This is for me: Any URLs 'splaining negative nos?

Des — I typed this all out at some length last year. Explanations in logical terms. Examples and everything. Four or six long posts. I lost my copies in the computer crash. Perhaps some nice person out there saved them down and can repost them here??
If not, I guess I should retype them anyway. If no answer in a day or two remind me and I’ll get at it.

Well if someone collected these… Otherwise LD online’s search is all but useless, pulled all sorts of unrelated stuff for ‘negative numbers’, including dealing with an 18 month autistic child (unless they have math savant skills)!

I did see some suggestions but they are for basic understanding, ie the no. line, etc. I have a basic understanding from my check book. I found a uk site and I *think* I have a better idea.

The uk site is: http://www.ex.ac.uk/cimt/mepres/book7/book7int.htm
Some good stuff on here.

—des

Welp, I found that some of my students got confused because sometimes when there’s a plus sign, you subtract and when there’s a minus sign, you add…
I’ve got some lessons for learning them on my site at
http://www.resourceroom.net/math/integers.asp
… but it’s geared towards the teacher. Peruse it for the concepts. Relating it to money helps most folks; if you’re already with a negative balance and you write another check (take more out; subtract)… you’re going to be that much *further* down the negative slope.
My seventh grade math teacher made us all cease and desist subtraction; henceforth, we were to add negative numbers. I’ve seen that work for some of my students.
Once the concept makes some kind of sense, what I recommend for when you’re in the testing situation and the brain becomes strange is (keeping in mind that this ONLY words for adding and subtracting) to sing, to the tune of row, row, row your boat:

Same Sign, Add and Keep
Different sign, SUbtract
Keep the sign of the bigger number
Then you’ll be exact.

If you run into a specitfic problem you have trouble with, fire it off :)

This is the first of four posts on negative/signed numbers.
Copyright V. M. Haliburton, 2004; please do not publish this post or any substantial parts of it (including photocopies and copies over the internet) in any form other than that printed here, FULL post including author’s name. Not for commercial use.

Part 1: introduction, meaning of negative numbers, examples
Part 2: addition and what it means, examples
Part 3: subtraction and what it meanss, examples
Part 4: multiplication and what it means, examples, division as “undoing” addition

Meaning of negative numbers:
There are two excellent models, the checkbook/debit card and the thermometer (especialy Celcius version). Also another productive model is altitude above nd below sea level.

Checkbook model:
money in the bank is positive, overdrafts that you owe back are negative. (“In the red” means negative; many years ago overdrafts were written in red ink.) When you use your card to put money in the machine to pump up the account, deposits are positive; when you use it to take money out and deplete the account, withdrawals are negative.
Examples:
Balance of $500 = +500
Overdraft of $200 = -200
Deposit of $150 = +150
Withdrawal of $80 = -80
Deposit of $346.75 = +346.75
Purchase (withdrawal) of $97.64 = -97.64

Thermometer model:
Well, with the Fahrenheit thermometer this works in really cold areas (come visit me in Montreal in January!) but with Celcius (formerly called Centigrade) thermometers the zero is set at the freezing point of water (which is 32 on the Fahrenheit scale) so you get lots of practice with negatives :). Moving up or increasing temperature is positive, and moving down or decreasing temperature
Examples:
Temperature of 10 degrees above zero = +10
Temperature of 10 degrees below zero = -10
Increase of 5 degrees in temperature = +5
Decrease of 5 degrees in temperature = -5
Increase of 3.4 degrees (accurate measurements necessarty in experiments) = +3.4
Decrease of 2.2 degrees = -2.2
Decrease of 3/4 of a degeee = -3/4

**A note on notation: “regular” numbers are positive; when you count eight sheep, that’s eight more than zero or +8. Positive numbers can be written with or without the sign. When playing with signed numbers, it is a good idea to put in the positive signs to remind yourself, and later when you are more comfortable you can relax and leave them off. Always put in a positive sign when you are doing things with possible sign changes (more on this later), to stress that yes this is a positive.

