Mesmerizing Manipulation Madness!

Mesmerizing Manipulation Madness!

Math can drive students mad sometimes. It helps to have a visual aid. But it's even better to have a physical aid that students can touch and manipulate. Manipulations are exactly that; they are physical representations of math concepts to help students learn. Geometric blocks can be used as manipulations, and I remember using them in my kindergarten class to learn about finding simple patterns, which laid a very important foundation for future math concepts. A big key in unlocking the mystery of math, after all, is finding and using mathematical symbols to represent patterns. So, it's very unfortunate that I never saw those blocks being used as a teaching tool in subsequent math courses again for the rest of my elementary and secondary school education.

Another great example of a manipulation tool are Algebraic Tiles. In class, we used these to explore manipulations to teach students about factoring quadratic equations. We also had a reading assignment on how to use Algebra Tiles for teaching: "Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts," by Annette Ricks Leitze and Nancy A. Kitt in Mathematics Teacher, Vol. 93, No. 6, September 2000 (pg 462-466 and pg 520). Students can do a quick internet search to find templates of these to print off and cut for class and their homework. Each term of the quadratic is represented: the tiny squares represent constants, the bars represent x, and the big squares represent x^2. For this blog, I will show how I would use these tiles to introduce quadratic expressions and factoring to my future students by using what I've learned from class and in the reading.

To slowly show how to use these tiles to represent linear and quadratic expressions, I would pick very simple equations for the students to represent with their Algebraic Tiles like x + 2 (Figure A) and x^2 (Figure B).

Figure B: x^2

Figure A: x+2

 
Here, I would make the connection that the bars squared equals the area of a square in Figure B, and that both x's are factors of x^2, which was a nifty trick Professor Golden showed us in class. Then, I would gradually make them harder like x^2+3x+2 (Figure C) and 2x^2+9x+4 (Figure D). Again, I would show that the sides represent the factors of the quadratic. I've circled the factors in blue (although, they aren't very visible). The factors for Figure C are (x+2)(x+3), and the factors for Figure D are (2x+1)(x+4). Here, students can clearly see the relationship between the factors and the quadratics they form. For both examples, students can count the number of big squares, bars, and tiny squares to get the quadratic expression that the factors create.

Figure C: x^2+5x+6

Figure D: 2x^2+9x+4

Next, I would introduce subtraction into the quadratic expressions. Tiles representing subtraction should be a different color or pattern from the tiles representing addition. To illustrate how the quadratics change when the + and - operations are flipped between factors, I chose quadratic expressions that have the same numbers in their factors. Figure E is x^2-x-6, and its factors, circled in blue, are (x+2)(x-3). Now the students might get confused and say but there are 5 bars, so it should be 5x. But the red bars mean subtraction, so for each red bar, a white bar gets paired up with it, and whatever is leftover is the value of the x-term in the quadratic expression. So, for Figure E, 2 red bars get paired with 2 white bars, and 1 red bar is leftover, so the x-term is -x. Now, some students might ask why the big square is white and not red, since the big square is flanked by a white bar and a red bar, and a positive x (+x) multiplied by a negative x (-x) should be a negative x^2 (-x^2). To clear the confusion, I would inform the students that the red bars represent an operation: subtraction. They don't represent a negative x. So, whenever there is a -x, I will say "minus x," and not "negative x." (This should also be done with plus x (+x), not positive x). Next, Figure F is x^2+x-6, and its factors are (x-2)(x+3). I would ask the students for the similarities and differences between the factors of Figure E and Figure F.  Then, I would reiterate that the numbers in the factors are the same, but the operations are flipped, which changes the quadratic expression they form. So, 2 red bars get paired with 2 white bars, and 1 white bar remains, which means the x-term is plus x (+x). Then, Figure G is x^2-5x+6, and its factors are (x-2)(x-3). Here, both factors have subtraction, so there are 5 red bars, no white bars. Then, the x-term is minus 5x (-5x).

Figure E: x^2-x-6

Figure F: x^2+x-6

Figure G: x^2-5x+6

Now, if the students seem comfortable with this exercise so far, -x^2 could be introduced. This will be completely contradictory to what I have just told the students: that red tiles means subtraction, not a negative x or number. But for this portion of the exercise I am going to break that rule; that we will now be multiplying a negative x (-x), and I would tell the students so. It will be harder for the students to factor, but I think it would be a good problem they can think through on their own. I also made the quadratic expression and the factors to be very similar to make it easier. And if they can't factor it, then I would do it together as a class. First, Figure H is -x^2+5x-6 and its factors are (-x+2)(x-3). Next, Figure I is -x^2+x+6, and its factors are (-x-2)(x-3). Since we have a mix of white and red bars, we need to use subtraction. So, 2 red bars get paired up to 2 white bars, and then 1 white bar is leftover, which gives us +x for the x-term. Then, Figure J is -x^2-5x-6 and its factors are (-x-2)(x+3).

Figure H: -x^2+5x-6

Figure I: -x^2+x+6

Figure J: -x^2-5x-6