**Tiling Pools
Learning Task**

This task is
similar to the introductory tiling learning task in that students will continue
to familiarize themselves with developing algebraic expressions to model
problems in context. The context for
this task is swimming pools. Students
will examine various pool designs and use their explorations to develop
algebraic expressions to model the area of the swimming pool(s).

**1. In the figures
below, there are diagrams of swimming pools that have been divided into two
sections. Swimming pools are often
divided so that different sections
are used for different purposes such as swimming laps, diving, area for small
children, etc. **

** **

**a) For each pool,
write two different but equivalent expressions for the total area.**

*Students should already be familiar with the properties of the
area of a parallelogram, i.e., that **. The
lengths of each of the rectangular pools are separated into two sections. Therefore, the students must realize that the
total lengths will be the sum of each of the sections. *

__Pool #1__: The area of the first pool is equal to its
length (x + 1) times its width (25), yielding an algebraic expression of its
area. An equivalent expression can be
determined by using the distributive property.
The equivalent expression for the area of pool #1 is 25 + 25x.

__Pool #2__: The area of the second pool is equal to its
length (22 + x) times its width (x), yielding an algebraic expression (22 + x) ∙
x of its area. Once again, the equivalent
expression is obtained by distributing, obtaining a total area of

**b) Explain how these
diagrams and expressions illustrate the Distributive Property.**

In order
to obtain the equivalent expressions for the area of pools 1 and 2, the Distributive
Property of multiplication had to be used.
The definition of this property states that and two equivalent expressions (as denoted by
the equal sign) are produced. For
example, for Pool #1, we found that or depending on how the student(s) defined the
length and width. In either case, we
used the Distributive Property, which tells us that 25(x + 1) is equivalent to

. We
used the same procedure to calculate the equivalent expression for Pool #2.

_____________________________________________________________________________________________________________________________________________________________________________________________________________

In-ground
pools are usually surrounded by a waterproof surface such as concrete. Many homeowners have tile borders installed
around the outside edges of their pools to make their pool area more
attractive. Superior Pools specializes
in custom pools for residential customers and often gets orders for square
pools of different sizes. The diagram at
the right shows a pool that is 8 feet on each side and is surrounded by two
rows of square tiles. Superior Pools
uses square tiles that are one foot on each side for all of its tile borders.

The
manager at Superior Pools is responsible for each job and needs an equation for
calculating the number of tiles needed for a square pool depending on the size
of the pool. Let N represent the total
number of tiles needed when the length of a side of the square pool is *s *feet and the border is two tiles
wide.

*Students should realize that the only thing that remains constant
in this problem is the dimensions of the border since this information was
given to us in the problem statement, i.e., each border tile is 1’x1’. Also, the students need to remember that
these pools are square pools. This is an
important clue as to how to determine the length and width of the pool. *

*Although there are multiple ways to solve this problem, modeling
the situation using a picture will help most students visualize the scenario in
order to address questions regarding dimension. *

**2. Write a formula in
terms of the variable s that can be
used to calculate N.**

*Many of the students will most likely attempt to draw the complete
picture of the pool, including the entire outside border, in hopes that they
can count the border tiles one-by-one.
This is not necessary if the students realize that the most important areas
of the pool border are the four outside corners of dimension 2’x2’ (see the
above diagram). The other sections need
not be included in the calculation because we already know that these pieces
are equal to side s. *

*Also, although it does not specifically ask the students to use
area in their calculations, they must be able to see that the only way to
calculate the number of tiles needed to create the border of the pool is by
subtracting the area of the pool from the area of the entire outside square that
is created by the border. What is left
will be the area of the border. Since
each of the border tiles are 1’x1’, the area of the border will also equal the
number of tiles, N, needed to create the border. See the calculations below to obtain N.*

Length
of the pool = *s*

Width of
the pool = *s*

Length
of the pool w/border = *s+4* (2
+ 2 + s + 2 + 2)

Width of
the pool w/border = *s+4* (2
+ 2 + s + 2 + 2)

Width of
the pool w/border = s+4

Area of
the pool =

Area of
the pool with the included border =

Area of
the border = Area of the pool with the
included border – Area of the pool

=

**= 8s + 16 = N**

*It is important to note that there are many possible answers that
the students could come up with depending on how they modeled the
scenario. Some of these possible answers
include:*

·

·

·

·

·

·

·

·

*_______________________________________________________________________________________________________________________________________________________________________________________________________________*

**3. Write a different
but equivalent formula that can be used to calculate N.**

*The students should see that both terms of their expression have 8
in common. Therefore, by factoring out
an 8 from each term and writing the result, the students will have created an
equivalent expression for N.*

