Kiss those Math Headaches GOODBYE!

Archive for October, 2008

Memorizing those Times Tables

Eegads! The times tables.

Is there any area of elementary math more fraught with stress and anxiety, save, perhaps, long division? Probably not. But for good reason.

Despite what a tiny minority of conceptual-learning purists might say, the times table facts ARE critical. Let’s face it: you really DON’T want your children to spend the rest of their lives reaching for the calculator to figure out 6 x 7; a certain amount of math simply needs to become automatic, to allow students to succeed at higher math skills and and to gain higher math concepts. Not only that, but knowing the times tables is widely recognized as a crucial milestone in children’s elementary math development.

In my work as a tutor, I’ve used many approaches to teach the times tables over the years, and each of them has one benefit or another. But I’ve settled on one technique as my “old-faithful” approach. This technique combines elements of both play and discipline, and it also melds both the “conceptual” approach and the “pure memorization” approach.

This technique relies on a three-step process, and it’s easy to learn and teach.

The first step is to simply isolate a particular times table fact set you’d like your child to work on, for example, the 4s. This act of isolating itself is critical. The child knows that she or he is required to memorize a limited set of facts for now (not the entire times tables), and that narrowing of the task decreases anxiety.

Once you’ve settled on the fact set, the second step begins, and it can be quite fun. In this second step there should be no mention even made of the times tables. All you’re doing in this step is laying the foundation for times tables facts. What you do here is work with your students/children to help them learn to first COUNT UP by the number you’re dealing with. So for example, if you’re teaching the 4s, you simply teach children how to COUNT UP by 4s. What that means is that you teach your children how to think their way through knowing and saying the following with speed and ease:  0 – 4 – 8 – 12 – 16 – 20 – 24 – 28 – 32 – 36 – 40 – 44 – 48. 

I’ve found that most children take well to this learning process if you approach it in the spirit of a game. You might, for example, start by saying 0 and then throw your child a ball. She or he will then say 4 and throw the ball back to you. You then would say 8, and then throw the ball back to your child. Keep going till you hit the peak number, 40, 48, or wherever you decide to stop. 

Another way to make this into a game for young children is to make it into a game like “patty-cake.” Make up a set of hand gestures to which you, very quietly, say:  1-2-3, and then clap hands and loudly say “4!” Then use the same hand gestures to quietly say:  5-6-7, and then clap again and loudly say: “8!” There are many ways to make this process of counting by 4s game-like. And if you’re short on ideas, ask your children/students what would make it fun for them.

In any case, once your children can accurately COUNT UP by 4s, work with them in the same fashion to COUNT DOWN by 4s. Same idea, but now you start by saying 48, or 40, and then help them count DOWN:  44 – 40 – 36 – 32 –  28 – 24 – 20 – 16 – 12 – 8 – 4 – 0. This takes a bit more time, but it can be done — and more easily than you might imagine.

Once your child can count both up and down, she or he has the mental “scaffolding” on which the times table facts are hung, as it were.

And so the third step involves combining this “scaffolding” with the actual times tables. Here’s how.

Have your children memorize what I call THE THREE KEY MULTIPLICATION FACTS:
 x 1,  x 5, and x 10.

For example, when learning the 4s, these key facts would be:
4 x 1 = 4
4 x 5 = 20
4 x 10 = 40

Once children memorize those three key facts, help them see that to find 4 x 2 and 4 x 3, they just COUNT UP by 4 once or twice, beyond the key fact of 4 x 1 = 4. Similarly, to find 4 x 6 and 4 x 7 they just COUNT UP by 4 once or twice, beyond the key fact of 4 x 5 = 20. And to find 4 x 11 and 4 x 12, they just COUNT UP by 4 once or twice beyond the key fact of 4 x 10 = 40. 

Work on this first, and have them master it before proceeding.

Once a child knows these facts, she or he has 9 of the 13 key facts (going from 4 x 0 through 4 x 12).

