Math Manipulatives

: I would like feedback about different math manipulatives? What do
: people like best? I am thinking about using the math-u-see
: program, but wondered about manipulatives such as cuisenaire rods,
: base 10 blocks, unifix, and others. Is it confusing to students to
: use different manipulatives? Are manipulatives color coded and
: would this be confusing to students used to one type of
: manipulative? If they are color coded could anyone tell me what
: color each number is to be able to compare similarities? Does
: anyone have a web address for the math-it program? Please let me
: know your opinions of this program.: Thanks!Good bunch of questions. Here are a couple of answers, not complete.I used to use a nice abacus, available in many toy storesm, that had ten rows of ten simple wooden beads. It was good for counting, simple addition, why the sums over ten work out as they do (eg 7 + 8 = one row of ten and 5 ones, or 15), base 10 up to 100, carrying in two-digit addition up to 100 (and with a second abacus, we could have gone further), and excellent for borrowing. I recommend this as a nice basic tool.I like base 10 blocks because they are simple and have very few distractors. The ones I have seen are wooden and nice and simple. Also the weight is a nice clue to the comparative size of the numbers. These are excellent for doing extended base ten, grade 2 and up. Especially good for carrying and borrowing, and if you can get a big enough set, good for multiplication and division.I have never used unifix, but the one teacher I did see using it was making a complete hash of things. Whether that was because she was just a confused and disorganized teacher or whether the fault was in unifix I don’t know. The kids were just snapping the blocks together and apart at random, doing no math.Cuisenaire rods are colour-coded. The one system I saw in any detail was a 1960’s system, and there is probably :-) a newer edition out by now. In the older system, the colour-coding was stressed far too much, and the kids were supposed to learn that cerise + green = orange ( not sure of the colours any more, but that was the gist of it). They wrote cute things on the board that looked like algebra, c + g = o etcetera. In this (older!) system, number values were not even supposed to be mentioned until the kids had learned how to do everything by colour. In practice, the teachers were not using the blocks by the book probably bbecause the plan was so far-out, and tended to fall back on the familiar worksheets while the blocks languished on the shelf.Cuisenaire rods could be a valuable tool if you used them to supplement almost any paper system; I prefer a “transparent” method, where you tell the child exactly what is going on and why, and start by finding counting equivalencies for the rods. I would even write numbers or tape number labels on them.If the child finds changing colour codes to be confusing, that is a sign that he/she has been leaning a little too hard on the colours as a crutch. I wouldn’t pressure a change too hard, but I would stress numbers rather than colours, and I would deliberately switch or combine systems to encourage looking at the essentials rather than distractors.Hope you get answers to the rest of these.

We have Math-U-See. The color-coding is green for units, blue for tens rods, and red for 100’s squares. 2 through 9 are each a different color. At the very beginning the child is encouraged to learn which color is which number, but once they learn the numbers it’s not important — except maybe to help grab the correct piece when laying out a problem.MUS pieces don’t snap together. To demonstrate subtraction you’re supposed to turn the pieces over so the empty side is up, in which case the smaller pieces do stay on top of the bigger pieces (not snapping together like lego, but they do fit together).At least with MUS, it would be possible to use differently colored pieces than they provide. I think it would be confusing to use multiple-colored rods for the same number — for example, tens rods in both blue and red. You’d want to keep all the tens one color, all the hundreds one color, etc.We haven’t gotten into the fraction overlays yet, so I don’t know how they would coordinate with other than MUS rods.Mary: I would like feedback about different math manipulatives? What do
: people like best? I am thinking about using the math-u-see
: program, but wondered about manipulatives such as cuisenaire rods,
: base 10 blocks, unifix, and others. Is it confusing to students to
: use different manipulatives? Are manipulatives color coded and
: would this be confusing to students used to one type of
: manipulative? If they are color coded could anyone tell me what
: color each number is to be able to compare similarities? Does
: anyone have a web address for the math-it program? Please let me
: know your opinions of this program.: Thanks!

