integers, adding positives and negatives

I posted a long and detailed answer to this several months ago. Please try searching the archives and see if you can dig it up, because the answers are all there at least to start. There’s one post on addition and subtraction of signed numbers and another one some days later on multiplication and division.

If you can find it, it would be a good idea to repost a copy here for anyone else who is looking.

If you can’t find it, please email me and I’ll see if I kept a copy on my end.

Victoria, thank you for this,
Bindiya

Victoria wrote:
Aaah!! Rub my hands in glee! A real question for me to get into!OK, first you need a really good mental model for negative numbers. Go back to this and work on it for a while until it’s really comfortable.(1) If you live in northern Canada or the US prairies, you have a perfect model of positives and negatives outside your kitchen window, the thermometer. Even in the rest of the US north of Florida it’s at least familiar. If you use Celcius, where zero is freezing water (32 Fahrenheit), all the better. It would be a good idea to get a Celcius thermometer (scientific supply houses, any Canadian distributor — try Scholar’s Choice, I’ll give website if interested)and do some experiments — temperature outside, temperature in fridge, temperature in freezer, etc. Good for science as well as getting a concrete grip on math.Draw some thermometers on paper. Practice locating numbers, +10, -10, +7, -7. zero, etc. After a while the thermometer becomes simplified to a vertical number line. Stay with vertical; going up and down has more physical meaning than left or right. Books use left to right number lines simply because they fit better into typography, not at all for any pedagogical reason.Note a couple of important facts: (a) +10 and -10 are the exact same distance from zero, but in opposite directions. Same with every pair of opposites. Make up lots of pairs of opposites and draw them.Note — be sure your spaces on your line are at least fairly even. Some kids will run out of space ansd squish the numbers 20 to 50 in the same space used for 0 to 10. Resist this. A pad of squared paper is helpful, or going vertically, just use one unit per line on your regular lined school paper. BTW, one of my hot buttons — my class is not garbage, our work in here is not garbage, and we don’t use paper fished out of the garbage, thank you very much. Yes, recycling is a very very good thing. But wasting years of your life failing math and re-doing it over and over out of a fear of wasting paper is not ecologically sound either. Please use fresh clean lined paper and draw your diagrams large enough to see and measure. And try to get over Miss Smith’s kindergarten commandments and use pen; smeary multiply erased pencil makes math papers so unreadable and distasteful that it’s no wonder kids feel ill at the sight of math. (b) +20 is clearly hotter than +10, because it’s higher. But look at -20 and -10 — which is warmer? Well, -10 is warmer than -20 because it’s higher up. We write 20 > 10 but -10 > -20, and similarly 10 -20 because 20 is bigger than 10 are not yet at a level of logic that can handle algebra. They need maturity and/or more work on thinking skills. Piaget — abstraction starts to be feasible in early adolescence, somewhere between ages 11 and 14 (highly variable; some of my college students never got there) If so, leave the algebra for a bit, because they will be filtering it through their own present misconceptions and developing worse ones. You don’t want to unteach later. Have the student do a lot of greater than and less than problems using the vertical number line / thermometer; keep the concrete guide visible and refer to it every time. Compare positives to positives, pos to neg, neg to neg, pos to zero, neg to zero. Avoid the math book temptation to make up verbal rules; the rules are impossible complicated and will never be remembered right — as above there are five separate cases, an that would mean five separate and contradictory rules — yuck. One rule; higher is warmer or more, lower is cooler or less. KIS — Keep It Simple.(2) A second model: read up about Death Valley and the Dead Sea. These are below sea level and have negative elevations. Make up stories about a prospector who goes through Death Valley and up the mountains (Sierra Nevada?? check map to find out …) and back down again. Example: If you start at -100, climb 400 feet, come back down 200, and climb up 50 again, at what elevation do you end up? Resist verbal rules and shortcuts! Draw it and look at it. This is vital; it’s actualy simpler and shorter, and draw it and look at it is a vital skill for any math past the elementary (I know a totally blind man who studied engineering and passed calculus — he draws graphs too, only on Braille paper in textures.)(3) A third and most important model. This model is vital for the arithmetic we do later; but since it is a more abstract, don’t be tempted to jump into it first, but take some time getting comfortable with the physical models above. Your bank check card/ ATM card with overdraft protection. Money in the bank is positive; overdraft and owing to the bank is negative (absolutely no mention at all of interest charges at this stage, please! Focus on our goal, which is understanding positive and negative numbers.)Depositing into the bank is positive; withdrawing is negative. Make up examples: you start the month at -100, having overdrawn. You deposit your pay of 700. What’s the balance? Again, use the number line, still easiest to stay with vertical, and just look at it. Now you write checks or make charges for -150.75 and -300.50. What’s the balance now? Note that fractions and decimals are perfectly easily represented on our model. Use up-pointing arrows for deposits, down-pointing arrows for withdrawals. These arrows are vectors, and are very useful in later work in math and science. Work on a number of examples like this.OK, now you’re playing happily with negative numbers. Now we start on arithmetic. First: addition. Addition means to put together. You put together 3 blocks and 2 blocks, and you get 5 blocks. When you put something positive together with a negative, there is a big temptation to say well, it’s “really” subtraction. Resist this. Say that yes, in some cases you end up with the difference, but please don’t call it subtracting because sometimes it is and sometimes it isn’t and you’ll give yourself a headache. Keep It Simple.