# Math question (subtracting negative integers) - I can't understand what I'm doing wrong!

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We are working on adding & subtracting negative numbers. The book is very procedural on this topic, so I have been trying to make it conceptual. The book is normally not this bad.

Adding integers - the book's only instruction is this: "Move right on a number line to add a positive integer. Move left on a number line to add a negative integer." We modeled equations on number lines and  I taught why adding a negative number is really subtracting that number ... he did fine.

Subtracting integers - again, the book drops the ball. "Move left on a number line to subtract a positive integer. Move right on a number line to subtract a negative integer." The examples ask the student to "find the difference" and show a number line. It says, "Find the difference: 7-4" and then shows how you can go up to 7 and then down four units to end up at 3.

They are using "find the difference" as code for subtraction, which works for positive integers. But then they say "Find the difference: -8 - (-2)" and the number line shows going down from 0 to -8, and then up two units to -6. As long as we are given straight computation problems, I can go with this ... ds practiced expressions that involve subtracting a negative number (which is really adding that number). All is well.

The problematic issue occurs in the word problems. When they ask him to find the difference between two temperatures (-40 and -30),  I am showing him the values on a number line and how they are only 10 degrees apart. However, the book says the answer is -10. In the real world, when you are finding the difference between two numbers, aren't you essentially finding the distance between them on a number line? Or am I wrong on that?

I was drawing bar diagrams and explaining absolute value to show him that the difference between 806 and -328 is 1134.

-328 ---------------------------- 0 --------------------------------------------------------806

|.................328....................||...........................................806.........................|

|.....................................1134..........................................................................|

And the answer in the book agreed with me on that one. But the next question is: "Which of the following is the difference between 247*F below zero and 221*F above zero?" The book says the answer is -468, while I say it is 468.

To the book, the order matters. 806 - (-328) = 806 + 328 = 1134. And by that same logic, -247 - 221 = -468. The book is treating using "find the difference" to mean "start with the first number and subtract the second number." But I can't see how that relates to the actual situation, like when you are comparing temperatures.

If I owe money to the bank and my balance is -247 dollars, and then I borrow 221 more, my total balance is -\$468. This uses the equation  -247 - 221 = -468. But if I say that my balance is -247 and my friend's balance is 221, then I would say the difference between our balances is \$468. I would need to deposit \$468 to equal what my friend has.

If I say find the difference between the numbers 10 and 2, the answer is 8. Yet if I ask you to find the difference between 2 and 10, isn't the answer also 8? When you are asked to find the difference between two numbers, aren't you calculating how "far apart" they are? Is it possible to "find the difference" between two numbers and get a negative number?

If I'm wrong, be gentle with me!

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You are right.  The book is stupid  wrong.

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The problem here is confusion of the term "difference" in normal English usage and "difference" in a narrower mathematical meaning. I think the word problems are ambiguous, but when they ask for the difference between -40 and -30 they want you to take -40 and the minuend, -30 as the subtrahend, and find the (mathematical term) difference.

minuend âˆ’ subtrahend = difference

Subtraction of numbers 0â€“10. Line labels = minuend. X axis = subtrahend. Y axis = difference.

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The problem here is confusion of the term "difference" in normal English usage and "difference" in a narrower mathematical meaning. I think the word problems are ambiguous, but when they ask for the difference between -40 and -30 they want you to take -40 and the minuend, -30 as the subtrahend, and find the (mathematical term) difference.

minuend âˆ’ subtrahend = difference

Subtraction of numbers 0â€“10. Line labels = minuend. X axis = subtrahend. Y axis = difference.

Right, but that isn't really finding the difference between two values. I found the lesson in BCM that asks students to find the difference in elevation between something above sea level and something below, and BCM does it right (adds up the absolute values to arrive a total difference between the values).

I looked online for some explanations, and all I found was a bunch of procedural garbage. When you see a negative sign, move the other way on the number line! Even Lial was very formulaic. It shouldn't be that hard.

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One way I think of subtracting negatives is using a parked car example. If you are parked in a parking spot and then drive forward, you will run into the car parked in front of you. If you are parked backwards in a parking spot (we call this "escape mode" in our family) and then you drive in reverse you will still hit that same car. So parked normal and then driving forward lands you in the same spot as parked backwards and then driving in reverse. Adding a positive is the same as subtracting a negative. You land in the same spot.

