Back to index of science and math pages by Donald Sauter.
(The first few paragraphs of introduction here are the same as in the companion page single-digit multiplication.)
If you've never noticed, almost everything you do in math is just some mixture of addition and multiplication (recognizing that subtraction is bound up with addition, and division with multiplication.) And since all addition and multiplication, no matter how big or complicated, is performed stepwise in the brain on two digits at a time, you'll see that the value of quick and sure answers to the single-digit additions and multiplications cannot be overstated.
There is simply no point in going any further in math without complete mastery of single-digit addition and multiplication.
But what's the problem? Isn't this kid's stuff? Didn't everybody learn addition in first grade and multiplication in third grade?
Not by a long shot. I've had much opportunity to observe the math skills of lots of people, from grade school to high school to college to intelligent, educated adults, and what I see is somewhere between heartbreaking and horrifying.
I have some definite notions of what the schools are doing wrong in math. For a start, calculators. Now, I can't swear no student has ever learned a bit of math in spite of calculators, but I assure you no student, anywhere, has ever learned math with a calculator. Grrrr... And there's a lunatic obsession with written explanations of "how you got your answer." Students are moved along on a conveyor belt regardless of how little has stuck. And, sorry about this, basic math is taught by teachers who were never taught basic math themselves. The insanity has been going on for several generations now.
But we're here to get down to addition and find a way to think about each basic fact. Obviously, the first grader himself won't be reading this, so the intended audience is anyone working with students in this critical area, or anyone wishing to fill in the gaps in his own basic math skills at any stage of life.
What we are gunning for is "quick answers" to all of the single-digit additions. It would be great if we could simply memorize them, but, being realistic about it, this is not an easy thing even for a "really smart" person. But "quick" is practically as good as "instant", and if a rapid-fire intermediate mental step or two is necessary, that is perfectly acceptable. But there must be no fumbling about. Counting up is NOT acceptable. It is too slow and risky; there's always a better way to think of it.
And it should go without saying, using fingers is NOT acceptable. There's no tomorrow in math if you can't get beyond fingers.
Here's the addition table:
+0 +1 +2 +3 +4 +5 +6 +7 +8 +9
-----------------------------
0 | 0 1 2 3 4 5 6 7 8 9
1 | 1 2 3 4 5 6 7 8 9 10
2 | 2 3 4 5 6 7 8 9 10 11
3 | 3 4 5 6 7 8 9 10 11 12
4 | 4 5 6 7 8 9 10 11 12 13
5 | 5 6 7 8 9 10 11 12 13 14
6 | 6 7 8 9 10 11 12 13 14 15
7 | 7 8 9 10 11 12 13 14 15 16
8 | 8 9 10 11 12 13 14 15 16 17
9 | 9 10 11 12 13 14 15 16 17 18
Some math curricula step through it column by column, from "plus 1" to "plus 2" to "plus 3" and so on to "plus 9". This may seem reasonable since adding something bigger has to be harder than adding something smaller, right? But a close look at the gears turning in your brain while you do addition indicates we can do better.
Here, then, is a more natural progression from easiest to hardest for the single-digit additions. As we step through, I try to suggest a way of thinking about each of the addition facts. For most people, there are just a few that cause trouble, so feel free to zero in on those.
Zero is the "do nothing" number when you're doing addition. Zero is nothing, and something plus nothing is the exact same something.
What could be easier? Nothing, except for the fact that when multiplication comes along, 0 does something every bit as easy - but totally different! In multiplication, it turns everything into 0. When the time comes, remember to be on your toes. Always give +0 and x0, as trivial as they are, a second thought.
(I'll devote a web page to it one day, but think of math as being divided into two "rooms" - the addition room and the multiplication room. Each room has its own set of tools, many of which correspond to, but are not identical with, tools in the other room. For example, each room has a "do nothing" number called the "identity element". In the addition room that number is 0; in the multiplication room it's 1. You must always be totally aware of which room you're operating in, addition or multiplication. No matter how complicated a math problem gets, your brain only ever operates in one room at one time. At any moment you are doing either addition or multiplication, nimbly stepping back and forth between the rooms as needed. And, to repeat myself, no matter how complicated the addition or multiplication gets, your brain only processes a pair of single digits at a time. In a nutshell: all of math is just a big mixture of single-digit addition and single-digit multiplication.)
But back to the plus 0's:
0+0=0
1+0=1
2+0=2
3+0=3
4+0=4
5+0=5
6+0=6
7+0=7
8+0=8
9+0=9
Of course we don't memorize these ten addition facts; we just know that adding zero doesn't do anything.
This is just the "next" number up. You heard me rant above that there should never be any need for counting up in single-digit addition. What I mean is that, if the addition fact isn't memorized, there is always a quicker, safer, better way than counting up. But I will admit that, in this case, there's not a whole lot of difference between instantly visualizing the next higher neighbor in the 0 through 9 cycle versus just counting up one.