(a) Put together (add) a +2 arrow and a +3 arrow (measure squares on your squared paper so the four examples will be consistent) Sure enough,starting at zero and measuring, you get to +5 on your number line, so (+3) + (+2) = (+5) Well, hey, we already knew that, but it’s good to see the system works. (b) Start at zero, draw beside your number line a +2 (up pointing) arrow, and from the end of it, draw a (-3) (down-pointing) arrow. You should end up at -1. Well, does it make sense that if you go up two degrees and then down three, you end up at one below where you started? You deposit 200 and take out 300 and you end up 100 in the hole? OK, model works. (+2) + (-3) = (-1) (c) Do the same for (-2) + (+3) = (+1) First draw, then make up a couple of models on the three systems above to make real-world sense out of it. (d) Do the same for (-2) + (-3) = (-5)Note that in two cases (b and c) you got the difference, but in two cases you got the sum. And the sign of the answer varies. Verbal rules are a quagmire; draw it and look at it, please.Now subtraction. This is where many students and alas far too many teachers throw logic out the window, a problem in math where you’re trying to teach logical thinking. There IS a sensible way to see this, promise. (a) In our usual positive minus (smaller) positive, we think of subtraction as “take away”. In this one special case (out of 11 possible, pos-neg-zero, smaller-larger-equal) “take-away” works well, and we can use it as a starting point for the logic to follow. Consider 8 - 5. We first draw a positive (up) arrow from zero to 8. The starting at the 8 we draw a down arrow — hmmm, that’s a negative, to take away the 5. And we end up of course at 3. So 8 - 5 is the exact same as (+8) + (-5) We see a rule here — to take away 5, add a negative 5. Makes sense. We rewrite our problem: (+8) - (+5) = (+8) + (-5) This problem we know how to solve- use our (vertical) number line. Of course we know the answer is 3 — that was the test case, to be sure we have a method that works. Now we move to harder cases. ** Make a point of actually rewriting the problems this way. Sure it takes ten seconds ande a bit of paper. It saves years of math nightmares — worth it. **The incredibly simple examples are given as stepping stones to help you. If you just skip all the steps because they’re too easy, then you won’t get the point of the lesson, which is the method, not the numbers. Take the time and work the easy ones out by the method; then when you get to the hard ones, you have a method to use.So we have evolved a general rule: to subtract a positive, add the same sized negative.(b) Let’s work out a harder problem by logic and then see if the method works. Suppose you are in debt and you deposit a check for 300. Now your balance reads -100. Oh-oh. The check bounces. You have to take the 300 back off from your account. Where will your balance be? Well, we see we have to go down by 300, so it must be at -400. Let’s try this by our system: (-100) - (+300) = (-100) + (-300) = (-400) Yep, it works. Make up a few more examples for yourselves and work them out by rewriting this way. Some mumbles from the peanut gallery, it’s too much trouble, look I have a faster way because it’s “really” subtraction — no, sorry, this is about to get messy, there’s a reason I’m doing all this.(c) OK, I deposited $100. into my daughter’s special college savings account. This is a no-fee account. Horrors! When I open up the statement, I see a balance of only $85. What happened? I see that they charged $15. which they were not supposed to do. They did 100 + (-15) = 85. Naturally I march down to the bank and demand that they take this charge off. How do you take off a charge, a negative? YOU GIVE THE MONEY BACK. That’s the big deal here, what I’ve spent four pages leading up to. To take away a charge of 15, they have to give back 15. We write: (+85) - (-15) = (+85) + (+15) = 100 and we get to the original $100 which is where we are supposed to be.(d) Let’s try another one. I charge $50 (written as -50) on my card. When I get the card, I see a bill for $20 (write -20) for insurance, for a total bill of -70. But I didn’t want the insurance and I said no to the telemarketer. So I call up and demand the charge be taken off. So they have to give my 20 back. Let’s work it by our system: My bill is -70, and I demand they take off a false charge of -20 (-70) - (-20) = (-70) + (+20) = -50 (by number line) Yep, we have it. That took me to the original bill of -50 which is where it should be. Make up some more examples for yourselves and work them out.So we have two rules for subtraction: subtracting a positive is the same as adding a negative. And to subtract a negative, give back the positive. These can be summarized even better: to subtract any number, add its opposite.A note: I cringe when I hear someone say a number “becomes” something else. Numbers are concrete, solid measures. They don’t writhe around and metamorphose. Yes, your teacher said this. She said a lot of other stuff, too. You still believe all of it? Even more painful is the FALSE “Two negatives make a positive” FALSE!! This fallacious “rule” is wrong more than half the time. Neg + neg = NEG, always. Neg - Neg = EITHER neg or pos, depending on absolute value (length of arrow) Neg x Neg = pos.But neg x neg x neg = neg. Neg over neg = pos. Neg to power of neg = EITHER pos or neg, depending on odd or even power. -(-a) = a, which is EITHER pos or neg, depending on how a is defined. This just isn’t a rule to work on.OK, back to some rules that really do work:______________________________________________ To add two signed numbers: start at zero on the number line, draw the first, and starting at the head of the first, draw the second. (pos = up, neg = down, zero = no motion) Wherever you end up is the sum.To subtract signed numbers: rewrite the problem; **add the opposite** of the number *being* subtracted (the number AFTER the minus sign). ________________________________________________ Short, sweet, simple, and really does work.Now go to the practice page in any algebra book where they give mixed practice on adding and subtracting positives and negatives, and work through it. Usually the answers to the odd numbers in the back. Practice until you can get 90% correct.I have good stuff on multiplication and division too — when you’re ready, just ask.

