Mr. McNamarra was my seventh grade pre-Algebra teacher. On the first day of class he went to the blackboard and wrote “3” with white chalk. He asked us what was on the board. Everyone tried the usual: three, a numeral, integer, a number, a digit… . Someone suggested a sideways “m” or a sideways “w”. No, came the answer. When we exhausted all the vocabulary we knew, Mr. McNamarra grinned and said, “That is chalk dust.” Cue the collective groans. I remember that lesson though because it made me feel comfortable and got me thinking. It was, after all, just a marking. We only recognized it as a three because we had learned it so and there would be more to learn about it. Yet, at the same time I was aware at how much we already new based on all the answers the class tried, and how much bigger my understanding was than I ever gave myself credit for.
So why is there a picture of cursive script in this article? Because those are just a bunch of squiggles, but they have a purpose. Writing in script uses efficient pen strokes; you can write faster. Even a person who primarily prints will slip into connecting letters when they are writing rapidly. If we have number sense, we can also do math more efficiently.
We recognize that we can write the letter A many ways. Print upper case, print lower case simple (a circle with a line), print lower case fancy (a written with a cap and tail), cursive upper case, and cursive lower case. We can read uncountable styles of writing and fonts and still pick out the letter A. The reason why early text scanners had such a hard time translating text was because you have to find a way to program the computer to learn how to recognize all the different styles of writing just one letter — something our brain can do without us thinking about it.
Part of Number Sense is understanding that a number can be written and said many ways, but its value stays the same.
The models and boxes and strange looking configurations our children are coming home with are meant to demonstrate number sense. How can we take this number apart? How can we put it back together? What are the properties of addition and subtraction? How does multiplication and division work?
If a person knows these basics and can manipulate the numbers, they can calculate what they need and have a means to check their work. They rely less on memorizing every fact possible, and instead begin to calculate what they want in the manner they find to be most efficient for the problem they are working on.
I want to share with you my favorite way to subtract large numbers which would normally have a lot of borrowing in another post.
It’s not the way I was taught in school. We were only shown borrowing (aka. regrouping or decomposition). However, it uses the same knowledge about how regrouping works and a few other tidbits that we all know…but… may not have noticed that we know! I taught myself the technique after I stumbled upon it accidentally while doing some research. It took me a bit to figure out what was actually going on enough to be comfortable doing more than just following steps and to be able to explain it to someone else (okay, a few days), but then there were rainbows and butterflies.
Before I show it to you, though, I want to go over some important concepts.
I want to show the number sense we all have, but may not be aware of. I don’t know about anyone else, but when I start borrowing/regrouping it can feel a lot like being on a conveyor belt on an on-the-rails, shooting-gallery game. I get so caught up in the steps that I block out everything else. Must do step 1, must do step 2, must get to answer… That’s what I mean when I said in a previous blog entry that you can do steps without ever really understanding what you are doing, yet get the answer.
To recap:
The term borrowing fell out of favor because it didn’t quite convey the process well enough. The term regrouping (in my opinion) doesn’t really explain what that means either. What you are doing when applying this algorithm is taking values of 10, 100, 1000 etc and moving them to the next lowest place value. This allows you to do the subtraction. You do need to be comfortable with higher subtraction facts like 14 – 8 (but there is a way to make that better).
It’s based upon the fact that you can take a number apart and write it differently, yet, its value is constant. Thus, 3 can be written as 2 + 1. Or 313 can be 100 + 200 + 10 + 3: one hundred, plus two hundreds, plus one ten, and three ones. We can write out 2456 out as all +1 if we want to!
We can decompose a value any way we darn well please as long as we keep its value constant. If you want to play around with place value and decomposition visually, you can find a virtual base 10 block manipulative. It’s a good refresher especially if you want to teach a child about place value.
When you start playing with decomposing numbers with addition or subtraction there are a few things you might miss if you always borrow. Instead of regrouping a number a little at a time, we can do it all at once. So if we have 832 – 474, we can do all the borrowing we need all at once instead of one at a time.
Or, we can also decompose the number in any way we want before we do any subtraction. Some decompositions we choose will be more efficient than others.
So in the image above. Instead of regrouping and rewriting 832 as 700 + 120 + 12 a little at a time (which works obviously), I decided to just expand 832 into 400 + 300 + 70 + 50 + 8 + 4. Why? I felt like it and I wanted to demonstrate something interesting. By making sure that I wrote 832 with the numbers 400 and 70 and 4, I was able to subtract off 474 in those chunks. It left me with 300 + 50 + 8 — the answer.
Maybe you knew that already or maybe you never noticed that you can subtract off in bits and pieces rather than all at once. When you are teaching a child how to subtract, it is an important bit of information to point out to them.
This information is important to remember for the method of subtraction I want to show you later! It took me a while to figure it out when I first saw it because I was “blind” to the process even though I knew it.
Here’s another interesting bit. You can change the numbers if you want to.
Did you know that you can ADD the same amount to both the minuend (top number) and the subtrahend (bottom) number and your subtraction value won’t change? It works for any number. If I have 5 – 3 and I decide to add 10 to both five and three at the same time, I get 15 – 13 which equals 2. The same result as if I do 5-3. No, I wouldn’t actually decide to subtract 5 -3 that way, but it does prove that it can be done. In addition when I see 15 – 13 we I can now also “see” the 5-3 in it better. I can choose to +5 and get 10 – 8 which is also 2. Any number we choose to add to both numbers at the same time will equal the same result.
Looking at the image above, my favorite configuration was adding 26 to both values. Getting all those zeros made for a pretty calculation. This equal distance concept is an important number truth.
We do do this quite often in our lives. We might add-in or subtract-off a value to make a nicer number such as turning $19.98 into $20.00. That extra two cents we added gets those nice zeros. Suppose we had 20.11 – 19.98. We could count-up to find the answer, we could borrow with pencil and paper, or we could change it to 20.13 – 20.00 = 0.13 We could, if we wanted to, do 20.11 – 20.00 and get 11 cents then add 2 cents. The point is, we can do a lot of different things to get our answer depending on the numbers, if we are using paper or our head, and how fast we want to do something.
Equal distance is not just confined to adding in numbers. We can subtract off an equal amount and the value won’t change. Showed this to my six-year-old and it blew her mind — in a good way.
Again, subtracting off 74 from both values created an interesting situation. This is a fun concept to teach with a number line or with scales. The more you can learn how to manipulate the numbers and arrive at the answer with small values, the more confidence it builds with larger numbers.
I am excited to show you my favorite subtraction algorithm in another post.
What did you think about decomposing a number and equal distance?
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Very good post!
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