1+1=2
2+1=3
3+1=4
4+1=5
5+1=6
6+1=7
7+1=8
8+1=9
9+1=10
We leave off 0+1 because we think of that as 1+0, which we already know from the "plus 0" table.
And this leads to an important point. To make life a lot simpler, and to halve the number of basic addition facts, you always "load" the two numbers in the same order in your brain, no matter how they appear on paper or the order in which your ears hear them. You will almost always load them up as bigger plus smaller. We generally think of addition as piling more onto the main pile, right? If you were counting up a pile of money, you'd start with the twenties, then the tens, then the fives, and so on down to the pennies; not the other way around. If you add 3 more steers to your herd, you would naturally think of that as "5491 plus 3", not as "3 plus 5491 more".
So when you see or hear "0+1", it lands in your brain as "1+0", at which point the "plus 0" rule kicks in: "zero doesn't change anything!" This beats counting up one from zero hands down. We might only be saving a millionth of a second in this case, but, in the bigger picture, always doing the same fundamental math fact in the exact same way has solid benefits in terms of speed, ease, and accuracy.
Adding 2 is just jumping to the next even or odd.
The very next thing in math after learning to count should be developing an instant recognition of even versus odd. I know the schools try to do this, but I don't think they are aware of how fundamental and important it is. I meet many students in upper grades who admit they don't see numbers as even or odd. It should be revisited as often as necessary throughout the elementary school years to make sure every student has it in his bloodstream. It's something that should require no thought, just as it takes no thought to determine whether someone you meet is a boy or girl; it's instantaneous.
In fact, I've always made the analogy of evens vs. odds with gender; the goody-goody, well-behaved evens are the girl numbers, while the unruly, untidy odd numbers are the bad boys. Don't ask me if the analogy still holds, or if we're allowed to think like that anymore.
So everyone must be able to mindlessly rattle off "two, four, six, eight" - with a total awareness that there's a 0 at both ends that counts as even. The odds are 1, 3, 5, 7, 9, but it's not necessary at all to memorize the sequence. When "two, four, six, eight (who do we appreciate!)" is thoroughly burned in, the odds are simply "the other ones", the ones in the gaps, the ones that aren't even.
2+2=4
3+2=5
4+2=6
5+2=7
6+2=8
7+2=9
8+2=10
9+2=11
I leave off 0+2 because it lands in our brain as 2+0 - "do nothing!"
I leave off 1+2 because it lands in our brain as 2+1 - "next number up!"
For the rest, we take one instantaneous step to the next even or odd, as the case may be. Note that 8+2 may be more naturally seen as a "pair that makes 10", and 9+2 as a "pair that makes 11" (see further down). But they're both so easy that this is just splitting hairs.
Let's take a short breather. "Why?" you ask. "We've barely started, and we've only done the baby stuff that everybody already knows anyhow." True, but you might be surprised to learn that what we've done so far - plus 0, plus 1, and plus 2 - constitutes more than half of the single-digit additions, that is, 51 out of the 100 problems in the addition table. Math is so easy!
And here you thought "plus 3" would follow "plus 2". Shame on you for never taking an inward peek at the gears cranking in your brain! It is most certainly not an easy thing, generally speaking, to instantly visualize the next number 3 up.
I put "5 plus a little number" in this slot as the next easiest basic addition. This stems from how we learned to count. You would go through the fingers of one hand, and, when they ran out, start on the fingers of the next for 6 through 10. So a young child might hold up one hand of outstretched fingers, plus the index finger of the other hand to indicate 6, and so on. Thus, "5 plus a small number" is burned in almost from the start.
IMPORTANT NOTE: When I make reference to hands and fingers as the basis for thinking of various basic addition facts, I am NOT suggesting using fingers to do addition. That is NEVER permitted. If it is observed, it MUST be stamped out. If I were in charge, anyone caught using fingers would be sent back to second grade, and as many times as necessary to break the habit. Going through math bogged down with finger calculations is like learning to dance with cinder block shoes.
As with all single-digit additions that don't go over 10, these must be instantaneous, with no intermediate steps to get to the answer:
5+1=6
5+2=7
5+3=8
5+4=9
5+5=10
We've already dealt with the general "plus 1" and "plus 2", but 5 is such a good jumping-off point that 5+1 and 5+2 may be just as quickly recognized and processed as part of this group.
But 5+3 and 5+4 are the important new ones. In your mind's eye you see a hand's worth of fingers plus 3 or 4 on the other. Bingo.
Why didn't I call out 5+5? That is more naturally handled in the next category.
