Traditionally, we teach 110 math facts to help students memorize the times tables from 0 – 10. My son was overwhelmed with the magnitude of the work required. But when I discovered I could teach just 28 and achieve the same goal? He still wasn’t thrilled about it, but it became doable. It turns out we’ve been making it way more complicated than necessary. My children are thanking God for helping me discover this; it’s forever changed how I teach math facts, and their experience learning them has improved substantially. I’m going to describe the process of learning the times tables, but all of these ideas apply to teaching addition tables, too. If you haven’t read part 1 of this series, How I Ended the Fight Over Memorizing Math Facts, you may want to start there.
Teaching the 0 – 2s
Let’s start with the zeros. To demonstrate how we multiply by zero, we’ll use some beans (or other small manipulative) or the Montessori bead bars. Give the child some beans. Ask her to put zero groups of one bean on the table. She’ll think about it, and place zero beans on the table. Then ask her to put zero groups of two beans, and zero groups of three beans, and so on. She’ll never put any beans on the table, and she’ll discover that when you multiply by zero, you get nothing. The answer is always zero. The important thing here is to allow the child to discover this. DO NOT TELL HER. This works the same way with the Montessori bead bars. You give the child the box of bead bars, with several of each quantity, and you follow the same procedure. It should end with the same discovery. And you’ve taught the zero times table. No flashcards required.
Now, the ones. We’ll use the same process, this time asking for one group of one bean, one group of three beans, one group of nine beans…and the child will discover that multiplying by one doesn’t change the number. They’ll know the identity property of multiplication. With the bead bars, do the same thing. Ask for one group of two (one two-bead bar), one group of five (one five-bead bar), one group of eight…and the child arrives at the same discovery. You’ve taught the ones table. No flashcards required.
Let’s go on to the twos. We’re assuming that the child knows addition before memorizing the multiplication facts. So we follow the same process again with the beans or the bead bars. Two groups of two, and we’ll see we’re adding 2+2. Two groups of seven, and we see we’re adding 7+7. We keep going with the examples until the child figures out the pattern. And once he discovers it, the twos are a piece of cake.
The commutative property
Now let’s leave the times tables for a bit, and teach the commutative property of multiplication. The easiest way to do this is to use the Montessori bead bars. These are made of round beads and wire, with a different color for each number. Red for 1, green for 2, pink for 3, yellow for 4, light blue for 5, purple for 6, white for 7, brown for 8, bright blue for 9, gold for 10. If you don’t have the bead bars, and can’t purchase them (I trust Kid Advance and IFIT for a purchase), a simple solution is to use pipe cleaners and pony beads. For other Montessori presentations it’s important to use round beads, but for this one it doesn’t matter and pony beads are fine. You can easily teach the commutative property with five bars of each color, though ten of each is ideal.
With the bead bars we’re going to demonstrate concretely that 6 x 4 equals 4 x 6. The bars allow the child to discover this commutative property on her own. When a child discovers it on her own, she never forgets it. She’s going to do examples of pairs of equations until she sees how it works, and will have a profound understanding of multiplication as a result.
I don’t want to explain and demonstrate the whole presentation here, because it deserves its own entry. I’ll link that up here when I write it later this week. For now, know that at the end of this presentation, it will be very clear that 3 x 5 equals 5 x 3. Knowing this, we can advance to the 3 – 9 tables.
Teaching the 3 -9 times tables
Generally in order to memorize the tables of 3 – 9, we have to help children memorize 99 different equations. But actually, we only need 28! What a relief, right? Let’s take the threes as an example. We already know that there’s no reason to teach 3 x 0, 3 x 1 or 3 x 2. We’ve covered those. So we can start with 3 x 3, and go through 3 x 9. Seven math facts. When we go on to the fours, we won’t have to teach 4 x 0, 4 x 1, 4 x 2, or 4 x 3 (we’ll know by the commutative property that 3 x 4 is the same!). So we can start with 4 x 4. Six math facts for the fours! And five for the fives, four for the sixes, three for the sevens, two for the eights, and just one math fact to teach the nines! Here’s a table that shows which facts we’ll have to memorize (and I’m including one in an easily-printable PDF at the end of this post, too!):
Now, the tens. Above all, remember this: NEVER, EVER TEACH THAT YOU ADD A ZERO TO THE END OF A NUMBER TO MULTIPLY BY 10. This works with whole numbers, but it’s not going to work with decimal fractions, for example, and you can avoid all kinds of future problems if you never start with this rule. The rule you want to teach is that when you multiply by ten, all of the digits move one place to the left. The zero fills the empty position, as a place marker. There are many ways to demonstrate this concretely, and they also deserve their own post, which I’ll link up when I write it soon.
The 11s and 12s
If you want to teach the 11s and 12s, easy! For the 11s, you just need to do some examples until the child finds the pattern, just like you did for the zeros and the ones. And for the 12s, you’ll add 12 x 2 to 12 x 12 to the list of math facts to memorize. That’s it!
Strategies for teaching the math facts that are left to memorize
So you’re left with 28 (or 39, if you teach the twelves, too) math facts to teach. How do you do it? There are several effective strategies. But before we talk about what they are, we need to talk about one strategy that is NOT effective: the timed test. No more Mad Minutes! That will be a relief for everyone, right?
I’ll be back with another post in this series. I still have to show why timed tests are not able to give us the result we hoped for, what can replace them, how to memorize the facts we have left, and share resources that will help you teach in this new way. And I haven’t forgotten I promised to show you the commutative property presentation in detail, as well as ways to show how to multiply by ten concretely. All of this is coming soon! I don’t know if you’re as excited as I am, but I am really enjoying sharing all of this with you!
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Want to print your own copy of this multiplication table? Just click on the image below to get it in easy-to-print PDF form!
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