Algebra Tiles come in three shapes, with two fixed but
undefined dimensions.
Unlike Dienes Blocks, the two different sides are not given
a value and neither is a factor of the other.
(note: most manufacturers of so-called Algebra Tiles do not
understand the importance of one length not being a factor of the other and
have unhelpfully created tiles that do factorise – this leads to pupils
stating, for example, x = 5 because five of the small ones make the
length. If purchasing Algebra Tiles, avoid
these types. The ones I use in the video
can be purchased from https://completemaths.com/teaching-tools/physical-manipulatives
)
The tiles can be used in a wide range of ideas from
counting, area, perimeter, arithmetic, through to solving equations, factorising
quadratics, algebraic long division and simultaneous equations. In this blog, I will look at a small number
of uses that have been helpful in introducing teachers to the use of Algebra
Tiles.
It should be noted that Algebra Tiles have been used in the
mathematics classroom for many, many decades (I recall many lessons as a child
using the tiles myself) and that this blog is not meant as a comprehensive
description of their use. It should also
be noted that Algebra Tiles are simply one representation of some ideas in a
wide range of representations. The intention
is not to replace ways of teaching ideas by the exclusive use of Algebra Tiles,
rather it is to augment the teacher’s current multiple representations with
another model and another way of discussing and thinking about ideas. The physical nature of the Algebra Tiles
makes them easy to manipulate – that is, to change mathematical systems and
structures easily and witness the impact of doing so. However, it is not the intention that pupils
are then expected to work with Algebra Tiles whenever faced with the ideas
outlined here. The use of the tiles is
one step in a scaffold towards efficient, symbolic representations and a way of
convincing pupils to accept an abstract idea.
Counting
Using just the small, square tile, we introduce pupils to
using Algebra Tiles at a very young age by using them as simple counters. We can ask pupils to represent a given
number, say, or to use the tiles as a representation of counting up or down
from a starting number. This is extended
to counting up or down in given steps, for example, “Here are 4 tiles, can you
count up in twos until you have 12 tiles?”
Arithmetic
The equals sign can be introduced by stating the law that
the same number of tiles must always appear on both sides of the sign. I would suggest spending a great deal of time
with young children repeatedly creating number sentences such as 4 = 4, 5 = 3 + 2, 7 = 10 - 3, etc. The equals sign is of such fundamental
importance, it cannot be overstated.
Adding and subtracting positive numbers is a way of
extending counting. The tiles can be
used to show addition and subtraction occurring and to give meaning to numbers
and symbols in number sentences.
Multiplication can be shown as repeated addition, arrays,
areas, collections of groups, etc.
Division can be shown as grouping, sharing, repeated
subtraction, etc.
Addition and subtraction of negative numbers requires the
introduction of the reverse side of the tile, which is the first step in
introducing the idea of Zero Pairs.
Zero Pairs
The most important idea in the use of Algebra Tiles is that
of Zero Pairs. Many teachers wish to
move quickly to using the tiles for algebraic expressions or equations and, in
haste, do not spend adequate time establishing the concept of Zero Pairs. Without this idea fully expressed and
appreciated, the efficacy of using of Algebra Tiles is limited.
Each tile is coloured red on its reverse. The red face of the tile is the negative of
the tile. Since the red tile has the
same absolute dimensions and area as the normal colour, combining a tile with
its red counterpart would result in an total area of zero.
Using the idea of Zero Pairs, we can introduce addition and subtraction
of directed numbers.
Now watch the video.
Linear Equations
We now introduce the rectangular tile. The tile has no labelled dimensions. Teachers should spend a good amount of time
working with these rectangles to discuss area before moving on to this
stage. Since there are no labels, we can
tell the pupils that the dimensions are anything we choose. Discussion might include:
“If this rectangle had a length of 6 and a width of 2,
what would its area be?”
“this rectangle has an area of 16, what might the length and
width be?”
before moving on to fixing the width to 1 and building up to
an unknown. To do this, we begin by
using specialised cases:
TEACHER: “The width is 1 and the length is 2, what is the
area?”
CLASS: “Two!”
TEACHER: “The width is 1 and the length is 3, what is the
area?”
CLASS: “Three!”
TEACHER: “The width is 1 and the length is 4, what is the
area?”
CLASS: “Four!”
TEACHER: “The width is 1 and the length is 5, what is the
area?”
CLASS: “Five!”
TEACHER: “The width is 1 and the length is 6, what is the
area?”
CLASS: “Six!”
TEACHER: “The width is 1 and the length is 7, what is the
area?”
CLASS: “Seven!”
TEACHER: “The width is 1 and the length is 8, what is the
area?”
CLASS: “Eight!”
You can see that, by keeping the width fixed as one, the
response to the area is always the same value as the value of the length, so we
can ask
TEACHER: “The width is 1 and the length is a, what is the
area?”
