Middle school algebra

Algebra Tiles

Pi

I once thought algebra tiles were stupid. But these days, I believe that they are essential for young children up to Grade 10 to understand how the factored quadratic is expanded, and to actually “get” the FOIL method when it’s introduced.

It is especially revealing when you understand the root of the word “quadratic” comes from “quadrat”, which is a device used to measure land area. Imagine buying a tract of land (x – 3) meters long by (x + 4) meters wide. Such a property must be rectangular, and I think for consistency sake, so should your algebra tiles. The placement of the tiles should reflect the dimensions of the rectangle. That is, an observer should be able to make out the factors of the quadratic in the finished product.

I actually don't own any algebra tiles, so I just drew a picture conveying the general idea. The square is the square of x; the lines each represent x, to make 8x; where the lines cross each represent one more number added to the constant term. 15 crosses thus make the number 15. This looks like a great idea for teaching how binomials can be multiplied to kids who haven't seen it before. This graphc was scrawled out in MS-Paint.

If you are like me, you probably don’t own these things, but to teach expansion of binomial factors to kids, you can certainly draw squares and lines.

Multiplying a pair of binomials will generate a quadratic upon expansion, of the form Ax^2 + Bx + C.

Along the horizontal, we can suss out a measurement of x + 5 units in length in the first illustration; along the vertical we see a dimension of x + 3 units. Draw the lines so that they continue past the square and extend so that they cross all of the lines going in the other orientation. Counting the number of crosses will give you the constant term (C); counting the total number of lines will give the coefficient before the x term (B); and counting the number of squares give the coefficient of the 

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x^2

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term. Knowing that MS Paintbrush does an ugly job when using the same colour, I used a different color for the horizontal lines. These different colors can become handy when the binomial contains a subtraction. For example:

This is an improved graphic, and does illustrate how the black represents subtraction.

This second illustration shows how 

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-2x

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may be represented by algebra tiles.  We see the middle term go to 

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3x - 2x = x

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. But what about the crosses? We need to make up a rule whereby if the crosses are of different signs (that is, different colors), the count is to a negative number (in this case, six crosses make 

*** QuickLaTeX cannot compile formula:
-6

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). Conversely, if they are of the same sign (both negative or both positive), the count is to a positive number.