# Teaching 100 x 100 times tables

First, why bother: I teach and play math with my son in the mornings. It's a lot about letting him enjoy learning, so the curriculum (my intentionally vague vision of how we'll proceed) is flexible. If he finds something interesting and wants to spend time thinking about it, we usually do. He thinks it'd be cool to memorize the times tables up to 100 - and it seems he'd be committed to doing it if we started. But the merits of doing it aside, are there any methods for doing this other than rote memorization?

Thank you

-Hal

• You mean other than the fact that after 10x10 you can do it algorithmically? Edit: Legitimate question, not snark. After 10x10 it would be easier to teach him how to do multi-digit multiplication in his head, and there are tricks for that, but I'm not sure if you want tricks like that or just exercises to do actual memorization. – LinearZoetrope Apr 17 '14 at 3:58
• – Markus Klein Apr 17 '14 at 6:58
• @Jsor yeah, he wants to memorize them so he could give the answer instantly. I think he thinks it would be cool. That said, I actually think it would be kind of neat - but I'm easily amused. – Hal Apr 17 '14 at 15:59
• @vonbrand I actually wrote a paragraph to obviate comments like that. He wants to learn it. He can judge for himself whether it's worthwhile when we get started. – Hal Apr 17 '14 at 18:19
• In fact, if your son wants to be really cool, he should memorize log tables instead. Then, he'll be probably able to do much more calculations in his head (though only approximately), including taking square and cubic roots with just a little practice. Now that is cool! – mbork Apr 20 '14 at 20:44

## 2 Answers

I realize that your question is about the $100 \times 100$ table: But since you ask about approaching the multiplication table in ways other than by rote learning alone, I thought I would leave you with a list of problems I generated based on the $10 \times 10$ table.

[Edit 5/9/14: You can find some of the problems below in an informal paper of mine; the citation is:

Dickman, B. (2014). Problem Posing with the Multiplication Table. Journal of Mathematics Education at Teachers College, 5(1).

You can also access a free copy online here.]

The inspiration for these problems can be found in a paper by Trivett (1980) in which he writes:

The recommendation here is that the multiplication table should be viewed, apparently for the first time by most people, as a dynamic synergetic combination of patterns, a veritable repository of mathematical relationships waiting as it were to gush forth from kindergarten through the secondary grades (p. 21).

Citation Trivett, J. (1980). The Multiplication Table: To Be Memorized or Mastered? For the Learning of Mathematics, 21-25. Link.

(For A, B, and G, see the concrete example section in my earlier post here.)

A. Estimate how many distinct numbers are in the 100 boxes of a 10x10 multiplication table.

B. How many of the entries in the 10x10 multiplication table are odd?

C. Starting at the 1 in a multiplication table, you can take steps to adjacent boxes: right or left, up or down, but not diagonally. What is the biggest number you can arrive at in 10 steps?

D. Consider the 2x2 sub-grid cutouts from a multiplication table. For which x can you fill in the remaining three numbers with complete surety?

E. Find the only 49 in the multiplication table. Add to it the six numbers above in its column, and the six numbers to the left in its row. What is the total?

F. Add up the numbers from all 100 boxes in a 10x10 multiplication table. What is the total?

G. Using a multiplication table, explain what it means for a number to be “prime.”

H. Look at the numbers on the south-east diagonal starting from either 2 entry in the 10x10 times table: 2, 6, 12, 20, 30, 42, 56, 72, 90. What pattern or patterns do you see?

I. Starting at the 3 in row one of the multiplication table, jump like a knight (from Chess) to move one square down and two to the right. Doing this repeatedly, you get the following numbers: 3, 10, 21, 36. What pattern or patterns do you see?

J. What kinds of symmetry can you find in the 10x10 multiplication table?

K. How would you extend the 10x10 multiplication table to cover the negative numbers?

L. Which numbers are missing from the 10x10 multiplication table and why?

M. How many one syllable number names are found in a 10x10 multiplication table?

• Yeah, I'm a fan of that. Good suggestion. – Hal Apr 17 '14 at 15:56
• +1 I wish someone had asked me these questions (or I thought of them) when I was a kid! They look very interesting. – Kevin Apr 18 '14 at 16:34
• I love this. Are these intended to be done with the table in front of the students or not? – PurpleVermont Apr 20 '14 at 3:20
• @PurpleVermont The couple of times in which I've done some or most of these problems, I've distributed a times table or drawn one up on a board. The problems change a bit depending on the audience. For prospective elementary school teachers, I'd present the four pictures from D, but I'd replace $x$ with a $4$ and then cover the blank squares with a red blob. Then I'd say someone spilled ink on this times table cut-out; can they tell for sure what the three covered numbers are? (Answer: Only in one of four cases.) For prospective high school+ teachers, I like to start with question A...(cont'd) – Benjamin Dickman Apr 20 '14 at 4:17
• ...but make the problem an $n \times n$ table and ask for a formula. Instead of having them solve it, I ask them to consider, first, what age group could explore this problem, and, second, what age group should be expected to solve the problem. Most tend to think it should be solved by middle school or high school. No one says college, and most laugh at the idea of graduate school or research mathematicians solving such a problem. But "research mathematician" is correct. (Click through the parenthetical link above for background information...) – Benjamin Dickman Apr 20 '14 at 4:19

If he wants to memorize them, then obviously there is no other way than to memorize them. (I would let him estimate the number of entries in the bigger table compared to the usual $10\times10$-table first, though.)

If he just wants to be able to quickly multiply two-digit numbers in his head, then I would start out differently: Memorize the squares, powers of 2, 3, 5, 7, then move on to tricks like $a(a+2)=(a+1)^2-1$, memorize products of primes $19\times 23$, etc.

A reasonable compromise is to memorize the table up to $25\times 25$ and exploit this to get the rest with arguments like $23\times 88 = 2300-23\times 12$.

• Last paragraph - great point. First paragraph - just to clarify, I meant to ask whether anyone knew of mnemonics for memorizing them, or a technique to accelerate the process of doing so. – Hal Apr 18 '14 at 21:02
• @Hal I do not know mnemomics, but I want to point out that up to $25\times 25$, you have three-digits numbers, you already know the last digit, and typically, you also know the first digit from approximate size, so you just have to memorize one digit (which will often be restricted by divisibility by 3, 9, 4, 8 or 25). – user11235 Apr 19 '14 at 10:21
• I am not sure that I understand your comment or that you have understood mine. When I try to calculate $19\times 18$, I know that it will be between $300$ and $400$ (because I know the squares by heart), I also know that the last digit will be two (because I know $9\times 8$ by heart, so the number is $3x2$, but since it is divisible by 9, I know $3+x+2$ is a multiple of 9, so $x=4$ and the product is $342$. A number like $17\times 19$ will not be divisible by any number in my list, though. – user11235 Apr 19 '14 at 19:21
• @Hal There's plenty. 49? 31? Every composite number betwen 1 and 100 is divisible by 2, 3, 5, or 7, but there's also plenty of primes. – Jack M Apr 21 '14 at 9:55
• @user11235 but the 18^2 - 1 trick wins in that particular case. – user1491 May 17 '14 at 22:09