To learn the four other facts, help children see that to find 4 x 4 and 4 x 3, they just COUNT DOWN by 4 once or twice, below the key fact of 4 x 5 = 20. And to find 4 x 9 and 4 x 8, they just COUNT DOWN by 4 once or twice, below the key fact of 4 x 10.

By breaking the process of learning the times tables into these steps, you make the process less daunting for children. By teaching students how to COUNT UP or COUNT DOWN by the number you’re learning, you help children develop many rich aspects of number sense. And by connecting the process of COUNTING UP or DOWN to the times tables, you help children learn these critical facts both solidly and with understanding.

My advice:  try it. I guarantee that you’ll like it.

Happy Teaching,

—  Josh

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Make Combining Integers a Lively Story

Who says you can’t have fun with integers?

For a number of years I’ve been using a fun story to explain the concept of combining integers with different signs. While I used to struggle at getting this concept across to certain kids, now, thanks to this story, there’s virtually no one who cannot grasp the concept when it is taught this way.

I’d like to share it with you now so you can use it, too.

The idea was first presented on p. 41 of my Algebra Survival Guide, along with a cartoon picture. The basic idea is that you can conceive of problems like:  – 3 + 9,  + 6 – 14,  – 9 + 4, etc. as a tug-of-war.

Here’s the situation — there are two teams, a positive team and a negative team. To see this kind of problem as made of two teams, you have to look at it in a certain way, a way that may be new to some of you. Take a problem like – 3 + 9. You DO NOT view this as “negative 3 plus 9.” Rather, you look at it as make up of two parts: a – 3 part, and a + 9 part. [This has always made more sense to me anyhow.]

So … the “– 3” part tells us there are three people pulling on the negative team, while the “+ 9” part tells us there are nine people pulling on the positive team.

To avoid false assumptions, tell students that all people pulling are equally strong. In other words it would be impossible, for example, for 3 on the negative team to beat 4 on the positive team. Whichever team has more people pulling must win. [I’ll discuss the situation with equal numbers of people on both teams at the end of this entry.]

Then tell students they need to ask and answer just two simple questions to find the answer:

Q#1:  Which team wins?  [In – 3 + 9, the positives win because they have more people pulling.]

Q#2:  By how many people does the winning team outnumber the losing team? [In – 3 + 9, the positives outnumber the negatives by 6, since they have 9 pulling compared to the negatives, who have just 3 pulling.]

When students put their answers together, they get the answer to the problem:  + 6. Amazingly simple, huh? All it takes is looking at things in this fun new way.

I’ve put together a little template that you can reproduce and use to teach this rule in this way.

What follows is an example that shows in step-by-step fashion how students would input the data that leads them to the correct answer.

First we’ll show this for the problem:    – 6 + 2

The empty template:

Then students input the problem, – 6 + 2, and they write in how many people are on each team, like this:


Next students make tick marks to show the number of people pulling on the each team, 6 marks on the Negative Team side; 2 marks on the Positive Team side:


Next students answer the two questions, right on the template.

Finally students write in the answer, based on the answers to the questions.


Here’s another model of how students would use the template, this time for a problem whose answer is positive the problem:  – 4 + 9




So that’s all there is to it.  If you find anyone who cannot learn it this way, let me know. I’ll be amazed.

I suggest using the template for a few days, and then, once students have the idea down cold, let them go off the template. The nice thing is that even after they go off the template, if they get a wrong answer, a really wrong answer, like:  – 3 + 5 = – 8, you can ask them to think this out as a tug-of-war, and they will virtually always get it right at that point. Cool, huh?

In a case where there are equal numbers of people on the positive and negative teams, the answer will be zero. Example:  – 4 + 4 = 0.  In terms of the tug-of-war, you might say this is a situation where neither team wins, and it ends as a tie. So a tie in the real world is a bit like 0 in the math world.

Feel free to leave comments on the blog on how well this works for you in your teaching.

And finally, if you don’t yet have my Algebra Survival Guide, it is loaded with analogies and metaphors just like this one. Teachers, parents, homeschoolers all enjoy and use this book.