PASSWORD>aa4um5Lp2CxdUCuisennaire rods and Math-u-see rods are color coded. The cuiesennaire rods are smooth, so until you get used to the colors, you can’t really tell the difference between a 6 and a 7 rod without lining it up next to a row of unit cubes. I think the math-u-see rods are marked at 1 cm. intervals, so you could just count the segments on a rod to figure out it’s length.Switching back and forth between math-u-see and cuisennaire might be a bit confusing because the colors are different Unifix cubes are simply individual cubes that you can snap together. They come in a pack of 100 cubes, 10 of each color, but they aren’t color coded like cuisenaire rods (1= white, 2= red, etc.)I like using unifix cubes with my boys (we’re homeschooling) but we aren’t doing anything fancy with them- Just 2 +3 =5, 3 x 4= 12 sort of thing. We could do the same thing with M & M’s, but there are limits to how much candy I want them to eat during math! I agree with Victoria that a set of base 10 cubes is great for borrowing, carrying, etc. In general, I’d say that different types of manipulatives are best suited for different tasks. You don’t need to spend a fortune on different manipulatives, but don’t feel you have to stick with the same thing ALL the time.Jean: I would like feedback about different math manipulatives? What do
: people like best? I am thinking about using the math-u-see
: program, but wondered about manipulatives such as cuisenaire rods,
: base 10 blocks, unifix, and others. Is it confusing to students to
: use different manipulatives? Are manipulatives color coded and
: would this be confusing to students used to one type of
: manipulative? If they are color coded could anyone tell me what
: color each number is to be able to compare similarities? Does
: anyone have a web address for the math-it program? Please let me
: know your opinions of this program.: Thanks!

PASSWORD>aaI221mi7wL3IVictoria,My 11 year old daughter has been doing hands on equations which teaches LD and gifted kids about algebra using manipulatives. Today her teacher approached me and said, “how do you teach subtracting negative numbers?” The way that hands on algebra does it is different and we understand what they are doing but it is confusing to teach it to my daughter. I told her to stop with the program until I got some advice. I also know my daughter needs to understand more about fractions and other pre-algebra concepts. Hands on Equations has been great for her self esteem and she has just been sailing through until we hit this glitch with negative numbers and subtracting them. Two negatives become a positive but WHY the child asks…. How do you teach the concepts of subtracting negative numbers? I memorized rules but I don’t understand the rationale behind the concept enought to explain it to a child.Thanks!

: Victoria,: My 11 year old daughter has been doing hands on equations which
: teaches LD and gifted kids about algebra using manipulatives.
: Today her teacher approached me and said, “how do you teach
: subtracting negative numbers?” The way that hands on algebra
: does it is different and we understand what they are doing but it
: is confusing to teach it to my daughter. I told her to stop with
: the program until I got some advice. I also know my daughter needs
: to understand more about fractions and other pre-algebra concepts.
: Hands on Equations has been great for her self esteem and she has
: just been sailing through until we hit this glitch with negative
: numbers and subtracting them. Two negatives become a positive but
: WHY the child asks…. How do you teach the concepts of
: subtracting negative numbers? I memorized rules but I don’t
: understand the rationale behind the concept enought to explain it
: to a child.: Thanks!Pattim,I’m a tutor currently working with a 7th grade student on pre-algebra skills like integers and rationals. The way the text explains it with manipulative is like this:Take the problem 5-(-3). You have 5 positive markers +++++ and you want to take away 3 negatives, but you don’t have any negative markers. So what you need to do is add what are called neutral pairs, (a positive and a negative marker) which don’t change the value of the original problem.+++++ (+-)(+-)(+-)Now you have 3 negative that you can take away which leaves you with 3 more positives than when you started. +++++ (+)(+)(+) or 8.Another way I’ve seen it explained is to think of a mailman bringing bills and checks to you. If he brings a check for $5 and takes away a bill for $3 then you are up by $8.I”d also like to know if anyone has some better ways of presenting the concepts.Lila

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.

One more way to present “negative of a negative is a positive” if the abstract concept of “negative means opposite” isn’t working, is with the VCR: If you’re watching a movie that is already backwards, and you rewind, it goes forward.Another more concrete presentation of the number line is an elevator in a building where the ground floor is zero, but I don’t like that because nobody numbers their floors like that (and what about that thirteenth floor?)Sue (who is *hoping* her website returns in the 24-48 hours promised, and looking at the thermometer hovering at the zero mark…)