OK, two ways that are logical and not excessively complicated:Verbal/graphic/financial people:A teacher, Miss June Smith, likes to go on educational tours in Europe. Every year she puts aside $200 per month from September to June in a special savings account.Then she goes on a six-week tour in the summer and spends $300 per week.Three basic ideas: time forward, in the future, is positive; looking back to the past is negative. Deposits are positive, withdrawals are negative. Increase or upwards is positive, decrease or downwards is negative. (All ideas which should be made clear in introductory work on signed numbers)Let’s sit down with Miss Smith on December 31 as she balances up her finances, checks over back records, and makes plans.(a) In six more months, will she have more money or less money? Obvious answer, more money, as she keeps putting it in the account. How much more, and how do we calculate it and why? We multiply when we have equal groups. 6 months ahead (pos) X $200 each month deposit (pos) = $1200 more in the account (Pos, increase) So (+6) x (+200) = (+1200) and pos X pos = pos (Stress pos TIMES pos; never say pos “and” pos; “and” either means addition or is too vague to be meaningful) We can graph this with a bar graph — write months of the year each one square wide, then a bar $200 for Jan, $400 for Feb, $600 for Mar, etc. The bars climb up like steps.(b) She checks back for errors in the past. Did she have more money or less money three months ago, at the end of September? Answer is clearly less; she hadn’t put it all in yet. How much less and how calculated? 3 months ago (neg) x $200 deposits (pos) = LESS by $600 (neg) so (-3) X (+200) = (-600) and neg X pos = neg (That’s neg TIMES pos)Graphical: re-copy the graph from (a) above, putting January in the middle so you have room to work back. Then look what we have to do to keep the stair-steps going leftwards. December must be at zero (this is where we started counting, end of December, so we started at nothing). November must be at -200 (200 less than Dec., 400 less than Jan, etc.) Oct. must be at -400, Sept. at -600. Draw it and look at it — the patern makes sense.(c) OK, now we visit Miss Smith on July 31, as she views the ruins of Rome. She’s halfway through her vacation, and she wants to check her finances. She is spending $300 per week now. In three more weeks, will she have more or less? Answer: clearly less, because she’s spending away and not putting in. How much less, and how to calculate? 3 weeks ahead (pos) X $300 withdrawal (neg) = $900 less (neg) (+3) X (-300) = (-900) so pos X neg = neg ((Please say TIMES, not “and”))Graph as above; start at zero end of July, and make bars getting longer and longer *downwards* as the weeks go on.(d) Now the punchline. Still on vacation, July 31, she looks back at her spending so far. Two weeks ago, did she have more or less money? Answer: **she must have had more, because she hadn’t spent it all yet! How much more, and how do we calculate it? 2 weeks ago (neg) X 300 withdrawal (neg) = $600 **MORE** **pos** (-2) X (-300) = (+600) so neg X neg = **pos** (negative TIMES negative)Graphical: re-copy the graph from (c) above with July 31 in the middle, and extend it leftwards. Since the bars get longer and longer downwards as you go forward in time and you keep stepping down each week, they must go upwards as you look back. To the right step down; to the left, step up.Summary: pos X pos = pos (please, please say TIMES) neg X pos = neg pos X neg = neg neg X neg = **pos*****************************************For division: sign rules come out to be exactly the same as for multiplication. You can do this several ways:(i) invent a story like the above if you wish. (ii) Look at division as multiplication by reciprocal: EX: 20 divided by (-4) is the same as 20 X (-1/4) = -5 so pos divided by neg = neg, etcetera, same as multiplication.(This assumes reciprocal of a negative is negative, but that makes sense) (iii) Look at division as “undoing” multiplication (a most productive way to do it) EX: (-5) X (-4) = +20 so just read backwards (+20) / (-4) = (-5) (using fraction slash / for division)pos X pos = pos so pos/pos = pos neg X pos = neg so neg/pos = neg pos X neg = neg so neg/neg = pos ** neg X neg = pos so pos/neg = neg*********************************** So there you have it. Now the hard part; get your algebra book and do a couple of pages of mixed practice (all four operations) and work on keeping them straight. Enjoy!

