Solving for Area (Rectangles)

Solving for Area (Rectangles)

It’s a peanut butter and jelly sandwich.

It’s also the set up for a math problem I presented my fourth grader, Simon, with.

Story problem goes like this:  Dad makes Si a PBJ using bread that measures 4 inches by 5 inches.  But before Si can eat it, Mom cuts a perfect 1 inch by 2 inch rectangle out of the soft yummy middle for herself (yes, sometimes I do mess with the boy-os like this).  What is the remaining surface area of that top slice of bread?

Remember solving for these “find the area of the shaded region” kind of problems? 

The answer we’re looking for here is that the surface area of “square doughnut” is simply the surface area of original sandwich face minus the surface area of small rectangular “bite.”  In numbers, the answer is (4 x 5) – (1 x 2) = 20 – 2 = 18 square inches.
Pretty straight forward, right?
Or maybe not.
There are multiple steps involved to solving this type of problem.  Let’s break it down.
  1. Recognize how to break a problem into manageable pieces.  Here, the thought would be:  There is no specific formula to solve for area of a “square doughnut.”   But I do know the formula for a rectangle… Oh!  Can I solve this problem by manipulating the area of two rectangles?
  2. Identify the applicable formula(s).  The area of a rectangle is length times width.
  3. If necessary, solve for missing values given initial set of information.  Not needed in this example.
  4. Perform the actual algorithms.  4 x 5 = 20.  1 x 2 = 2.
  5. Know how to manipulate calculated numbers to arrive at final answer.  It is not enough to just crunch out “length times width.”  In this problem, I need to subtract area of smaller rectangle from area of larger rectangle.
  6. Clearly state the complete final answer (including units of measurement).  The answer is not just “18.”  At the very least, I need to write “18 square inches.”  Or better yet, I could write something like “the surface area of the remaining sandwich face is 18 square inches.

This is not easy stuff.  Think of multiple part math problems like patting your head, rubbing your tummy, marching in place while singing a tune all at the same time.  Each individual piece may not be that big of a deal, but pulling it altogether can be tricky. 

Here’s another area problem.  I worked on something similar with a bunch of fourth graders the other day (I volunteer at school once a week – math workshop with a 4th grade class is one of my stops).

Want to give it a try?

The kids at school had a terrible time with this problem…. which then really piqued my interest on how to help them (or help their parents help them) understand how to attack it.

Some comments about this problem & some pitfalls you might want to keep in mind:

  • There is some missing information that will have to be solved for.  I.e., I did not label every length in the original figure.  Assume all corners are right angles.
  • There is more than one way to solve this problem.  In the above picture, I’ve drawn out the four ways the original shape could be broken up.  While I think it most natural to sum the area of two small rectangles (Method #1 or Method #2), it seemed that many of the children gravitated towards finding the area of the large rectangle first and then subtracting out the area of the corner rectangle (Method #4).
  • Kids seem quick to dismiss their own approach to a problem in favor of following the other children’s calculations.  I encourage them to try their own way first and then compare final answers with their neighbors.
  • I ask students if their answers agree with each other.  If not,  I ask them to figure out what they did differently from each other.  If yes, I confirm if the answer is correct or not.  Always possible that the children all made the same mistake.
  • Solutions can go awry anywhere along the 6 steps specified above.  Patience, organization and ability to compute those basic algorithms super important.

After working with the other fourth graders, I was curious to see how my own boy would approach this type of math problem.  Hence my quizzing him.

We worked out the second problem independently of each other.  Si & I each arrived at the answer, 203 square centimeters.  Does your answer agree with ours?

Math Meets Reading Streak

Math Meets Reading Streak

It was Ethan’s turn to choose the read aloud book.  Little boy went rummaging through our home library and came back waving The Number Devil by Hans Magnus Enzensberger.

Well, why not?  I thought.

So, in our tenth read aloud book, Math meets Reading Streak!