*** A note on “integers”: Integers are just whole numbers zero, one, two, three, etc., *with* signs positive or negative. Some integers are +5, -7, +139, -3421, 0 (note that 0 is *neither* positive *nor* negative; it is the *dividing line* between the two — draw your thermometer and look at it.)
In real use, signed numbers do NOT have to be integers; we use fractions and decimals with signs all the time, a couple of examples given above. The rules for signed fractions and decimals are *exactly the same* as the rules for integers. So there is absolutely no need to teach integers only as a number of textbooks try to do; in fact it is very confusing to students who have just started to work with the complete number system and who cannot understand why they have to drop the fractions and decimals when they use signs — the fact is you *don’t* have to drop them, so this is a distractor from the main ideas.

**** A note on language: Technically and super-correctly, -5 should be read as “negative five”. In general use we say “minus five” and that is all right — just be super-correct in testing situations.
However please learn to use the verb “to subtract” and the noun “subtraction”; there is NO verb XX”to minus”XX or XX”minusing”XX and besides making you sound bad, these usages lead to great confusions with signed numbers.

Altitude model:
Positive is above sea level and climbing upwards; negative is below sea level (Death Valley ant the Dead Sea) and climbing downwards.
This one is good to draw diagrams.
Draw a line in the middle of your sheet of paper to be sea level or zero. Use the scale one line on your paper = 100 feet. Now draw the path of a geologist as she surveys the Death Valley area and surrounding hills. She starts at a parking lot that is 100 feet above sea level or +100 (draw a short dash one line above your zero line) (Point P). Then draw a steep sloping line to show her going down into the valley to an altitude of -100 feet, one line below your zero line (Point A). Question: *look* at your diagram. How far did she go down from +100 to - 100? (Important idea!!) Then draw a steep upward-sloping line to show her climbing up 550 feet from that point to a ledge (Point B). Question: *look* at your diagram. At what altitude is she now? (Point B) Now draw a line showing her first climbing down 250 feet (to Point C) and then up 150 feet (to Point D). Question: *look* at your diagram. At what altitude is she now? (Point D). Then draw a line showing her returning back to the parking lot (Point P). Question: *look* at your diagram. How far did she have to climb down from her last point (Point D) to the parking lot (Point P)?
Answers: P = +100, given.
A = -100, given. P *to* A = -200 (negative, going downwards). If you didn’t write the sign, yes, it is very important, that is the point of this whole thing, to learn to use signed numbers, so please make a point of noting those signs; on tests you would be marked wrong without it.
B = +450.
C = +200, D = +350
D *to* P = -250 (negative, going downwards)

Number lines:
The thermometer model above IS a number line. It happens to be a vertical number line, which is very good for v
isualizing since it makes sense that up is more or higher and down is less or lover. It makes sense especially at the beginning to use vertical number lines.
Horizontal number lines are most common in textbooks for the very simple reason that it is easier to print horizontally than vertically. There is NO mathematical reason for the preference, just practicality.
On a horizontal number line, many people get confused about directions — this is why it’s easier to start with a vertical number line. Remember that we are working in English. The NORMAL FORWARD DIRECTION of English is LEFT TO RIGHT. (Leave Hebrew and Arabic for another day). Going *forward* in the normal direction left to right is *positive* or *increasing*; going backwards, right to left, is negative or decreasng. You put zero in the middle and count forwards +1, +2, +3, and so on; then from zero you count back or down -1, -2, -3 and so on.
Since this is typed I have to use horizontal lines, although I’d really prefer vertical.

Here’s a number line
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
______________________

A very important note: A number line is a measuring device, a special kind of ruler. And for a ruler to be useful, the spaces on it have to be even. It wouldn’t be much use if one inch (or centimeter) was ten times big as onother inch (or centimeter). When students draw number lines, at first they may not realize how important it is to space well. Also, it is a temptation when you run out of paper to shrink the sm=paces and squish in more numbers; this is counterproductive. Stress even spacing so things make sense.
On the other hand, number lines are a tool, a means to an end, not an end in themselves. Don’t go overboard and spend half an hour drawing each one. Take a minute and do a rough but clear sketch, and that is all that’s required. Fussing with rulers is usualy too time-consuming and distracts you from the *ideas* which are what is important here.

