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.

5 Principles of the Evil Teaching Guru August 13, 2012

People who don’t teach have no idea.  Even non-teachers who are all “ra-ra-teachers” have only a slightly better idea.  Broadway performers and circus sideshows have a clue.  Only a teacher can know what it’s like to be on 24/7 for 10 months a year. 

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So as a teacher, I was curious about the auto-notification that popped into my inbox recently “Five Principles of the Evil Teaching Guru” by Maxwell’s Demon.  I mean, another person calling me evil?  Haven’t teachers been beaten down enough already?  But since it’s summer and teachers don’t work in the summer (right?) I had a few minutes to read.  At first I read just the five principles:-

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1: Know that teaching is impossible (wait, what?)

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2: Believe that it is important to impose yourself (be a [buzzword] bully?)

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3: Do less (um, fired?)

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4: Confuse and take risks (“I’m confused.” No, you just need to remember your pencil.)

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5: Learn (ok, this one is good already)

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I was more or less horrified by these five principles.  However, it was already a much different post than the title lead me to believe.  Since the last one was “Learn”, I thought that there may be more to the story so I decided to keep reading.  You know the old “Don’t judge a book by its cover”?  Well, this should be recrafted to something along the lines of “Don’t judge a blog post by its subtitles” or something.  You get the idea.

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“Five Principles of the Evil Teaching Guru

1) Teaching is impossible

The idea of teaching implies that you can be the active party in someone else’s learning. This is not really the case if you want to go beyond a little rote recitation and rule following.

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2) Impose yourself

Once you have accepted that you are engaged in a fool’s errand get arrogant, unless you are confident that you can persuade, cajole and trick people into learning for themselves, you will not be able to. In order to do this you must be able to gain some control, getting a classroom or individuals to listen to you. Without some form of control you will be ignored or even humiliated. Once you can gain control, however, please do not stop there. Many do, and they become the legends people complain about for years to come. Instead…

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3) Do less

Remember that what you do really does not matter. It is what your students do that matters. If you have opened up a class discussion and it is going well and on topic, let it be. The best state for anyone learning is when they go for it on their own, the teacher silent.

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4) Confuse and take risks

Now we are into the essential, but dangerous skill. There is certainly bad confusion, but there are good forms too. Again this is about what the student does, more than about you. The simplest thing is to simple “be less helpful” but you can take it further and take a risk. Make your students confused, make them fail, it can really help their journey to learning independent of you.

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5) Learn

I have used the term students throughout this piece, but that is wrong, try to drop it from your thinking. Take every chance to learn from the people you work with, make it a two-way engagement. Also never forget to consciously hone your craft. You might have explained how to do a certain problem hundreds of times, is there a new way to try?”

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How great is this post?  I mean, I’m a bit hesitant to purposefully confuse my students as some find Math to be confusing enough, but this post reminds me that people appreciate more the things they find on their own and that it’s my job to make that happen.  It’s the difference between finding a dollar on the street and having Mom give you one.  The one from the street is way more awesome than the one Mom just hands over.  What can you buy with a dollar these days, anyway?

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Is Common Core meeting its Goal? May 21, 2012

Is the original goal of Common Core being lost in the upper grades?

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One major difference between the U.S. and so-called ”A+ Countries” is, while we focus on breadth, they focus on depth.  While there is a natural progression throughout a student’s school years from one math topic to another in these high-achieving countries, in the U.S. we seem to have a “throw at the wall and see what sticks” mentality.  For example, in grade 8 we cover 32 unique mathematical topics.  In high-achieving countries this number is just 18. 

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The new Common Core curriculum aims to bring our focus back to depth in the lower grades but seems to miss this mark once the abstract maths are reached.  While it is true that more topics have been cut out than added in most grade levels, topics traditionally covered in Algebra 2 (and some may say pre-Calculus and above) – piecewise functions, limits, logarithms, areas under curves, Algebraic proofs, and rational function graphing to name a few – are now part of Algebra 1.  Does adding so many advanced topics to the Gateway of Higher Math (ok, I’m biased) do what Common Core initially set out to do? 

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Below is a comparison of the math topics taught each year in A+ countries (first chart) and those covered in the U.S. (second chart) each year (compiled by Professor W.H. Schmidt).  These comparison charts were created before, and as a support for, Math reform in the U.S.  Still, to meet the new upper-grade Common Core Standards, school districts are turning to hybrid-type courses: “Algebra/Geometry/Stats Year 1″, etc. (Yes, that’s ONE year’s course) to meet all of the new high school requirements.  While the Common Core sets out for mastery at the Elementary level, did it really mean to hybridize high school math?  If depth is more important that breadth, what are we doing?

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Encryption and Algebraic Code Breaking! (Prezi + Activity) May 9, 2012

 Code breaking is so cool.  Above is a short Prezi on Kryptos in Washington, DC.  Below is an activity that allows students to encrypt, and decode, their own secret messages using Algebra. 

