ACTIVITY 5
Area Models
Goal
To help students build meaningful concrete and visual models to demonstrate the distributive
property.
Materials
 dienes blocks, graph paper, overhead blocks, overhead graph paper (paper models will do)
 algebra tiles
 scissors
Links to Research
Arrays can be added and compared to build not only fact recognition and recall but also
understanding and application of the commutative, associative and distributive properties of
multiplication.
Area models help students build the mental constructs they need to conceptualize multiplication
as a two-dimensional operation. Many students can quickly and fluently transfer their
understandings of whole number multiplication to fractions, decimals and algebraic expressions
when they have internalized a visual image for area.
When asked what multiplication is, students invariably answer “repeated addition” or “a fast way
to add.” If that were all they have as a construct for multiplication, then why would any student
think it was important to learn to multiply? Addition can always do the trick. The problem with
this thinking is that it is not true. Multiplication is much more than repeated addition. It is also
dimensional. It creates areas. Even though this is an algebra workshop, we need to consider how
an understanding of multiplication can lead to success with polynomial work in algebra.
Instructions
Area models are briefly introduced with whole numbers both single and double digit.
Area models highlight the commutative and distributive properties of multiplication.
1.
Draw an area model for the multiplication fact 3 × 4 on the overhead.
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This is a 4 by 3 or a 4 × 3 array. It covers an area of 12 squares. Its side lengths or dimensions
are 3 and 4. I can use the multiplication 3*4=12 to describe it or the multiplication 4 * 3 = 12 to
describe it.
2.
If I turn my array what do I have? (Still 3 * 4 or 4 * 3, both of which still cover an area of
12.)
3.
This shows (3 × 4)+ (4 × 4)
12 + 16 = 28
(3 × 4)+ (4 × 4)
7 × 8 = 28
Where does the 7 come from?
Where does the 8 come from?
If you can’t remember a fact, you can always break it apart:
3×8+3×8= 6×8
24 + 24
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What property are we practising?
13 × 12
3
+
10
10
Can you see 10 × 10?
Can you see 10 × 2?
Can you see 10 × 3?
Can you see 2 × 3?
+
100
20
30
6
2
(10 + 3) (10 + 2) = 13*12
Can you see 13 × 10 = 130?
Can you see 13 × 2 = 26?
This way of thinking quickly becomes shortened when students practise it. They create variations
of visual and mental models that help them make sense of the multiplications.
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Build 13 × 14.
There are two ways to build this model. We can lay out the side lengths and build the new area
inside them, then remove the sides.
We can use the side lengths as the starter points for connecting to the new area.
Alge tile trays allow a trough for students to build the sides in, and the new area is inside the
boundary of the trough.
However, when students work without the tile tray, they often simply construct around the side
lengths.
So you can begin by outlining the sides and building inside them:
How does each of these pictures relate to the equation 13 × 14 = ?
14
×
14
13
13
13
14
13
14
Alge tiles focus on the second diagram.
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This way of thinking quickly becomes shortened when students practise it. They create variations
of visual and mental models that help them make sense of the multiplication.
Students begin to think in visual ways like the following Overhead 11.
36 * 45
might become:
10
10
10
10
10
(40 × 30)=
1200
10
10
40
5
40 × 30 =1200
40 × 6 = 240
5 × 30 = 150
5 × 6 = 30
1200
30
4 × 60
5×6
6
This is a visual representation of the distributive property that allows us to jump from whole
number multiplication to manipulating expressions in algebra.
(10 + 4) (10 + 3) has a four step solution that can be shortened once students develop a
conceptual understanding of what is being multiplied.
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14 × 13 is the same as (10 × 10) + (10 × 3) + (10 × 4) + (4 × 3).
Can you find each of the pieces?
13
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Where is the 13 × 14 in each drawing?
Can you see it in your mind’s eye?
1 0
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What multiplication fact is this a representation of?
Write the face in expanded notation.
Can you name the partial products?
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Learning About Algebra Tiles: Activities for Teachers1
Algebra tiles provide a useful way to introduce operations on polynomials to students of all ages.
The following can be used both as a self guided tour of teaching with algebra tiles, and as a set of
activities for use with students. Examples of instructions to students are indented. The remainder
of the text is addressed directly to the reader who is assumed to be a teacher.
