We’ve been talking about the best way to teach multiplication tables to elementary students. In the first part, I shared three discoveries I made that allowed me to stop the fight over multiplication facts. In the second part, I identified the only 28 equations that have to be memorized in order to know all of the multiplication facts. And today, we’re going to see how to teach the commutative property so that students truly understand why it works; this is essential for allowing memorization of the whole times tables with just 28 multiplication facts. It’s also crucial to advanced mathematics.
The traditional way to teach the commutative property
The commutative property tends to be taught like this: the teacher goes to the board, writes “commutative property”, and says, “Look, students. In multiplication, it does not matter in which order you multiply the factors. The product will always be the same. 5 x 4 = 4 x 5. 6 x 3 = 3 x 6. Does everyone understand? Here’s a worksheet to practice.” And the teacher hands out a worksheet full of pairs of equations so that students can show their “understanding”.
Lots of us learned the commutative property this way, and overall, it works. But it doesn’t lead to an understanding of WHY it works. We just memorize the definition of the property, and we can recognize and use it when presented with a worksheet of pairs of equations, but most of us have trouble putting it in practice in context, and using it fluently when we get to algebra.
Fortunately, there’s a better way to teach this property. And it’s a way of teaching it that allows students to DISCOVER IT, instead of just memorizing something that is presented and spoon fed to them.
When to introduce the commutative property
It’s important for students to understand the concept of multiplication before presenting the commutative property. For a child in a Montessori program, they will typically already have experience with long multiplication with the golden beads, will have the multiplication tables memorized, will be beginning long multiplication on the large bead frame, and will be about six years old. For a child in a traditional program, it will be presented after beginning to learn simple multiplication, as they are beginning to memorize their multiplication facts. They will probably be in third grade, and around eight years old.
The necessary materials
For this lesson, or presentation as it’s called in Montessori, you will need the Montessori bead bars, specifically, what is called the decanomial. This is a box of 55 bead bars for each number 1-10. Each number has a distinct color: red for one, green for two, pink for three, yellow for four, light blue for five, purple for six, white for seven, brown for eight, dark blue for nine, and gold for ten. These bead bars are used in many, many presentations for a wide variety of concepts, so it is worth purchasing them. For just this lesson, you could also make bead bars with Pony beads and pipe cleaners, and ten for each number would be enough. I made my own bead bars with 6mm beads, wire, a wire cutter, and pliers; this is cheaper than purchasing the box, but you will pay in time and tedious work.
If you wish to purchase the bead bar decanomial, I recommend purchasing from Kid Advance (EE.UU.), IFIT (Canadá y EE.UU.), Jaisa (España), or Montessori Educativos (México). (Note that I am not affiliated with any of these companies; they are my recommendations based on a balance of price and quality.) And if anyone has a recommendation for a Montessori materials company in South America, I’d love to hear it, because I don’t have a recommendation there yet.
You will also need: small white cards (about 5cm by 5cm, though they do not have to be this precise size) with the numbers 0-9 on them, one number per card; small white cards with the multiplication symbol on them; graph paper; and colored pencils that are the same colors as the Montessori bead bars. The number cards and the multiplication symbol cards are kept in an envelope, unless you have space for them in your decanomial box like I do in the photo above.
I’m going to show you the presentation in detailed form, with lots of pictures. At the end, I’ll share a PDF with the presentation in an easy-to-print form. (You can go ahead and jump to the PDF if you are familiar with the presentation already!)
Put the decanomial and the envelope with the number and multiplication symbol cards on the table. Invite one to five children to the table for the presentation (perfect for doing during guided math, for example, in traditional schooling). Take out one four bead bar and the card with a seven before beginning. Note that the bead bar always represents the multiplicand, and the card always represents the multiplier. Then follow this script to present the lesson. Montessori presentations always use a script so that we are giving only the necessary information to children. You can, of course, change words or phrases so that they fit your style, but you want to be careful NOT to add any more information to the presentation, so that the children can concentrate on just one concept at a time, and make discoveries for themselves. Their learning will be deeper and more powerful if they are permitted to discover the commutative property, rather than being told how it works. Resist the temptation to let them in on the secret!
Guide (Teacher): Suppose we have this (lift up the four bar) and we want to take it this number of times (show the seven card). (Put the multiplication symbol card between the bead bar and the seven card to form the expression.)
Guide (Teacher): This symbol means that we will take this four bead bar seven times. Let’s do that. (Place seven of the four bars horizontally below the expression.) What do we have?
