Marsbarn Designs...

Multiplication work arounds

There are a lot of methods of double digit multiplication.  Personally, I don't feel that they are all created equal.  Some are cumbersome, some are confusing, and many don't carry forward.  Consider the following three methods: the Familiar, the Frustrating, and the Fabulous.

The Familiar. Yes, there is the traditional algorithm that  most of us learned in school.  This method is still taught, but it's one of several methods.

And yes. When done correctly, it works.  But the honest truth is that it isn't always done correctly.  Things go wrong (often it's that pesky zero in the second partial product that represents our multiplying by a factor of 10). So, having alternatives is a good thing. 

But then there are alternatives like this: 
The Frustrating:

Yes, it looks vaguely familiar, but upon closer inspection, it's just confusing.  Why am I starting with my 10's? How am I supposed to proceed?  What is that cross hatch thing in the middle supposed to be?  Can I expand this to more digits?

Sure, I can create an explanation for this method, but why?  Does it follow into any other methods?  Will my student see this again?  And the answer to both of those last questions is "No."  As far as I can tell, this is a stand alone method. It doesn't help with understanding the traditional algorithm.  It isn't useful in algebra or other higher math. And, trying to expand it to three digit multiplication is migraine inducing.  In my opinion, it is a dead end which confuses both parents and students.  

I vastly prefer this method:

The Fabulous:

This is another evolution of our array method.

The box on the right has the two numbers we are multiplying on the exterior of the box.  They have been separated into 10's and ones for ease of figuring.  Each interior box has a set of partial products, which added together (on the left) give us our total.  (As the student advances, the factors being multiplied in each little box don't need to be written.  This significantly speeds up this method.)

If you are having trouble understanding this method, think of it as an area problem.  You have a rectangle that is divided into smaller rectangles.  By calculating the areas of the interior shapes, you can find the total area. 

Why do I like this method?  Well, it builds on earlier visual methods.  The individual computations are easy. You can easily expand it into more digits (you just add more cells to your box).  And, most importantly for me, this method is built upon in algebra.  It replaces (or acts as a partner to) foiling in algebra. 

Ah yes, our frenemy, foiling.  The old "First, outer, inner, last" method that plagued my high school brain.  

If someone had shown me the box method back in the day,  I would have been a much happier camper.

The box method?  So much sweeter.  Notice how the x factors that can be combined line up on a diagonal?  So nice.  My high school students do not know the panic foiling used to give me, because they use this method instead.  I love a multi-use tool.  It slices, it dices, it reduces mistakes in multiplying and combining. 

So hang in there.  Some of the methods being taught will work for your student, and some won't,  but there is light in the spectrum! 

 

 

eggs & legos & clocks: fractions for the fourth grade set

I have an ongoing fraction-positive crusade with my older pupils.  Embrace
the fraction!  Learn to love it, and it will be your bosom friend.  I'm not sure
what it is about fractions that frighten and intimidate, but they do, and while
calculator created decimals might seem nicer, they aren't.  Decimals can be
messy, tricky, inexact creatures that can get a person in trouble. 

At it's base, a fraction is simple:  The bottom (denominator) number tells you
how many pieces you've cut your item (cookie, pie, cake) into, and the
top number (numerator) tells you how many of those pieces you have.  
Sure, there are rules about adding/subtracting, multiplication, and division,
but there is just so much you can do with a fraction!

In the hands-on, visually driven world of Common Core,  fractions are
introduced using egg cartons and clocks. Why? Well, they are comfortable
items most children have seen, they are cheap, and they contain units of twelve.
Twelve is a very friendly number when it comes to fractions and
equivalent fractions (different looking fractions which have the same value). 

(This blog post is part of an on-going effort to given parents an understanding
of how mathematical elements are being taught currently.)

To start, we'll practice dividing a dozen eggs into equal parts.
Here they've been divided into two equal parts (or halves): 

Three equal parts, or thirds:

Four equal parts, or fourths:

Six equal parts, or sixths:

And twelve equal parts, or twelfths:

Now, you'll notice that I've skipped, 5ths, 7ths, 8ths, etc. 
That's because 12 can't be evenly divided into those parts. 

Once students can recognize the shape of the different fractions,
we can start playing with equivalent fractions.

The fraction 1/2...

is the same as 2/4. 

Or 3/6.

When I work with my students, I often use legos, instead of
eggs.  I have more freedom to use fractions not included 
in the family of twelve (Hello, 8ths!). Also, they take up less
space, and all of my students like playing with legos. And, I have 
bricks that are set units, so it is easier to pick up 1/2 and hold it in
my hand and manipulate it.  It's also easier to show that 1/3 and 
1/4 are different sizes, so I can't directly add them.

Here I'm using the same basic set up of twelve as the 
egg carton, but I've expanded how I've represented the fractions.

Halves:

Thirds:

Fourths:

Sixths:

and twelfths:

Now, what else can I do with this model?
I can add fractions.  I can take a 1/4 piece and show that
if I'm going to add it to a 1/3 piece, I need to break both
sections into twelfths to do it.  Once I do that, I would 
have 7/12 for an answer. Similarly,  subtraction can be 
introduced this way.  

We can also explore the idea that 1/4 of 12 is 3.  Or that 
5/6 of 12 is 10.   To do this, I count the eggs or the pips (the
circles on top of the legos) contained in each fraction.

Improper fractions (where the numerator is larger than the
denominator) and their relationship with mixed numbers 
(whole numbers with fractions) are also easy to manipulate
with this modeling.

This method can also be used with clock faces, as they are
conveniently divided into twelfths.  That pathway opens up
pie shaped graphs, geometric elements, and time work.

I find that my students really like this method of fraction 
manipulation and exploration.  It works for most of them.
They have a deep understanding of why you need common
denominators to add, and how a single fraction can have
multiple representations.  There is often a stumbling point about
why 1/4 is 3 eggs, and 1/3 is 4 eggs (out of twelve), but we sort
that out.

Down sides?
My  main complaint with this model (and with many other
visual based methods) is that I wish the mathematical 
part of fractions was more integrated.   Most of my 4th and 5th
grade students memorize the equivalent fractions in an egg carton.  
They don't actually know  how to mathematically reduce fractions.  
Their knowledge of fractions is tied directly to the fraction families
you can make in a base of 12. If you give them 8ths, they don't
know what to do. Do they have a "deep understanding"? Yes.
But without the numerically driven (non-model) base, it isn't
as flexible as it needs to be. In my experience, this void isn't 
addressed effectively in 5th grade, and then in 6th grade I have 
students who want to draw fraction out in a model.   None of 
my high school students (those who fear fractions, and turn 
to decimals as their (faulty) saviors) had the complete Common
Core base.  It will be interesting to see if the students who 
grow up completely immersed in Common Core, and who
have that model based knowledge at their heart, are more 
comfortable with fractions.  I hope so.