A recent question (@Namaste) made me realize that it would be good to pull together the best resources for learning the multiplication facts. When seen as a rote memory task, this can turn students off or give students the wrong impression of what math is. When seen as finding patterns, it can be joyful and provide deep learning.

My son loved doubling numbers when he was young. I think many kids enjoy this. That's the 2's.

There are images that go with certain facts; like 2x6 is the shape of an egg carton, which holds 12 eggs.

2s and 5s are easy because of their relation to 10. And, though it's less obvious, 9s lovely patterns exist because of its relation to 10, as seen in @Namaste's post

4 is doubling twice.

What are good strategies for seeing 3s, and 6s through 8s? Also, what are good images for individual facts?

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    $\begingroup$ Note that once one covers all you name (1, 2, 4, 5, 9), then, when covering 3, a student knows the time table for $3\times 1, 3\times 2, 3\times 4, 3\times 5, 3\times 9$, provided students are taught commutativity. Similarly, for the other remaining numbers, even fewer facts need to be explicitly taught. $\endgroup$ – amWhy Dec 16 '19 at 18:15
  • $\begingroup$ Also, what you term "doubling" actually presupposes mastery of the multiplication table for $2$. So you can't really write off the need to learn multiplication by 2, else, the understanding of doubling means little. $\endgroup$ – amWhy Dec 17 '19 at 22:01
  • $\begingroup$ Not writing it off, just saying that kids seem to especially enjoy it. (And the connection with adding is very clear in doubling.) $\endgroup$ – Sue VanHattum Dec 17 '19 at 23:00
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    $\begingroup$ This is about double-digit numbers, so I'm not posting it as an answer: I like to note that 50, 60, 70, 80, 90, 100 are the values of 10x5, 12x5, 14x5, 16x5, 18x5, 20x5, respectively. Quickly recalling these "mid-points" allows (1) to know that 55, 65, 75, 85, 95 are the values of 11x5, 13x5, 15x5, 17x5, 19x5, and (2) to compute (say) 17x8 in two steps: as 17x3 + 17x5. So, I only need to quickly recall half of the multiplication tables for 11 to 20. $\endgroup$ – user7990 Dec 19 '19 at 11:41

As for 3's, I always found it easy as a student to think it as $3=2+1$. So, in order to multiply by 3, one has to double and then to add what they've found.

For instance:

$$3\times7=2\times7+7=14+7=21,$$ which is much easier, since douling is carried out relatively easy by most students and addition is, in general, easier than multiplication.

Similarly, multiplication by 6 is made easy since $6=5+1$.

In general, if $\mathcal{N}$ is the set of all numbers whose multiplication table is easy, then $n-1$ and $n+1$ have relatively easy multiplication tables, for any $n\in\mathcal{N}$.

Edit: Summarizing the presented tricks, we have that the multiplication tables of: $$0,1,2,3,4,5,6,9,10$$ are easy or relatively easy to find/explain/remember. Consider that the tables of $7$ and $8$ are almost complete but for $7\times8$. So, the problem of rote learning the multiplication table from $0$ to $10$ has been reduced to rote learning $7\times8$.


When it was my turn to learn the multiplication table, one of the broadcast television stations in the United States would play animated music videos between the cartoon shows on Saturday morning. This was Schoolhouse Rock.

The Multiplication Rock videos continue to this day. They are generally 70's American folk rock, although they sample a wide variety of musical genres. The Grammar Rock songs like "Conjunction Junction" are more famous, but I personally think that they have aged relatively well. (I am also in my 50's, so don't take my musical tastes as gospel.)

Overall, I think the pedagogical tack they were taking was rote memorization by getting these earworms stuck in your head. At the same time, there are often visual aids like your egg carton example that can spur mental connections. And they also describe things like the distributive property and the tricks for 9, 10, and 11.

I doubt they're right for everyone, but it certainly helped me as a kid with auditory intelligence to memorize nine songs instead of one table.

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    $\begingroup$ Lovely example! My first criteria is joy. Some things may have to be learned by rote, and music makes that fun. I think it was Row Your Boat that helped me memorize the quadratic formula. (When I began teaching, I had to have it in my notes, because I still wasn't sure I had it right.) $\endgroup$ – Sue VanHattum Dec 16 '19 at 19:54
  • $\begingroup$ @SueVanHattum Wait, the only quadratic formula song I know is to the tune of Pop Goes the Weasel. I demand a concert to choose the right song! $\endgroup$ – Matthew Daly Dec 17 '19 at 14:37
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    $\begingroup$ @SueVanHattum On a more serious note, I agree. My educational philosophy is that the best way to remember things is to understand them. But there doesn't seem to be much to understand in the multiplication table aside from step-counting, so our best strategies might be supplying students with a large number of hooks appealing to various intellectual and sensory modes knowing that different students will be helped by different combinations of those tips. $\endgroup$ – Matthew Daly Dec 17 '19 at 14:41
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    $\begingroup$ I think there's more "understanding" available in the multiplication table than we might see at first. Like the 9s thing, which might look like a trick, but is based on a pattern. There might be patterns for other things that we haven't mentioned here yet. Like you say, let's appeal to students in as many ways as possible. $\endgroup$ – Sue VanHattum Dec 17 '19 at 18:58
  • $\begingroup$ And I can't get the Pop Goes the Weasel one, because it's too close to Row Your Boat. I think I've heard of others too... $\endgroup$ – Sue VanHattum Dec 17 '19 at 18:59

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