Well Chosen Ones explain dividing fractions

Invert and Multiply? Why?

Invert and Multiply? Why?

Well-Chosen Ones give us an empowering mathematical why

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Poets sing that one is the loneliest number…

I say, if we choose our “ones” carefully, they are one of our best friends in the wild world of mathematics!

Wisdom from a homeschool-pioneer

First, to give credit.  The phrase “well-chosen one” is something my mom used constantly when I was growing up.  I am indebted to her for her zeal in making sure we understood and cared about the “why” behind what we were doing in mathematics.

Once my math book told me to “invert and multiply.”  I rebelled and proclaimed that this “trick” better be proven to me before I would deign to use it.  Apparently my cynicism (or maybe just pride) started early.

Well Chosen Ones help us understand dividing fractions

My mom could easily have fought back with the investigative-dousing “because I said so.”  Remember, this is before the internet, when Khan Academy and plenty of stellar teachers make their information available on youtube.  Instead, my mom called a friendly math guru, worked out the proof, and came and passed it along to me.  Any errors that follow, though, are totally my own!

What is 1?

Let’s start with a bit of background.  We all know what “1” is, right?  1 apple, 1 pizza, 1 pan of brownies (yum).

Now let’s look at a pan of brownies a bit more closely.  Here is 1 pan of brownies.  But let’s say I have friends coming over so I cut it into 6 equal pieces.  1 pan of brownies is the same as saying 6 pieces out of 6 pieces, or 6/6.  Or we could imagine we have more friends over, and cut the pan of brownies into 18 pieces.  1 pan of brownies is the same as saying 18 out of 18 pieces (18/18).

We could look at some more names for “1.”  1 whole pizza is the same as 5/5 pizza.  It’s also the same as 8/8 pizza.

In other words, 6/6, 18/18, 5/5, and 8/8/ are all just different names for the same idea of “a whole” or “1.”  You might call me “Amy,” “Mommy,” “Daughter,” or “Chocolate fiend,” and it doesn’t change me at all.  They’re just different names for the same idea.

Multiplying by 1

Ok, so with that in mind, let’s look at fractions other than 1.  Hopefully everyone can agree before we start that if you multiply any number by 1, its value stays the same.
5 * 1 = 5.  4 * 1 = 4.  783 * 1= 783.

It’s kind of magically soothing. Or maybe that’s just me.

Now let’s think about the fraction 2/3.  What is 2/3 * 1?  Why, 2/3!  Its value doesn’t change!

Would a 1 by any other name multiply as sweet?

And here is where we get to start picking some renamed-ones (a “1 by any other name”).

We’ve already decided that 4/4 = 1.  So what if this time I multiply 2/3 by 4/4?  Remember, 4/4 is just a different name for 1.  So the value of the original fraction “2/3” should not change.  But while its value doesn’t change, its name does!  2/3 * 4/4 = 8/12.  2/3 = 8/12.

Let’s try this again.  7/8 * 1 = 7/8.  7/8 * 2/2 = 14/16.  7/8 = 14/16.

fractions multiplied by well chosen one

Choosing the name of our “1” carefully when adding fractions

Now this is not a useless exercise.  We obviously need to choose our ones carefully when we are adding fractions which have different denominators.

Let’s take 2/3 + 1/4.  If I just choose a “1” randomly, let’s say multiplying 2/3 times 7/7, I’m not really helping myself.  Instead, I need to pick a “well-chosen 1,” a 1 that will help me solve the problem.  In this case, I need to pick 4/4 and 3/3.

Why we invert and multiply

Ok, really, all this background is just an excuse for the big reveal: the WHY behind “invert and multiply”!!  Like I said, this has such a special place in my heart as a childhood memory.  Who says math can’t be good and true and beautiful in our relationships?  (Ok, it can also cause a lot of angst, I know.  We have tears over math sometimes at our house, too.)

We have some fractions to divide: 2/3 divided by 1/4.  You’ve all been taught, “invert the 2nd fraction and multiply.”  Ok, let’s do that.  Great.  We got the right answer.  Who cares?

Well, obviously, I care that we get right answers eventually, but frankly that matters a whole lot less to me than understanding the process.  Sometimes my son makes a careless arithmetic error in the midst of a complicated Geometry or Algebra problem.  I don’t mind half as much that the final answer is wrong if I can go back and see that he understood the whys and wherefores and followed the steps as necessary.

Fixing a careless error is a lot simpler than fixing an understanding-level error.

So, ta-da, let us behold the magic of the well-chosen one!  Watch carefully!  We’ve already discovered that 4/4 = 1.  What happens when we apply this further?

First, let’s write our original problem as a gigantic fraction.  A fraction is literally just another way to write a division problem (for example, 10 divided by 2 can be written 10/2).

Now, let’s multiply this gigantic fraction by (4/1)/(4/1).  Keep in mind, in the fraction (4/1)/(4/1) our numerator and denominator are the same, so this is just an exceptionally well-chosen and slightly cumbersome “1.”

Our numerator is now (2/3)* (4/1) and our denominator is (1/4)*(4/1).  The denominator becomes 1!  And now, all we are left with is (2/3)(4/1).

Hey!  Wait a minute?  Doesn’t that look just like our “invert and multiply” equation?  Why, yes.  Yes it does.  Confetti everywhere!

Let’s try it again to see if that was just a coincidence.  This time, let’s try (2/5) divided by (3/6).  Write it as a huge fraction and multiply by our well-chosen one (this time it’s (6/3)/(6/3)).  Remember, our goal is to get the fraction on the denominator to become one; that’s why we’re choosing the reciprocal.    Yep, once again we’re left with (2/5) * (6/3)!

It works every time.  That is why I love math.  It can be so comforting on those crazy days, if we can view the wonder and “magic” correctly!

Empowered through understanding the Why

I think many of us find math much more exciting when we understand why it works.  Knowing the “why” also empowers us to be problem-solvers with more complicated equations.  And as parents, empowering our children with a math education full of wonder and understanding gives them the opportunity to think of themselves as Mathematicians!

Now it’s your turn.  Try to find other places where these “well-chosen ones” are hidden throughout your math books, even in higher math.  Let me know what you discover!

Video Version

I discussed these concepts last year in a video as a guest on the Homeschooling Without Training Wheels facebook page:

*Grammar Nerd alert: in the video, at one point I said, “I could care less.”  All fellow grammar nerds inwardly cringed. 😉  I should have said, “I could not care less.”

Fraction Manipulatives

 Brownies are the best math manipulatives ever!

All the kids at my house can play this game, even the very smallest ones.  It gives a concrete feeling to fraction equivalencies.

These are similar to the fraction overlays I use. Great for demonstrating equivalency and teaching fraction multiplication!

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