## Sunday, 18 March 2018

### An Introduction to Algebra Tiles for Teaching Mathematics

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.

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.