I promised to give two ways of doing the multiplication, did one and forgot to type the other.A theoretical mathematical approach:This is true, correct, and leaves a bad taste in most people’s mouths because they feel they’ve been conned by some sort of shell game. It’s just too quick and slick. However, if you have someone who has grasped the basic rules and likes logic tricks, here it is:(a) multiplication by a positive can be viewed as repeated addition.(+3) X (+4) = (+4) + (+4) + (+4) = 12 OK, we knew that and I think we all agree. Pos X pos = pos(b) Negative multiplied by a positive can be viewed as repeated addition of the negative: (+3) X (-4) = (-4) + (-4) + (-4) = -12 Makes sense, and we get pos X neg = neg(c) We believe firmly in the commutative law of multiplication, which says that multiplication in either order comes out the same. For example 5 X 7 = 35, and 7 X 5 = 35. We want our signed numbers to cooperate in this way too, or we will get very confused.so since (+3) X (-4) = -12then (-4) X (+3) = -12 alsoand we must have neg X pos = neg.Can we figure this out another way? Multiplication by a negative can’t be seen as repeated addition — that doesn’t make sense, and anyway wouldn’t give the result above. How about viewing multiplication by a positive as repeated addition, and multiplication by a negative as repeated subtraction?(-3) X (+4) = -(+4) - (+4) - (+4) = (-4) + (-4) + (-4) = -12 Yes. This gives neg X pos = neg, and agrees with our commutative law, so it will work.(d) Apply the repeated subtraction rule, above, to neg X neg(-3) X (-4) = -(-4) - (-4) - (-4) = (+4) + (+4) + (+4) = +12and we get neg X neg = posAnd ladies and gentlemen, there is nothing up my sleeve, the quickness of the hand deceives the eye …