Personally, I like to keep things as simple as possible for my kid who can get very confused over wordy or complex explanations. By all means, spend lots of time on the concept of what a negative number means. Adding negative numbers is relatively simple to visualize. Stress the concept that the subtraction operation is the same as adding a negative number 10-5 = 10 + (-5)If you think of the subtraction operation as reversing addition then you can use that operation like a kind of “opposite switch”.if a problem reads 10 - (-10) look at the opposite problem first 10 + (-10)It is easy to see using markers that the solution is 0 since the negative and positive 10 cancel each other out. In order to solve the problem you had to subtract 10 from 10.When you change the operation from addition to subtraction, you have flipped the “opposite switch”. To get the correct answer, you must do the opposite of what you did before so instead of subtracting 10 from 10 you must add the two 10s together.As you get into multiplication and division of negative numbers the process gets more and more difficult to visualize and think about logically in terms of real life situations. I have seen attempts by college proffessors to come up with real life models to show why the rules work. It is very amusing and not very helpful. The very best explanation is the simplest to my mind.Our mathematical system is invented. We could have done it another way. The only thing critical in a mathematical system is that the rules MUST be consistant. They must follow a pattern and maintain that pattern throughout the system or the system will not work. You can show the patterns by making a chart if you want to.Subtraction is the opposite of addition. A negative number is the opposite of its positive. Therefore, changing the operation *OR* the sign of a number causes you to do the opposite thing. If you change both, then you have changed nothing because you have flipped the opposite switch TWICE.Once that is understood, memorize the rules and use them.When you get to complex algebraic equations, you won’t be trying to visualize what is happening when you solve an equation. The understanding is required when writing an equation to solve a real world problem, but the actual solving is just “following the rules”. That’s why algebra was invented. To simplify a complex problem down to it’s barest elements so you can eliminate confusion and use the “rules” to get a final solution. Mathemeticians found eliquent shortcuts by manipulating equations. They call these “proofs”. These proofs created new rules to follow that allowed eqautions to be solved faster and easier.For a kid who is not going to be a mathematition, keep the explanation simple enough for them to follow. They may or may not internalize that explanation, but at least you have shown them that there is a reason for what they are doing. Then let them use the rules. Those who are not initially able to internalize the explanation *may* find a deeper understanding later after much practice applying those rules. If they don’t, at least they can solve the problems. That’s what the rules were created for in the first place.Personally I LOVE understanding what I am doing thoroughly. In fact, it drives me nuts when I don’t. I have finally come to the conclusion that some folks aren’t built to have that kind of understanding. My daughter and my husband will never “see” math like I do but that doesn’t mean they can’t “do” math. My husband is at least as good (if not better) at calculating than I am. He is very good at applying those rules he has learned. He gets along in life just fine.My poor daughter is going to school at a time when we pretend that “understanding” is THE most important thing. It doesn’t matter nearly as much whether or not you get the correct answer (try telling that to the IRS!) They are presenting complex mathematical concepts before she is ready to understand them, glossing over the “doing” and then moving onto the next topic with lightning speed. They have spiraled through math topics in this way for her entire 7 year school career. This means that not only does she not understand the math concepts, but she can’t “do the math” either. She is the type of learner that has to DO before she can UNDERSTAND. I can’t seem to get this through to anyone at school.: Victoria,: My 11 year old daughter has been doing hands on equations which
: teaches LD and gifted kids about algebra using manipulatives.
: Today her teacher approached me and said, “how do you teach
: subtracting negative numbers?” The way that hands on algebra
: does it is different and we understand what they are doing but it
: is confusing to teach it to my daughter. I told her to stop with
: the program until I got some advice. I also know my daughter needs
: to understand more about fractions and other pre-algebra concepts.
: Hands on Equations has been great for her self esteem and she has
: just been sailing through until we hit this glitch with negative
: numbers and subtracting them. Two negatives become a positive but
: WHY the child asks…. How do you teach the concepts of
: subtracting negative numbers? I memorized rules but I don’t
: understand the rationale behind the concept enought to explain it
: to a child.: Thanks!

PASSWORD>aaI221mi7wL3IWhat a bunch of incredibly smart women I can count on and consider friends! Thank you so much!! I will be reading and devouring your counsel..inbetween my neurology and child psyche classes this morning. I don’t live in the midwest so I can’t use a thermometer outside. Where I live it rarely gets below 30 degrees, I am in southern California about 2 miles from the ocean. But I am just amazed at the answers, the more I know the less I know!!

Thanks for all the information! Does anyone know of any web sites that discuss different ways to use manipulatives?

EMAILNOTICES>noHey, Victoria! How’d you like to become my math tutor? =)Kathy G.

: Hey, Victoria! How’d you like to become my math tutor? =): Kathy G.I’ll teach math to anyone if I can hold them down long enough.Seriously, this is my profession; short questions answered for free, longer help available as distance learning on a basis of you pay for photocopies, S&H, and a little extra for my time.