Getting Started
Work with either the basic commercial set (small white & red squares, large white & green
squares, and long white & green rectangles) or cutouts from one of the attached sheets. The
commercial sets are nice and sturdy, but tempt students to negative numbers too quickly and cost
too much to give away a set to every kid. The sheets are not as sturdy, but can be made available
to kids to take home and delay the idea of negatives until the teacher is ready to cope with it.
You may also have a commercial set with other colours for a second variable, and some more
squarish rectangles. Hide them for the moment. The third sheet in the cutout set provided fulfills
the same functions, so ignore it for the moment, too.
Naming
I call the small squares “nameless ones.” This captures two important ideas: They aren’t named
when we talk about polynomials, and their value is 1. The reason for the name isn’t something
you’d tell kids on the first day, but having reasonable names for things makes the going easier
later on.
Nameless ones behave like the bingo chips or whatever from learning integers, and like the bears
the kids counted in kindergarten. One of them stands for 1, and five of them stands for 5. If you
are using the cutout ones, they are one square centimeter, and you can make a connection to
measurement if you want.
What do you want to call the other pieces? Settle on names within groups, and within the class.
Naming conventions are important to mathematical communities. You might find you have
pieces which look similar to another group’s but not exactly the same (rectangles that are longer
than yours for example). It’s OK to give these the same names because they are used in the same
way.
Some names which students have proposed in the past are: Longs & Big Squares, Rectangles &
Rectangle squares, and X and X2. The point to emphasize with names is not that there’s a correct
name for each piece, but that we need names to be able to talk about them. The mathematical
concept here is that a variable is a name for something else (usually an unknown or variable
quantity).
1. From http://plato.acadiau.ca/courses/educ/reid/Virtual-manipulatives/tiles/tiles.html, accessed September 19,
2005.
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Collections, part 1
Make a collection of pieces, using all three kinds, and write down its name. Settle on a
convention for writing collections which shows that all the pieces are added together in one
collection. Abbreviations might help. You could call this collection:
2BS + 3R + 5 (for two big squares, three rectangles and five nameless ones)
or 2R2 + 3R + 5 (for two rectangle squares, three rectangles, and five nameless ones)
or 2x2 + 3x + 5 (for two X2, three X, and five ones)
Here you want to try to get the names sorted out, so that groups can describe their collections.
Have the students draw their collections, and write the names of the collections with symbols.
Collections, part 2
Combine the collection you made with a collection another group member made. How would
you name the resulting collection? How would you draw or write the action of combining them?
Now make a collection and take some pieces out of it. How would you name the resulting
collection? How would you draw or write the action of taking out pieces from a collection?
This is introducing the idea of adding and subtracting polynomials. Not much attention needs to
be paid to this right now. All we are doing is agreeing on some more notational conventions:
(2R2 + 3R + 5) + (R2 + 2R + 6) = (3R2 + 5R + 11) means:
+
=
and
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(2R2 + 3R + 5) - (R2 + 2R + 3) = (R2 + R + 2) means:
=
where crossing out the picture of a tile denotes removing it from the collection.
Again, have students perform the action on the tiles, and then draw the actions, and then write
the symbols for the actions. If some of your students want to stop drawing everything and just
write symbols, that might be OK, but don’t let them get the idea that drawing pictures and
moving tiles is the “bad” way to do things.
Making rectangles
Make a collection with two big squares, seven rectangles, and six nameless ones. How would
you name that collection? Now arrange the collection so it makes a rectangle.
Both of these are rectangles made with the same pieces. I like the one on the right better. Why
not introduce some aesthetic rules of Algebra tile rectangles? My first rule would be:
 Big squares can’t touch little squares.
My second rule (which I am more lenient about) is:
 Little squares must all be together.
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Measuring
Use rectangles and nameless ones to “measure” the length and width of your rectangles. For
example:
the length and width of this rectangle are 2R+3 and R+2. It is sometimes confusing to have the
measuring tiles so close to the rectangle. To clarify we can use “crossed lines” to separate them.
(The X in the corner is because they are “crossed”)
Spend some time on making rectangles, drawing them, measuring them, and writing down the
collections and their measurements. For example, I could write (2R2+7R+6) = (R+2)x(2R+3) for
the rectangle I made above.
If your students are getting tired of drawing you might introduce them to this notation:
X
2R
R
2R
2
2
4R
3
3R
6
This shows how the rectangle is made, and its measurements.
Now you have the idea of rectangles sorted it you might give the students a break and do
something easy for a while:
Given two measurements, what tiles are in the rectangle? Start by putting the tiles for the
measurements along your crossed lines, and then fill in the rectangle. Draw your result (or use a
symbol table).
