A Time When a Little Instruction is Better than a Lot

The other day, I had a young friend show me a method of factoring that he favoured.  He was told that "nobody" knew the reasoning behind this method; it just worked.  I couldn't let that go.  There is no mathematical method that just "always works" and no one knows why.

Not having been able to find any explanation on the internet, my friend thought I should post an explanation.  I thought about it.  I even began to prepare a post.  But I couldn't do it.  To show him a full explanation would be like giving the answer to a problem at the first sign of struggle.  I don't want to rob him of an opportunity to work through some algebraic thinking.  Instead, like someone stuck while climbing a wall, I am confident he simply needs a bit of a leg up.

Here then, is my challenge:  I will show you my work, along with a couple of questions to think about.  Can you reason your way toward a proof that your method of factoring will work for any quadratic expression (and why is it an expression, rather than an equation??) that factors?

First, let me remind you of what happens when we multiply two binomials with a leading coefficient of 1.

Next, let's factor the quadratic expression you came up with the other day. 

Why is the 4^2 important?  Did each step make sense?  Which ones do we skip when we apply the method?  Will this method always work?  Why?

Golden Ratio and Fibonacci Sequence

Monday, November 23 was, apparently, Fibonacci Day.  I Googled it, and it apparently is really an official "Day."  Who knew?  Someone on my Facebook feed acknowledged the special day by posting about Fibonacci numbers found in nature.

http://io9.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature

Of course I saved it to show my little ones, who love this sequence.  What matter that it's bedtime when there is math to be explored?  The article mentioned the golden ratio too.  I haven't brought it up before, but since they now have the background in operations on fractions and decimals, they were ready. 

I think this is the first time we have brought out a calculator for math.  The one that comes on the latest version of windows even records your calculations as you go along.

We took consecutive numbers in the Fibonacci Sequence and divided the larger by the smaller:  2, 1.5, 1.66667, 1.6, and so on.  Every so often, we stopped to see what patterns the kids noticed. 

At first, they said things like, "They all have a six.  Well, most of them have a six."  While true, we discussed whether this was a mathematically useful observation in this situation.  Does that tell us anything about the whole number?

The answers all begin with 1.6.  Except for the first ones.  That's something, isn't it.

Then they noticed that the answer gets smaller, then bigger, and smaller, bigger, smaller, and so on.  That's interesting.  There's a pattern in the sequence of answers.  Anything else?

Spider Girl mused, "What's happening?"  A good question!  It's not just the resulting numbers, but there is a sequence, a chain of events. 

We decided to graph to get a better sense of "what's happening."  Off to Excel!

BatBoy pointed out that the graph has a couple of bumps at the beginning and then makes a flat line.  It looks like a flat line, yes.  But we know the numbers are not exactly the same.  What's happening?  They stared at the graph a moment as I tried to figure out how to expand it vertically.  (I never did.)  Oh!!  They're (the results are) getting closer to a number!

Does the pattern continue if we keep going?  SpiderGirl wanted to know.  I added a couple of columns and modeled how to use formulas on Excel.

The sequence really seems to tend towards a number.  We had to take more and more decimal places to see the differences between terms higher up in the sequence.  This number that we are getting closer and closer to is called the Golden Ratio.

BatBoy wanted to keep going to find the exact number.  No, I told him, the Golden Ratio doesn't repeat and it doesn't end, just like pi.  We aren't going to be able to find the exact value.  But we can approximate it.  Why is it interesting outside of the Fibonacci Sequence?

Because, shells and galaxies, hurricanes, the human face, human fingers and animal bodies, reproductive dynamics and health, animal flight patterns, even DNA molecules.   Wow.  Both SpiderGirl and BatBoy were floored.  It was worth staying up a few minutes late, I think.

Alternatives to Acceleration in Math

For awhile now, I've been getting to know many parents of gifted children, a population where there seems to be a disproportionate interest in math.  I identify a lot with many of these kids.  I had a propensity for math, and it was a subject in which I took pride because I could get the high scores that made me stand out.  At the same time, I didn't like the math that I learned, not until calculus.  When I look at how the math-gifted kids are dealt with, I am somewhat befuddled.  Here we are, gifted and many of us homelearning:  We acknowledge that there is no one size that fits all, not in pacing, not in interests, and not in learning styles.  Yet, it seems to me that nearly all the math-gifted kids are offered the same accommodation, differing only in pacing.

In the schools, both brick and mortar and online, advancement in math seems easy to deal with.  The student takes some tests and gets accelerated to the appropriate grade.  Now, for many kids, that is a good way to go.  Acceleration means more difficult material delivered in a neat package.  The package is important because many parents are unsure of their own knowledge in math.  Many kids thrive on the challenge.

Sometimes, acceleration isn't possible or it isn't enough.  Then students might be offered contest problems or lateral thinking puzzles.  Very occasionally, they are shown math from other cultures or given some math history.  These topics and exercises are fun and interesting, but they're also piecemeal and lacking in the area of actually educating in math.

Additionally, what about those whose thinking is divergent, who learn through problem solving, or whose language or development makes them unready to tackle coursework at their mathematical level?

How about depth instead of pacing?  Making connections?  Cultivating intuition?

BatBoy is 7 now.  His placement tests put him somewhere in the middle of grade 5 for math.  He doesn't know all of his multiplication tables.  He can do short division but not long.  Multiple digit multiplication is messy for him.  He has a need to know numbers in depth.  He loves prime numbers, families of exponents, roots, and factor trees.  He senses there is more to know about fractions and wants to dig in.  He is not ready for the procedural task of traditional algebra.  Contest problems freak him out.  So what do we do?  What have we done?

I write this because we have delved into deeper math with BatBoy, and I am convinced that other children would also love to explore math in this way.  BatBoy's past year has looked like this:  Along with regular skills, such as listing combinations, and surface explorations, such as of fractals, he's also continued to explore numbers in depth.  The topic of prime numbers led us to explore 0 and 1.  0 and 1 are special; in what other ways are they special?  What are the roles of 0 and 1 in multiplication, fractions, exponents, negative numbers?  What is interesting about other numbers, such as 6, 7, 8, 9, 36, 49?  What happens when we try to take 0 to the power of 0?  What does it mean to get one answer when you look at it one way (anything to the power of 0 is 1) and another answer when you look at it another way (0 to any power is 0)?  It means there's no solution, because in math, a solution is valid if you can reach it no matter how you get there!  How do we divide by fractions?  The math books will say that you multiply by the reciprocal.  This is a huge pet peeve of mine, because it is around this time that many students seem to throw up their hands and decide that math is meaningless and arbitrary.  Who can blame them really?  Nobody can explain why we multiply by the reciprocal; nobody even attempts an explanation.  BatBoy would not accept (nor remember) such a procedure.  What do we do?  We make a foray into the idea of mathematical proof.  Turn a concrete example into a representation that could be any numbers m and n, and we prove that dividing by 1/2 is the same as multiplying by 2.  (For older students, I have begun with the idea of the fraction as a division.  Once they see that dividing by 2 is the same as multiplying by 1/2, the reverse translates easily.  BatBoy did not accept the reverse as obvious.)

BatBoy may not extend these mathematical habits of mind into other problems right away.  However, the seeds are planted.  By filling out the world of mathematics beyond definitions and skills, we create context.  This is the kind of context a "mathy" kid needs.  This world of math is the context in which connections between numbers and concepts are made.  And if we can be patient with the slow and unsystematic way a child makes these connections, the foundation is laid for encouraging the development of mathematical intuition.