Multiplication

Quick note on pacing: I'm putting out a lot of information really fast because I am on vacation and have time now that I might not have later.  Please don't try to teach your kids at the pace that I am teaching you!  Think of this as a reference guide that gets a little longer every day.  Anything that's new or unfamiliar is going to take time to sink in, practice, and incorporate before it begins to stick.  That's totally normal.  I'm not trying to rush you at all.  Take these at your own pace.

Today, MULTIPLICATION! -- It's just adding in bulk!

Sometimes counting numbers up one at a time is great!  But other times... oy.  Too many numbers!

So we count by twos.
Or we count by tens.

Or we really save time and use multiplication to count as many groups of whatever number we want!

The most basic intro is to start with groups of counters.

Here we have three groups of four beads each.  This algorithm is "3 times 4" and if we count them all up, the answer is 12.

Multiplication has several symbols that all mean the same thing.  Multiplication can be expressed with an x or a dot or a star, or even with no symbol at all, just jamming the numbers up against each other.  I'm having trouble accessing all my symbols in this blog program, but here are some ways I can express this multiplication:

3 x 4 = 12
3 * 4 = 12
(3)(4) = 12

Kids, understandably, can get confused by so many different symbols, so take the time to decide together which one you're using today.  By the time I get to algebra, I heavily favor the dot or the parentheses and discourage use of the x because, in algebra, we use variables, and I don't like confusion.

NB, for those with high schoolers or college students, if a child is learning about vectors and matrices in math or in physics, the "dot product" and "cross product" are different operations, so some of the ambiguity disappears as things get complicated and different types of multiplication branch off from each other.  Some high school seniors, especially those taking AP physics, get a taste of this in their senior spring, but most people don't have to worry about it unless they go on to be scientists, mathematicians, or engineers in college. 

One of the coolest things about multiplication is that it has a certain symmetry to it.  I can take the exact same 12 beads and split them into 4 groups like so: 

And when I do... 4 groups of 3 beads = 12, too!  

4 * 3 = 12
3 * 4 = 12

It works forwards and backwards!  You should definitely encourage this kind of exploration for your children and let them see the pattern that develops.  When kids come to the pattern organically, through their own curiosity and exploration and play, the pattern sticks much better than when they just learn it as a rule. 

We're demonstrating The Commutative Property of Multiplication -- but I do not give the tiniest care if your child knows or remembers that name.  For the development of number sense and delight in math, the only thing that matters is that your child plays around with groupings and finds this pattern for himself.  

Perhaps your child will notice that addition has a similar pattern to it!  The Commutative Property of Addition.   It just means that: 

2 + 3 = 5
3 + 2 = 5 also.

It isn't silly to play around with this, forward and backwards and backwards and forwards.  When we talk "math intuition," it's this kind of play that built intuition.  

If your kid is already mad intuitive around multiplication, and PREDICTS that the pattern will work backwards, too -- AWESOME!  Don't stop there.  Ask them to explain their reasoning.  Explaining reasoning is a developmental milestone that my own eldest daughter had not hit in 1st grade, was developing in 2nd grade, and now she has pretty strongly in 4th grade.  Understanding the math is one thing.  Understanding AND communicating it is another.  And it's a skill worth learning because it entrenches the understanding much deeper when you have to put it into words, images, and patterns for someone else. 

One way I love to demonstrate this principal is using rectangles.  

Instead of grouping on post-its in little piles, I can lay out my beads in rows and columns, like so: 

3 rows of 4 beads each = 12, arranged in a wide, short rectangle
4 rows of 3 beads each = 12, arranged in a tall, narrow rectangle

It's the same rectangle!  We just turned it around.  

This kind of tactile play builds intuition and familiarity with geometry and linear algebra (think, matrix math) years, decades before it's developmentally appropriate to be working proofs or multiplying matrices. 

And it is never, NEVER too late to get back to the beads.  Nobody is too old, too cool, or too advanced for number play.  

This is the perfect moment to explore the concept of SQUARE NUMBERS with your child.  When working with little rectangles, it's just the right time to point out that some rectangles are squares!  They have the same height as width.  And just start to notice what it means to be a square, what a square looks like, and what are some square numbers.  Consider doing a 10 x 10 grid.  Kids love the number 100!  One hundred feels like it holds a special kind of magic when you're working in a base 10 system. 

Above all, let this be playful.  If your child says, "I think 40 will be a square!" Don't contradict him.  Help him count out 40 beads and try to make that square.  Let your child learn by doing -- mistakes and all.  We want to build ALL this learning on a sense of comfort with failure.  

In math, there are often clear right and clear wrong answers.  Kids who feel comfortable getting a dozen wrong answers on the path to the true solution will thrive in math.  Kids who don't feel safe to mess up will get stuck.  Getting stuck is fine and normal and part of the learning process, too -- but we don't want it to be the end of the learning. 

Let getting stuck be a bump in the road.  Messing up and getting the wrong answer isn't a bump.  It's just the road. 

*****

A note on math facts: I don't push for memorization of much when it comes to math.  However, once the concept of multiplication is strong, it is EXTREMELY HELPFUL to learn one's multiplication tables by heart for speed and ease in computing. 

I strongly encourage my own middle school students to re-drill the basic multiplication tables, at least up to 12, even better up to 15.  It just helps with fluency.

I hear good things from fellow teachers about XtraMath.  I know it can be frustrating to students at first, as there's time limit, but it does drill effectively.  You might check it out if you're not into old-fashioned flash cards.  And if you have a math drill app you like better, please do share in the comments.