# Why is $a+b = b+a$?

In primary school, it's extremely easy to show that $$a \times b = b \times a$$, as follows:

The surface of a rectangle can be calculated using the formula $$\text{Basis} \times \text{Height}$$, as in following examples:

(Flipping the rectangle does not change its surface, you can even elaborate that by drawing unity squares inside the rectangles and count them.)

But how can you show that $$a + b = b + a$$?

• Depends on what $a$ and $b$ are supposed to be ... Commented Mar 7 at 15:59
• In primary school, let's start with natural numbers. Once they get $a+b=b+a$ for natural numbers, they will easily go further for more complicated number types. Commented Mar 7 at 16:00
• Thanks for clarifying. And I would say all the "more complicated" number types have been carefully designed so that this property still holds, because it's really handy ... Commented Mar 7 at 16:03
• @MahdiMajidi-Zolbanin: although this might be correct for students of higher age, no way you get this explained to scholars of primary school (typically 6-10 years old). Commented Mar 11 at 7:05
• Why is the formula for the rectangle basis times height? It seems to beg the question. But if ti's allowed, then take a line segment and split it into lengths $AB=a$ and $BC=b$. Then from $A$ to $C$ the length is $a+b$ , and from $C$ to $A$ the length is $b+a$. Commented Mar 16 at 19:16

Suppose you have $$a$$ crayons in your left hand and $$b$$ crayons in your right hand. All together, you have $$a+b$$ crayons.

Now you switch hands. You now have $$b$$ crayons in your left hand and $$a$$ crayons in your right hand. All together, you have $$b+a$$ crayons.

You haven't put any of the crayons down, so the number of crayons you have hasn't changed.

• Sure, you haven't put any crayons down, but when the four year old tries it they'll be all over the floor :) Commented Mar 7 at 15:53
• The funny thing is, nobody has ever actually tried it with $a=5378, b =979$, and yet almost everyone is convinced it would work. Commented Mar 7 at 16:01

If you're happy with the given demonstration for multiplication, then take two connected line segments (rulers) and flip them 180 degrees.

• I'm afraid that's not going to work: first you need to explain that $a+b=b+a$, only afterwards you arrive at $a \times b=b \times a$. I'm afraid that this geometrical approach might be confusing for the children. Commented Mar 7 at 14:18
• @Dominique The line segments are supposed to be stacked on top of each other in the same direction, like -- ----, which gives ---- -- rotated 180 degrees Commented Mar 7 at 15:49
• @MichaelBächtold: I understand the example, but I don't like it that much. Justin's answer, however, is very clear for children. Commented Mar 7 at 15:53

It's pretty straightforward to demonstrate:

You can also get the distributive law $$(a+b)c=ac+bc$$ and others this way.

I'm not directly involved with elementary ed, but my kids (USA, recent) were taught such explanations.

• I saw some downvotes, but it's unclear why if no comment is left.
Commented Mar 14 at 14:49
• Not the downvoter, but personally this feels more like an example than a proper explanation: If I didn't believe in the commutative property of additition, I don't think I'd find your first diagram convincing. It would be more effective if you offered a convincing justification for why the two line segments are of equal length. E.g., matheducators.stackexchange.com/a/27567/1233 . Commented Mar 18 at 20:01
• @Brian OK, I guess, but I think that it's pretty evident that I drew exactly the thing that you referenced. (Literally using copy, paste, and rotate tools) In the spirit of how a sufficiently general example often makes a proof, but concretely, so that students aren't lost.
Commented Mar 18 at 23:06

Perhaps this drawing will make the answer of @DanielR.Collins clearer. The person on the left sees 3+1; the person on the right sees 1+3. They're looking at the same thing.

Edit: Oops, it seems that @Taladris has already mentioned something like this. Anyway, I'm keeping my answer so that the drawing can make things clear.

• My answer slightly differs from those given in the sense that no movement (like swapping or flipping) is necessary.
– JRN
Commented Mar 16 at 1:47

I would encourage you not to "teach" this (neither multiplication nor addition). Kids discover these things on their own, and it can be an exciting discovery for them.

Designing activities that help them to see it is great.

Quite similar to @JustinSkycak's idea: take $$a$$ blue marbles and $$b$$ red marbles. First, put the marbles in line, the blue ones on the left and the red ones on the right. Then swap them.

Add a third color and you can show associativity as well (without needing a third arm!)

Note: I read Comment les enfants apprennent a calculer? by Remi Brissiaud (How children learn to calculate?, I don't think there is an English translation) a long time ago. It is about the psychological development of the concept of numbers and calculation in young infants, and I was impressed by how complex the process is. If the goal of the questions is to teach your own child, be careful not giving him wrong mental images.