I’m pretty sure E chose this book because he liked the pictures; he finds the little red guy with horns amusing.  I was a bit skeptical about whether he could actually follow along.  However, my rule of thumb is to talk math with them whenever they request it of me, so if the kid wanted to read “a mathematical adventure” and discuss it, then I’d be happy to oblige.

I find the actual story line rather so-so.  This boy, Robert, has reoccurring bad dreams about being swallowed by stinky fish or endlessly sliding, sliding down.  Robert also hates math, or, more specifically, he hates the dull arithmetic algorithms he is forced to solve at school.  One night, his string of bad dreams is broken by a visit from the little red number devil.  Robert and the number devil embark on a series of 12 nights of dreams discussing what the number devil considers interesting basic mathematical concepts.

I dunno… math… devil… dreams… the story around the math feels disjointed to me.  However, the math part of the book is really interesting.

The concept of infinity.  The importance of zero.  Prime numbers.  Repeating decimals.  Fibonacci sequence.  Irrational numbers.  Permutations.  The idea of a proof.  And lots more.

There’s good meaty stuff with lots of potential for discussion and experimenting in this book.

Currently, we’re working our way through the fourth night (chapter 4).  It’s slow going because we’re approaching this read aloud with pencils, paper and calculators in hand.  We frequently stop to verify calculations and to try to extrapolate before reading on.  For example, does (11 x 11) = 121?  Now check to see if (111 x 111) = 12,321?  Who sees the pattern?  How long do you think this can continue on?

So far, both boys are enthusiastic about this interactive read aloud.  Do they understand everything?  Probably not.  I’d say that Simon (9 1/2) is getting more out of this than Ethan (almost 7).  But even so, he’s only absorbing a fraction of what these concepts mean.  And even that, I’d guess, he will forget again shortly.  Does this matter to me?  No, not at all.  I first introduced the Fibonacci sequence to Simon about 3-4 years ago.  He got all excited about it.  Then he forgot.  It came up again in conversation a year or two later, I reintroduced him to it.  He thought it was great.  Then he forgot again.  No big deal, because it took only a brief reminder to jolt his memory.  This time, when I said Fibonacci & Ethan looked quizzically at me, Simon was the one to explain it to his brother.

So overall?  I like this book.  I welcome the chance to talk math with the boys.  And of course, it fulfills the reading requirements for our continued reading streak (day 93 today if anyone wants to know).  Not to mention, I’m also learning and rethinking a few things myself.  It’s been about 20 years since I earned my undergraduate degree in mathematics.  Nowadays, I wishfully title myself “mommy-extraordinaire,” “crafter,” and “wanna-be-gardener.”  But a long time ago, I studied to be a mathematician… and I guess there must still be a little part of her around because I felt all warm and fuzzy when I saw the words “elegant” and “proof” in one book together.

So it was the little boy who picked this latest read aloud.  But quite possibly, I’m the one who’s happiest to see our reading streak cross paths with all these neat mathematical ideas.

Estimating Square Roots

Estimating Square Roots

Simon and I had a really satisfying conversation about estimating square roots. 

Before I tell you about it,  let me clarify that square roots are not part of his third grade math curriculum.  And our conversation had nothing to do with school work.  Sometimes Simon just gets in a mood where he wants to talk math.  Of course, Jin & I are always happy to oblige.

 So, our big discussion about square roots broke out into the following segments:

  • Talking about the definition of a square root
  • Practicing finding square roots of perfect squares
  • Estimating the square root of 10
  • Using our prior estimate to estimate the square root of 40
  • One more square root question to think on…

Definition of a square root

My first request was for Si to tell me what he remembers about square roots.  He struggled over the exact definition so he gave me an example instead.  He stated that (5 x 5) = 25, so the square root of 25 is equal to 5.

A correct example.  I then went over the actual definition of a square root as well.

For any (positive) number n, we say that k is the square root of n if (k x k) = n.

Si wrinkled his nose at me.  He hates it when I use letters in instead of numbers.  But he rolled with it and we moved on to some quick drills with square roots.

