Algebra constitutes a major part of math curriculum, specially in high school and middle school. It forms a crucial base for several equations and operations to be dealt later in higher studies. Hence, it becomes important to get a good grip of it in early phases of learning. For students with learning disorders such as dyscalculia, it can be nightmare.
We already talked about the strategies that could work for teaching algebra to students with dyscalculia. Algebra Tiles plays an important role in achieving this goal. If you are still not aware, Algebra Tiles are widely used mathematical manipulatives to help students better understand algebra. Algebra Tiles gives a better visual perspective to solve the problems.
Although, made to simplify the algebra, but can be confusing sometimes if you are not aware of each corner of the equation you are dealing with. There are several limitaions to where algebra tiles can be used and upto what extent. Keeping this mind, Today we will be discussing 5 core type examples of Algebra Tiles to make you understand it easily. We will reach the extent upto which use of algebra tiles to solve the equations seem meaningful.
In Algebra Tiles, square and rectangle shaped colored tiles are used to represent numbers and variables. As per general rule, Each small square tile, the unit tile, represents number one. If we have two tiles, then we have the number two. Red colored square tiles are used for negative numbers, while any other color could be used for positive ones.
1.) Case of simple addition and substraction
We will exhibit an example of simple addition and substraction of two numbers. We will not going to take variables for this example to keep it simple and easy.
Let’s solve the operation, 5-7+4
We will arrange algebra tiles for this but first, we will restructure the numbers and their signs a bit to further simplify the operation.
The operation can also be written as, 5 + 4 – 7 => (5) + (4) + (-7)
Let’s arrange the tiles and solve it.
It will look something like the tiles arranged in the pic below.
Let’s arrange the tiles after addition of positive numbers as below
Since, positive tiles are more than negative tiles, the answer would be positive number. After rearranging, we will take 1 red color with 1 yellow color tile till the no red ones are left. Whatever we are left with yellow ones will be the number.
2.) Case of simple multiplication and division
We will exhibit an example of simple multiplication and division of two numbers. Again, we will not going to take variables for this example to keep it simple and easy.
Let’s solve the operation, 8 * 3 ÷ 4
We will arrange algebra tiles for this but before, we will restructure the numbers and their signs a bit to further simplify the operation.
The operation can also be written as, (8 * 3) ÷ (4) (Make sure you do not break the BODMAS principle while rearranging)
In some cases, the value is negative. Do remember that a negative value times a positive result gives a negative result.
Let’s arrange the tiles and solve it.
For Multiplication, we will make a grid with the bigger number in the columns and smaller in the rows (You can do as per your choice). Count the total number of spots in the grid, which is the outcome of the multiplication.
Let’s do the division. For this we will take the grid and rearrange with rows equal to the divisor. Now pair it up as such given below
The number of rows in the column will be the outcome of the division.
3.) Finding x
We will jump to solving equations in this bit. To model the equations, The rectangle represents the variable, x. The large square represents x2 (x to the power 2). Solving expressions via. algebra tiles is a two way process – modelling the equation and equating it. We got to balance the expression. What we do on the one side of equal sign, we must do the same on the other side as well.
The approach is to isolate x and get the x alone on one side of the equation.
Let us solve the expression 3x + 5 = 14
First we will model the equation. To achieve it, Divide the sheet or board in two parts. Keep the 3 yellow rectangle tiles (representing 3x) with 5 blue square tiles (representing number 5) on the left side and 14 blue square tiles (representing the number 14) on the right.
Now that we have modeled our equation, we will going to solve it. For our equation, Our first goal is to get the 3 yellow rectangle tiles by themselves. We need to move the 5 blue square tiles .
To move them, we need to pair them up with a different colored square. As the tiles are positives, we need to pair them up with the negative number. We, thus, need equal number of red colored tiles. We remember that whatever we do to one side, we also must do to the other side. So, we also add four blue square tiles to the right side.
Now, we can take out the blue and red tiles on the left tiles as they cancel out each other. Similar we take 5 blue and as well the red tiles from the right side.
We are left with 3 yellow rectangle on the left and 9 blue square tiles on the right. Split and rearrange the 9 blue square tiles in three qual groups. Now you can get the answer by looking at the tiles i.e each yellow rectangle represents 3 blue square tiles.
4.) Distributive Property
To model the distributive property of the equation, we will use the expression 2(x+3) = 2x + 6,
On algebra tiles, it can be represented as
5.) Working with polynomials
Solving polynomials may get a little bit tough. Proceeding with algebra tiles can make it really easy though. Addition and subtractions of the two equations work same as before. In this we add or subtract the value with the same variable degree. It could be represented on algebra tiles as such
To do the multiplication, we will take the example of (x+3) (x-2). We will draw two lines, one horizontal and other vertical. We will place the the terms (in terms of tiles) on the lines as given in the image below.
Every value in the term will multiply with the value in the other terms and represented on the tiles as such. Be sure that straight lines are maintained with grid at all the times.
For division of polynomials, we will drive the procedure in reverse.
Let us the expression -> (x2+2x-3)/(x-1)
We will fill the internal grid with numerator. Then we will place the denominator on on of the lines as given below
The result will be whatever the outcome on other line of the grid as given below.
So we discussed 5 different types and examples of expression that can be solved using algebra tiles. This post covered what most students will face in early algebra. Good practice with these will help you build a good base in advance for lengthy equations to come in the senior years. If you still have any doubts, you are most welcome to comment your questions or queries to this post or use our contact form.
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