A pause to get our bearings.
You now know how to teach adding and subtracting polynomials (as long as there are no
negatives involved), and factoring and multiplying polynomials with positive coefficients.
Dividing is just factoring with one factor already known, so I skipped it. From here you can go a
couple of directions.
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You can introduce a new variable, and pull out those other tiles. If you are using the cutouts you
need to get the kids to choose a new label for one set to distinguish it from the other one. We’re
going to use them differently now, so it’s time for a new name. You also need those squarish
rectangles. What do they stand for?
Another direction you can go introduces negative coefficients. This is a bit tricky, so I’ll head
that way now, and let you figure out extra variables by yourself.
Back to square ones.
How would you take seven ones from three ones? Remember your work with integers. It’s OK to
turn the ones the other way up now. If you are using the commercial tiles they change colour, so
you can use them for negative ones. If you are using the cutouts nothing happens at all, so this
might be a good time to shade one side of each of your cut out tiles. Now, how do you find
(2R2 - 3R - 7) - (R2 - 5R - 3)?
Sorry about all those symbols, but by now you and your students can probably handle them. Did
everyone get
? Good.
Crossed-lines with a flip.
Set up your crossed-lines and figure out how these new negative pieces work there when you
multiply. Draw some examples, using symbols if you want. This part is pretty easy after you’ve
mastered integers and multiplying polynomials with positive coefficients.
If your students are a bit shaky on multiplying integers you might want to remind them that
positive numbers are used to count, so 1x1 means “one times one,” which means one 1, which is
1; and 1x(-1) means one (-1) which is (-1). Once you have that sorted out, you can look at this
multiplication of zero times zero (with zero represented as (+1 + -1)):
?
Figuring out what to put in three of the four positions is easy. The fourth one is harder. But if we
think about what we are multiplying (zero times zero) then we know what the answer is (zero),
so the fourth position must be occupied by (+1).
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Now for the tricky part
Let’s factor (R2+ R - 6). Put the pieces between your crossed lines. There aren’t enough. Try
anyway, and put things where you learned they belonged when you were working with positive
coefficients:
We know the R2 and the nameless ones can’t be in the same row or column because that violates
one of our aesthetic rules. That leaves four spaces which look like they ought to have rectangles
in them. Remember about adding zero? Try it. Now try some others.
Have students make up some by multiplying and give them to other people in their group.
You might have discovered already that it isn’t always clear where to put the rectangles you add.
It’s easy in the above example to know you need to add two positive rectangles and two negative
rectangles, but you can’t put them just anywhere. For example, if you put them like this:
You’ll find you can’t measure your rectangle. We need a new aesthetic rule:

The squares in a column or row must either all be the same colour as the rectangles, or all be
the other colour.
Given this rule, we can see that we should put our four new rectangles in like this:
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In conclusion
Polynomials are unlike the other “numbers” students learn how to add, subtract, multiply and
divide. They are not “counting” numbers. Giving polynomials a concrete referent (tiles) makes
them real.
Implicit in the treatment I have given above is an important educational principle:

Learning, at any level, is easiest if it starts with concrete actions, then moves to iconic
representations of those actions, then to symbolic representations of those actions, and finally
to formal actions on symbols.
In the case of the tiles these four stages are:
1.
2.
3.
4.
Actions on the tiles themselves
Drawings of those actions
Using symbols in or instead of the drawings
Traditional algebraic manipulations of symbols.
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How Can a Box Help My Students with Multiplying Polynomials?
For most of my students, multiplying polynomials is
not a difficult procedure to understand. Students
quickly grasp the idea that each term in the first
polynomial must be multiplied by each term in the
second polynomial. The difficulty, however, lies in
the organization of the product’s terms for ease in
combining like terms and in the accuracy of having
multiplied all of the appropriate subproducts. The
purpose of this article is to demonstrate how a box
can help students with both of these aspects.
x
x
4
Fig. 1 Setting up the box to multiply (x + 4) and (x + 2)
Multiplying Binomials
Consider the example of finding the product of
(x + 4) and (x + 2). Since each polynomial has two
terms, begin by drawing a 2-by-2 box and placing
the terms of the binomials on the outside (see fig. 1).
To complete the box, multiply each row term by
each column term and fill in the corresponding cell
of the box (see fig. 2). Once all cells of the box are
filled, the contents of the cells represent the product
of the binomials. Combine like terms to arrive at the
product of x2 + 6x + 8. A second example,
(2x – 5)(x – 3), is worked in a similar fashion but
requires that we pay particular attention to the
operation signs (see fig. 3).
