A twist on multiplication flash cards.

I read a short ebook this week by Renee Ellison called Teach Math Faster.  (Quick read, only $0.99 on Amazon Kindle.)  She has some very good hands-on ideas for helping children with math basics.  One of them was a twist on the tried-and-true multiplication flash cards, in which she added a visual representation of each quantity instead of simply writing the numerals.

Why is this helpful?  Because the more often a child can use concrete ideas when learning math, the better he will understand what the numbers are doing. Math educators call this "number sense."  For example, instead of rotely memorizing the words and symbols of
6 x 4 = 24,  the child using Ellison's flash card design sees an array of 24 blocks, arranged in four rows of six blocks each.

This is the front of a
Teach Math Faster flash card.

Now of course, children who are old enough to learn multiplication will have long mastered the concept of number symbols standing for quantities.  However, many children have only a shallow understanding of what multiplication actually means.  They will use "times" as a verb-- "I can times six an three"-- in the same way that my children used to think that verse was a verb meaning "to defeat"-- as in, Godzilla versus ("verses") Megalon.  (They would say, "I'm going to verse those bad guys!")  But when the front of the flash card shows a graphic representation of what  "six times four" means, the concept is reinforced at the same time that the memorization is taking place.

An added dimension of concept reinforcement occurs when a range of multiplication facts is studied-- while "8x7=56" uses the same number of digits as "2x5=10," there is a clear difference between the space occupied by ten squares compared to fifty-six squares.  The child's number sense increases when she sees just how much bigger 56 is than 10.

On the back of the flash card, Mrs. Ellison repeats the array, this time without the numbers, and provides the product:

This is the back of a Teach Math Faster flash card.

So when the child is using this flash card, he is reading the numbers on the front and multiplying them in his head just like a traditional flash card.  He checks for the correct answer on the back, just like a traditional flash card.  But his brain is experiencing something totally new: the abstract symbols are gone, replaced by a picture of what's happening in the problem.

Genius!

Now for the bad news:  as far as I know, this style of flash cards is not commercially available.  Like the other manipulatives in Mrs. Ellison's little book, they have to be home made.  She recommends using graph paper to keep the size of the squares consistent-- otherwise, there is less visual impact in the size difference that I explained above.  So you-- or your child-- write the numbers in the squares and cut the arrays out, then glue them onto index cards.

One suggestion I would make:  if you are using color-coded math manipulatives such as Cuisenaire rods or Math-U-See blocks, match the flash cards to whichever set you're using.  It's as simple as adding lines of the appropriate colors to the edges of the array on the front side of the card, like this:

This array matches Cuisenaire rods, with
forest green for 6 and magenta for 4.
Math-U-See would use purple and yellow.

Cuisenaire rods and Math-U-See blocks
use different colors to signify quantities.

Be sure not to color-code the back side of the card, however, because that would too quickly give away the numbers that the student is supposed to be memorizing.  The array itself already shows the answer if the student counts the blocks, but he still has to work for it. Color-coding the front side makes sense because the number symbols are also written in, so nothing is being given away.

As mentioned in a previous post, adding color provides one more level of connection for the brain.  If your child already associates certain quantities with specific colors, tapping into this association will increase his number sense as he learns his multiplication facts.

Teaching with Games

I came across a helpful resource last month, but am just now finding time to post.  (Ah, the holidays!) This is another offering from the people at the Institute for Excellence in Writing-- aka IEW-- and I have already used it with my tutoring students as well as my Sunday morning Bible class.  It's that versatile!  And, no, this is not a paid or compensated endorsement.

The book is called Teaching With Games, by Lori Verstegen, and can be purchased either as a book only or as a dvd workshop course with the book and video examples of teachers using the games with actual students. I'm cheap, so I just bought the book.  It has reproducible games and templates for customizing your own games, plus ideas for easy "no-prep" games.  Granted, many of them are old standards like Bingo, Hangman, and Jeopardy, but having them all together, with ideas for using them in the various subjects, is very helpful for teachers like me who get stuck in a rut.  It's like reading a recipe book when you find yourself making the same dinners over and over: sometimes you just need to be reminded that other meals exist.

One of the first games I tried was "Fun Times" with my math kids.  It's pretty simple-- the game board is a one-page reproducible table with columns labeled 5, 6, 7, 8, 9, and 10 across the top, and rows labeled 2, 3, 4, 5, 6, 7 down the side.  Within the table are the products of the numbers; e.g., where 4 and 7 intersect, the square is labeled 28, and the square where 9 and 6 intersect is labeled 54. Also, some squares are grey and others are left white.  To play the game, students use two dice that have been modified so that one has the numbers 7-10 and the other has the numbers 2-7.  The player rolls the dice, and multiplies the resulting numbers, placing his marker on the appropriate square on the board.  The second player then goes.  If he rolls a number that has already been played, he can replace it with his own marker if it is not a gray "safe" square.  The object of the game is to get four squares in a row (like Bingo, in any direction).  Any markers can be used-- I used pennies and covered half of them with pieces of paper to distinguish the two players.  The game is easy, fast, and my students enjoy it more than regular multiplication drills.

My Bible class has also recently played Jeopardy and Wheel of Fortune as suggested in the book.  They loved both, and for Jeopardy I was able to reuse game cards I had made for a previous board game that we had all grown tired of.  Funny how a different format can renew their excitement over old material!

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.

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!