Tuesday, January 28, 2014

10: Negative Numbersssss.

Can anyone explain in a clear and concise way, in English that is understandable by middle schoolers, why a negative number times a negative number is a positive number but a negative number plus a negative number is always negative?

e.g., ( -41 )( -10 ) = +410  but  ( -41 ) + ( -10 ) = - 51 



  1. A negative number -n is really the sign (-) applied to a number n. Signs introduce a new concept to number, namely direction. (-) reverses the direction of a number* along some arbitrary ordering (credit vs debt, warmer vs colder). Multiplication is just repeated addition, and this truth survives the introduction of signs: forget the signs and multiply the absolute values, then compute the net direction of the signs involved. How do we do that? If we reverse direction twice we are going in the same direction. So two minuses leave us going in the same direction. Note that I did not say "in the plus direction": if I apply two minuses to -42, I get -42. Note that this is precisely what adept students do: see if the number of minuses in a long sequence of multiplicands is odd or even, decide the result's sign accordingly, then compute the product of the absolute values.

    * Interestingly, eulers identity e^(i*pi)=-1 translates to a rotation of 180 degrees (turning around) in the complex plane, ie turning around is equivalent to mutliplying by -1.

    As for a real world analog, if a vendor's software goes haywire and charges me (charges are -) ten (+) times for one $5 order (so I owe (-) $50), my CC company will undo (-) nine charges (charges are still -) of $5, so I am up (+) $45 from the point of the error.

  2. I don't know any physical, practical example but look at a the sequence
    (....6, 4, 2, 0, -2, -4, -6.......) and compare it to the integers
    (....-3, -2, -1, 0, 1, 2, 3.......) you can generate the first sequence from the integers using the rule -2n only if multiplying a negative by a negative yields a positive

  3. With my students I grab 3 five dollar bills (that are fake drawn on a 3x5 card). I say giving you things is positive, so I'm going to give you 3 groups of 5 dollars. The student now has 3x5 = 15 dollars. Then I'm going to take things away from you. Taking back the 3 groups of 5 dollars makes the student feel 15 dollars poorer (-3)x5 = -15. Now introduce IOU's, three 5 dollar IOU's are given to the student who now owes me 15 dollars. You say, I just gave you 3 IOU's you now owe me 15 dollars. Are you happy about that? 3x(-5) = -15 Finally, you say, "Oops, I made a mistake. You don't owe me that money. I'm going to take away those IOU's." Taking away the negative value three times is (-3)x(-5) = 15. Ask the student are you happy about me cancelling that debt? Taking away negative things is actually a good thing. Now if you want to discuss adding tweak the example and make a 41 dollar IOU and a 10 dollar IOU does giving more IOUs make the student feel richer? -41 + -10 = -51 or if the student has 10x-41 IOUs and you say I'm going to take away that debt -10x-41. Hey, they suddenly feel 410 dollars richer in knowledge.

  4. Sorry, I forgot to address the difference with negative plus negative. Adding a negative makes my result smaller, it does not change its direction. If I lose two $5 bets when my horses place second I have lost $10. If, however, the winning horses both fail their drug tests I am up $10 because I have gone from losing (-) to winning (+) two (+) times.

  5. Let the numbers represent lengths. Then multiplying two together gives the area of a rectangle. Apply this to (a - b) * (c - d) with a picture. Expand the brackets to get
    a * c - b * c - a * d + (- b) * (- d)
    From the areas in the picture it is clear that the second and third terms in the expression have subtracted the bd rectangle area twice, so it has to be put back once, and so (- b) * (- d) = b * d

  6. I have 2 great lessons involving double-sided counters and walking on number lines. I'll blog about it over the weekend and post the link.

  7. For the first part, just ask students to complete these sums and ask if there are patterns in the answers.


    No explaining needed. Many students will defend the pattern they create: "it goes down by 3 every time", "because 3x3 can't be the same as -3x3".