One thing I did as a teacher, I would make a giant number line on the floor with masking tape. I would have the students come up and we would solve the problems by walking on the number line. The problem is 0 + 5. You start at 0 you face the positive numbers because 5 is positive and walk forward 5 notches because it is addition. You land at 5. The next problem is 0 - (-5). You start at 0, the 5 is negative so you face the negative numbers, the operation is subtraction so you walk backwards 5 notches. You still land at 5. Any integer addition and subtraction problem can be solved using these rules. One more example -3 + (-5). You start at -3. You face the bigger negative numbers since 5 is negative. You walk forward 5 notches because it is addition. You land at -8. The first number tells you where to start on the number line, the second number tells you which direction to face (positive or negative) and how many steps to take. The operation (addition or subtraction) tells you which way to walk (forward-addition or backwards-subtraction). The spot you land on is your answer.

HTH!

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One way I think of subtracting negatives is using a parked car example. If you are parked in a parking spot and then drive forward, you will run into the car parked in front of you. If you are parked backwards in a parking spot (we call this "escape mode" in our family) and then you drive in reverse you will still hit that same car. So parked normal and then driving forward lands you in the same spot as parked backwards and then driving in reverse. Adding a positive is the same as subtracting a negative. You land in the same spot.

One thing I did as a teacher, I would make a giant number line on the floor with masking tape. I would have the students come up and we would solve the problems by walking on the number line. The problem is 0 + 5. You start at 0 you face the positive numbers because 5 is positive and walk forward 5 notches because it is addition. You land at 5. The next problem is 0 - (-5). You start at 0, the 5 is negative so you face the negative numbers, the operation is subtraction so you walk backwards 5 notches. You still land at 5. Any integer addition and subtraction problem can be solved using these rules. One more example -3 + (-5). You start at -3. You face the bigger negative numbers since 5 is negative. You walk forward 5 notches because it is addition. You land at -8. The first number tells you where to start on the number line, the second number tells you which direction to face (positive or negative) and how many steps to take. The operation (addition or subtraction) tells you which way to walk (forward-addition or backwards-subtraction). The spot you land on is your answer.

HTH!

Those are great examples, but he's not having trouble with understanding adding & subtracting negative numbers. My problem is with the book's use of the term "find the difference" in real-world problems. In the real world, "finding the difference" does not automatically mean you start with the first number on the number line and then go up or down a certain number of steps. It means finding the distance between two temperatures, two elevations, or two other values.

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The book is outright wrong. You are correct. I would skip those problems in the book or use the actual correct answers.

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Right, but that isn't really finding the difference between two values. I found the lesson in BCM that asks students to find the difference in elevation between something above sea level and something below, and BCM does it right (adds up the absolute values to arrive a total difference between the values).

I looked online for some explanations, and all I found was a bunch of procedural garbage. When you see a negative sign, move the other way on the number line! Even Lial was very formulaic. It shouldn't be that hard.

As I said, I think the usage is ambiguous. I don't think either the book's way or your way are wrong, but because of ambiguity I think the word problems should be stated differently. Ordinary definitions of the word difference support your interpretation, but mathematical term definition lists I have been able to find almost exclusively support the "result of subtraction" usage. Of course, in many cases those would be the same...

http://www.barcodesinc.com/articles/mathematical-terms-dictionary.htm

Difference â€“ That which results from subtraction.

difference The number that results from subtraction.
`	3  	minuend-  	2 	subtrahend    __________        1	difference`

http://www.mathwords.com/d/difference.htm

Difference

The result of subtracting two numbers or expressions. For example, the difference between 7 and 12 is 12 â€“ 7, which equals 5.

