Musings on Multiplication

Some of us remember the days before claculators were common in schools.  [ The "Good Old Days" - formerly know of as "These Hard Times"! ]  I am not going to suggest that we should halt all use of calculators in elementary and secondary classes.  I would like to point out a few observations I have made about computation and particularly about multiplication.

First, let us consider that terrible burden: Learning the Multiplication Table - Memorizing It.  I don't recall what grade I was in when that was required.  I know that I did it along with all the members of my class.  No, we did not all get it easily nor did we all attain quick use of those facts.  At the time, many of us still had to stop and think a few moments when we needed a multiplication fact.  We might still make mistakes on some of the "hard" combinations.  Still, as a group, we did well and most of us got to be quite fluid in our use of the table.

Current practice in many schools seems to downplay the need to memorize the table.  Along with not memorizing it comes the side effect of not becoming fluid in multiplication.  Too soon the calculator takes the place of the intellect in performing those computations.

So, what did we get from our practice with the table that calculator toting young people do not have today?

The first thing we got was quick answers that were usually correct.  Give a problem that required us to multiply 6 x 8, we could immediately put down the result as 48 and move on to the next section of the problem or to the next whole problem.

Consider what a student must do who does not have that multiplication fact instantly accesible in their memory:

They look at the problem and know that they don't know the answer but they may feel that they should know it.  That is dis-spiriting.

Next, they look for their calculator which may not be immediately at hand.  Once they have it, they look back at the page and find the problem again and then find the numbers.  They enter the first number and hit X.  They look back at the page to find the second number and pause to decide if it was the one they just entered.  Deciding that they have the right one, they enter it and hit =.  The answer comes up and they get ready to write it down.  Then they look at the problem again to decide if it is the answer they needed or if it is part of the solution and requires more work.

Some of you reading this will think I exadurate.  I have seen students reach for a calculator to multiply 3 x 4.  Why?  Because they have learned to ALWAYS reach for a calculator to do multiplication.  Those students don't know that they can do the work in their heads because they were never shown that it is possible.

My second observation has to do with the way students are taught to multiply.  I recently happened upon a translation of the earliest Arithmetic text printed in Europe.  I was rather surprised to find that is taught the "lattice" method of multiplication.  On thinking about it, I should not have been surprised.  At the time it was printed, few people knew even the most basic mathematical skills.  The lattice method has the advantage that it keeps track of all the sub-products as one or two digit numbers so the carrying is all done at the end.  I often see students use this method today and, if it is done well, it gives good results.

However, it has a major disadvantage for students who only know how to use this method.  It takes approximately three times as long to perform the operation using lattice as compared with the traditional method.  Aside from the sheer time element, there is the fact that it takes the student's attention away from the nature of the problem for all that time.  By the time they have done the multiplication, they will have to re-read the original problem to find out what they need to do with the product they just calculated.  ( Although there is a simple method for keeping track of decimal points, I have found that many students do not understand how to use it correctly in the lattice calculations. )

So, what do I suggest? 

Students should be taught the multiplication table and should use it until they are fluent in its pattern and accurate in their results.  In the process of this, they should not simply memorize the table as 144 unrelated facts.  If they are shown the patterns within the table, it should help them later with division and with factoring.  For example:  the number 48 shows up as 4x12, 6x8, 8x6, and 12x4.  36 shows up as 3x12, 4x9, 6x6, 9x4, and 12x3. 

Also, although I do not take issue with showing students the lattice method when they start multiplying larger numbers, once they have the general idea and they have been shown that the lattice keeps track of the place values of the sub-products, they should all be taught the standard algorithm until they are fluent in its use.  This will save them great amounts of time in later courses and allow them to come back to the essentials of the problem quicker.

There will be more.

Multiplication Presentation - Complete.ppt

Here is more taking a different tack on matters of multiplication.

We are all (I hope) familiar with the process for multiplying a two digit number by a single digit number.

 

34

x 7

 

We start by multiplying 7 * 4 = 28

 

 34

x 7

 28

 

Oops, we don’t generally write the 2

down there.  We often put it up near the 3

to remind ourselves that we are going to

add it to the next product.

I’m going to show it a little differently.  Bear with me here.

We are ready to multiply 7 * 3 = 21

Well really that is 7 * 30 = 210

  34

 x 7

  28

210

238

 

Now if I write it this way, the carried “2”

lines up with the “1” and it will just add without any

“carrying”.

Lets look at that again.  34 = 30 + 4.  So         

34

x 7

becomes

30 + 4

       x 7

210 + 28

 

And that is 30*7 + 4*7 = 210 + 28 = 238

 

Remember the process of Distributing multiplication?

 

It was probably shown as A(B+C) = AB + AC, that is A*B + A*C

  Here, A = 7, B = 30, C = 4

B+C = 34

There is a limited possibility that you were shown distribution as (B+C)A = AB + AC. 

It works the same way.

 

Suppose you know the numbers inside but you do not yet know the number outside?

A (30 + 4) = 30A + 4A       

We like to put the numerical part, the coefficient first and follow with the variables, the letters.

 

What if we then find out that the number A is really a two digit number?  Let us use 56.

Then what happens to our algebraic multiplication scheme?

A(30+4) = 30A + 4A  becomes

30*56 + 4*56

I’m going to change this just slightly to make a point.

10*3*56 + 4*56

I just want to move it back to where I have a single digit multiplied by a two digit number.

The multiplication by 10 can be done easily after we do the harder part.

So

 

56

x 3

 

and

56

x 4

 

 

Again I will do this without the floating carry digits.  (You have probably figured out by now that I don’t know how to type them!)

 

56

 x 3

  18

150

168

 and

56

 x 4

  24

200

224

 

 

So A(30 + 4) = (30 + 4)*56 = 10*3*56 + 4*56

                    = 10*168 + 224 = 1680 + 224 = 1904

If we set the whole thing up in the traditional way (but without showing the carries)

 

Notice that all those partial products look very familiar from the numbers in the boxes just above.

 

   56

 x  34

      24

    200

    180

  1500

  1904

 

 And what we just did was exactly the same as the FOIL system in multiplying two binomials

(50 + 6) (30 + 4) = 50*30   +  50*4       +      6*30      +  6*4

                                 firsts   +   insides   +   outsides  +  lasts

     56       

 x  34

      24     lasts

    200     insides

    180     outsides

  1500      firsts

  1904    The order is different but the process is the same.

And since the order of addition does not matter, the answer is the same.