Well, similar to the stories for multiplication. First another financial one, then a physical one:time forward = pos, time back in the past = negmoney gained or profit = pos, money lost or spent = negdeposit = pos, withdrawal = neg.upward or gain in height = pos, downward or loss of height = negFinancial —(a) An investment is predicted to be worth $1200 more in six months. How much is it predicted to gain per month over the next year? (+1200)/(+6) = +200, pos/pos = pos(b) A dotcom investment is losing money. It’s predicted to be worth $1000 less in two months. How much on average is it expected to lose per month? (-1000)/(+2) = -500 neg/pos = neg This negative makes sense because the value is decreasing as time goes forward.(c) Same bad dotcom investment as in (b). If the investment is losing value, then it must have been worth more in the past, before the losses. If it was worth $2400 more twelve months ago, how much did it lose per month on average last year? (+2400)/(-12) = -200 pos/neg = neg This negative makes sense because the value is decreasing as time goes forward.(d) Go back; same good investment as in (a), still increasing in value. It was obviously worth less in the past (before you earned the money). If it was worth $900 less three months ago, how much was it gaining per month last year? (-900)/(-3) = +300 neg/neg = pos***Note used to be worth less = neg, time past = neg, but monthly change = **pos** because it was *increasing*)Physical— Note: All of these benefit very very much from sketch illustrations/diagrams. I would add some but the medium doesn’t cooperate.(a) A balloon is rising in the air at a fairly steady rate. Standing on top of a tall building, you start a stopwatch as you measure its height the first time. 5 minutes ahead of that (+5) it has risen 200 feet more. What is its speed?(+200) rise/(+5)time advanced = +40 feet per minute increase in height, positive speed/velocity because it is going higher over time.pos/pos = pos(b) Other students are sinking things in water. A student down in a pool starts a stopwatch underwater as a weighted balloon passes him; 4 minutes later the balloon is 8 feet lower. What is its speed of sinking?(-8) sunk/(+4)time advanced = -2 feet per minute, negative speed/velocity because it is going lower over time.neg/pos = neg(c) The water-sinking group reports that three minutes *before* the balloon passed the underwater student, it was 9 feet higher. What was its speed of sinking in that part of the pool?(+9) higher/(-3)time *before* = -3 feet per second, negative because it is sinking.pos/neg = neg(d) The air-rising group meet on the ground and the other members tell the student from the roof that 6 minutes *before* the balloon passed her, it was 300 feet below her. What was its speed of rising on that part of the journey?(-300) below/(-6)time *before* = +50, positive speed/velocity because it is *rising*neg/neg = pos.

Victoria,

hello. I searched the archives and found lots of good strategies! Thank you.
I have reposted the 3 main ones.

Bindiya

Somehow in the copy-repost system a part of another post got copied in the middle of one of these, and all the carriage returns got lost so the concepts are kid of strung together.

I will try to re-copy these later with the carriage returns so they’r easier to follow.

Thanks very much for finding these, and I hope they help.

Please do note that each of these posts is a very succinct summary of weeks or months of teaching — do lots of practice and don’t rush it!!

HI,
I have found a great way to teach this to students. A fellow teacher actually gave me the idea. Use a number line and have them start on the number which the problem starts with. They should face the positive direction, because remember to always start with a positive attitude! Then for every negative sign they turn 180 degees. So for 5 - (-2) they would start on 5 then turn 180 degrees twice for the 2 negative signs. Then they walk the number of steps which would be 2.

My students love this we even make a big number line outside, in high school, and go out to walk the line. In the classroom they use a pencil to simulate themselves when working with a numberline at their desk.

This also works for multiplication and division. Start on zero, and turn 180 degrees for each negative sign. Then multiply or divide the answer.

Hope this can help.

This is actually mathematically correct.

I would very strongly recommend doing real-life examples, as noted above, to develop number sense. To repeat a 2500-year-old quote, there is no royal road to mathematics; you absolutely must spend some time thinking about and modelling and comparing things in order to use them in any application.

Remember, success is not measured in number of blanks filled in or poundage of paper covered with “right answers” or speed in “covering the material” in a text; success is measured in retention and transfer, as in using negative numbers successfully in chemistry and physics and advanced trigonometry two to four years down the road from now. For these you need visualization and number sense and measure and graphing and vectors as well as quick calculation tricks.