Tag Archives: subtraction

A. N. Whitehead on negative numbers (1911)

This is really worth reading. It is from his book, “Introduction to Mathematics”, published in 1911.

whitehead intro to math negative nos






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Subtraction and the “standard” algorithm

CCSSM talks about “the standard algorithm” but doesn’t define it – Oh, how naughty, done on purpose I suspect, since there are varieties even of the  “American Standard Algorithm”. Besides, if it is not defined it cannot be tested (one hopes!). I checked some internet teaching stuff on it, and as presented it won’t work on for example 403 – 227 without modification.

Anyway, I was thinking about subtraction the other day (really, have you nothing better to think about?), and concluded that subtraction is easiest if the first number ends in all 9’s or the second number ends in all 0’s. So, fix it then, I thought, change the problem, and here are the results

.Two simple algorithms for subtraction

I am quite sure that some of you can extract the general rule in each case, and see that it works the same in all positions.

While I am going on about this I would like an answer to the following-

“If I understand subtraction, and can explain the ideas to another, and I learn the standard algorithm and how to apply it, and I have faith in it based on experience, WHY THE HELL DO I HAVE TO BE ABLE TO EXPLAIN IT?”

I guess this post counts as a rant!

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Another Common Core Math Horror

I thought I had found them all, but NO.

Subtraction. Read this
Operations and Algebraic Thinking
• Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
What has subtraction got to do with taking apart ???
(The examples are all of the form 9 = 3 + 6 and so on).

Also there is NO mention at all of subtraction as a way of finding the difference between two numbers, or of finding the larger of two numbers (anywhere).

While I am in critical mode I offer two more, less awful, horrors from Grade 1:

“To add 2 + 6 + 4,…”  and  “For example, subtract 10 – 8″.

The poor symbols are clearly in great pain at this point. Just read aloud exactly what is written…..


Filed under algebra, arithmetic, language in math, operations, teaching

Subtraction in algebra – let’s use algebra !

I have seen some heavy handed ways of explaining the identity

a – (b + c) = a – b – c

Let us use algebra. Give the left hand side a name, say d .Then

a – (b + c) = d

This is an equation, so add (b+c) to each side and get

a = d + (b + c), then a = d + b + c as the parentheses are now superfluous.

Now subtract  b  from each side

a – b = d + c

Now subtract  c  from each side

a – b – c = d

so  a – (b + c) =  a – b – c

or is this too simple ? Look, no messing with p – q = p + -(q) stuff,

and no appeal to the famous distributive law.

You can do this, and other stuff, with numbers as well.


Filed under algebra, arithmetic, teaching

Double negatives, or the meaning of -(-2)

In the extended number system of signed numbers, that is, the positive and negative numbers I see a lot of heart searching over the meaning of -(-2). This can be put to rest in one or both of two quite satisfactory ways:

1: Signed numbers are directed numbers, used for position, temperature, voltage etcetera. The basic question is “How far apart are the two numbers  A  and  B ?”, or more useful in a practical situation “How far is it from  A  to  B ?”.

This is a subtraction problem with direction and the answer is  B – A
For A=3 and B=7 we get
Distance from A to B = B – A = 7 – 3 = 4
For A=-3 and B=7 we get
Distance from A to B = B – A = 7 – (-3) = ???????????????
But a quick look at a number line shows that the distance is 10
So 7 – (-3) = 10
But 7 + 3 = 10 as well
Conclusion: -(-3) = +3

2: A simple and more abstract approach:
Starting with 7 – (-3) = ??????????????? we give a name to the unknown answer. Call it D.
Then using the basic fact that 12 – 4 = 8 is equivalent to 12 = 8 + 4 we have
7 – (-3) = D is equivalent to 7 = D + (-3)
7 = D + (-3) is equivalent to 7 = D – 3
7 = D – 3 is equivalent to 7 + 3 = D
which says that D = 10
So subtracting -3 is the same as adding 3

A meaningful example is as follows:
My friend from Anchorage calls me and says “It’s cold here this morning, -5 degrees”.
Down here in Puerto Rico it’s 68 degrees this morning.
How much warmer is it here than in Alaska?


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Commutative, associative, distributive – These are THE LAWS

Idly passing the time this morning I thought of a – b = a + (-b).
Fair enough, it is the interpretation of subtraction in the extended positive/negative number system.

I then thought of a – (b + c)
Sticking to the rules I got a + (-(b + c))
To proceed further I had to guess that -(b + c) = (-b) + (-c)
and then, quite ok, a – (b + c) = a – b – c

But -(b + c) = (-b) + (-c) is guesswork.
I cannot see a rule to apply to this situation.

The only way forward is to use -1 as a multiplier:
So a – b = a + (-1)b = a + (-b),
and then -(b + c) = (-1)(b + c) = (-1)b + (-1)c = (-b) + (-c)
by the distributive law.

It’s not surprising that kids have trouble with negative numbers!

Do we just assert that the distributive law applies everywhere, even when it is only defined with ++’s ?

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Fraction subtraction construction

So, you want to do a fraction subtraction. Here’s how, as a geometrical construction. You will need a piece of paper and a ruler.

Draw three number lines through a common point, which is the zero. Pick a nice point on the middle line to be the 1, say 6 inches away from the zero. Label the other two number lines 1,2,3,4,5,6,7 at equally spaced points, scale completely immaterial.

Now do what is shown in the picture below. (the pairs of lines are parallel)


Now measure the distance with the ruler, and divide by 6 (if you put the 1 at the 6 inch point).


A simpler version of this (2 number lines) can be used to locate the point on the number line corresponding to any (relatively simple) fraction.

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Another way to do subtraction

No more borrowing and paying back, or the new alternative

Use addition and then simple subtraction, naturally there are several steps

Example     234 – 187


-187  (should be lined up!)

Now 7 is greater than 4,  so we add 3 to both numbers, giving a zero in the units position of the second number

234 -187 = 237 – 190   (must be able to add 3 to 187 !!!)

moving to the 10’s position, and very formally, 90 is greater than 37 so add 10 to both numbers, giving

237 – 187 = 247 – 200, which is now simple, with result  47

Of course, if the lower digit is less than the one above just subtract it from each number

234 – 117 = 237 – 120 = 214 – 100 = 114

It is best to keep the “one above the other” layout, but I have fought with the editor enough today.

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