Hi,
I am new to this forum and really really hoping you can help me! I am only university age (in the UK) but with an A Level in Maths, I spend some spare time tutoring friends' children. I am currently tutoring a 16 year old girl studying for her GCSEs. Obviously it is important that her knowledge of the curriculum is near perfect, but she struggles with her basic number knowledge and mental maths - this therefore affects her question answering time.
I provide her with simple worksheets and time her whilst she completes them.. but I have recently provided a sheet that has me STUMPED! Any suggestions to the following problem would be greatly appreciated...
Basically the problem is "long subtraction", which doesn't really exist, it's simply subtraction, e.g:
10
- 8
= 2
For a very young child to do this, you could suggest "taking" the 1 from the first column and making the 0 a 10, then 10-2 is 8. This is obviously a very simple question to prove my point.. please try the following, imagining you are a young child who has just learnt this method...
34
- 38
= ?
We know that the answer is -4 but how do you show this using the subtraction method. If i take "1" from the "3" in the top left column then the 4 becomes 14, 14-8=6.. then 2-3=-1.. the answer would show as -16???? VERY CONFUSED! Is this even possible?
Help!!! Thank you for reading
Thank you, I got that I just hoped there was a simpler explanation that could show it in the same form.. As per usual maths isn't a one way street..
I haven't told my student that I am confused but thank you for your concern.
here is the simplest explanation i can think of:
think of "positive numbers" as instructions to "go left x spaces", and negative numbers as instructions to "go right x spaces" (you may substitute "go up" versus "go down", east/west, north/south, to me/from me, or any other pair of "opposites").
then to subtract A-B you do the following:
if A is bigger, take B from A.
if B is bigger, take A from B, and "change the sign"
if A and B are the same, you can do either one, you get 0 either way.
thus 34 - 38:
38 is bigger than 34, so if we are left of "the start" (0) by 34 spaces, and we go right for 38 spaces, 34 of those 38 spaces take us back to 0, and we still have 4 more spaces to use (38 = 34 + 4), so we wind up 4 spaces right of "the start" (0), at -4.
that is: a negative sign really doesn't refer to an "amount" but a DIRECTION. the idea is to visualize numbers as lying on a line (applying facts about how things in space are: that is, geometry, to arithmetic), an intuition well worth developing as it will make more complicated kinds of numbers that one encounters later on make MUCH more sense.
a bonus is that one can "draw" these things, and kids often readily understand pictures better than abstract squiggles (like the squiggle 2, or 4). never underestimate the power of visual presentation (we are, after all, highly visual beings).
EDIT: as far as you example goes, yes, you are muddled.
what you wind up with is -1 in the 10's column, and +6 in the 1's columns.
(-1)*10 + (6)*1 = -10 + 6 = -4
you can't add (-1)*10 to the 6 you get in the 1's column to get -16, the "1" and the "6" don't have the same SIGN. that is:
-16 = (-1)(16) = (-1)(10 + 6) = (-1)*10 + (-1)*(6)*1 = (-1)*10 + (-6)*1
so after finishing your "first subtraction", you get:
...34
..-38
-----
..-10
....6
we "borrowed" 10 from the 34, but since we have 3-3 = 0 in the "10's column", he (it? the 34 i mean) has none to actually lend, so he winds up "10 in debt" (-10). the 10 borrowed is "positive" so we get 14-8 = 6 in the ones column. so our "total" is -10+6 = 6+(-10) (because a+b = b+a, for any two integers). and:
6+(-10) = 6-10. this is now the second subtraction we have to do "to get the answer":
....6
..-10
------
oh dear. see the problem? trying to apply the same methodology leads us again to 6-0 = 6 in the 1's column, and 0-1 = -1 in the 10's column. we're stuck.
the deeper question being glossed over here is: why does "borrowing ones" work?
let's look at an easy example to see what is REALLY going on:
22 - 8.
one first begins by trying to subtract 8 from 2. well that doesn't work so well (8 is bigger than 2, but we feel that 22 is bigger than 8, so we shouldn't need "negatives").
but what, exactly do we MEAN when we write "22"?
it's an ABBREVIATION, the actual "long form" is:
22 = 2*(10) + 2*(1).
writing 8 the same way:
8 = 0*(10) + 8*(1)
then:
22 - 8 = 2*(10) + 2*1 - [0*(10) + 8*1]
= 2*(10) + 2*1 - 2*(10) - 8*1 = [2*(10) - 0*(10)] + [2*1 - 8*1] (the fact that we can re-arrange the sum and the parentheses are two laws of numbers called associativity and commutativity. these laws are "hidden from view" in ordinary arithmetic, but they are why it WORKS).
now we take advantage of the single most important rule in all of arithmetic: the distributive law. the stuff inside the brackets becomes:
[(2 - 0)*(10)] + [(2-8)*1] (this is the whole POINT of "adding/subtracting in columns", we can separate the 1's, 10's and 100's (etc.) columns and add them "one at a time" using SINGLE DIGIT operations, which are easier to work with).
now the 2 - 0 gives us no trouble: 2 - 0 = 2. the 2 - 8, on the other hand, leaves us with 20 + -6 = 20 - 6, which leaves us in a similar position as where we started.
the common solution to this is to "borrow 10" from the 20 (leaving it as just 10). that is:
22 = 10 + 12, so:
22 - 8 = 10 + 12 - 8 = 1*(10) + 0*(1) + 1*(10) + 2*1 - 0*(10) - 8*1
= [1*(10) - 0*(10)] + [1*(10) + 2*1] - 8*1 = [1*(10) - 0*(10)] + 12 - 8*1 = [1*(10) - 0*(10)] + (12)*1 - 8*1
= [(1 - 0)*(10)] + [(12 - 8)*1] = 1*(10) + 4*1 = 10 + 4 = 14
now, if THAT seems complicated to you, you're right, it IS. arithmetic is very complicated, numbers have a lot of "internal structure" (the technical term that the pros use is: "a euclidean domain (ring) of characteristic 0, additively generated by 1"). arithmetic involves keeping careful track of "what happens in every column" (and perhaps it's clear why accountants get paid so much).
learning to do it "by rote" hides most of this structure from view..understandable, small children cannot be expected to grasp the subtlety of abstract algebra. but YOU, as a teacher, should realize "there is more going on here than meets the eye".
trying to subtract the way you envision it, is needlessly complicated, take for example:
999 - 1000:
you'll have 3 separate "mini-subtractions" to work out, before you can get the answer, whereas writing:
99 - 1000 = -(1000 - 999)
allows one to just subtract 999 from 1000 "the normal way" and stick a minus sign in front. much simpler.
Thank you very very much! That's incredibly helpful I will definitely use that kind of description to point out her mistake and explain the correct way in doing things.. I guess this is where your knowledge of numbers is really tested! It's crazy to think that such a simple question actually goes so deep into maths!
Thank you very much again for all of your time and effort, it is really appreciated!
Take care.