*Again, the equivalent expressions will vary depending on how the
students modeled the scenario.*

*_______________________________________________________________________________________________________________________________________________________________________________________________________________*

**4. Give a geometric
explanation of why the two different expressions in your formulas for the
number of border tiles are equivalent expressions. Include diagrams.**

*The student in the above example found the two equivalent
expressions to be **. Ironically enough, this student used the fact
that the area of the border is the same as the number of tiles needed along the
border due to the fact that each tile’s dimension was 1’x1’. By understanding that it was an area problem
and coming up with his expression from the above diagram, he, in turn, gave a
geometric explanation of his expression.
To make things interesting, let’s say that another student came up with
the same set of expressions but came up with them in a different way. How would this student represent the scenario
geometrically?*

Let’s
look at the second student’s diagram…

This
student recognized that the entire border was comprised of 8 shapes, 4 squares
of dimension 2’x2’ and four rectangles of dimension 2’x s’. This can be translated into the following
expression: . This
student also realized that, due to the dimensionality of each border tile being
1x1, the area of the border is the same as the number of tiles, N, needed to
create the border.

*The most important thing in any of the examples that students come
up with is that they are able to explain their reasoning and connect the
expression to the figure!!*

*_______________________________________________________________________________________________________________________________________________________________________________________________________________*

*5. Use the Commutative, Associative and/or
Distributive properties to show that your expressions for the number of border
tiles are equivalent.*

*Students should be familiar with the above mentioned
properties. But, for the sake of being
clear, here are the basic property definitions that students should already be
familiar with by this point. *

__Commutative Property of Multiplication and Addition__*: ** and *

__Associative Property of Multiplication and Addition__*: ** and *

__Distributive Property__*: *

Let’s
revisit the different possibilities that some of the students could come up
with for expressions for the number of border tiles, N.

·

·

·

·

·

·

·

·

Using as our
starting point, let’s see if we can get the other expressions to equal this
one.

: The
distributive property tells us that this expression is equivalent to .

: The distributive
property tells us that this expression is equivalent to

:
Expanding this expression gives .

·
*If asked
how to proceed at this point, many students will respond that they need to
“FOIL” the product in order to combine like terms. This is a great opportunity to show the kids
that “FOILING” as they may or may not know it, is just a fancy way of learning
how to distribute a product of two binomials.
*

* *By
distributing the s and the 4 of the first factor to the s and the 4 of the
second, you will simplify the above expression to . By combining like terms, a further
simplification yields . *Note:*
*The Commutative Property tells us
that **!*

*: * The distributive property tells us that
Combining like terms gives the equivalent expression .

: The
distributive property tells us that .
Furthermore, by applying the Commutative Property, we know that .

: The
distributive property tells us that .
Combining like terms gives **8s+16**.

: The distributive property tells us that
Combining like terms gives

**By the transitive property, since every expression is equivalent
to **** every expression is equivalent to each other!**

*The next exercises in this task require that the student is fully
comfortable with the concepts, procedures and solutions of the first 1-5 parts
of the task. For those students who are
still having difficulties with any of the skills in parts 1-5, refer them to
the supplemental text based on individual student need.*

*6. How many 1-foot square border tiles are
needed to put a two-tile-wide border around a pool that is 12 feet wide and 30
feet long?*

*This is a very similar problem to the above. However, instead of a square pool, the pool
is now rectangular. Modeling the
scenario using a picture or diagram is still a great way for the kids to get a
visual grasp of what is going on and what is being asked. *

As in the
above problems involving the square pool, the number of 1-foot square border
tiles needed to put a two foot wide border around the a 12’x30’ rectangular
pool is found by taking the area of the outer rectangle (formed by the outer
edge of the border) and subtracting the area of the pool.

Using
the same diagram, you can also calculate the number of tiles needed to form the
border by looking at the shapes that comprise the pool border. A student could look at the geometry in one
of two ways:

(1) two rectangles of dimension 2’x32’ and two
rectangles of dimension 2’x14’:

__OR__

(2) four
squares of dimension 2’x2’, two rectangles of dimension 2’x12’ and two
rectangles of dimension 2’x30’:

In all
three scenarios, the student should find that the number of tiles needed to
form the 2-foot-wide border around the rectangular pool is __184 tiles__.

______________________________________________________________________________________________________________________________________________________________________________________________________________

**7. Write an equation
for finding the number N of border tiles needed to put a two-tile wide border
around a pool that is L feet long and W feet wide. Explain, with
diagrams, how you found your expression.**