# Staircase method of small integer addition

I found the Math Games: Math for Kids to be excellent (if not extremely, excellent) at teaching elementary school math.

But it made me think of some tricks, that could be displayed when teaching at a blackboard or on some apps.

Here is one of them:

I was wondering whether anyone can improve on the following method of doing addition for someone who has learned multiplication tables.

Suppose one needs to do

7+3

The user draws a statisíce on top of a staircase. The staircase at the top is there steps long, the one at the bottom seven steps long.

The user sees that the bottom three steps of the three step staircase are stacked upon the first three steps of the bottom step staircase.

So the user can see the staircases coming together, as though, being, superimposed.

They can see that

3+3 = 3's timetable carried forward two steps (3*2) = 6

Just like the first three stairs can be visually seen to be 3, the remaining stairs can be visually seen to be 4

So the user must do

6+4 = counting up yields 10

The full equation is

7+3=(3+4)+3=6+4=10

...

Second point, another trick is to find the complement of 6 in ten, in this case it is just 4, which together with 6, fits in exactly in 10. Noticing this, you can just fill in the sum.

But if you had to do 6+5 you would use the staircases method to do 1+(5+5)=1+10=11.

I wonder whether anyone works like to implement any of these tricks and others in a fast and mentally comfortable math trick teaching app for elementary school kids I was thinking. I would like to invite anyone with tricks and ideas and feels of what feels comfortable (not just efficient) and, hopefully, fast, to do so.

Meanwhile, this question would like to ask for any observations or improvements to the staircase method.

Thank you.

• This seems like nonsense. If I were a child, to perform 7+3 I would use my fingers and count up. This seems like lots of overhead..reduce 7+3 to 6+4, and all for single digit addition? Jan 21 at 4:41
• If you don't know Vedic, check it out. It's important that a method for this be very simple, and help lead students towards retention of these basic facts. Jan 22 at 21:57
• @SueVanHattum Vedic is a religion, right? I don't get the connection. I have read the OP over many times and tried to understand. But just don't get the point. Jan 22 at 22:08
• @user52817, it may be a religion. I don't know. But if you look up vedic math, you'll see the numbers one to ten in a circle. I know of one person who found it very enlightening. Jan 22 at 23:44
• @SueVanHattum ok I read the Wikipedia page about Vedic mathematics. It’s very negative…makes it seem dubious. en.wikipedia.org/wiki/Vedic_Mathematics Jan 23 at 0:01

I would like to invite anyone with tricks and ideas and feels of what feels comfortable (not just efficient) and, hopefully, fast, to do so.

Maybe this is an unpopular answer nowadays, but honestly I think the best "trick" is to just memorize your single-digit addition facts (while conceptually understanding that addition is ultimately just "counting up" and the order doesn't matter).

Yeah, it takes some up-front work to memorize them, but it's really not that much work, and once you can recall these automatically, addition in general becomes super easy, comfortable, efficient, and fast.

On the other hand, if you don't memorize them, then your limited working memory will continually get hijacked and wasted on low-level processing, which will make it difficult to build more advanced skills of which single-digit addition is a component. At best, you'll take longer to learn/execute more advanced skills and have an elevated error rate while executing them; at worst, you'll experience total cognitive overload and not be able to learn/execute more advanced skills at all. Everything becomes super hard, uncomfortable, inefficient, and slow.

• It is hard to see what you can do if you cannot even do single digit addition at sight in your head. Jan 21 at 13:30
• I think it is more important to know that 7=||||||| and 6=|||||| and that we write numbers using 10-base system, so ||||||||||'|||=13. Dumb memorization of "7+6=13" is not as useful, will it help to quickly realize, for example, that 7+26=33 ? Jan 22 at 6:17
• @RustyCore see the parenthetical in the first paragraph of my answer -- I never endorsed "dumb" memorization of 7+6=13. Also, having 7+6=13 memorized WILL help to realize that 7+26=33 because you'll know instantaneously that 7+6=13 and 2+1=3 (tens place). If you don't have instant recall on single-digit facts then you're going to be slow on multi-digit facts. Jan 22 at 13:44
• I think you're very negative. I, actually, use, this method (but with your negativism I might not be able to do so Jan 23 at 12:20
• @JoselinJocklingson: Negative? How about "realistic". Jan 25 at 7:05

I thought of the concentric rotating polygon method.

In this method, which could prove extremely strong, for some users, in some cases, or for an extremely vast variety of enthusiasts and enthusiastic aficionados, also intriguing, and you might think gotta love and try it:

In this method, you envision one larger polygon with a smaller polygon inside it. The inner polygon rotates and covers the edges of the outer polygon.

As it does so, you can see where it touches, to paint the outer polygon with all of it's sides paint.

The rest of the sides is the rest you need to add.

And, then, you get

3+7 = (3+3)+4 = 6+4 = 10

.

You must get comfortable with the case n=2, as well. Because, here, you can't have a polygon (you must consider a two sided line segment).

It, turns over itself, twice (painting, twice).

So, you may do

2+7 = (2+2)+5 = 4 + 5 = 9

You, might, get more entusiastic, doing it this way (no, you wouldn't want, to just add two at the end).

Thank you.

• Although your image might indeed explain what you are doing, they are very chaotic. Can you make some clearer drawings? Jan 22 at 7:18