Square roots of perfect squares

I didn’t call them “perfect squares” while we were practicing.  But I’m giving us the correct term & link in case you or I ever need a quick Wikipedia refresher on this stuff.

To Simon, I just threw out drill questions like:

  • What is the square root of 4?
  • What number has a square root equal to 4?
  • What’s the difference between those last 2 questions I just asked?
  • What is the square root of 64? 
  • What is the square root of 121?
  • Who has a square root of 10?

Once Simon demonstrated he could handle basic square root problems, I upped the ante….

Estimating the square root of 10

… I asked him what the square root of 10 is.

Complete silence.  And then… huh?!

Up to this point, Simon had simply been referring to the multiplication tables he recently conquered. 

Since (11 x 11) = 121,  11 is the square root of 121.

Similarly, because (10 x 10) = 100, the square root of 100 is 10.

But the square root of 10?  (1 x 10) = 10.  And (2 x 5) = 10.  But what is the number, k, where (k x k) = 10?

Si’s face took on a wrinkled scowl that means he was stumped.  And irritated.  And curious.  Ah-ha!  Exactly where I wanted him.

Listen Si.  Here comes the interesting part.  And I walked him through the steps to estimate this square root.

  1. The square root of 9 is 3.  Agree?
  2. The square root of 16 is 4.  Yes?
  3. So does it make sense to you that the square root of 10 must be bounded between 3 and 4?
  4. What number do you think our square root is closer to: 3 or 4?  Not sure?  Well, think about this, is 10 closer to the number 9?  Or is 10 closer to the number 16? 
  5. Let’s make a guess.  You think our square root must be a little bigger than 3.  How about we try 3.1.  What is (3.1 x 3.1)?  Do you remember how to multiply with decimals?
  6. We slipped in a little multiplication practice and calculated  (3.1 x 3.1) = 9.61. 
  7. Getting close.  We tried (3.2 x 3.2) and calculated this product to be 10.24.
  8. Since (3.1 x 3.1) = 9.61 <10  and (3.2 x 3.2) = 10.24 > 10, this means the square root of 10 must be somewhere between 3.1 and 3.2.  Do you agree?
  9. We could keep going, but let’s stop here and estimate our square root to be the midpoint between 3.1 and 3.2.  
  10. That makes our estimated square root of 10 to be 3.15.  Pull out the calculator, kiddo – let’s see how we did.

Our calculator said the square root of 10 is 3.16227766016838…  Not bad.

Estimating the square root of 40

Simon wasn’t finished with me yet.  He wanted another square root problem.  So I gave one final twist to our estimation problem.

What about the square root of 40?

We could have repeated the steps outlined above.  But instead, I suggested we work with the square roots we’d just been playing with.

  1. We know 40 = (4 x 10). 
  2. Does it make sense to say that the square root of 40 is equal to the square root of 4 times the square root of 10?  (in fact, the rule is sqrt(ab) = sqrt(a) x sqrt(b))
  3. We know the square root of 4 is 2.
  4. We estimated the square root of 10 is 3.15.
  5. So our estimate for the square root of 40 is (2 x 3.15) = 6.30. 
  6. Let’s check our estimate against the calculator.

The calculator spat out that the square root of 40 is 6.324555320336759….

One last square root question to think on

When Jin came home from work that evening, Simon proudly shared our square root discussion with him.

Jin and I were grinning ear to ear.  We want our boys to think and reason things out.  Not just memorize.  So this particular development was a giant step in the direction we’ve been working towards.

We don’t necessarily expect Si to remember every step I took him through to get to that final estimate of 6.3.  But we were impressed he made it through the conversation and were confident he followed most of it.

Jin, of course, couldn’t resist twisting the knife just a little bit more.  He ruffled Si’s hair, told him good job and then leaned in and asked….

So what would you say the square root of 0.9 is?

Simon fell for Jin’s “trick question” and blurted out the expected incorrect answer of 0.3.  His face took that fiercely scrunched up thinking look again as Jin and I delightedly said – No, Si.  (0.3 x 0.3) = 0.09.  We want the square root of 0.9. 