: Personally, I like to keep things as simple as possible for my kid
: who can get very confused over wordy or complex explanations. By
: all means, spend lots of time on the concept of what a negative
: number means. Adding negative numbers is relatively simple to
: visualize. Stress the concept that the subtraction operation is
: the same as adding a negative number 10-5 = 10 + (-5): If you think of the subtraction operation as reversing addition then
: you can use that operation like a kind of “opposite
: switch”.: if a problem reads 10 - (-10) look at the opposite problem first 10 +
: (-10): It is easy to see using markers that the solution is 0 since the
: negative and positive 10 cancel each other out. In order to solve
: the problem you had to subtract 10 from 10.: When you change the operation from addition to subtraction, you have
: flipped the “opposite switch”. To get the correct
: answer, you must do the opposite of what you did before so instead
: of subtracting 10 from 10 you must add the two 10s together.: As you get into multiplication and division of negative numbers the
: process gets more and more difficult to visualize and think about
: logically in terms of real life situations. I have seen attempts
: by college proffessors to come up with real life models to show
: why the rules work. It is very amusing and not very helpful. The
: very best explanation is the simplest to my mind.: Our mathematical system is invented. We could have done it another
: way. The only thing critical in a mathematical system is that the
: rules MUST be consistant. They must follow a pattern and maintain
: that pattern throughout the system or the system will not work.
: You can show the patterns by making a chart if you want to.: Subtraction is the opposite of addition. A negative number is the
: opposite of its positive. Therefore, changing the operation *OR*
: the sign of a number causes you to do the opposite thing. If you
: change both, then you have changed nothing because you have
: flipped the opposite switch TWICE.: Once that is understood, memorize the rules and use them.: When you get to complex algebraic equations, you won’t be trying to
: visualize what is happening when you solve an equation. The
: understanding is required when writing an equation to solve a real
: world problem, but the actual solving is just “following the
: rules”. That’s why algebra was invented. To simplify a
: complex problem down to it’s barest elements so you can eliminate
: confusion and use the “rules” to get a final solution.
: Mathemeticians found eliquent shortcuts by manipulating equations.
: They call these “proofs”. These proofs created new rules
: to follow that allowed eqautions to be solved faster and easier.: For a kid who is not going to be a mathematition, keep the
: explanation simple enough for them to follow. They may or may not
: internalize that explanation, but at least you have shown them
: that there is a reason for what they are doing. Then let them use
: the rules. Those who are not initially able to internalize the
: explanation *may* find a deeper understanding later after much
: practice applying those rules. If they don’t, at least they can
: solve the problems. That’s what the rules were created for in the
: first place.: Personally I LOVE understanding what I am doing thoroughly. In fact,
: it drives me nuts when I don’t. I have finally come to the
: conclusion that some folks aren’t built to have that kind of
: understanding. My daughter and my husband will never
: “see” math like I do but that doesn’t mean they can’t
: “do” math. My husband is at least as good (if not
: better) at calculating than I am. He is very good at applying
: those rules he has learned. He gets along in life just fine.: My poor daughter is going to school at a time when we pretend that
: “understanding” is THE most important thing. It doesn’t
: matter nearly as much whether or not you get the correct answer
: (try telling that to the IRS!) They are presenting complex
: mathematical concepts before she is ready to understand them,
: glossing over the “doing” and then moving onto the next
: topic with lightning speed. They have spiraled through math topics
: in this way for her entire 7 year school career. This means that
: not only does she not understand the math concepts, but she can’t
: “do the math” either. She is the type of learner that
: has to DO before she can UNDERSTAND. I can’t seem to get this
: through to anyone at school.Agreeing with you! Hmmm, I was trying to keep it as simple as possible — work out a simple idea of what a negative number is; then add by draw-and-look, subtract by add-the-opposite. The examples are there to make this real and meaningful, not just another page of drill added to the landfill.My personal experience as both student (up to grad school math) and teacher (K to university) is that doing and understanding absolutely have to go hand-in-hand. Doing without understanding is cookbook, and retention is absolutely zero. I got A+ in Differential Equations for two semesters and now cannot do anything beyond chapter 1; the professor was treaching cookbook and formula memorization. If this can happen to someone who likes math and is good at it, imagine how bad it is for those who struggle. On the other hand, “understanding” without doing is a heap of BS. A popular quote among math professors: Math is not a spectator sport. Take the analogy — would you say someone knows about swimming because they had read books and talked about it and watched videos, if they had never gone in the water? Sure, they can talk a good line, but they do not know the sport. Would you want that person working at your pool? Well, in math, you have to be able to DO something with it, or it’s not math. ALL math up to third year university (advanced calculus, linear algebra, differential equations, etc.) is applied!! Just talking about it is not math.In teaching, understanding/modelling and doing/applying/problem-solving have to go hand in hand at all times. You introduce a problem you might want to solve, look at things you know that apply to it, and develop new techniques that will solve it efficiently and generalize to other areas. Please read up on the Third International Math/Science Study (TIMSS) for some frightening illustrations of how far North American education has gone wrong, and how math can be better taught.The spiral curriculum is one of the worst disasters to hit the school systems of this continent. To quote one parent (inaccurately), the only thing I see spiralling is my kid’s learning and motivation, going down the drain. TIMSS blasts spiral systems, politely but thoroughly. They point out that other systems teach four or five main topics per year, maximum, and are two or more years ahead of ours in achievement; our system tries to teach everything every year and succeeds in mastering nothing.Hint to homeschooling parents and teachers: you can take the textbooks and decide which four topics are central to this year, and teach them in depth, adding supplementary work; then skim over only the high points of the other topics as either review or enrichment. For example, in Grade 5, main topics are multiplying large numbers, long division, and fractions — measurement/number line/equal/ beginning addition/fractions over 10 and 100 are decimals. You can get supplementary work, your own or other texts or reproducibles etc. and stress these topics for 80% of your time. The chapters on addition, counting, etc. are review, and can be run through quickly or tested out of. Area, beginning statistics, measurement are applications of multiplication, division and fractions and can be taught as such. Number theory is needed only in a limited way for finding common denominators. The chapters on graphing, more statistics, algebra, recognizing geometrical shapes, geometry constructions, etc. are enrichment and can be omitted in some cases or taught quickly for interest only in June. This leaves you with a teachable and learnable curriculum. If your system is worried about “covering the material”, spend June or Friday afternoons or similar traditionally waste time on the enrichment stuff, and let it count for extra credit (Your gifted kids will take off in this un-stressed situation, and your others won’t be hurt.)If most of the teachers in your school focus like this every year, your students’ achievement (both real and test scores) will shoot up. The main thing is to make sure that every topic that is needed in high school gets to be a major focus at least one year, preferably two. One of the worst disasters of the spiral curriculum is that some topics always have to be left out because there isn’t time to do everything in the world — and the topics omitted are always those that the teacher thinks are “hard”, so our students never learn fractions and never do any problem-solving — and why are our high schools having so much trouble with math?