50% OFF Sale for both the Algebra Survival Guide and Workbook at my website:

Algebra Survival Guide: Regular Price $19.95 — Sale Price $9.95

Algebra Survival Workbook: Regular Price $9.95 — Sale Price $4.95




You can also purchase both the Guide and Workbook at

You’ll be able to download a sample chapter that includes the concepts in this post at or

You’ll find that the Algebra Survival Guide has 173 Amazon customer reviews, with a 4.5 star rating!

Happy teaching!

—  Josh

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“Cruel” Sequences Puzzle

Here’s a fun math treat, a great puzzle to pose when you have just a few minutes of class left. It’s guaranteed to drive your students “to distraction.”

As we know, one important area of math involves the study of sequences — those strings of numbers or terms that have some order to them. Often your task in studying sequences is to figure out the next number or term in the sequence — or to figure out the pattern on which the sequence is based.

Here’s a simple example. What would be the next number in this sequence: 

1, 4, 9, 16, 25, ?

The answer is 36, since the elements of this sequence are simply the perfect squares of the natural numbers.

Algebra, as we know, involves using letters in place of numbers, so what better time to introduce a sequence involving letters.

Ask someone if s/he can figure out the next letter in this sequence:

O, T, T, F, F, S, S …

Kids will puzzle over this agonizingly. But the answer is remarkably simple. The answer is E.

The reason:  the sequence of letters are the first letters of the first numbers in the Natural Numbers:  One, Two, Three, Four, Five, Six, Seven. The next number is Eight, so the next letter is E. Isn’t that just cruel?

Supposedly, as the story goes, this problem was given both to MIT graduate students and second graders. Guess who got it faster? The second graders.

If there is a moral to the story, it may just be that sometimes we over-think situations. It’s important to keep in mind that the answer may be right in front of our noses.

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How to Find the GCF — FAST!

Time-saving tips are great, right? So I’d like to share a time-saving tip for math.

This tip SIMPLIFIES the process of finding the greatest common factor (GCF) for two numbers, a good thing to know when simplifying fractions, reducing proportions, etc.

First, have you ever noticed that when students search for a GCF, they sometimes don’t know when to stop searching? This tip alleviates that problem, for it tell students exactly when they can stop testing numbers.

It turns out that students can stop testing when they reach the DIFFERENCE between the two numbers whose GCF they’re trying to find.

As an easy example, let’s say you need to find the GCF for 16 and 20.

All you do is subtract 16 from 20, to get the difference, 4, and this number — 4 — is the largest number that could POSSIBLY go into both 16 and 20 evenly.

Once you know that, just test 2, 3, and 4 to find the highest one that goes into 16 and 20. Of course that would be 4, so you got the GCF right off the bat, in this case.

Keep in mind that that greatest possible greatest common factor is not necessarily the true, greatest common factor. But it does set an upper limit for GCFs, and having that upper limit really reduces kids’ stress.

Another example:  find the GCF for 25 and 35.

35 – 25 = 10, so 10 is the greatest possible GCF. But of course 10 does not go into 25 and 35, so 10 is not the GCF. Check the numbers less than 10, and you’ll see that 5 is the GCF. But no more checking above 10, as kids are likely to do, unless you tell them when to stop.

I have dubbed this mathematical object the GPGCF, for Greatest Possible Greatest Common Factor, and I’ve found that students really appreciate learning it’s there — to alert them when it’s “quitting time.”

Try it out yourself, whenever it next flows with your lesson. Let me know what kind of reaction you get from the kids, and good luck.

By the way, if you’d like to explain to your students why this trick works, here’s a way to look at it. If you think about this situation via the number line, the GPGCF is simply the distance between the two numbers whose GCF you’re trying to find. Let’s go back to our first example: searching for the GCF for 16 and 20. The difference between 20 and 16, 4, is the distance between 16 and 20 on the number line. So if any number does go into both 16 and 20, it cannot be larger than 4, since that’s the space between the numbers.