We left the boy just where we wanted him.  Puzzling.  Wondering.  Figuring.  Thinking.

Adding Strategies

Adding Strategies

4 cards from the game rat-a-tat-CAT

I noticed my little boy, Ethan (6), counting on his fingers when working through his addition homework for first grade.

Counting his fingers was fine when he was first learning to add, but it’s time to work on adding without counting out every value on fingers, toes and other manipulatives. 

What about that number scroll I created and have shown my kids using along with snap cubes? I do think using manipulatives is a valuable learning tool because it helps kids visualize what’s going on.  I also  think it’s necessary for them to practice other strategies so they can figure out answers more efficiently.  This becomes more relevant when the numbers they work with get bigger.  We can still count out (9 + 8) on our fingers, but what if we want to know what (36 + 29) equals?

My goal for Ethan was for him to be able to figure out something like (36 + 29).  I don’t expect him to grind out problem after problem.  Rather, I want to help set the stage, if you will, for the math I know he’ll be doing in 2nd grade.

The steps we’ve been working through to meet this goal are:

  1. Be comfortable adding a sequence of numbers.  For example:  (30 + 6 + 20 + 9) or  (10 +10 + 10 + 10 + 10 + 9 + 6).
  2. Know how to break or combine numbers to make more manageable pieces.  If Ethan is not comfortable working with the numbers 36 and 29, he may choose to use 36 = (30 + 6) and 29 = (20 + 9) instead.
  3. Given any two numbers to sum up, be able to combine steps 1 and 2 to figure out the answer.  Given the sum (36 + 29), Ethan chooses how to break down each number (step 2) and then adds up the rearranged sequence of numbers (step 1).
  4. Know how to check his answer using our number scroll and snap cubes.
  5. Commit to his answer!

A surprisingly fun game 

Adding sequences of numbers

We started by just adding sequences of numbers; we didn’t worry too much about how E was getting to the answer.  We just wanted to get comfortable adding more than 2 numbers at a time.

As always, the more fun we can have with it, the better.  One way we practice adding sequences is by playing rat-a-tat CAT.

When one of the teachers at school recommended this game to me, I couldn’t see the appeal of it.  Each person plays 4 cards, face down.   Players exchange cards with the goal of obtaining the lowest score (sum of the 4 cards).  And that’s it.  I thought the game sounded a little boring and not so mathematically challenging….

Until I realized that adding a sequence of 4 numbers can be a real challenge for little kids.

Until we discovered we could mess with the other players’ cards when we started swapping for their good cards (those are the ones they’ve replaced).

Until I found out how fast this game can really move and how excited the kids get when playing it.

Now we play this game just for fun at the end of the day and I give myself a little pat on the back because I can check off “quality time with kid” and “practice addition” on my mental check-list.

If I wanted to add a little more drill here, I’d deal out 6 or 7 cards and ask Ethan what the sum of the cards are.

How to add 8, 2, 5 and 9 without counting out each value?

Rearranging the numbers by breaking and/or combining them

Now, let’s say we’re comfortable with the idea of adding sequences of numbers and our end combination of cards for rat-a-tat CAT are the ones you saw pictured above: 8, 2, 5 and 9.  How can we add these numbers up without counting fingers or drawing tick marks or counting out each of the handy-dandy snap cubes I love so much?

 Regrouping the numbers to make sets of “10’s”

Most of the kindergartners and first graders I know seem very comfortable counting by 10’s.  I’m pretty sure most first graders know that (10 + 10) = 20, (10 + 10 + 10) = 30, and so on.

They are also not bad about figuring what two numbers add up to 10.  So I think it’s natural to start with these two skills in mind.