PASSWORD>aa4um5Lp2CxdUI love your advice about focusing on a few major topics for each year and covering them thoroughly, rather than dabbling with every possible mathematical concept. With my older son, not only has this enabled him to gain a solid understanding of math, but this approach will actually allow us to cover MORE material in the elementary years than he would cover if we just went through a typical curriculum as written. Since the amount of “review” material in the text increases each year, and since he learns it the first time around, we have weeks of time available to focus on other things, like problem solving skills, and at the rate he’s going, I think we may end up almost a whole year ahead by the time he gets to 6th grade. That gives us a whole year to cover a lot of “extra” stuff before we get into pre-algebra.Jean.

Boy do I agree with everything you said. Just letting you know that I did not read your first response before I posted mine.I do agree that understanding and doing go hand in hand, but kiddos can be extremely individual in the way they process information. My own two are prime examples. They process in completely opposite ways.Kiddo number one is extremely verbal. She LOVES to talk a topic to death and look at it from a variety of angles. She picks up concepts with the speed of light. She abhors repeated practice of boring material. If she had her way, she would almost never pick up a pencil. If she is not constantly stimulated by new and interesting material that makes her think she goes into zombie mode and makes careless errors right and left. A bad grade is not enough of a consequence to make her attend to material that has been repeatedly presented to her every year of her school career. She got it the first time and doesn’t feel she should have to keep proving it! Now of course if she needs to use it in real life with a real life consequence for getting a wrong answer she is more careful.Kiddo number two is miss independant. She doesn’t want to discuss anything. She likes to jump in and do things by herself. Complex paper and pencil math topics do not come easily for her but she is very good at following a proceedure. After many years of working with her I finally figured out that all my wonderfully eloquent “explanations” and diagrams were falling on deaf ears and blind eyes. I was so used to working with kiddo number one with the verbal learning style, and my own style which is a combo of visual and verbal (heavy on the visual,) it never occurred to me that neither of these techniques would work well with kiddo number two.She simply was never ready to hear the “WHY” until she had mastered doing the wrote proceedure. She could not think about the concept and the proceedure at the same time. Teaching the “why” first just didn’t work no matter how many times or how many different ways we went over something. She is perfectly happy doing a page of meaningless calculations because she can be successful and get all the right answers. When she has memorized a process and can do it by wrote, THEN and ONLY THEN, can she handle learning how and why it works. If an explanation HAS to come first it has to be as brief as possible. She just HAS to get her hands dirty DOING something ASAP. I don’t have experience with enough students to know how unusual she might be. I just know that she IS that way and I can’t imagine she is alone on this earth.Many topics in math really lend themselves to everyday real world examples and some just don’t. Negative numbers can be easily demonstrated with real world examples up to a point. Number lines (my kids hate them), thermometers & borrowing money all are great for addition of negative numbers. They also work for subtraction but not so intuitively.I love hearing the different things people use to try to get kids to visualize multiplication and division of negative numbers. Those are the ones I find an amusing stretch for many (if not most) students. The most elegantly simple explanations that work across the board are the “opposite” concept and logical extension of patterns shown in a chart or a table. For highly visual kids you must, of course, throw in some kind of visual diagram or use manipulatives whenever practical but it gets pretty rough as soon as you go beyond + x + and - x -.For my kiddo number two, I chose to start with a method that will be simple and consistant across all operations because experience has shown me that giving her more than one way to do or understand something initially only confuses her. When she is able manipulate negative numbers in all operations I will attempt to show her more ways of visualizing why and how the rules