Guide (Teacher): Here we have 4, 8, 12, 16, 20, 24, 28. Let’s make that with the smallest number of bead bars possible. (Get two ten bars and one eight bar from the box, and put them side by side, vertically, under the horizontal four bars.)
Guía (Docente): Now we can see clearly that four taken seven times is 28. Now, let’s write that down. (Write the equation on paper or on a whiteboard/chalkboard.)
Guide (Teacher): Now, let’s try one more. (Take out a seven bar and the four number card.) Let’s say we have this (lift up the seven bar.) and we’re going to take it this number of times. (Lift up the four card.) Let’s make this expression. (Place seven bar vertically, then another multiplication sign card, and then the four card.)
Guide (Teacher): What does it equal? (Take out four seven bars and place horizontally below the expression.)
Guide (Teacher): It’s 7, 14, 21, 28. Let’s make that with the smallest number of bead bars again. (Get two ten bars and an eight bar form the box, and put them side by side, vertically, under the horizontal seven bars.)
Guide (Teacher): Now we see clearly that seven taken four times is 28. Let’s write this equation down too. (Write the equation on the same piece of paper or whiteboard/chalkboard.)
Guide (Teacher): Now, let’s do a few more! (Return all bead bars to the box, and number cards to the envelope. Do not erase the 4 x 7 = 28 and 7 x 4 = 28 equations.)
From here, you repeat the process, allowing children to choose the multiplicands and the multipliers. Keep making pairs of equations. If they choose a multiplicand of three and a multiplier of nine, do that equation, and then repeat, using a multiplicand of nine and a multiplier of three. WE WANT THEM TO DISCOVER THAT THE PRODUCT WILL ALWAYS BE THE SAME IN THE PAIR OF EQUATIONS. DO NOT TELL THEM!
It is possible that they will have tired after the first example in the presentation. In that case, you stop and leave the next step of doing more examples for the next day. They should also have the opportunity to do some examples on their own.
When they’ve done several pairs of equations with the materials, you can introduce this next way of representing the equations on graph paper instead of using the bead bars and the number cards.
This can be introduced in a short presentation, like this:
Guide (Teacher): Instead of getting out the bead bars, we can also do this exercise on graph paper. We just need the paper and colored pencils. (Point to graph paper an colored pencils.)
Guide (Teacher): If I want to take three six times, I can draw the six three bars like this. (Draw the bars, using the pink colored pencil, as that is the color of the three bar beads.)
Guide (Teacher): I can write the equation under the drawing. Then I’ll find the product: 3, 6, 9, 12, 15, 18. And I’ll write the product as well. (Write the equation under the drawing with the product.)
Guide (Teacher): I’ll do the same thing to the right, switching the multiplicand and the multiplier. So now I’ll take six three times. I’ll draw it like this. . (Draw three six bars, using the purple colored pencil, in accordance with the bead bar colors.)
Guía (Docente): Here I have 6, 12, 18. Now I’ll write the equation below the drawing. (Write the equation below.)
Guía (Docente): Now you can do some more pairs of equations on your own.
And this is the follow-up work for this presentation: the children continue exploring on their own, inventing their own combinations of multiplicands and multipliers, and using either the bead bars or the graph paper method to show the results. After enough experience, they will discover that the product is always the same for a pair of equations, that you can interchange the multiplicand and the multiplier without affecting the product. When they begin to express this discovery, we can give them nomenclature to describe what they’ve found.
Guide (Teacher): I see that you’ve discovered that when you multiply two numbers, no matter which is the multiplicand and which is the multiplier, the product will be the same. If you take three five times, or five three times, you will end up with the same product: fifteen. As you’ve shown with your work, this is always true. And this is called the commutative property of multiplication. Let’s write that down in our notebooks, so we don’t forget what it’s called, along with a description and an example or two. (Write it in your notebook as an example, and encourage them to write in their notebooks. They can, and should, write their definition in their own words, and use their own examples based on their experiences, instead of just copying what you write.)
And that’s it! When you follow this process, you guide the discovery of the child. And as it is their discovery, they will remember it. You won’t have to teach it over and over again. They will understand it deeply, and will be able to put it into practice in context, and in advanced mathematics. This is exactly the result we want to achieve.
Now, so you don’t have to try to print all of this from the blog in order to use it, I’ll share a copy of the presentation in PDF form so you can print it easily. Just click on the photo to download your own copy!
And if you have any questions, please do not hesitate to get in touch. I’m always willing to help!
Esta entrada también está disponible en: Spanish