Swimming Pool Tiling
Background:
A
first year math teacher in a small middle school, Tak Hui
enjoys studying pure mathematics, and he enjoys teaching pure
mathematics even more.

Created
by Tak Hui
Location: P.S. 127Q in Region 4
Grades: 7th through 9th
Subject: Math
If you have any questions
regarding this activity, please contact Tak Hui at: ps127math@hotmail.com

Aim:
How can we use algebraic equations to represent a visual
model?
Instructional
Objectives:
1.
To use a strategy to determine the number of tiles needed to form
the border around a square pool.
2. To articulate
such strategy in words as well as using a mathematics sentence,
this will later be translated into an algebraic equation.
3. To describe
any noticing patterns that may help students in generating the algebraic
equations.
4. To use algebraic equations to generalize the problem as the lengths
of the swimming pool vary
5. To justify that algebraic equations are equivalent by substitute
specific values to the variable.
6. To define
equivalent equations as equations which represent the same quantity.
7. To define
algebra as a generalization of arithmetic
Terms
Used
border, algebra,
generalization, algebraic equations, equivalent algebraic equations
Procedure
A rectangular
swimming poor has a length of 10 '/2 feet and a width of 8 feet.
What are the area and perimeter of this pool?
Solution:
Area: 10 1/2 x 8 = 84 ; 10 x 8 = 80 and 1/2 x 4 = 4 ; 80 + 4 = 84
Perimeter:
10 1/2 + 10 1/2 + 8 + 8 = 21 + 36 = 37 (10 1/2 + 8) x 2 = 18 1/2
x 2 = 37 10 1/2 x 2 + 8 x 2 = 21 + 16 = 37 (Elicit responses from
students and focus on their strategies in finding the perimeter.
Leave those strategies on the board.)
To look
for: If students still misunderstand perimeter as area and
area as perimeter. A strategy used to find the perimeter and area
of a rectangle.
Lesson
Motivation:
How many of you have been to swimming pools? Did you notice that
sometimes these swimming pools are surrounded by borders of tiles?
What is a border? (A border is a strip around an edge or a side.)
 Draw a rectangular
pool.
 Being a math
teacher, I am always thinking how mathematics works in our day
to day lives. I wonder how the construction workers know how many
tiles to put around swimming pools. What do you think?
 I have a
5 by 5 square swimming pool here surrounded by the square border
tiles. These tiles measure 1 foot on each side.
 Draw of a
5 by 5 square pool.
 Determine
the number of border tiles used to surround this 5 by 5 square
pool. (Elicit responses from students without getting into their
strategies.) In this problem, you will explore this question:
If a square pool has sides of length s feet, how many tiles are
needed to form the border? Before we can answer this question,
we need to collect some data.
Instructions:
1. Make sketches
on graph papers to help you figure out how many tiles are needed
for the border of square pools with sides of lengths 1, 2, 3, 4,
5 and 6 feet. Record your results in a table. Describe how you figure
out the number of border tiles needed in words and in a mathematical
sentence.
2. Describe
any patterns that you notice.
This is a partner
activity. However, students will work independently first. One of
you will work on square pools 1, 2 and 6 feet and the other will
work on 3 and 4 feet for 3 minutes. Then you will discuss your findings
with each other for about four minutes. At the end of seven minute,
we will elicit strategies that you used to determine the number
of border tiles needed and discuss the patterns that you come up
with.
During this
time, I will be collecting evidence on:
.
The different ways students organize their data.
. The
mathematical sentences
. The
different strategies
. Any
students who miscount the number of tiles.
. Any
students who forget to subtract the four corners that are counted
twice.
Anticipated
Strategies: (Use swimming pool length 6 as an example)
. 6 + 6 + (6
+ 2) + (6 + 2)
. 4 x
6 + 1 + 1 + 1 + 1
. (6 +
(6 + 2)) x 2
. 4 x
(6 + 2)  4
. (6 +
1) x 4
. (6+2)^2
6^2
* * These
strategies should be on the board or on easel pad.
Anticipated
Patterns:
. As the length
of the square pool increases by 1 foot, the number of tiles needed
for the border increases by 4.
. The
shape of the border is also a square.
. The
length of the border is always two feet more than the length of
the swimming pool.
Pose Question:
Is there an efficient way to calculate the number of border tiles
needed for a square pool, no matter what the lengths of the sides
of the pools are?
We will then
proceed to the next two questions.
3. Without drawing
the pictures, and using what you have so far, determines the number
of border tiles needed when the lengths of square pools are 10 feet
and 26 feet.
4. Write an
algebraic equation for the number of tiles, N, needed to form a
border for a square
pool with sides of lengths feet.
Try to write
at least one more equation for the number of tiles needed for the
border. How could you convince someone that your equations for the
number of tiles are equivalent?
(At this
point, most of the students should have some entry points to complete
parts 3 and 4. The strategies and patterns from part 1 and 2 would
be good tools to assist students who are still struggling. They
will have 5 minutes to complete these two parts.)
Class
Discussion:
1. What strategies
did you use to determine the number tiles used for 10 feet? 26 feet?
(Students would probably use one of the strategies stated previously.)
2. What algebraic
equations did you come up with for s feet as the length of the square
pool?
Anticipated
equations:
N = s
+ s + (s + 2) + (s + 2)
N = 4s
+ 1 + 1 + 1 + 1
N= 4s
+ 4
N = 2(s
+ (s + 2))
N = 4(s
+ 2)  4 {counted the four corners twice)
N=(s +
2)^2  s^2
N= 4(s+1)
(Encourage students
to use the visual model to explain how they come up with the equations.)
3. How are these
equations similar to the mathematical sentences that we came up
before?
Instead
of using a specific length of the swimming pool, we use a variable
to represent the any length of the swimming pool.
Explain
to students that algebra is basically a generalization of arithmetic.
4. How can you
convince me that all of these equations would get me the same answer?
Anticipated
Responses: substitutes with a value, set up a table for each
of the equations, or graph the equations. In mathematics, we said
that these algebraic equations are equivalent because they represent
the same quantity. In this case, all these equations would be able
to help you to determine the number of tiles needed to surround
a square pool whose length is 10 feet.
Summary
Go back to our
aim for today. Take about 3 three minutes to answer the question.
Elicit volunteers to share their thoughts.
Extension
1. For example,
the only thing you know about the rectangular swimming pool is that
the length of the swimming pool is 6 feet longer than the width.
Is it possible to find the number of tiles needed for the border
using an algebraic equation for this swimming pool? Draw a picture
to help you. Be sure to explain what the variables stand for.
2. What if you
don't know anything about the swimming pool other than it is a rectangular
pool? Is it now possible to determine the number of tiles needed
for the border with an algebraic equation? Draw a picture to help
you. Be sure to explain what the variables are.
Standards
Addressed by This Unit
 New York
City Math Standards: Be familiar with assorted two and threedimensional
objects.
Determine
and understand length, area, and volume.
Model
situations geometrically to formulate and solve problems.
Discover,
describe, and generalize patterns, and represent them with variables
and expressions.
Represent
relationships
Overall
This is a investigative
unit on introductory algebra. It draws upon interest from the real
life construction of a swimming pool
Tips
Right before
or after summer vacation is the optimal season for the introduction
of this lesson. 