work. The minute she starts to get confused I will know I’ve thrown too much at her before she’s had time to digest. If she can only understand one way that is fine with me, at least she will be able to work with negative numbers.With my kiddo number one I discussed negative numbers every which way from Sunday from square one. We did all four operations in one afternoon. She caught on to most explanations right away, but she was never able to follow the number line method which is my own personal favorite way of visualizing negative numbers. As soon as we finished our discussion she could solve any negative number problem you could throw at her. If she had been taught with only a number line (the way I was taught,) she might never have gotten the concept at all.I guess the real lesson here is that no one way will work for everyone. Trying to force one way on a class is a big mistake, but it can be equally bad to insist that kids learn the whole gamut of different explanations and methods. I really feel for math teachers today since this is what school systems seem to be insisting that they do. Not only that but they aren’t given an adequate amount of time to do it.My kiddo number one is bored in class and has learned to hate math because of repeating the same topics over and over. She will quit taking math as soon as she is allowed. I find that sad but at least she understands the concepts and can do the math when she really needs to. My kiddo number two does not learn math in school, she learns frustration. I have to drag her through her homework all year. She passes most tests only with lots of “coaching” from her teachers during the tests. She failed the high-stakes state test. In the summers I start from scratch and teach a topic or two using the process that works for her. Then she goes back to school and everything goes out the window again. ARGGGHHH!There HAS to be a better way.: Agreeing with you! Hmmm, I was trying to keep it as simple as
: possible — work out a simple idea of what a negative number is;

My kiddo number one was taught in school using that marker method. She had no trouble with it but it drives me nuts. I do wonder what would have happened if we hadn’t done that afternoon of negative numbers the summer before her 7th grade year though. It works, but there are so many simpler ways to represent this concept I cannot figure out why on earth they chose that one. I know I have to teach kiddo number two negative numbers BEFORE she is presented with them in school. Thank goodness kiddo number one came first so I always know what’s coming for number two!: Pattim,: I’m a tutor currently working with a 7th grade student on pre-algebra
: skills like integers and rationals. The way the text explains it
: with manipulative is like this: Take the problem 5-(-3). You have
: 5 positive markers +++++ and you want to take away 3 negatives,
: but you don’t have any negative markers. So what you need to do is
: add what are called neutral pairs, (a positive and a negative
: marker) which don’t change the value of the original problem.: +++++ (+-)(+-)(+-): Now you have 3 negative that you can take away which leaves you with
: 3 more positives than when you started. +++++ (+)(+)(+) or 8.: Another way I’ve seen it explained is to think of a mailman bringing
: bills and checks to you. If he brings a check for $5 and takes
: away a bill for $3 then you are up by $8.: I”d also like to know if anyone has some better ways of
: presenting the concepts.: Lila

I am a middle school 8th grade math teacher for students who Learn Differently (LD) and they too have difficulty with negative and positive numbers. Ironically, I also have a “regular” 8th grade homeroom and those students also have the same problem. I have explained negative numbers and the relationship of positive numbers as the ground being zero. Now, when you dig a hole each shovel full of earth is a negative number and as you put the earth on a pile next to the hole you are creating a positive number. So, when you have dug 3 shovel fulls of dirt (-3) and you dig down 3 more shovel fulls + (-3) you get -6. -3 + -3 = -6. Now lets say you have dug 5 shovel fulls (-5) and you decide to fill in your hole with three shovels of dirt from the pile of dirt next to the hole (+3) how many shovels have you dug? -5 + 3 = -2. You could even go outside with your students and dig with shovels to provide a real “hands on” experience!!! Hope this helps.Petra

: I have good stuff on multiplication and division too — when you’re
: ready, just ask. I would be interested in hearing about your multiplication and division tips.Thanks

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!