Remember that we've got even vs. odd burned into our psyche? Well, the little doubles, meaning the doubles of the numbers that can be held on one hand (1 through 5), are just the even numbers up to 10.
1+1=2
2+2=4
3+3=6
4+4=8
5+5=10
We don't need another way to view 1+1, which could be omitted here.
We could also omit 2+2 since we've done the "plus 2"s, but I believe it's more natural to view 2+2 as "two 2's" than as stepping up 2, or, as I suggest, jumping to the next even.
And consider how natural it is for a child to hold up two fingers on each hand to indicate 4; three apiece for 6; and four apiece for 8. We all hold up two whole hand's worth of fingers for 10; there's no other way!
In other words, even though these problems are presented on paper as addition problems, the brain really processes them as multiplication problems - very basic multiplication problems that we were doing even before we heard the word "multiplication".
Again, we take a mental glimpse at our fingers. There are 10 of them, which is so fundamental a number that they based our whole number system on it! And very early on you had to have noticed that, if you hold up a certain number of fingers to show a certain number, and count up the ones held down, those two numbers make up 10. (Uh oh, here we are, not even finished the basic additions and we're touching on subtraction.) Or, if you didn't notice, now's the time to burn in the pairs that give "two handfuls":
1+9=10
2+8=10
3+7=10
4+6=10
5+5=10
6+4=10
7+3=10
8+2=10
9+1=10
Now who could memorize a complicated mess like that? Well, after we weed out the ones we've already got burned in, the only new ones are:
6+4=10
7+3=10
As always, I'm counting on you to automatically do the mental flipflop if you're given "4+6" or "3+7". Big plus little...
Now that we have the pairs that make 10 burned in, we can see in an instant when a pair just goes over 10 by one.
5+6=11 (Instantly seen as one more than 5+5=10)
7+4=11 (Instantly seen as one more than 7+3=10)
8+3=11 (Instantly seen as one more than 8+2=10)
9+2=11 (Instantly seen as one more than 9+1=10)
We met 9+2 in the "plus 2" category but I think it's a slightly more natural fit here.
Also note the presentation of "5+6" rather than "6+5". Certainly this problem is much more easily seen as "two 5's plus one more", rather than "one more than 6+4".
These are the doubles of the numbers that require two hands, 6 through 10. They should be instantaneous in spite of the fact that the answers are double-digit. As was the case with the little doubles, the big doubles are actually processed as multiplications in our brains. Here, the answers are all the even numbers from 12 to 20 - and we're pros at even numbers!
6+6=12
7+7=14
8+8=16
9+9=18
10+10=20
So two 6's are instantly seen as the first even number after 10; and two 9's have to be the biggest even number before you get to 20. It's kind of funny to hear students, who may have a few weak spots remaining in their addition table, rattle off 9+9=18 without a moment's hesitation. Bigger doesn't mean harder!
Then 7+7 and 8+8 are the remaining two evens between 12 and 18. What else can they be but 14 and 16, respectively?
Nine is the biggest, clumsiest single-digit number. (I won't say crankiest or most eccentric - that's 7.) But what 9 has going for it is that it's right next to 10, and 10 is a pure joy to work with. The suggested thought process for "9 plus something" is to steal one away from the "something", which gets you from 9 to 10, and then what's leftover from the "something" jumps you to the final answer, lickety-split.
Distilling that down into a quick and easy "recipe":
9+3: Drop down one from 3 to 2. Slap a "1" in front. Bingo, 12!)
9+4: Drop down one from 4 to 3. Slap a "1" in front. Bingo, 13!)
9+5: Drop down one from 5 to 4. Slap a "1" in front. Bingo, 14!)
9+6: Drop down one from 6 to 5. Slap a "1" in front. Bingo, 15!)
9+7: Drop down one from 7 to 6. Slap a "1" in front. Bingo, 16!)
9+8: Drop down one from 8 to 7. Slap a "1" in front. Bingo, 17!)
I left 9+0, 9+1, 9+2, and 9+9 off the list since we've nailed them in much more basic ways.
If you ask me, these "9 plus" facts never need to be memorized. You can get the answers fast enough using this "9 plus" recipe. Remember that we used good, old 10 as a stepping stone once before, in the pairs that make 11. Think of 10 as sort of like first base in baseball. It makes a nice, safe, comfortable place to get to, and depart from, on a trip from the single digits up into the double digits. We'll meet a few more examples of that in our final, mop-up category below.
Here are the single-digit additions that do not fall in any of the categories above. I've separated them into related pairs, and I left the clutter of the answers off to make the problems themselves more readily visible.
4+3
6+3
8+4
8+6
7+6 (or 6+7 better?)
8+7 (or 7+8 better?)
7+5 (or 5+7 better?)
8+5 (or 5+8 better?)
Going down the list . . .