    For the second part, ask students which of these things are equivalent:
    2 lots of $3
    3 lots of $2
    $3 and $3
    $-3 and $-3
    2 times $3

  8. I love this 'hot air balloon' explanation from NRICH

  9. I find the real hang-up is interpreting "-" as both a way to modify a number (making the opposite) and an operation (subtraction).

    In our curriculum, we model "+" and "-" in two main ways to start: "Dollars and Debts", and the "Balloons and Sandbags" mentioned above.

    Students have no trouble seeing "3*4" as "3 + 3 + 3 +3" or "4 + 4 + 4". We have them create that product with green and red pieces of paper (dollars and debts) and find the result. ["One of the factors counts the number of groups, and the other factor tells you what is in each group." "Getting 4 groups of 3 dollars makes 12 dollars total."]

    When they have to create "-3*4", they generally see (or are convinced by classmates) that this is just the same as before, except that with near unanimity, they take "4" as the number of groups and "-3" as the contents of each group. ["Getting 4 groups of 3 debts makes 12 debts total."]

    When confronted with "-3*-4", I usually let them pattern this with the calculator first. They can create the rule by just seeing what happens. The thing I need to convince them of is why/how this fits the "dollars and debts" model.

    I have tried this approach with mixed success:

    Write down the number 4 on your page. Did you know that you just did an addition problem? You just added 4 to the nothing on the page. In other words, by writing 4, you just did the addition problem "0 + 4". Sounds silly...I agree!

    Now write the number -3 on a new line. Did you know that you just did another addition problem? You just added -3 to the nothing on that line. You just did the addition problem "0 + -3". Do you also see that you just did a subtraction problem?

    Now write "-3*4" on a blank line. We know the answer to this problem, but think of it as a problem involving zero. You have added this product to the nothing that was on the line, so you really did the addition problem "0 + -3*4". Can you convert this to a subtraction problem? [Answer: "0 - 3*4"]

    Now for the tricky part. Write "-3*-4" on a blank line. Write it as an addition problem involving zero, just like before. Then write it as a subtraction problem involving zero, just like before. Simplify your result using what you already know.
    [They would write "0 + -3*-4" for addition, then "0 - 3*-4" for subtraction", then
    "0 - -12" after simplifying, and finally "0 + 12".]

    The upshot is in the interpretation: Writing "-3" means that you are either adding 3 debts, or subtracting 3 dollars. Writing "-3*4" means that you are either adding 4 groups of 3 debts or subtracting 4 groups of 3 dollars. Writing "-3*-4" means that you are subtracting 3 groups of 4 debts, and experience would tell us that subtracting debt is a good thing...you've really added 12 dollars!

    I found that this really helped about a third of my class articulate "why" the product of two negatives is a positive, if only though the punchline. They didn't seem to want to believe the pattern the calculator was giving them, but this grounded the idea in what addition and subtraction (and positive and negative numbers) are all about.

  10. Usually I use a physical concept with a football game and a DVR. If you watch a player get sacked (pushed back lets say -10 yards), but then if I'm watching it on my DVR and I play the sack backwards, it's like he's going in the positive direction. Kids somehow get this concept... undoing a negative thing is positive.

  11. pushed back let's say, 10 yards (which would be a negative 10)..... why can't I proofread my texts/notes prior to uploading! ;o)

  12. Similar to Mrs. B's comment above... you could use a video of water getting pumped into / out of a tank.

    (+)(+) = +
    Water gets pumped in (+), we watch the video forwards (+) = We see the tank filling up (+)

    (+)(-) = -
    Water gets pumped in (+), we watch the video backwards (-) = we see the tank emptying out (-)

    (-)(+) = -
    Water gets pumped out (-), we watch the video forwards (+) = we see the tank emptying out (-)

    (-)(-) = +
    Water gets pumped out (-), we watch the video backwards (-) = we see the tank filling up (+)