To see this clearly, imagine that you wonder for a moment if 8 might be the GCF for 16 and 20. Well it is true that 8 does go into the first of these numbers, 16. But the next number that 8 goes into evenly must be 8 greater than 16, or 24. In other words, 8 is going to “leap past” 20, by hitting 24, when it goes into its next multiple. So the space between the numbers — 4 in this case — gives you the biggest number that could possibly fit into both numbers.

Now, to help your students get used to this tip, here are some problems.

DIRECTIONSs:  Given each pair of numbers, first find the GPGCF. Then use the GPGCF to help you find the GCF.

a)  6, 10

b)  8, 12

c)  12, 15

d)  12, 20

e)  14, 28

f)  18, 26

g)  27, 36

h)  36, 48

i)  42, 60

j)  72, 80


a)  6, 10   GPGCF = 4   GCF =  2

b)  8, 12   GPGCF = 4   GCF =  4

c)  12, 15   GPGCF = 3   GCF =  3

d)  12, 20   GPGCF = 8   GCF =  4

e)  14, 28   GPGCF = 14   GCF =  14

f)  18, 26   GPGCF = 8   GCF =  2

g)  27, 36   GPGCF = 9   GCF =  9

h)  36, 48   GPGCF = 12   GCF =  12

i)  42, 60   GPGCF = 18   GCF =  6

j)  72, 80   GPGCF = 8   GCF =  8

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on  Just click the links in the sidebar for more information! 

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Multiplication Trick #1 — Fun with the 5s



That’s my advice to teachers and parents who see students getting bored or frustrated as they try to learn their times tables. 

As you help students learn these critical facts, it helps, from time to time — to work on multiplication in a fun and relaxing way.

This is the first in a series of blogposts that make it more pleasurable to learn multiplication facts — by teaching multiplication tricks. Each post will contain a complete lesson plan:  instruction, practice problems, and all answers. 

The first such trick is for multiplying by 5.


  Multiply numbers by 5.

When multiplying an even number by 5, just take half the value of the even number, then put 0 at the end.  Ta da … that’s your answer.


Example:  5 x 14

Half of 14 is 7.

Put down the 7, then put a 0 after it, and you get 70.

That’s the answer:  5 x 14 = 70.

Can you believe that it’s that easy? Watch how you can do the same feat with larger numbers…


Another example:  5 x 48

Half of 48 is 24.

Put down the 24, then put a 0 after it, and you get 240.

That’s the answer:  5 x 48 = 240.


PRACTICE Set A:  (Answers at bottom)

5 x 8

5 x 16

5 x 4

5 x 28

5 x 36

5 x 84

5 x 468

When multiplying an odd number by 5, first subtract 1 from the odd number, thus making it an even number. Then use the trick (above) for even numbers. And here’s the new thing to know — instead of putting a 0 after the result, put a 5.

Example:  5 x 13

13 – 1 = 12

Half of 12 is 6.

Put down the 6, then put a 5 after it, and you get 65, That’s the answer:

5 x 13 = 65.


Another example: 5 x 29

29 – 1 = 28

Half of 28 is 14.

Put down the 14, then put a 5 after it, and you get 145. That’s the answer:

5 x 29 = 145.


PRACTICE Set B:  (Answers at bottom)

5 x 7

5 x 13

5 x 9

5 x 15

5 x 23

5 x 47

5 x 685



5 x 8  = 40

5 x 16  = 80

5 x 4  =  20

5 x 28  =  140

5 x 36  =  180

5 x 84  =  420

5 x 468  =  2,340




5 x 7  =  35

5 x 13  =  65

5 x 9  =  45

5 x 15  =  75

5 x 23  =  115

5 x 47  =  235

5 x 685  =  3,425



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October Problem of the Month


A)   Each of the variables x, y, and z, stands for a unique

natural number from 1 thru 9 inclusive.


B)  The following, therefore, represents a number in the millions place: 3,x2y,1z3


C)  This number:  3,x2y,1z3   is divisible by 9.


