Multiplying: from single- to multi-digit numbers in 10 simple steps

Multiplying large numbers by other large numbers is a process that normally is taught over several years.  In second or third grade, students begin to learn the single-digit multiplication tables, and then gradually the difficulty of their calculations in increased.  By sixth grade, most will be multiplying multi-digit numbers with ease.

But if you are working with a student who got a late start, or has some learning gaps and needs to catch up, you don't have "years" to work with.  Happily, the older student will likely not need years.  She (or he) will just need the gaps filled in and can then make quick progress.  But first make sure those multiplication tables are learned, at least through the 10's.

Once your student has her math facts down pat, it's time to learn how to work with multi-digit numbers.  Ready?  Let's Go!

4 x 3 = 3 x 4

Step one:  Be sure the student can visualize what it means to multiply.  Use manipulatives (math rods, or beads, popsicle sticks, toothpicks, beans)  to model for her, and then have her model for you, that 3 x 4 means four groups of three things, and 4 x 3 gives you the exact same product with three groups of four things.  Have her show you all the 1-9 times tables until you are sure she understands what is happening.

Step two:  Have her compare the 1 and 10 times tables.  With the 1's, one group of one, two, six, or any other number will be that same number.  With the 10's, ten times any number is the number with a zero after it.  This is important.  If she doesn't see it immediately, have her write down the 1's and 10's beside each other until she sees the pattern.  

Modeling 2's and 20's

1x2=2     10x2=20
1x3=3     10x3=30
1x4=4     10x4=40
1x5=5     10x5=50

Step three:  Have her model with manipulatives the 2's tables and then 20 times each number:

2x2=4     20x2=40          
2x3=6     20x3=60
2x4=8     20x4=80
2x5=10   20x5=100

She may see the pattern before modeling.  If so, have her just write out the products.  Once she sees the pattern, have her tell you what 30, 40, 50, 60, etc multiplied by each digit will be.

Step four: Show her how to write the problems vertically.  When she writes the answer, have her write the zero first and then the rest of the product.  Tell her we write the zero first to hold the ones' place and then we can multiply the other digits-- in this case, 6 x 4:
60            60               60
x4            x4               x4
                   0             240

Step five: When she can quickly do problems like 40 x 6 = 240, 70 x 3 =210, 50 x 9 = 450, show her a problem like 62 x 4.  How do we work this out?

All we have to do now is understand what 62 means:  60 + 2.   6 tens and 2 ones.  So have her work the two parts of the equation separately:  60 x 4 = 240,  and 2 x 4 = 8.  Then she adds them together:  240 + 8 = 248.

Step six: Show her how to work through the multiplication in one vertical problem.  62 x 4: This time instead of writing the zero, she multiplies the 4 by the 2 and writes 8 in the one's place.  Then she immediately goes back to the 4 and multiplies it by the 6.  Even though it is actually 60, she doesn't need to write 240 because the 8 is holding the one's place.

 62             62                62
x 4            x 4               x 4
                    8              248

Step seven:  carrying tens.
If she is familiar with quarters and dollars, the student may already know that 25 x 4 = 100.  If not, remind her that 25 means 20 + 5.   2 tens and 5 ones.  So have her work the two parts of the equation separately:  20 x 4   and 5 x 4. Then she adds them together.  20 x 4 = 80,  5 x 4 = 20,  80 + 20 = 100.

But how does that work vertically?  In this case, remind her that 20 means 2 tens and 0 ones.  So we put zero in the one's place.  Just as when we add, the 2 gets carried over to the tens' place, to be added in after we multiply 20 x 4.

So, then we try something like 67 x 4.

She should have a lot of practice with this step before going on to the next one.  You can find practice problems here. 











Step eight:  2 digit x 2 digit multiplication.  This is similar to steps 5 and 6.  Once she has 2 digit x 1 digit multiplication down, show her a 2 digit x 2 digit problem.  For example, 62 x 47.   Now, ask her to split up 43:  40 + 3.  So now she can do two multiplication problems: 62 x 7, and 62 x 40.  If she isn't sure about the 62 x 40, remind her that it's just like 62 x 4 with the "0" place holder on the end:

   62            62                 62
x 47            x 7             x 40
                  434            2480

Now she can add the products together:

     434 
+ 2480
   2914

Step nine:  2 digit x 2 digit multiplication in one vertical problem.  Step eight works, but it takes a lot of time and space.  Happily, we have a short cut!  We can split the numbers and still work with them one at a time, without rewriting them.  And then we can add the products together without rewriting them!

The operational steps are the same-- multiply the ones, multiply the tens, then add the products.  But since we know we're going to add them in our final step, we go ahead and write them that way to begin with.  Emphasize the importance of using that place-holding zero in the second line to keep the digits lined up correctly.

  62            62               62           62
x47          x47             x47         x47
                434             434         434
                                 2480     +2480
                                                2914

Step ten:  infinity and beyond.  Your child will probably understand immediately how to do 3 digit by 1 digit (452 x 7), and then 4 digit by 1 digit (4276 x 3).  Give her some practice with those.  When she can do those, give her 3 digit x 2 digit (452 x 27) and 4 digit by 2 digit (4276 x 53).

Finally, introduce the idea of multiplying by 100.  How many place-holding zeroes do we use?  Two.
So when we have a problem in which a number is multiplied by a three-digit number, we can use our short cut with a third line for the hundreds, and use those two place-holding zeroes before adding all three rows:


  162            162               162           162
x247          x247             x247         x247
                  1134             1134         1134
                                       6480         6480
                                     32400    + 32400
                                                      40014

At this point, the student should see the pattern, but if not, you can explain to her that for every digit you are multiplying the top number by, we use a separate row for the product and the appropriate number of place-holding zeroes for every row after the first product.  The final answer comes from adding the individual products together, being sure to keep the numbers lined up.

With practice, this pattern will become second nature and your student will be successfully multiplying as many digits as she needs to.