Writes and Wrongs #10: Subtraction Without Borrowing – The Nines Plus 1 (Number Sense)
My mother-in-law wanted my husband to hang up a picture. When she couldn’t find the hammer she brought a ping-pong paddle. We all laughed. After a time searching, she produced a ball-peen hammer since the other couldn’t be found. A picture nail is quite delicate and a ball-peen hammer is heavy and meant for pounding out dents in metal. You can imagine — that while it worked — it was too much tool for the job. I wish I had a picture of our expressions and of my husband banging in a tiny nail with a hand held wrecking ball.
Math is like that. There are different techniques that fit certain situations better than others. I didn’t know this when I was learning math. I thought that you had to use a borrowing all the time or do numbers in a certain order all the time. Whenever I deviated I felt like I was doing something wrong. Knowing that there is a tool you can choose to fit the problem makes a big difference in building confidence and gaining math sense.
Subtraction Without Borrowing: The 9’s plus one
I have no idea what this technique is called, but it is my favorite of all the subtractions. I can’t find my original resource. I call it “Nines Plus One.” It is based on three simple truths:
- 9 is the largest single digit with which you never need to borrow in a subtraction
- You can decompose a number in many ways
- You can subtract in chunks if desired rather than all at once
When I first encountered this method, it only gave the steps and not the explanation. I could follow the steps and it worked, but I really didn’t get it until I began to explore more subtraction-without-borrowing techniques.
It’s pretty basic and does not require the use of complements or higher subtraction facts like 17 – 8 which makes it quite fast. Not for every problem, but just like hammers, it has its place in the toolbox.
With this tip you are only ever going to work with ten, single-digit facts: (9-9), (9-8), (9-7), (9-6), (9-5), (9-4), (9-3), (9-2), (9-1), (9-0).
First, let’s explore proof of concept.
1000 – 777
If you have a number like 1000 and from it you subtract 777, that is a lot of borrowing with zeros if you choose to regroup/borrow. You can add up to find the answer, of course. Another way is to decompose 1000 into 999 + 1. Then you subtract 777 from 999 and add back in the extra +1. Since all the digits in the minuend (number being subtracted upon) are 9, the subtraction is easy.
1000 – 777
1000 becomes 999 + 1
(999 – 777) + 1
(222) + 1
Answer: 223
1234 – 777
Take 1234 and decompose by separating all digits after the first to create an easy number with zeros: 1000 + 234
Then decompose that value by separating 1. 999 + 1 + 234
Perform the subtraction 999 – 777 to get 222.
Finally, add back in the decomposed values: 222 + 235 = 457.
Answer: 457
1234 – 777
(1000 + 234) – 777
(999 + 1 + 235) – 777
(999 – 777) + 1 + 235
(222) + 235
457
435 – 378
If you prefer, you may take a shortcut by focusing on the first digit. Decrease it by one. Change all of the trailing digits to 9. Perform the subtraction then add in the trailing digits plus 1.
435 – 378
399 – 378 = 21
21 + 35 + 1
21 + 36
Answer: 57
Once you know how it works you can quickly pick out problems it is best suited for where it far exceeds the speed of standard American regrouping/borrowing.
There are few things to be aware of, such as, you have to remember that when you decrease the digit it must be larger than the number in the subtrahend. So for 6887 – 6738 I wouldn’t change 6787 to 5999. I would look to the next digit that I could decrease ignoring the first digit which would give 6-6 = 0. I would change 6887 to 6799. This video explains better visually.
What did you think about using 9’s to subtract?
4 comments on “Writes and Wrongs #10: Subtraction Without Borrowing – The Nines Plus 1 (Number Sense)”
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Ooh. This is going into my next MAT 103 class.
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I think it is a fun way to subtract. Oh, it will blow their minds. The fist time I showed students everyone was like, “Wait. What did you just do! Do that again!” haha 🙂
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This is very interesting.
Never heard of it before.
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I’ve only learned it about three years ago.
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