Back to our example.  We want to figure out (8 + 2 + 5 + 9).  The (8 + 2) part shouldn’t be a problem.  That’s just 10.  but what about the (5 + 9) piece?  Well, we know that (1 + 9) = 10.  So let’s try breaking the 5 into (4 + 1).  Then (5 + 9) = (4 + 1 + 9) = (4) + (1 + 9) = (4 + 10)

So (8 + 2 + 5 + 9) can be rearranged to look like (8 + 2) + (1 + 9) + 4 = 10 + 10 + 4.
We know (10 + 10) = 20.  We still have 4 to go.  Now go ahead and count the last 4 out (on your fingers if necessary) – that’s 21, 22, 23 and 24.  So (8 + 2 + 5 + 9) = 24.

That was a whole lot of discussion to add 4 small numbers, wasn’t it?  Did it feel like a waste of time?  Keep in mind that the goal is to set up a strategy to deal with summing bigger numbers.

Practicing steps 1 and 2

Once we’ve walked through a few examples with our kids, it’s time to see what they can do on their own.

Deal out a set of rat-a-tat CAT cards.  Hand your child a pile of snap cubes or legos.  If we are adding up the numbers 9, 9, 7 and 4, I’d arrange the snap cubes so we have something like 9 red, 9 orange, 7 green and 4 blue.  Ask your child to explain how he/she is breaking and grouping the numbers to figure out the sum.

If you don’t want to use cards, you can always roll a die.  Or just assign some numbers for your child to add together.  Use bigger numbers as your child becomes more adept at this skill. 

Today I asked Ethan how he would add the numbers 43 and 24.  He told me he would break 43 into (40 + 3) and 24 into (20 + 4).  Then 40 and 20 make 60.  And he still has 3 and 4 left over.  But (3 + 4) = 7.  So that makes 67 total.  Correct answer.  But even more importantly, a definite understanding of how to break down the problem and solve it.

Using the number scroll to check that (8 + 2 + 5 + 9) = 24
(See post about Val’s Number Scroll  for more details) 
Checking answers on the number scroll

I don’t always confirm if Ethan’s answer is correct or not.  If I know he’s close to the correct answer, I’ll often suggest he check his own work using our number scroll and snap cubes.  Assuming he sets his snap cubes up correctly, he can usually find his own mistakes.

Committing to his answer

Initially, Ethan used to finish every problem with the question, “Mommy, is this the right answer?”  Now he’ll say “Mommy, the answer to this problem is ….”
I want my kid to have the skill to break down these addition problems and to have the confidence to stand by his solution.  Even if his initial solution is incorrect, just having him commit to his answer changes the paradigm a little.
Another example of this is when we’re reading together.  I’ll often stop to check if Ethan knows the meaning of some new vocabulary.  He used to just shrug his shoulders and say “I don’t know.”  But then I asked him to at least try.  It was ok if he didn’t know the exact meaning.  Just start by saying “I think this word means….” and maybe he’d get it correct.  But if he didn’t, no big deal, we could just talk about it.  We tried again.  He gave his explanation of his understanding of new words.  Sometimes he was correct.  Sometimes he wasn’t.  But with each try, I saw him gain confidence in his own ability and understanding.
Same thing math.  That’s why I make him commit to his answers.
And this is what has been going on with Ethan, addition and me. 
Multiplying 2 digit numbers

