Use systematic procedures to measure, describe, and compare the
area of rectangles and right triangles and the volume of rectangular prisms – identify
area or volume as the attribute to be measured, select and use appropriate unit for area
or volume, select and use appropriate unit for area or volume, select and use tool that
matches the unit chosen, count to determine the number of units; use formulas to find
the perimeter and area of rectangles and right triangles. (GLE 1.2.4 and 1.2.5)
Item Characteristics:
Items may ask students to use and/or describe procedures for measuring length,
perimeter, and area of a rectangle or right triangle and/or the volume of a rectangular
prism:
• identify the attribute to be measured;
• select and use appropriate unit of measurement;
• select and use a tool that matches the unit chosen;
• count or compare the attribute to the units on the tool to determine the number of
units.
Items may ask students to use concepts and procedures from two or more mathematics
concept strands-number sense, measurement, geometric sense, probability and
statistics, and algebraic sense.
4-point response demonstrates mathematical content and communication by doing the
following:
•
•
•
•
Includes all four (4) components: cost of edging/bullnose, cost of tiles, cost of
adhesive, and cost of chemicals
Presentation has a layout that is clear, organized and work that supports
answers.
Labels for feet, square feet and cubic feet are used sufficiently to demonstrate
understanding and appropriate use.
There are no errors in item calculations (total cost is correct)
3-point response includes three of the four listed above.
2-point response includes two of the four listed above.
1-point response includes one of the four listed above.
0-point response shows little or no mathematical understanding of task.
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The Pool
Part C
Teacher Copy
TC-1: Part C is meant to allow students to practice what they have learned in Part A
and Part B in new situations to expand their understanding.
Your parents want to look at other possible options before they commit to the exact
construction plan, so they want you to make a few last calculations for them.
TC-2: Students will need their previous work to complete Part C.
1) If they decided to double the width of the pool, what would the new surface area
and volume of the pool be?
Original perimeter (of pool) _________60________ feet
New perimeter (of pool)
__________80_______ feet
Original surface area
New surface area
_________440_______ square feet
_________720_______ square feet
Original volume
New volume
_________800_______ cubic feet
________1600_______ cubic feet
TC-3: Check to make sure students understand what a side view diagram is and make
sure that they understand that the steps would go clear across the pool (10 ft.) in #2.
2) What would happen to the volume of the pool (original pool volume) if seven steps,
a half foot high (6 inches) and one foot deep (12 inches), were added across one
end of the pool? See the side view diagram below for help.
a)
How much of the volume of the pool would be eliminated by the steps?
_____140_____ cubic feet
b)
What would the remaining volume of the pool be? _____660____ cubic feet
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TC-4: #3 can be used to differentiate instruction even further for advanced students.
Ideas? Sloped entry into the pool – surface area and volume of more complex figures.
Tile designs, tessellation . . .
3) What other design ideas do you have? How would they affect the surface area and
volume of the pool? Explain using words, numbers, and/or pictures.
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