Visualizing Binomials

 As is well known, it is great if we can provide a way to visualize math concepts so students understand the topic better. Sometimes, it is more difficult to do this because algebra tends to be taught without providing ways to visualize concepts. I've looked for something to use but haven't had any luck in figuring it out so I had to come up with my own version.

So the way I do it is to begin with a base 10 manipulative. I talk about x being a set length as shown to the right.  So if we see X + 1, this says we are adding one unit to the length we started with. I tend to add the extra one in a different color as seen with the next one.

This is a physical way to show the adding one length. I also talk about this using a few numbers such as X represents 10 units and we just added one more to it.

On the other hand, when showing X - 2, start with the basic 10 strip, just like we did before. So instead of adding one, we take away two units. This shows a strip that is two units shorter than the original.

 

Unfortunately this doesn't translate well when it comes time to teach binomial multiplication, so the other way I show X - 2 is to do it like X + two negative values so it looks like this 

This just show that you have two small units that are negative but this one sets it up to slide into binomial multiplication visually.

To visually show binomial multiplication such as (X - 2) time (X + 1) I set it up this way.

I have the X - 2 along the top and the X+ 1 along the left side.  I didn't bother showing the X's in any color other than blue because they are both positive and I wanted the -2 and +1 part to stand out so students could see what was up.

So the next step is to show that X times X gives you X^2 or visual it shows a square with each side that is X units in length.  This allows the students to see where the X^2 term comes from.  

  The final two steps shows that the X times a negative gives a negative so X times - 2 gives you a -2x while the +1 times X results in a plus one X, easy to see.

Finally, the + 1 times -2 gives -2.  From here it is easy to show the parts of X^2 - 2x + 1x -2 or as we see when its all done, X^2 -x -2.  I like using this model for introducing binomial multiplication to my lower performing students because they can see the parts much better and do not have to remember the "FOIL" method.  

Let me know what you think, I'd love to hear.  Have a great day.  See you on Monday.