That was an excellent explanation. But, I always was told that if you had an even number of negatives, you’d have a positive result; an odd number of negatives, you’d have a negative result. Does that hold up?: 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!

The answer to your question is a definite maybe. Sorry. Not to be rude here, just trying to give accurate and meaningful answer:The rule you quote is incomplete — and this is a huge headache in teaching math; I have found that more trouble is caused by incomplete rules applied incorrectly than by not knowing any rules at all.This is why when giving examples I keep stressing negative TIMES negative is positive. You can’t say neg “and” neg — what does “and” mean? In math it either means addition, strictly and always, or else it’s just a junk word that means whatever you want it to mean, and you will get any old answer, which most likely won’t agree with anyone else’s or the answer key.”If you have an even/odd number of negatives” hmmm … what are these negatives doing or having done to them? What is the mathematical operation?If you ADD a bunch of negatives together, it doesn’t matter whether even or odd, you will get a negative result. Think of all those withdrawals on your bank card — we can’t turn them into a positive balance no matter how hard we try.If you SUBTRACT a bunch of negatives, life gets complicated — depends if your first number is negative and how big, and whether you have any parentheses in the problem. The answer has to be worked out piece by piece and can be either positive or negative. ====================================================If you MULTIPLY a bunch of negative factors, YES —multiply an odd number of negative factors together and the result is negativemultiply an even number of negative factors together and the result is positiveThese rules are easy to prove by putting parentheses around each pair of factors and using the rule neg X neg = pos=================================================== If you DIVIDE a bunch of negative factors, same problem as subtraction; depends on the sign of the dividend and on parentheses etc.If you have EXPONENTS, a positive exponent tells you how many times to multiply the same factor and a negative exponent tells you how many times to divide the same factor under a numerator of 1; even integer exponents give positive results and odd integer exponents (pos OR neg) on negative bases give negative results. This is an extension to your rule.Some things just get complicated. Four basic operations, four basic sets of rules. Then advanced operations such as exponents. Oh, well.: That was an excellent explanation. But, I always was told that if you
: had an even number of negatives, you’d have a positive result; an
: odd number of negatives, you’d have a negative result. Does that
: hold up?

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 …

Can you please give us your best invented story for visualizing division of negative numbers that will account for all the possible variations (+/+, +/-, -/=, -/-)?: ***************************************: 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!

Thank you so much for the clarification. It just came 25 years too late for me to benefit at school! I’ll save your patient explanation so that I can share with my kids if this question ever arises.The answer to your question is a definite maybe. Sorry. Not to be
: rude here, just trying to give accurate and meaningful answer: The
: rule you quote is incomplete — and this is a huge headache in
: teaching math; I have found that more trouble is caused by
: incomplete rules applied incorrectly than by not knowing any rules
: at all.: This is why when giving examples I keep stressing negative TIMES
: negative is positive. You can’t say neg “and” neg —
: what does “and” mean? In math it either means addition,
: strictly and always, or else it’s just a junk word that means
: whatever you want it to mean, and you will get any old answer,
: which most likely won’t agree with anyone else’s or the answer
: key.: “If you have an even/odd number of negatives” hmmm … what
: are these negatives doing or having done to them? What is the
: mathematical operation?: If you ADD a bunch of negatives together, it doesn’t matter whether
: even or odd, you will get a negative result. Think of all those
: withdrawals on your bank card — we can’t turn them into a
: positive balance no matter how hard we try.: If you SUBTRACT a bunch of negatives, life gets complicated —
: depends if your first number is negative and how big, and whether
: you have any parentheses in the problem. The answer has to be
: worked out piece by piece and can be either positive or negative.
: ====================================================: If you MULTIPLY a bunch of negative factors, YES —: multiply an odd number of negative factors together and the result is
: negative: multiply an even number of negative factors together and the result
: is positive: These rules are easy to prove by putting parentheses around each pair
: of factors and using the rule neg X neg = pos: =================================================== If you DIVIDE a
: bunch of negative factors, same problem as subtraction; depends on
: the sign of the dividend and on parentheses etc.: If you have EXPONENTS, a positive exponent tells you how many times
: to multiply the same factor and a negative exponent tells you how
: many times to divide the same factor under a numerator of 1; even
: integer exponents give positive results and odd integer exponents
: (pos OR neg) on negative bases give negative results. This is an
: extension to your rule.: Some things just get complicated. Four basic operations, four basic
: sets of rules. Then advanced operations such as exponents. Oh,
: well.