Multiplying 2 digit numbers

 Calculating (31 x 24) the “old school” way
How are we supposed to help our kids with something like (31 x 24)?  Do we pull out all the snap cubes to set up a grid with dimensions 31 by 24 and then count each cube to solve this multiplication problem?  I’d prefer not.  But, if it was necessary, setting up the grid for (31 x 24) is still doable (click here for post on introducing and visualizing multiplication with 2 numbers).  
What happens when they start working with problems like (794 x 81) or (1273 x 306)?  
Can we please just start doing multiplication the way we learned it already? You know… the “old school” way?
Actually, I did teach my son (Simon, grade 3) how to do multiplication the “old school” way.  But I also want him to understand why it works.  So, we continue to
  • Visualize multiplication of 2 whole numbers as a grid
  • Practice breaking down multiplication of 2 bigger numbers into more bite-sized and manageable pieces (still in the grid form)
  • Double check our answer by recalculating using the “old school” method
  • Discuss why the “old school” algorithm works (it’s just a specific way of breaking down the applicable grid)
Example 1 – Visualizing why (7 x 21) = (7 x 11) + (7 x 10)
The Grid & Breaking it down – Example 1
We start with the assumption that everyone knows the basic multiplication tables; from (0 x 0) through (12 x 12).
Consider (7 x 21).  We can visualize this multiplication problem as a grid with 7 rows and 21 columns. How should we handle the 21 columns?
Let’s break the 21 columns into 2 smaller sets.  In the above picture, I split our 7 by 21 grid into 2 smaller grids:  the green grid with dimensions 7 rows by 11 columns and the purple grid with dimensions 7 rows by 10 columns.  If we zoom in and look only at the green grid, we see this represents (7 x 11) which we know equals 77.  Similarly, the purple grid represents (7 x 10) = 70.  Combining the green and purple grids back together brings us back to our original problem of (7 x 21).  
We can see that (7 x 21) = (7 x 11) + (7 x 10) = 77 + 70 = 147.
Can we split the 21 up another way?  Yes!  We can choose to partition the numbers however we like.  The key is to remember that multiplication is simply a special case of addition.  When we say (7 x 21), we really mean “add 7 to itself 21 times” (or equivalently, “add 21 to itself 7 times”).  When adding a sequence of numbers, it doesn’t matter in which order we add or how we group the numbers.  We just need to account for every value exactly once.
We decide how to rewrite 21.  So we can say 21 = (20 + 1) or 21 = (12 + 9)  or 21 = (10 + 11) and so on.  And instead of working with (7 x 21), we can work with equivalent problems of [(7 x 20) + (7 x 1)] or [(7 x 12) + (7 x 9)] or [(7 x 10) + (7 x 11)], etc.
By the way, when we break (7 x 21) down to the equivalent form of [(7 x 11) + (7 x 10)] we are using the Distributive Property of multiplication.  That’s the property that says for any numbers a, b and c, we know that  a x (b + c) = (a x b) + (a x c).

Example 2 – Visualizing why (11 x 16) =  (6 x 10) + (6 x 6) + (5 x 10) + (5 x 6)
The Grid & Breaking it down – Example 2

Now let’s work through (11 x 16) and see what happens if we decide to rewrite both the 11 and the 16. 

Start by setting up a grid with 11 rows and 16 columns.  Our goal is not to count every element in the grid but to partition our grid into smaller bite-sized chunks where we already know the multiplication problem associated with it.

I decided to break down the numbers as 11 = (6 + 5) and 16 = (10 + 6).  Now, instead of working with the entire grid as a whole, we are looking at (6 x 10) represented by the red grid, (6 x 6) represented by the blue grid, (5 x 10) represented by the green grid and (5 x 6) represented by the purple grid

We once again apply the distributive property to see that 
(11 x 16) = (6 + 5) x (10 + 6) = (6 x 10) + (6 x 6) + (5 x 10) + (5 x 6)60 + 36 + 50 + 30 = 176.
Connecting what we know to the “Old School” algorithm 
Next, let’s recalculate (11 x 16) = (16 x 11) using the “old school” method we all know:  
  x    11
  +  160
How does our method described in the last 2 examples tie back to this one? 
In fact, the “old school” calculation follows the same reasoning except the rules are 
  1. We break down only one number and 
  2. We have no choice of where we break the number apart.  The one number is always broken apart at the 1’s place, then the 10’s place, then the 100’s place and so on.
So (16 x 11) = 16 x (1 + 10) = (16 x 1) + (16 x 10).
Let’s do another calculation using the “old school” rules.
(240 x 45) = 240 x (5 + 40) = (240 x 5) + (240 x 40) = 1200 + 9600 = 10800.  
But this format isn’t aligned nicely for addition, so we rewrite this in the form of our old friend:
  x     45
  +  9600
Now what?

As I said earlier, Simon and I continue practicing multiplication with larger numbers using both the methods described above.