4+3: As with all single-digit additions that fit on the two hands, this must be instantly accessible from the memory banks. I'm not aware of 4+3 causing trouble, but if so, maybe it would help to point out it has to be an odd number (use chips to show that a number plus his neighbor has to be odd), and that this one obviously has to fall between 5 and 10. But I'd like to think that's overkill for 4+3 . . .
6+3: Again, this has to be instantaneous. I was startled once to hear an intelligent, educated adult do 6+3 by quickly murmuring, "seven, eight, nine." No counting up! You can get it instantly from the latter part of the familiar 3, 6, 9 "counting by threes" sequence. Even though that comes from higher order math (multiplication), I suggest that jumping ahead to play with counting by threes as the way to nail 6+3=9. Sooner or later, the student must come to see those three swirly, open-ended 3, 6, and 9 digits as a nice, warm, little family that absolutely "drips with threes." It might as well be sooner.
Another approach may be to point out how 6+3 is clearly not quite big enough to reach 10, which requires a 6 plus 4.
8+4: I can't quite tell whether this is accessed from hard memory in my brain, or whether I almost instantly see the answer 12 as forming a perfect symmetry with 8 around good, old 10. The problem cries out "evenness"; the 4 practically breaks itself into two 2's, one of which gets you to 10, the other to 12.
8+6: Again, I can't be sure if this one is accessed in one instant step from memory. It might come from a lightning-fast process of elimination. We have two numbers which are as even as even can be, forming the upper half of "two, four, six, eight!" Thus, the sum has to be even. The answer has to spill over 10, but not by a huge amount, and it's certainly not 12, and it's certainly not 16, so what else can it be but 14? The whole thought process taking a fraction of a millisecond . . .
7+6 and 8+7: These two have always been instantaneous for me, with no hint of intermediate steps. But I've met many students who stumble on them. Maybe it will help to mention, as we did with regards to 4+3, that a number plus his neighbor has to be odd. Still, that only limits the choices; it doesn't slap an answer in our face.
For 6+7, the best I can suggest is a quick doubling of 6 to 12, and add 1 to get 13. After doing it a few times, hopefully the 13 will burn into memory.
For 8+7, something similar can be done, but I think there's something better. Notice that these two numbers are "in the middle" of 5 and 10. Everybody knows two 5's are 10, and two 10's are 20, so it should be no surprise that 8+7 falls smack dab in the middle of 10 and 20, which is 15. Pointing this out to a student may help him burn in the 8+7=15 fact.
7+5 and 8+5: Here are the two killers in the single-digit additions. I'm not proud to say that for much of my math life, I used a quick mental procedure (and there may be echoes of it yet.) I'd take the smaller number, 5, and break it into two parts. Since 5 is odd, we know it won't break evenly; it snaps off-center into chunks of 2 and 3. One of those chunks gets you up to good, old, safe 10, and the second chunk jumps you to the final answer.
So, for 7+5, after splitting 5 into two chunks, 7 (the smaller of 7 and 8) needs the big chunk (3) to get you up to 10; and 10 plus the remaining chunk (2) jumps you to the answer (12). Likewise, 8 (the larger of 7 and 8) only needs the little chunk (2) to get you up to 10; and 10 plus the remaining chunk (3) jumps you to the answer (13).
That's how it played out in my mind, probably because of my unbreakable habit of automatically arranging the addends as "big plus small" in the brain. By making an exception in this case, and seeing the problems as 5+7 and 5+8, a much more elegant procedure emerges. Remember our "5 plus a little number" category? That was one of our simplest and most basic cases because of its connection with learning to count on our fingers. Instead of breaking the smaller 5, let's break the 7 and 8 into chunks. Seven is a handful plus two more fingers; 8 is a handful plus three fingers. So 5+7 is easily seen as two handfuls (10) plus 2 more, equals 12. Likewise, 5+8 is seen as two handfuls (10) plus 3 more, equals 13. Beautiful!
Here is the addition table:
+0 +1 +2 +3 +4 +5 +6 +7 +8 +9
-----------------------------
0 | 0 1 2 3 4 5 6 7 8 9
1 | 1 2 3 4 5 6 7 8 9 10
2 | 2 3 4 5 6 7 8 9 10 11
3 | 3 4 5 6 7 8 9 10 11 12
4 | 4 5 6 7 8 9 10 11 12 13
5 | 5 6 7 8 9 10 11 12 13 14
6 | 6 7 8 9 10 11 12 13 14 15
7 | 7 8 9 10 11 12 13 14 15 16
8 | 8 9 10 11 12 13 14 15 16 17
9 | 9 10 11 12 13 14 15 16 17 18
Rather than bashing through it column by column, I have proposed a more comfortable approach, from easier to harder. Here it is again, in condensed form.
Congratulations! You are now an adding machine!
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