Difference of Squares (and binomial multiplication) With Pictures! January 12, 2013

We’re starting to see a difference of squares emerge…

Multiplying binomials.  FOILing.  Whatever you call it, and however bad we want it, there’s no real shortcut.  So why does (x + 5)²   ≠   x² + 25?  Let’s take a look:

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Above is a representation of (x + 5)².  We can see along the top edge “x 1 1 1 1 1”, representing x + 5.  Whenever we square something, we multiply it by itself, so we see the same x + 5 along the left edge.  Since (x + 5)² = (x + 5) times (x + 5), let’s multiply to find the area of each colored region:

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If we put all the pieces together, we get:

(x + 5)²   =   x² + 10x + 25

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When we say that (x + 5)²   =  x² + 25, we miss out on all of those little blue 1x’s.  Multiplying two expressions together will always give us an area.  For example, a rectangle with length 5 and width 3 will have an area of 15.  Multiplying two binomials together, like we did above with (x + 5)(x + 5), usually yields a trinomial.  I say usually because there is one case when this is not true…

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Let’s multiply (x + 5)(x – 5).  A great way to do this is with the Box Method:

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Above, we see (x + 5) along the top of the Box and (x – 5) along the left.  If we multiply these two binomials together:

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and then combine like terms, we get:  x² – 25.  Since both x² and 25 are square numbers, and they are being subtracted, we literally have a difference of squares.  There is no middle term because the +5x and the -5x cancel each other out.

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To see how this problem translates into areas like our first example (x + 5)(x + 5), let’s start at the end and work our way back to the beginning….

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Here we see two squares: one is green and one is white.  The white one is being subtracted (difference) from the green one.

Since “difference” means subtract in the language of Math, we quite literally have a difference of squares.  Above, we see 5² being subtracted from x².  To make things more interesting, let’s overlap the regions:

Because the green shape is pretty lopsided now, let’s draw some dotted lines to think about the green shape in terms of three nice, regular shapes:

And now let’s multiply to find the areas of each of the nice, regular shapes:

If we simplify each of the white expressions, we get:

5(x – 5)  =  5x – 25

5(x – 5)  =  5x – 25

(x – 5)(x – 5)  =  x² – 5x – 5x + 25   =   x² – 10x + 25

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And then if we add them up:

(5x – 25)   +   (5x – 25)   +   (x² – 10x + 25)   =   x² – 25   It’s a difference of squares!

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But can we express this x² – 25 as the product of two expressions, like we did with x² + 10x + 25  –>(x + 5)(x + 5)?  When we ask this question, we’re asking if we can go backwards; we’re asking if we can factor the expression to find out where it originally came from.

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In the first example, x² + 10x + 25 factored to (x + 5)(x + 5).  Can we do the same with x² – 25?

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Let’s go back to our overlapped picture to find out:

Maybe if we break up the green region:

And begin to rearrange the pieces, first sliding one rectangle up:

and then chopping that bottom part, rotating it 90° and putting it on the left:

We made a rectangle!  And what are its dimensions?

(x + 5)(x – 5)!

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So x² – 25 came from (x + 5)(x – 5).  In this situation we didn’t get a middle x term when we multiplied the two binomial expressions together.  Instead, we got a difference of squares, which makes sense since that’s where we started!

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Here’s a video that shows why (a + b)² ≠ a² + b:

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Contact this blog’s author at shanadonohue@gmail.com.

New (free) ZeroSum ruler – for teaching addition with negative numbers September 30, 2012

Below is a new version of the ZeroSum ruler.  This one needs no hardware to construct, just scissors and glue.  You can download, print and use this proven tool right now by clicking on the picture, which will bring you to the PDF file that contains 2 ZeroSum rulers. 

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Geometric Transformations (video) July 18, 2011

Super Cool Math Tricks: Be a Human Computer! June 29, 2011

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Also see Selling Math to Students – number tricks, cell phones and fractals

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The Coolest Math Class Ever. Seriously! June 6, 2011

I was lucky to get into Oliver Knill’s class Teaching Math With a Historical Perspective, one of the choices within the Harvard Extension School’s ALM in Mathematics for Teaching program.  It changed the way I think and the way I teach.  He explained complex topics, such as code breaking and non-verbal proofs, with such ease.  He was inspiring and made me look at math in a way I had never before – from a historical perspective!  His site is worth checking out, especially if you can not get into his class!  (Click on the red circle to go to Oliver’s site)

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Let’s grow some grass! November 7, 2010

 

Who knew?  Students love growing grass.  Let me elaborate…. Students love taking care of their grass as they watch it grow- enough to get them to do some pretty complicated algebra.

 

What started out as a simple week-long project that incorporated a bit of environmental sciences (my undergrad background) into algebra became a 10-week long project spanning the curriculum from ecology to linear extrapolation.  It’s been a long time in the making, but beyond a shadow of a doubt this lesson is one of the most engaging that I have created.   My students love the life aspect of the project and hardly complain about doing some pretty complex algebra.   

 

Now, with the magic of a WordPress widget called ”My Shared Files    BOX” (on the sidebar), I was able to upload the Growing Grass Project files onto my blog for all to use! 

 

All three files are important, but the excel workbook includes everything the student needs to create a final portfolio piece, including a formatted final excel sheet that the student can type into and cell directions. 

  

I’m very excited about this and hope that if you do use the project, that you will add a comment to this post on how it went.  I also have other files that go along with the starter ones I posted. 

 

I guarantee that your students will be engaged in their learning and that you will find ways to link most of algebra 1  to what comes up along the way.  

 

WARNING!  This lesson takes on a life of its own!  Proceed with caution! 

 

Now Let’s grow some grass!

 

 

p.s. For supplementary files, you can email me at ZeroSumRuler@gmail.com.  The files include ones on scatter plots and lines of fit as well as a PowerPoint and activity on linear inter- and extrapolation.