I figure he’ll be responsible for that “old school” algorithm sometime in the next year or two.  So it can’t hurt to practice a little now.

But what I really want is for him to understand that he is not limited to that one set of rules when multiplying larger numbers.  By playing with regrouping grids and breaking down numbers we can explore different ways to solve a multiplication problem.

Today, I asked him to to calculate (8 x 26) by breaking down the 26.  I expected him to rewrite the problem as (8 x 26) = (8 x 20) + (8 x 6).  However, Simon declared he preferred working with 10’s. Simon’s solution to the problem was (8 x 26) = (8 x 10) + (8 x 10) + (8 x 6) = 208.  The kid challenged my suggestion, decided he would solve the problem another way and then proceeded to work it through to the correct answer.  Not bad.

Introducing and Visualizing Multiplication of 2 Whole Numbers

Introducing and Visualizing Multiplication of 2 Whole Numbers

Simon (3rd grade) is now expected to know his basic multiplication facts; from (0 x 0) all the way to (12 x 12).  You know what that means, right?  That means we’ve been practicing a lot at home and now I have multiplication on my mind.

How many of us enjoy helping our children with their math homework?

I admit it.  I do.  I’m guessing I’m in the minority.

But, like it or not, parents do have to step in and get, at least, a little bit involved with the homework process.  And sometimes, more than a little bit.  Sometimes, we need to help our kids breakdown a concept, visualize it, drill it and own it.  Be it reading, writing or math.

Or, I suppose, you could hire a tutor.

But for us, well, I am the tutor (and the cook and the storyteller and the…).  And this is how I break down learning basic multiplication.

  1. Understand what (n x k) looks like
  2. Drill
  3. Punch through the target
  4. Reiterate
  5. Trouble shoot
 Visualizing (12 x 4) using Snap Cubes

Visualizing (12 x 4) using Snap Cubes & Val’s Number Scroll
Step 1:  Understand what (n x k) looks like

We use snap cubes as math manipulatives.  But you can use any set of same size objects to set up multiplication grids (e.g. pennies, 2×2 sized duplo, m&m’s, etc).

Consider (12 x 4) = (4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4). 

Or more succinctly, (12 x 4) equals 4 added to itself 12 times. 

We can visualize this as a grid (or matrix) of 12 rows and 4 columns.  Or we can consolidate the 12 sets of 4 into one long line along our number scroll (see Val’s Number Scroll post if you want more details about the number scroll.  Not on Etsy yet… sorry…).  We used the single line along the number scroll more when Simon was learning to count by k’s (where k = 1, 2, 3, …, 12).  Now we usually just put everything into grid form.

When I’m working with my kids, I always set up the first example so they know what I’m asking of them.

Next to my grid of 12 rows by 4 columns, I may set up another grid with 4 rows and 12 columns.  Do we agree that (12 x 4) = (4 x 12)? 

 (4 x 6) = (6 x 4)

Now what does (6 x 4) look like?  Does it matter if it’s with 6 rows and 4 columns?  Or 4 rows and 6 columns?  Let your child set up the manipulatives and have him/her explain the answer.

Set up grids to represent (3 x 8), (2 x 1), (1 x 1), (11 x 9) and so on.  By the way, what does (0 x 4) look like?  How about (0 x 500)?

I believe this initial step of visualizing multiplication is invaluable.  Once our children understand what (5 x 8) looks like, then they’ll have a starting point to figure out something like (15 x 32) (more on multiplying 2 digit numbers next time).

Step 2:  Drill

Now it’s time to drill.  That means practice, practice and more practice.  Flashcards.  Verbal quizzes (what’s 8 x 4 equal to?  How about 11 x 11?  7 x 9?).  A speed worksheet.  Any type of game you can come with that will make your child practice this new math skill.