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I would probably just explain to the student that the negative sign in front of the answer just shows which direction one is going.  So if a word problem talks of the temperature dropping from -30 degrees to -40 degrees, and asks the difference between the temperatures, although the difference between them is 10 degrees, the temperature went down, thus the answer would be "down 10 degrees" or in a short cut version, -10 degrees.  If you had the problem of -4 minus 5, the difference between the two numbers is 9, but the act of subtracting 5 from -4 is going in a negative direction, thus the answer is written -9.  (This would be easy to show on a number line, by starting at -4 and then going backwards 5.)  If it were the other way around, 5 minus -4, the difference between the two numbers is still 9, but the act of subtracting -4 from 5 is the same as adding 4 to 5, thus one is going in a positive direction and the answer would be positive.

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Perhaps the mathematically correct term for difference as you are understanding it is absolute difference?

http://en.wikipedia.org/wiki/Absolute_difference

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I would probably just explain to the student that the negative sign in front of the answer just shows which direction one is going.  So if a word problem talks of the temperature dropping from -30 degrees to -40 degrees, and asks the difference between the temperatures, although the difference between them is 10 degrees, the temperature went down, thus the answer would be "down 10 degrees" or in a short cut version, -10 degrees.  If you had the problem of -4 minus 5, the difference between the two numbers is 9, but the act of subtracting 5 from -4 is going in a negative direction, thus the answer is written -9.  (This would be easy to show on a number line, by starting at -4 and then going backwards 5.)  If it were the other way around, 5 minus -4, the difference between the two numbers is still 9, but the act of subtracting -4 from 5 is the same as adding 4 to 5, thus one is going in a positive direction and the answer would be positive.

OK, that I can see. Thanks!

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You are right.  The book is stupid  wrong.

+1.

That being said, I would teach him that even though I think the way the book phrases it is stupidly ambiguous, here is what they want so let's give them what they want. And then I'd tell him to write 'find the difference between = first - second' on his homework so he'd remember. It will not be the first time he needs to figure out the directions.

Subtracting is the adding of the negative. He gets that. Great. So what's the negative of the negative? The negative of x  really means the additive inverse of x -- that is, the number we add to x to get 0.

2 + (-2) = 0. x + (-x) = 0.

So ... what's the additive inverse of -2? Well, what do we add to (-2) to get 0? We know that 2 + (-2) = 0, and since addition is commutative, (-2) + 2 = 0. So the additive inverse of -2 (that is, - (-2)) is 2.

Another way if you've studied the distributive property, worked with equations a little, and know that (-1)(x) = -x:

What is -(-2)? We know that 2 + (-2) = 0. What happens if we multiply both sides by (-1)? Well, then we get (-1)(2) + (-1)(-2) = (-1)(0). This gives us (-2) + (-(-2)) = 0. If we add 2 to both sides of the equation, we get 2 + (-2) + (-(-2)) = 2. Since 2 + (-2) = 0, this gives us 0 + (-(-2)) = 2, and so - (-2) = 2.

ETA: Ah, I thought he was having trouble with the concept as well. I'm gonna leave the explanation since I did all the work of typing it up :D

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The difference between your bank account and your friends is -\$468, indicating that your balance is that much BELOW hers. Alternatively, the difference between your friend's account and yours is \$468, indicating that that her balance is GREATER than yours.

The difference between -40 and -30 is -10. -40 is 10 degrees BELOW -30. The sign tells you the direction of the difference.

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While I see what the other posters are saying, I still disagree. If the question was phrased:   the temperature changed, it went from -30 to  -40. What is the difference in temperature? I would answer -10 to indicate a drop in temperature. If it just asked what the difference in temperature was between -40 and -30, I would answer 10. I would understand it to mean absolute distance. Btw, on the ACT and SAT, they will expect you to give the absolute difference when phrased that way.

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I agree that the problem comes with the wording of the story problems because if you are just working with numbers - and an equation, order DOES matter for subtraction.

I also agree that you are looking for 'absolute difference' and they giving you the answer to a strict subtraction problem.
-247 - 221 = -247 + (-221) = -468 NOT +468
-40 - (-30) = -40 + (30) = -10  NOT +10

The wording could be clearer  -  but the math itself isn't the problem.

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Our Pre-Alg program uses the same language with the same results- the difference is expressed as a negative number.  I am having to learn to do it that way as I've come across it in two different curricula now.  But I too would have sought out the absolute value left to my own devices and had to go online for further explanation.

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