INTRODUCING VARIABLES
AND WORKING TOWARD
ALGEBRA
Math Smart TLQP Workshop @ Binghamton University
December 14, 2013
Dr. Rachel Bachman
Weber State University
I have more than impression – it amounts to a certainty – that algebra is
made repellent by the unwillingness or inability of teachers to explain
why we suddenly start using a and b, what exponents mean apart from
their handling, and how the paradoxical behavior of + and – came into
being. There is no sense of history behind the teaching, so the feeling is
given that the whole system dropped down ready-made from the skies,
to be used only by the born jugglers. This is what paralyzes – with few
exceptions – the infant, the adolescent, or the adult who is not a juggler
himself.
~Jacques Barzun, Teacher in America, 1945
THE PROBLEM
s = ½ at2 describes the relationship between time and distance
fallen for a free falling object, where
s = distance fallen in feet
a = acceleration due to gravity (about 9.81 ft/sec2)
t = time in seconds
What are the variables represented
in this function?
This issue with this problem is that students traditionally thinking that s, a, and t are all
variables because they are letters. However, a is not a variable because it is a constant.
THE STRATEGY
Real Activity 
Extending a Pattern to the General Case 
Verbal Communication of Pattern 
“Syncopated” Communication of Pattern 
Symbolic Communication of Pattern 
TODAY’S AGENDA
 Clap
 The
Your Name
Border Problem
 Total
Wall Area
 Introducing
Algebra Tiles
Math Dance can be ordered only from mathdance.org. If you have any further
questions, you can contact Erik Stern at estern@weber.edu
CLAP YOUR NAME
Schaffer, K., Stern, E., & Kim, S. (2001). Math Dance with Dr. Schaffer and
Mr. Stern. Santa Cruz, CA: MoveSpeakSpin
RACHEL
-Start off by clapping your name for the students. Clap your hands for consonants and slap your legs
for vowels. See if they can figure out the pattern of clapping and slapping.
-Once you have the clapping/slapping pattern established, have the students practice clapping your
name with you.
NOW IT IS YOUR TURN….CLAP YOUR
NAME
 Clap
 Slap
Y
for Consonants
for Vowels
 You get to choose clap or slap
 Have
the students practice clapping their own
name. Then have a few students
demonstrate. Pick up on any variations they
are doing (syncopating, pausing, etc.). Teach
the students to repeat their clapping
sequence with equal beats.
-Once the students can repeat their sequence, you pick one of their names to clap
but this time capitalize their name by clapping/slapping louder on the first letter of
their name. See if they can pick up on the change you made. Have them practice
capitalizing their name while repeating their pattern.
-Choose two students in the class (with different length names). Have one of the
students work with one half of the room to clap his or her name (repeating and
capitalizing). Have the other student do the same thing with the other half of the
room. Once both groups are well practiced, perform the two names together.
Introduce a discussion about what the sequences sounded like together. Ask if the
capitalizing was always occurred together.
After how many beats will the two names
begin their sequence at the same time?
1. Create a drawing of the two clapping
sequences happening at the same
time.
2. How many times does each sequence
need to be played till they begin at the
same time? How many beats is that?
3. With your partner, use your method of
drawing to answer for the same
HOW ABOUT FOR THREE OR FOUR
NAMES?
THE BORDER PROBLEM
Boaler, J., & Humphreys, C. (2005). Connecting mathematical ideas:
Middle school video cases to support teaching and learning. Portsmouth,
NH: Heinemann.
Without
counting
one by
one,
determine
how many
squares are
in the
border of
this 10x10
grid.
The chart below shows the six methods for the Border Problem
10 x 10
6x6
15 x 15
231 x 231
10 + 10 + 8 + 8
6+6+4+4
15 + 15 + 13 + 13
231 + 231 + 229 + 229
4(10) – 4
4(6) – 4
4(15) – 4
4(231) – 4
4(8) + 4
4(4) + 4
4(13) + 4
4(229) + 4
4(9)
4(5)
4(14)
4(230)
10 x 10 – 8 x 8
6x6–4x4
15 x 15 – 13 x 13
231 x 231 – 229 x 229
10 + 9 + 9 + 8
6+5+5+4
15 + 14 + 14 + 13
231 + 230 + 230 + 229
Step 1: Figure out the methods for a 10 x 10 grid
Step 2: Edit the methods for 6 x 6, 15 x 15, and a 231 x 231 grid
Step 3: Pick one of the six methods. Write directions for calculating the number of squares in
the border for any square grid using that method. Use only words, no symbols.
Step 4: Work on one of the methods with the class to be as specific and precise as possible.
Step 5: Edit the instructions by swapping out numerals and symbols, leaving only the words
“number of squares on one side of the grid” in the instructions.
Step 6: Introduce a symbol to take the place of the words “number of squares on one side of
the grid”
Step 7: Write all of the methods symbolically
THE BORDER PROBLEM EXTENSION

Verify all the strategies are algebraically equivalent.

