Writes and Wrongs #11: Subtraction Without Borrowing – Partial Differences (Negatives and Positives)

I hated math in school. It gave me a lot of stress and anxiety. Dear Mrs. Morris (my first grade teacher) wrote on one of my official assessments that I didn’t see until I graduated high school: “Can do the work. Tries very hard.” I was doing the work, but getting answers to certain concepts wrong.

Looking back now, it wasn’t the math I hated. I hated being lost and confused and muddling my way through. I could do the steps, but I often didn’t understand the work. I didn’t have math sense in the way that I had a sense of words. I was a precocious reader. I could read before I went into Kindergarten. Words made sense. Letters had shapes, those shapes represented sounds, I could put those sounds together and a word with meaning appeared like magic. I didn’t even mind the freaky spelled words like “guess”. Contractions delighted me. I despised dictionary drills, but I loved, loved, LOVED words. Math was a dark cloud. I had difficulty telling digital time, counting money, doing word problems, and working with fractions outside of the steps. Don’t even get me started with my frustrations with rulers. It was all about the steps. If I couldn’t do the steps, I choked. I had no grasp of the relationships between the numbers or even how to read the examples properly so that a word problem wouldn’t seem so — to be blunt — WTF. This may have been the reason I ended up liking Algebra a lot, but struggled endlessly with Geometry proofs — and later — statistics (ugh).

Subtraction Without Borrowing: Partial Differences

Last post, I shared with you my favorite subtraction method without borrowing. I tend to fall back on borrowing out of ingrained habit, but it’s not the one I like to do. This time I want to share another method I enjoy, though, I didn’t realize I’d been doing it for years in Algebra. It never occurred to me to use it for something like three-digit subtraction until I was in my thirties.

At first, I was like, “What!? Wait. You can do that?” I studied it and practiced it. Realization dawned. “Duh. Self, you do this all the time you fool!”

Does this look familiar: -40xy + 20xy – 4xy   ?
We focus on the numbers: – 40 + 20 is – 20   then – 4    is -24.
Don’t do algebra? Well, it’s that same thing going on with your credit card statement (income and debt.) You spend forty (-40) and you pay twenty (+20).   You still owe twenty (-20). You spend another four dollars (-4) and you now owe twenty-four (-24).

As I said before, you can become blind to your own skills.

A few things before I show you Partial Differences

You must have knowledge of positive and negative values such as: – 11 + 6 is the same value as +6 – 11
You must have a strong grasp of place value
Though you can work in any direction, mentally it’s better to work left to right.

What is nice about Partial Differences is that it takes advantage of place value so that there are a lot of zeros to make the arithmetic easier. The ones place value doesn’t become something unwieldly like 12 -9. The subtraction facts will always be simple and focused: 400 – 300 or 40 – 30 or 4 – 3.

Let’s try 632 – 491

We can work left to right and subtract starting from the hundreds place and working our way to the tens and finally to the ones.

600 – 400 is 200
30 – 90 is negative 60 ( 3 – 9 = -6)

Because it is negative, we subtract 60 from 200 and get 140

2 – 1 is 1 (one is positive so we add it in)

The answer is 141.

And we just did that without borrowing.

435 – 378

400 minus 300 is 100

30 minus 70 is negative 40 (-40)

Subtract 40 from 100 is 60.

5 minus 8 is negative 3 (-3)

60 minus 3 is 57

For a little bonus I dug up one of my math tests from 1983.
Why do you even have that, you might be asking.
If you’ve been reading my blog for a while, that shouldn’t surprise you. For the new folks, I’m weird.

Anyway, notice anything interesting except for the horrible readability of the mimeograph (and no this is not age degradation. Mimeograph copies were notorious for being as hard to read as they were fun to sniff when fresh)?

I could reduce fractions just fine. Then I seemingly forgot/didn’t understand how to divide fractions. Only then to be able to divide mixed numbers (which is harder). Then I choked on the word problems. This is what I mean about being able to do the steps, but not to understand what it was I was doing.

From a Yahooligan Yahooligan: Bill Bryson in his memoir The Life and Times of the Thunderbolt Kid writes, “Of all the tragic losses since the 1960s, mimeograph paper may be the greatest. With its rapturously fragrant, sweetly aromatic pale blue ink, mimeograph paper was literally intoxicating. Two deep drafts of a freshly run-off mimeograph worksheet and I would be the education system’s willing slave for up to seven hours.”

What did you think about Partial Differences?

2 comments on “Writes and Wrongs #11: Subtraction Without Borrowing – Partial Differences (Negatives and Positives)”

Tony Burgess
March 15, 2016

Math is my worst subject ever.

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- Laissez Faire
  March 15, 2016
  
  I think a large number of Americans feel that way.
  
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