Skip-Counting Beads (Pickle Beads)

This post was inspired by a bright young lady I have "met" only online, who is affectionately known as The Pickle.  She was skip-counting 7's on her math assignment last week and not enjoying it at all. I can relate.  Some students find that skip-counting songs are very helpful when they are trying to learn the number sequences, but for those who prefer a hands-on or visual approach (my daughter loathed memory songs), I offer this activity.

A set of 10 Skip-Counting Beads

If you're unfamiliar with the term, skip-counting is a valuable step in learning to multiply.  It's basically counting by a number: 7, 14, 21, 28, 35, 42, 49, 56, and so on.  Not everyone learns to skip count, other than the usual counting by 2's, 5's and 10's, but those who do will find that later memorization of multiplication facts comes almost effortlessly.

A set of skip-counting beads is a hands-on tool for skip-counting-- helpful not only for learning to skip-count, but also for visualizing multiplication as a concept, and comparing the quantities involved.  As an added bonus, it is also a handy illustration for Least Common Multiples.

If your student is just starting out in the lower grades, it would be beneficial to have him string his own beads, perhaps one set a day.  Older children might make more strings per session.   Some students may lack the fine motor skills or patience to complete the set without a meltdown;  it's not a total loss if you end up making them yourself, as long as the student uses them when you're finished.

Materials for making the strings.

To make a set of 10 (through 10x12) or 12 (through 12x12) skip-counting bead ropes, you'll need a couple of large bags of pony beads of a single color (shown here as white)-- 530 beads for a set of ten strings, or 771 for a set of 12), plus 12 beads of a different color for each rope.  (The set above includes 10 ropes, modeling 1x12 through 10x12.)  You will need about 24 feet of string or cord (18 if you're making only ten ropes), and a pair of scissors. An extra-fine point permanent marker is optional; it allows you to write the numbers on the beads.  Numbers are helpful, and can reduce error when learning to count.  If your child is very visual, I would definitely recommend writing the numbers on the beads.

Bottom to top: One, Two, Three, Four, and Five strings.

To make the set, make the first rope by stringing 12 of the white beads.  Tie off the ends.  (If you're using para-cord, you might want to melt the ends in a flame to prevent raveling.)  Next, make a second string alternating white and a second color, using 12 of each.  The third string uses 2 whites for every one of a third color; the fourth string uses 3 whites for every bead of the fourth color.  Continue until you have made a string to count by 5's, 6's, 7's, and so forth.

To label the beads, write the numbers 1-12 on the beads of smallest string, and then write the

Labeled beads.  (Needed a finer point marker.)

appropriate numbers on the colored beads of
each string.  (Writing the numbers on all of the white beads on every string is not a bad idea, but it may take the emphasis off the colored beads. However, some children may prefer having every bead labeled, and that's okay.)

To use the beads, the student holds a string in his hands and touches the colored beads as he counts aloud.  If he needs help. he can either count up using the white beads or read the number written on the colored beads.  The student should practice until he can rattle off the numbers easily.

Later, when the student is learning his multiplication tables, you can show him how it works on the string:  for the 3 times table, for example, use the Three string on which every third bead is colored. Touch the first colored bead as you say, "Three times one is three."  Then touch the second colored bead: "Three times two is six," and so on, up the string to "Three times twelve is 36."  Then demonstrate the facts out of order: count up 5 colored beads, and say, "Three times five is fifteen."  Count up nine: Three times nine is twenty-seven."

Lining up Three and Four string.

You can show how the facts work out the same, regardless of the order of the numbers being multiplied (Commutative property of multiplication), by lining up two strings with their beginning white beads together and the rest of their beads matching up beside each other.  Here we have the Three string and the Four string. Point out to the student that 3x4 and 4x3 have the same answer: both strings have colored beads at 12.

To illustrate Least Common Multiple (LCM), explain to the student that the colored beads represent multiples of a number.  When you skip count, you are listing the multiples of a number in order.  To find the LCM of a pair of numbers, line up strings representing the numbers you are comparing.

Lining up Four and Five string.

Here we have the Four sting and the Five string. When their beads are lined up together, the first time two colored beads line up next to each other is at 20.  This is the least, or lowest, multiple that the two numbers have in common.  If you put Two and Five together, the LCM will be 10.  Two and Four have an LCM of 8.  The student may be tempted to conclude that the LCM of two numbers is the same as the product of the two numbers; however,  sometimes the LCM is less than the product:  The LCM of Ten and Five is 10, not 50.  The LCM of Nine and Six is 18.

You can use the strings to show LCM of more than two numbers; just line up three or more strings together to point out the alignment of the colored beads.  Five, Ten and Two align at 10;  Three, Six, Two, and Nine align at 18.  (Some number combinations cannot be shown given the length of the strings; for example, the LCM of 9, 5, and 10 is 90, but the Five string stops at 60.)

Once the student has mastered the use of the skip-counting beads, he is well on his way to understanding-- and mastering-- multiplication.