What does the expression mean in reference to the Border
Problem?

What does the sentence mean in reference to the Border
Problem?

What does the sentence mean in reference to the Border
Problem?

What does mean in reference to the Border Problem?
TOTAL WALL AREA
Bachman, R. M. (2013). Building conceptual understanding in a remedial
college mathematics classroom: A study of effectiveness. (Order No.
3596992, State University of New York at Binghamton). ProQuest
Dissertations and Theses, , 324. Retrieved from
http://search.proquest.com/docview/1449822624?accountid=14168.
(1449822624).
What
is the total wall area in a room
8 feet high, 12 feet wide, and 16
feet long?
Suppose
a painter wanted a
formula to use to calculate the wall
area of any rectangular room.
Develop such a formula for the
painter.

The most common expression is 2LH + 2WH (finding the area of each kind of
wall, doubling each to get the wall just like it, and adding all the wall areas
together). Once we come up with one expression, I ask the class to interpret
the following expressions with reference to the room…
a. H(2L + 2W) -- perimeter times height…one long wall
b. H(L + W + L + W) -- same as above
c. 2H(L + W) -- the length of half way around the room times height gives
half the wall area and then times 2 gives the total wall area
d. 𝐻 × 4
𝐿+𝑊
2
-- find average wall length, multiply by four to find the
perimeter and then multiply by height to find the wall area.
ILLUSTRATIVE MATHEMATICS
PROBLEMS
Illustrativemathematics.org
Some of the students at Kahlo Middle School like to ride
their bikes to and from school. They always ride unless it
rains.
Let d be the distance in miles from a student's home to
the school. Write two different expressions that represent
how far a student travels by bike in a four week period if
there is one rainy day each week
DISTANCE TO SCHOOL - 540

Suppose P and Q are two different animal populations, where Q
> P. For which, say what the expressions mean and then decide
which expression is larger.
a.
𝑃+𝑄
AND
2𝑄
b.
𝑃 + 50𝑡
AND
Q + 50𝑡
c.
𝑃
𝑃+𝑄
AND
0.50
d.
𝑃
𝑃+𝑄
AND
𝑃+𝑄
2
e.
𝑃
𝑄
AND
𝑄
𝑃
P AND Q - #436

A company uses two different-sized trucks to deliver sand. The first
truck can transport x cubic yards, and the second y cubic yards. The
first truck makes S trips to a job site, while the second makes T trips.
What quantities do the following expressions represent in terms of the
problem's context?
DELIVERY TRUCKS – #531

A candy shop sells a box of chocolates for $30. It has $29 worth of
chocolates plus $1 for the box. The box includes two kinds of candy:
caramels and truffles. Lita knows how much the different types of
candies cost per pound and how many pounds are in a box. She said,
“If x is the number of pounds of caramels included in the box and y is the
number of pounds of truffles in the box, then I can write the following
equations based on what I know about one of these boxes:
𝑥+𝑦 =3
8𝑥 + 12𝑦 + 1 = 30
Assuming Lita used the information given and her other knowledge of the
candies, use her equations to answer the following:

How many pounds of candy are in the box?

What is the price per pound of the caramels?

What does the term 12y in the second equation represent?

What does 8x+12y+1 in the second equation represent?
MIXING CANDIES - #389
1. Player E and Player F are two different baseball players. On average, Player E
scores more homeruns per season than Player F. Let e represent the average
number of homeruns scored by E each season and f represent the average number
of homeruns scored by F each season.
a. Write a variable expression that represents the number of additional homeruns
Player E makes on average compared to Player Fin a season.
b. Write a variable expression that represents the ratio of the average number of
homeruns made by Player E to the average number of homeruns made by
Player F this season.
c. Explain which expression is larger: 𝑒 + 𝑓 or 2𝑒
d. Interpret the expression
𝑒+𝑓
2
in reference to E and F.
e. Write a variable expression that represents the number of total homeruns
made by E and F in their entire careers. Define the variables you use.
IN THE SAME SPIRIT…
INTRODUCING ALGEBRA TILES
Bachman, R. M. (2013). Building conceptual understanding in a remedial
college mathematics classroom: A study of effectiveness. (Order No.
3596992, State University of New York at Binghamton). ProQuest
Dissertations and Theses, , 324. Retrieved from
http://search.proquest.com/docview/1449822624?accountid=14168.
(1449822624).
LET’S REVISIT THE BORDER PROBLEM…
-You can use base 10 blocks to model the border problem strategy of doing
four times the side length minus four for the corners. Then transition to a
multi-base block other than ten to show the method for a different grid.
Finally introduce the algebra tiles to show the method working on any
square grid.