*The following was translated from this post by Jane Kats. Translation and translator’s notes are by Yulia Shpilman.*

(Translator’s note: I taught a lesson inspired by this post to my math circle for 1st and 2nd graders on Sunday and we had a blast. We spent about 20 minutes filling out the multiplication table (some kids finished the whole thing and some about half but everyone was able to get through something) and spend another 20 minutes looking for various numbers and patterns.

Everyone had their own strategy of filling it out (just as Jane describes) and, to my great surprise, everyone really enjoyed doing it. One of my favorite by-products of this activity is that it allowed the students to find and correct their own mistakes – if someone were filling out the 4s row, for example, and goofed up, she saw that something didn’t work when she was filling out the 8s column because the numbers didn’t make sense! It’s a rare activity where the mistakes are so obvious and easy to fix – it was a great lesson for the kids!

And last note – if you’re going to give your kids a visual multiplication table, please use this version – not the version that lists out all the equalities such as 2×2=4, 2×3=6 and so on – that is NOT a table and it’s not at all useful for seeing the beautiful patterns in the actual multiplication table!)

*And now on to Jane’s post.*

At our math circle, we don’t do many arithmetic exercises, but we try to show our students some rules and patterns. For example, last week, we explored the multiplication table.

We gave the kids a blank table and discussed that the first row is numbers 1, 2, 3, 4 and so on – we count by 1s and can keep doing it indefinitely. In the second row, the numbers are 2, 4, 6, 8 – we are counting by 2s. In the third row, we count by 3s and so on.

So how do we will out the table? In which order?

Someone was filling it out by rows. Someone else preferred columns.

Many were actively using their fingers, adding by 3s, 4s, 6s, 7s.

Some kids filled out the the 10s row and column, leaving the 7s, 8s and 9s for later.

Some kids filled out the same row and column together

Someone started with squares

Everyone had their own strategy.

Everyone chose the “most convenient” cells and filled them out first.

Actually, from my experience, different people find different multiplication problems challenging and everyone finds their own way of remembering the answer.

Let’s say, for example, someone memorized the correct answer to 7+8. Someone else might check himself, adding 7+7+1, and yet another person might get to 10 first, 7+3+5. And everyone finds their own way most convenient and natural!

The tables are filled out. And, might I point out, they go up to 14, not up to 10 (translator’s note – ours went up to 15 😊). That’s because I know a lot of people who think that the multiplication table only “works” through 10!

Now we can play with it:

Color the number whose neighbors are 5, 8, 12 and 15. How many numbers like that did you find?

Now color the number, whose neighbors are 8, 9, 15 and 16. (Translator’s note: this was a challenging activity for our students, but they slowly got the hang of it with practice. I also showed them a “magic trick” – they gave me the neighbors and I guessed the number without looking at the table).

Now I give out a filled-out table and ask the kids to color in all the 12s, all the 16s, all the 72s – and to understand how many there are of each number.

Then I ask them to cover a part of the cells in the top left corner with a small square sheet of paper – and I guess, how many are hidden.

After that, I write the numbers 3, 6, 9, 12, 15 and 18 on the board. The challenge is this – find all the numbers the sum of whose digits is one of the numbers on the board, and circle them.

everyone has a different strategy

many start by looking for the same numbers and circling them first

I walk around, admiring their work, and wait till someone notices a pattern.

but it won’t happen right away.

not yet

this is getting closer

even even closer

Got it! Someone figured it out!

I ask him not to tell the others and we quietly discuss the found pattern.

I think this is a much more interesting way to discover the rule about divisibility by 3, rather than just feeding it to the kids as something to be memorized.

Do you remember when you learned the multiplication table? Which problems were the hardest to remember?