CLASS: “a!”
TEACHER: “The width is 1 and the length is b, what is the
area?”
CLASS: “b!”
TEACHER: “The width is 1 and the length is m, what is the
area?”
CLASS: “m!”
TEACHER: “The width is 1 and the length is x, what is the
area?”
CLASS: “x!”
This perhaps seems a trivial exercise, but we are using the
specialised case to move to a generalised case and the conjecture that, if one
of the dimensions of a rectangle is equal to 1, then the area will have the
same number value as the length of the rectangle.
Now we have a way of speaking about the rectangular tile as
a tile with area x and the small square tile, now a 1x1 tile, as having an area
of 1.
It is important to establish with pupils that we do not know the length of x. It could be any length, could take on any value, could even be the same length as the small tile. We are simply using the tiles to help the discussion of resolving the equation. This is worth spending plenty of time on. We need pupils to understand this is simply a representation - the actual physical size of the tiles is not related to their unknown values, nor are the dimensions in proportion.
It is important to establish with pupils that we do not know the length of x. It could be any length, could take on any value, could even be the same length as the small tile. We are simply using the tiles to help the discussion of resolving the equation. This is worth spending plenty of time on. We need pupils to understand this is simply a representation - the actual physical size of the tiles is not related to their unknown values, nor are the dimensions in proportion.
This allows us to build systems of expressions and
equations. Teachers should start with
expressions, just as they did with asking pupils to show a number. So, for example, we can ask pupils, “can you
show me 4x + 6?” and so on.
Then, because we have established earlier the importance and
meaning of the equals sign, we can move to equations. Prior to this, we would have also used
balancing scales with pupils to show a range of objects balancing and the
impact of doing the same thing to bost sides of the balance.
Now watch the video.
Note, as mentioned earlier, it is not the intention that
pupils always solve equations using the tiles.
We will move them away from the concrete to the symbolic over time as
they gain greater appreciation for the need to keep equations balanced. Pupils will also, quite quickly in many
cases, move themselves on to solving without the tiles.
Factorising
Although this blog is a discussion of Algebra Tiles, it is
not possible to introduce their use in factorising without showing the previous
step, factorising with Dienes Blocks.
Now watch the video.
The video shows a number of examples of using Dienes Blocks
in base 10. It is crucial that pupils
become adept at working with other bases too.
Multi-base arithmetic should be taught from early in learning mathematics
so that the ideas of place value, arithmetic and base 10 itself are fully
understood. The exercises in the video
should be repeated using multi-base Dienes Blocks too. This ensures a sound transition to working in
base x.
We now introduce the final tile, the large square tile.
As before, it is worth taking pupils from the
special to the general case:
TEACHER: “Here is a square.
If this side is length 1, what is its area?”
CLASS: “One!”
TEACHER: “Here is a square.
If this side is length 2, what is its area?”
CLASS: “Four!”
TEACHER: “Here is a square.
If this side is length 3, what is its area?”
CLASS: “Nine!”
TEACHER: “Here is a square.
If this side is length 4, what is its area?”
CLASS: “Sixteen!”
TEACHER: “But squares are such lovely, special rectangles, I
am going to give their areas special names.
I am going to say that the area of a square of length one is ‘one-squared’
and the area of a square of length two is ‘two-squared’”
TEACHER: “Here is a square.
If this side is length 1, what is its area?”
CLASS: “One-squared!”
TEACHER: “Here is a square.
If this side is length 2, what is its area?”
CLASS: “Two-squared!”
TEACHER: “Here is a square.
If this side is length 3, what is its area?”
CLASS: “Three-squared!”
TEACHER: “Here is a square.
If this side is length 4, what is its area?”
CLASS: “Four-squared!”
TEACHER: “Here is a square.
If this side is length a, what is its area?”
CLASS: “a-squared!”
TEACHER: “Here is a square.
If this side is length b, what is its area?”
CLASS: “b-squared!”
TEACHER: “Here is a square.
If this side is length p, what is its area?”
CLASS: “p-squared!”
TEACHER: “Here is a square.
If this side is length x, what is its area?”
CLASS: “x-squared!”
Building on our work with multi-base Dienes Blocks, we can
now move on to factorising algebraic expressions.
Now watch the video.
You can see that we are removing the tiles and introducing
an image in their place. Again, the
intention is to move to purely symbolic representations over time. It cannot be overstated that the use of
physical or digital maniplulatives is simply one step in a scaffold to
efficient, symbolic and abstract methods.
This short blog was intended to be no more than a brief
introduction to using Algebra Tiles.
There are many more mathematical ideas that the tiles can be used to
represent, but I hope this has given you a sense of how they can be used to
augment a range of representations to bring yet another perspective and opportunity
for insight and generalisation.