Students should be given the opportunity to explore
the properties of the 2-by-2 boxes. Of particular
interest are the following items:
 It does not matter which binomial is placed
along the top of the box and which is placed
along the side.
 The terms that lie on the outside of the box are
actually the greatest common factors for each
row and column.
 The products of the diagonals are equal. To see
why this is true, use the box to multiply
(a + b)(c + d) and compare the products of the
diagonals.
 In general, like terms will lie along a diagonal
that runs from the lower left to the upper right.
Multiplying Higher-Order Polynomials
The box can easily be extended for use with higherorder polynomials.
Consider the example of
(2x2 + x + 7)(3x3 + 6x2 – 4x –5). Since this product
involves multiplying a three-term polynomial by a
four-term polynomial, we will need a 3-by-4 box
(see Fig. 4). Having multiplied the terms, students
will notice that the like terms lie in diagonal cells of
the box and should be encouraged to investigate
whether or not this will always be true.
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2
x
x
x2
4
4x
2
2x
8
Fig. 2 The solution to the product of (x + 4) and (x + 2)
x
2x
2x2
–5
–5x
–3
–6x
15
Fig. 3 The product of (2x – 5) and (x – 3)
2x
3x3
–6x5
6x2
–12x4
–4x
–8x3
–5
–10x2
x
3x4
6x3
–4x2
–5x
–28x
–35
2
7
3
21x
2
42x
Fig. 4 Extending the box to find the product of
(2x2 + x + 7)(3x3 + 6x2 – 4x –5)
[Try, for example, (x + 7)(x5 + x4 + x2 + 1) and see
what happens.] Students should be asked to explain
when this property will be true of the terms in the box.
Conclusion
By providing structure to the process of multiplication,
the use of the box has enabled my students to become
more organized in their approach to the problems and
has reduced the number of errors. Your students will
appreciate this alternative representation for the
multiplication of polynomials.
From Mathematics Teaching in the Middle School, Vol. 9, No. 9,
May 2004.
Algebra, Junior High Workshop Series
SOLVING EQUATIONS
Solve the following equations with the algebra tiles. Model and draw them out. Then solve to
check your answer.
You will work with a partner and take turns doing the following:
a)
You first need to decide who will be the scribe and who will be the person who
models with the algebra tiles.
b)
The one partner starts by modelling with the tiles and telling the other person the
steps to solve the equation. The other partner knows what to write by the first
partner telling them step-by-step what to do.
c)
Then switch roles. If help is needed, take turns guiding each other. Do not have one
person explain how to do everything.
1. 3x = 5 + 1
2. 2x + x = -4 + 1
3.
x 42
3
4. x + 4x –3x = 3 + 1 – 4
Reprinted with permission of Jackie Ratkovic, Cold Lake, AB.
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Multiplying Binomials by Binomials
Cut out the binomials and organize them into groups, then:
1. Discuss and write down why you sorted the binomials as you did.
2. How are the groups the same? Different? Explain.
3. How is multiplying binomials similar to finding the area of a rectangle or square?
Explain by drawing a picture using algebra tiles, e.g., (x + 2)(x + 3)
If you were given the area, could you find the dimensions of the following rectangle?
x 2  5x  4
4. Choose 3 binomials, write them down and multiply them together.
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Factoring Polynomials
Cut out the polynomials and organize them into groups, then:
1. Discuss and write down why you sorted the polynomials as you did.
2. How are the groups the same? Different? Explain.
3. How is factoring the polynomials similar to finding the length and width of a rectangle or
square if the area is given to you? Explain by drawing a picture using algebra tiles, e.g.,
A = ( x 2  4x  3 )
E.g., A = ( x 2  4)
4. How is factoring a polynomial similar to multiplying a binomial? How is it different?
Explain.
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Final Reflection:
5 3
1
On your own, list 5 important or key ideas from today that will impact your
teaching.
With a partner, share your five and between the two of you pare it down to 3 key
ideas, phrases or words to summarize your learning today.
Across the table share your 3 with another pair. From that discussion can you pare
down your six to 3?
Share with the whole group.
Can you summarize today in one statement or key idea?
What is the one thing that will change or impact your teaching when you return to
your classroom?
That will be your goal from today. Write it down. Commit yourself.
What data will you bring to the next workshop to share?
Write it down. Commit yourself.
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