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.

Memorizing multiplication facts

If you haven't seen Times Tales, you should check it out.  It is the quickest way I know to get those multiplication facts to stick in the brains of kids who are having trouble with memorization.

But if you can't buy one more piece of curriculum, here's the system in a nutshell:  turn each digit into a picture, and then combine them in scenarios that help you remember the products.  If you buy Times Tales, the work is already done for you, and there are illustrations and flash cards and other helps.  You can do the same thing yourself, though, with a little imagination.

For example:  Draw a 5 and doodle it into a duck.  Draw a 9 and turn it into a balloon with a string.  Then make up a scenario, such as, "The duck had a balloon that blew away in the wind, and he chased it for 45 minutes."   5 x 9 = 45

Draw an 8 and turn it into an owl:  "The owl saw the balloon stuck in a tree.  He hopped across 7 branches and pecked it 2 times before it popped."  8 x 9 = 72.

Draw a 6 and make it into a mouse with a long tail.  "The owl tried to catch the mouse but those 4 tiny feet ran 8 yards to his hole and he was safe,"  6 x 8 = 48

Draw 7 and make it a flag.  "The duck saluted the flag from 3pm to 5pm." 5 x 7 = 35

Anyway, you get the idea.  You don't have to make up a story for all of them, just the ones your child finds hard to remember.  If you can get your child to make up the scenarios, all the better!

Solving Memorization Problems

What if I gave you the sign below and asked you to post it up on the wall-- but I didn't give you any tape?   Or if I asked you to put it up on a bulletin board without tacks or staples, or on a refrigerator without a magnet?  You could put that paper in the same place over and over, and it would never stay.  No matter how many times you tried, it would fall to the ground every time.

Your child's memory can be that way, too.  Have you seen your child practice those math facts over and over, and nothing seems to stick?  Or maybe you've been helping her with those states and capitals, and she still can't remember that Concord is the capital of New Hampshire?

Your child can be very bright and still have trouble with these things.  In fact, sometimes it is because your child is bright-- her brain is busy learning new concepts, thinking through ideas, making sense out of the world.  It is busy making connections between one concept and another.  And unfortunately, busy brains tend to discard what feels like random facts that are unconnected to anything else, no matter how many times they may be exposed to these facts.   Endless repetition of flash cards or other drills, if they're not working, will only frustrate your child and drive you both insane.

 What the brain needs is just what you needed when I gave you the sign-- something to make it stick.  And that "something" is not the same for every brain-- some brains need tape, some need staples or pins, some need magnets-- some may need glue, Velcro, or putty!  Metaphorically speaking, of course.  Using the wrong something can be as effective as trying to stick that sign on the wall with a magnet.  So if your child is getting discouraged over his failure to memorize something important, maybe it's time to try a different tactic.

Mnemonic devices ("a pattern of letters, ideas, or associations that assist in remembering something") are helpful for many students.  The sign itself is an example-- using "PANDA" as an acronym for classroom supplies.  Associations, the sillier the better, are among my favorite memory aids. A great product for learning multiplication facts through associations is Times Tales.  This is a set of silly one-liner "stories" that introduce children to characters that represent numbers, and then present a situation for each multiplication fact that helps tie the numbers together. For example, 'Mrs Week (7) and Mrs Snowman (8) were in a car driving one mile over the speed limit (56). ' 7 x 8 = 56.  And I love Yo, Sacramento -- a book that teaches states and capitals with the silliest pictures ever: I will never, ever forget the capital of New Hampshire after seeing the illustration of a Concorde jet being driven by a Nude Hamster!

Music is a great memorization tool for a lot of people.  Audio Memory has been around for decades and has taught thousands of children everything from math and grammar to geography and history through songs. Back in the 70's we had Schoolhouse Rock-- now available on dvd or YouTube.  In fact, there are countless math fact drills set to music-- in pop, rap, even country styles-- that can be found on YouTube.  Same with states and capitals.  After 25 years, I can still recall the countries and capitals of South America from this song.  Even the quadratic formula from algebra class has several songs in its honor.  You can Google just about anything that commonly has to be memorized and find a song for it: just type in (the thing you have to memorize) and "song."  Like, "layers of the atmosphere song" or, for a real challenge, "periodic table song."

But songs don't work for everybody.  Some students remember things best when they build, or draw, or in some other way interact with the information. Maybe working a map puzzle, or labeling a plastic-covered page with a dry erase marker, or matching up a set of cards, would help those states and capitals stick.  Maybe linking all the pairs of numbers that add up to 10 or 12 or 16 with a set of Cuisenaire rods would help with addition and subtraction facts.  Maybe creating models of molecules and breaking them into atoms to reassemble into new molecules is the key to acing that chemistry test.

The point is, if something isn't working for your child and she's given it a good, honest effort, it's probably the wrong approach.  Like using a thumbtack on a refrigerator, it's not going to stick.  Try something different!