“Unschooling”: A Movement? May 23, 2011

I recently came across what I thought was a rogue blog about “unschooling”. Out of curiosity, I Googled “unschooling” and was aghast to find that it’s an actual movement. This to me feels as legal as polygamy and kidnapping. Maybe you think my analogies are a bit extreme, but you should know I looked into “unschooling” further to see what kinds of parents “unschool” their kids. I found that parents who have strong educational backgrounds, often with advanced degrees, “unschool” their kids.

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On the surface, this sounds reasonable. These parents are well-educated and therefore can give their kids a strong education. There is no doubt kids with educated parents grow up to be smarter and more educated themselves. In fact, there is research that links a kid’s mother’s education level with how well the kid does in school. Why? Research points to the number of words the kid hears as a youngster even before kindergarten.

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But thinking more about “unschooling” and extrapolating out to when a kid is, say, 18 and deciding whether or not to go to college, a few kinks surface. The kid will have to take the SATs. Unschooler’s philosophy on math is to teach just the basics one needs to cook a meal (measuring) and a few other things. With a poor math SAT score, the kid doesn’t stand a chance of getting a score any higher than what they for bubbling in your name (will the kid even know how to bubble? Frightening.). Thinking further into the future when the unschooled kid has a kid of his own, how will this now unschooled parent teach his children? He can unschool them, but he definitely wouldn’t be able to homeschool or answer homework questions brought home from a traditional school. The grandkids then of the original parents who decided to “unschool” are left with the extreme disadvantage of having no useless school knowledge, which is, consequently, what gets you further in life.

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It seems to me, and correct me if I am wrong, that “unschooling” is a selfish, thoughtless decision made by people whose parents gave them an excellent education. “Unschoolers” are setting their kids up for limited career choices and failure. The unschooled kids may be well-spoken and seemingly intelligent, but there’s a big difference between well-spoken and well-educated, and as a teacher I know that a kid who is well-spoken is often hiding a learning deficiency. It is sad to me that these “unschooled” kids will be hiding a learning deficiency imposed on them by their parents, who no doubt had good intentions- ”let’s keep our kids with us and teach them what they really need to know about the world”- but who fail to realize that someday those kids may want to make decisions for themselves, like going to college, and will be left with no tools to do so. They will do poorly on the SATs thereby having limited college choices. They will then never be able to get the strong college educations their parents had. And this says nothing of LSATs, GREs, or any other exam that goes along with deciding to further one’s career.

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“Unschooling” to me is a one-generation dead-end decision that is rooted in a selfishness to keep kids home, like one would keep an indoor cat. “Unschooling” parents pride themselves on letting their kids make their own education decisions, but the decision to [not] go to college is made for these “unschooled” kids way before they have a voice.

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Already ill from the thought of rich kids all over the US “unschooling” their kids, I get an email from the tutoring company WyZant from a parent in the South End of Boston who’s looking for help for her 11-year old son:

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Our son is 11 years old and attends the Sudbury Valley School. We would like tutoring for him on the weekend, starting in September. He has not had any academics at all, as this is the philosophy of his school. He taught himself to read and his word comprehension has been tested at a 5th grade level (by an outside evaluator). His reading comprehension is not good, however. The main point of what he has read often eludes him! He has never had formal math and only knows the plus and minus signs, but is quite intelligent and does the math in his head for daily life easily and effortlessly. He has matured and thrived at Sudbury Valley, but we want some formal academics taught so he has choices for attending traditional schools, should he so choose. His weakest point is writing, and is at kindergarten level or less. He has a mild case of NonVerbal Learning Disorder, according to the evaluator, and it is clear that writing is agonizing for him. We live in the South End.

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For those of you who don’t live in Boston, the South End should be called the “High End” in that it’s price-prohibitive for 99% of Bostonians. What struck me about this email is 1) the kid’s age. He’s 11. He’s on the verge of being a teen and his mom is freaking out that 2) he writes like a kindergartener even though he 3) GOES TO SCHOOL in 4) Framingham! That really sucks. This parent drives her kid 30 miles every day to a school that costs $7000 per year (I looked it up) only to have him be 11 years old with a kindergarten education.

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But the “GOES TO SCHOOL” thing is what really struck me. Is the “unschooling” movement becoming an institution?

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Oooo, that shiver just cooled me off a bit.

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(To be sure, I know that unschooling is much different from homeschooling. Homeschooling involves a curriculum and parents who care to teach their kids subject matter that will prepare them for work in the real world. I applaud any parent who has decided to take on this huge endeavour! I was in a Calculus class at Harvard with kids who had been homeschooled; they were half my age and scored twice as many points. I wanted their parents to homeschool me.)

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