I recommend a little practice everyday.  Think of it like brushing your teeth.  You could just brush your teeth for one hour straight once a week.  But we all know it’s much better to brush at least twice a day for a minute or two instead.  Drilling multiplication tables doesn’t have to be a formal thing.  Why not brush up on the multiplication tables too?  Quiz each other while you’re on the subway or in the checkout line at the grocery store.  Let your child see you get confused over (8 x 7) and then watch you work it out (for some reason, (8 x 7) is the one that I always have to double check). 

Step 3:  Punch through the target

I learned a really important lesson when I started working out on a punching bag.  In order to have real power in my punch, I need to punch through the bag, not at the bag.
Same thing goes for learning multiplication.  It’s not sufficient to just memorize and drill the times tables.  It takes applying these new math facts to story problems and games to really make it stick.
I’m still learning about the myriad of educational games available.  One game Simon and I are currently playing is Check Math.  It’s checker-type game but dialed up a whole lot because you have to know your multiplication facts and practice strategy.  The goal is to play the game, not to recite times tables.  But in the process of playing, you can’t help but brush up on your multiplication.  I found this game while shopping at Kidding Around (15th street by 6th avenue), but I’ve also located on-line  here.
About half way through our first game of Check Math, Simon made the observation that “1” is the most powerful checker on the team because every number on the board is a multiple of “1” (a number can only travel one row at a time and must land on a multiple of its own value).  Then a few minutes later, he commented that the “10” is too big so it can only move to 20, 30, 40 and so on.  But the “5” can make its way to every number that “10” can plus 15, 25, 35, etc. 
Frankly, I was floored by the connections that boy made during that game.  Punching through the target indeed.

Step 4:  Reiterate

Just because Simon made some darn astute observations during our math game doesn’t mean he’ll correctly answer me if I suddenly blurt out what is (12 x 7)?

We continue to drill. 

Sometimes, we take it all the way back to a pile of manipulatives and a simple product of 2 whole numbers. 

Sometimes I ask him a challenge question like is it possible for an even number times an even number to equal an odd number?  Why or why not?

Today, I counted out 24 snap cubes.  I asked him to show me all the ways that 2 whole numbers can be multiplied to equal 24.  I think tomorrow I’ll have him work out all the possible values of “n” and “k” where (n x k) = 56.  He’ll just love that…

Step 5:  Troubleshoot

There are all sorts of reasons our kids struggle with understanding and learning multiplication.  We never know what might be tripping our kid up.  Maybe it’s just a matter of more drilling.  Or a little less carelessness.  But maybe there’s something else going on.

I think it’s super important to watch our children work out problems and listen to what they are saying as they explain their work process.

Here are a few things I’ve noticed that can cause trouble:

  1. Addition versus Multiplication.  Does the child know the difference between (3 + 5) and (3 x 5)?  How can you tell?  I made up a little addition versus multiplication drill for Simon to make sure he could visualize the difference between the two.
  2. Addition Strategies.  Not sure if that’s the correct term for it.  But here’s my point.  If a child can’t add numbers like 8 + 8 + 8 + 8, then this will really slow him/her down when it’s time to figure out what (4 x 8) equals.  Although we start by setting up a grid, the last thing we want to do is to count every single counter in that grid.  If (8 + 8 + 8 + 8) is too intimidating, does the child know that it can be rewritten as [(5 + 5 + 5 + 5) + (3 + 3 + 3 + 3)]?  Or rearranged into another more palatable form?
  3. Forgetfulness and poor handwriting.  Seriously.  Sometimes they just can’t read what they wrote down or drew.  Or they drew a picture to match story problem, but then forgot what the question actually was.  Say the story is that we have 5 boxes of crayons.  Each box contains 16 crayons.  How many crayons in total do we have?  Ask your child this.  There’s a good chance he/she will start drawing boxes of crayons (which is a good thing!), but I think he/she should write out (5 x 16) as well.  Because we want them to make the connection between the picture of the crayons and the fact that (5 x 16) = 80.

Multiplication tables.  We all had to learn it.  And our kids will too.  But if we can help them understand and own these first steps, then I believe we help them springboard into the next phase…. like is 186 divisible by 3?  And how do we break (21 x 11) into bite sized pieces?