: Can you please give us your best invented story for visualizing
: division of negative numbers that will account for all the
: possible variations (+/+, +/-, -/=, -/-)?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.

This sounds hazily familiar. If I were going to draw this, wouldn’t I be plotting this on a grid as slope? Maybe something stuck, after all?: Well, similar to the stories for multiplication. First another
: financial one, then a physical one: time forward = pos, time back
: in the past = neg: money gained or profit = pos, money lost or spent = neg: deposit = pos, withdrawal = neg.: upward or gain in height = pos, downward or loss of height = neg: Financial —: (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.

: This sounds hazily familiar. If I were going to draw this, wouldn’t I
: be plotting this on a grid as slope? Maybe something stuck, after
: all?YES! Those were all examples of how fast something is increasing or decreasing, or RATE of increase/decrease, or *rate-of-change*.Rate of increase/decrease, rate of change, or how fast something is rising or falling, is indeed slope. Congratulations for catching it.If you did graph the questions I gave as examples — a very good plan — you would find that, reading from left to right in the normal English direction, all the positive quotients/slopes climb uphill (increasing functions) and all the negative quotients/slopes slide downhill (decreasing functions).One side note — when you talk of rate of increase/decrease or slope in this form, you are assuming that the change is *linear*, that is, that its graph is a *straight line*. I got around that in the questions by asking for *average* rate of change. If the change wasn’t steady, the graph was curved rather than dead straight, then there would be different slopes at different places. And that leads us to calculus . . (as my audience runs away screaming)

EMAILNOTICES>no: One side note — when you talk of rate of increase/decrease or slope
: in this form, you are assuming that the change is *linear*, that
: is, that its graph is a *straight line*. I got around that in the
: questions by asking for *average* rate of change. If the change
: wasn’t steady, the graph was curved rather than dead straight,
: then there would be different slopes at different places. And that
: leads us to calculus . . (as my audience runs away screaming)EEEEEEEEEEEEEEEEEEEE!!!!!! =)

My long-suffering algebra teachers Mssrs. Guzo, Davis and Tylka would be proud and thrilled to know something stuck after all! :-)So, if a chart of average rates of increases/decreases over time result in a curved graph, would it be fair to say that a stock chart that plotted average yearly results over a decade or two would be a reasonable example of curved slope?Maybe I should revisit algebra. My case may not be as hopeless as I once thought?

EMAILNOTICES>no: Maybe I should revisit algebra. My case may not be as hopeless as I
: once thought?[sigh] I am so dumb in math.Kathy G.

: [sigh] I am so dumb in math.: Kathy G.HA! You’re reading this board and writing to it — you can’t be THAT bad.You also, I presume, manage money and a household.So you got a bad experience of so-called math on silly school worksheets — tell me, when was the last time you walked into a business and heard someone saying “Oh dear, we just have to get all these worksheets done?”Throw away the built-for-failuire school attitude, and work on your strengths.

: My long-suffering algebra teachers Mssrs. Guzo, Davis and Tylka would
: be proud and thrilled to know something stuck after all! :-): So, if a chart of average rates of increases/decreases over time
: result in a curved graph, would it be fair to say that a stock
: chart that plotted average yearly results over a decade or two
: would be a reasonable example of curved slope?: Maybe I should revisit algebra. My case may not be as hopeless as I
: once thought?You’re actually getting ahead of yourself here. If you plotted stock changes over the year, that would be an example of a *derivative* function (not related to derivative investments, sorry). The derivative of a function/graph is a NEW function/graph that tells how the original one is changing. For example, if we took our old friend the parabola y = x squared written y = x^2 in typing format, then the slope (steepness) at every point is 2x. How do we get slope of a curve? Well, we get out our magnifying glass and take a very very tiny piece of the curve, small enough that it’s pretty straight, and take the slope of that bitty piece. So y = 2x is the derivative of y = x^2. And that is the beginning of calculus.See, you learned lots of math.For more ideas and hints, read Sheila Tobias “Overcoming Math Anxiety”. (Not any other book or website by same or similar names.)