Question:  List three other 7-digit numbers containing all of these

same digits:  3, x, 2, y, 1, z, and 3   that are also divisible by 9?



To get credit, you must not only provide the correct

answer, you must also explain your reasoning.

ANSWER: July Problem of the Month

a)  How much is the discount, under the terms of this sale, for a bundle of clothes with retail value of $250?

Discount = $130. You get this as follows:

1st)  Take a 20% discount off $250 by multiplying 250 by .8.

Answer is a $50 discount, making the new selling price $200, $250-$200.

Take a 40% discount off the new selling price of $200 by multiplying 200 by .6.

Answer is an $80 discount, making the new and final selling price $120, $200 – $80.

Total discount is the sum of the two discounts:  $130 = $50 + $80.


b)  How much would the discount be if Al’s had just run a straight 60% discount sale?

At a straight 60% discount, the dollar value of the discount would be $150, which you get by multiplying $250 by .6.


c)  Why do you think that Al’s is running the sale in this way?

Many possible answers, but the main point is that Al’s strategy is to make customers think they are getting 60% off, while he actually gives them less than 60% off. He actually gives them a 52% discount.

Algebra mistakes due to Negative Signs? Use COLOR to EXPLAIN.

Have you ever noticed how much trouble students have figuring out the difference between these two kinds of expressions:

The key, I’ve found, is to use the Order of Operations to decode the meaning of each expression.

Ask your students to think through how many Order of Operation steps there are in each expression.

If they’re on the right track, they’ll see that the top expression has just one operation:  working out the exponent by multiplying (– 3) by itself twice. This looks like this:

As for the bottom expression, this is more difficult because of that “loose” negative sign, a negative sign that is not enclosed inside parentheses. Just like a “loose canon” can be a danger in the military, a “loose” negative sign can spell trouble on the algebra playing-field. Students will make all kinds of weird mistakes in trying to figure out what to do with this little varmint.

But if you tell students to view that “loose” negative sign as meaning (– 1) times the expression that follows it, they can use the Order of Operations again to do the right thing first. 

Specifically, they notice that there are two operations to be performed: multiplication and exponents, and they remember that they do exponents before multiplication, following the Order of Operations.

Following these steps, the simplification looks like this:

And while this is fine and correct, notice that you can “kick it up a notch” by using color to separate out the two operations. I use blue for decoding the negative sign and working it out; I use red for the 3-squared term. Check it out:

I’ve yet to meet a student who cannot follow this procedure, when color is used.

Try it out and see what kind of results you get.

Color Your Way to Integer Success

We all know that one of the trickiest subject for many students is the subject of combining integers.

I’ve hit on a new way to help students with this topic, a way that involves using color.

Using different colors helps students relate similar concepts and separate different concepts. Color works faster than underlining or drawing rings around numbers, it’s more attractive, and it makes a student’s page fun to look at, too.

Here’s just one example of how using color can help students make sense of those oft-bewildering positive and negative numbers.

Take a problem like:   + 3 – 7 + 6 – 9

Many students get confused by a problem like this because they don’t have the least idea of what to do first.

But when you use color, tell students that the first step is an easy step:  just re-write the problem, making the positive numbers red, and the negative numbers blue, like this:

The next step is equally easy … group the positives on the left and the negatives on the right. I use my handy-dandy “double-slash” divider to show the separation, like this:

After that, use the rule for combining integers with the same sign (here you can say “with the same color!”), like this:

At this point you depart from color, as you combine the integers with different signs, and you get the answer, like this:

All together, it looks like this:


I have found this approach extremely helpful for those students — and you know the ones I’m talking about — who just struggle endlessly with these rules. 

By the way, if you’d like to see the chapter of my Algebra Survival Guide, which explains the “Same-Sign” Rule and the “Mixed-Sign” Rule, just go to this page and scroll down about half-way down till you come to the link for downloading chapters:

Please try this out yourself and feel free to let me know how it goes. I’m always open to feedback.