Second note: make your number lines big enough to do some good. Use the whole width (or height) of your paper. Paper is cheap, failing a year of school is extremely expensive; use the paper! This is *use*, an investment in your education, and *use* is NOT waste. If the number line is so small you can’t see the numbers marking the scale or tell where the dots are, it’s too small, no good for anything, and *that* is a waste.

Section 2 coming shortly.

Thanks Sue and Virginia. For my own uses, negative nos do make sense, hence I wouldn’t really need all the stuff on the thermometer, etc. I have enough experience with a checkbook and keeping track of my business.
I mean I understand the concept of them (withouth doing anything to them, that is).

Here goes:

A negative no. is less than 0. It could be a negative balance in a check book; a negative no. on the thermometer (ie 20 below— hey I lived in IL).

If you ADD negative nos together, they go down. You have
-$5 and you “add” another -$6, you’ll get -$11. That seems pretty clear.
Since multiplying is “fast” adding, if you have 3 (-3s), you’ll get -9.
Ok I think I have that too. My guess, correct me if I’m wrong, if you multiply a -3 to a -3, it is NOT 3 (-3s), so it would NOT be -9. It is a +9. But I don’t exactly know how it works.

I *think* I understand substracting negatives. I can do it. It seems like subtracting negative nos. The example somone used was Bobby kicks, hits and swears (-3 behaviors), if he stops doing one of them he has only -2 behaviors.But it helps me get it a bit. I don’t believe that this is an actual mathematically valid example though. I suppose using the temperature example if it is -20 and you take away -10 of those degrees it is -10, but that doesn’t make it clearer to me, as you are “adding degrees”. Division is “fast substracting” so it makes sense then to
that -9 divided into 3 groups is -3. But I dont’ exactly get the 2 negatives making a positive.

I suppose for a testing situation, though, now that I know 2 negatives make a positive, I have a little less to remember now that I understand how 3* (-3)=9 and the reverse.

Is this clear what I get and what I don’t now? BTW, this is more than I have ever “gotten” before. Now that I understand math a bit better with the MathUSee and On Cloud Nine, I can get this alll a bit faster.

BTW, I think the reason I am having more trouble with certain types of operations, just more chance to make arithmetic mistakes.

—des

Des — I’ll retype the stuff on adding and subtracting shortly, tonight or tomorrow. I’ll do the one on multiplication and division the day after. It;s a “canned” lesson that I’ve done so many times I can do it on autopilot, just a heck of a lot of typing. Hang in there and it will come up.

Gee, it’s very nice of you, but it is just for a test in my situation. Maybe I should just memorize Sue’s little poem. Anyway, how many negative nos questions could they actually have on a test where only 33% of it math.
The test is 4 hours and there are also tests of reading comprehension; English usage; and an essay.

Other stuff covered in the math section are decimals, fractions, graphs, algebra (though I recognize that is where it is likely to come up), area and perimeter, word problems, etc.

Just a guess, but I would estimate maybe 2-3 questions with a negative no. in them.

OTOH, if you do it, you should definitely hang onto it. You could do a very nice educational site with info and sales of some material ala Sue’s site.

—des

P’raps the automobile description for taking away negatives… A car is worth 3,000 less each year… so if you go back three years, you are subtracting that negative — and adding to the value of your car. (Visually — a photo of your car in all three years, each with it’s price tag and a little more or less wear & tear, depending on the year ;))

Here is a site that explains very well. http://www.purplemath.com/modules/negative.htm

des,

if you have hit a brick wall with the integers…which I doubt if you have heeded Victoria’s and Sue’s advise…but if you are still unsure, then find out if you can use a ti 83 or not. i would not know how to do the -5+7 thingy if not for the ti 83, thankfully i do not see that in my banking, just in math.

I had the Big Test yesterday, and I was right not getting too worked up about it. There were maybe 2 questions on negative nos.

—des

Three cheers on doing the Big Test.
Now we’re all waiting with bated breath for your marks, you know.

Des,
Congratulations! We can’t wait to hear the results!