Teaching negative/positive integers (add/subtract)
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Teaching negative/positive integers (add/subtract)
Right now we are at a point in the curriculum where my DD needs to learn and understand how to add/subtract negative integers. The lesson in the book is rather confusing (using a football analogy) so I am looking for ways to teach her this concept. Anyone have some tips, websites, ect.?
My son started this a couple of years ago. The first thing I did was made a number line up and put in on the wall. As far as teaching it. Well I told him to ignore the + sign. All numbers are positive unless they are negative. If you are doing an addition problem and the bigger number is pos. and the smaller neg. It is essentially a subtraction problem. The knowledge of how you get this will have to eventually be understood to do more complicated problems but I found this was a good place to start. Good luck!
Here are a couple of sites that help us:
http://www.mathsisfun.com/positivenega ... egers.html
http://www.math.com/school/subject1/les ... L11GL.html
http://www.factmonster.com/ipka/A0876848.html
Here are a couple of sites that help us:
http://www.mathsisfun.com/positivenega ... egers.html
http://www.math.com/school/subject1/les ... L11GL.html
http://www.factmonster.com/ipka/A0876848.html
I was taught with credit cards.
If the shirt you want is $45, but you only have $30 on you, how much do you have to put on the credit card if you want the shirt? Or, hey, if you're getting a rebate of $30, how much is that going to take off your credit card bill of $100?
Putting it in real world perspective helped a lot (and tamed my credit impulses as an adult!)
If the shirt you want is $45, but you only have $30 on you, how much do you have to put on the credit card if you want the shirt? Or, hey, if you're getting a rebate of $30, how much is that going to take off your credit card bill of $100?
Putting it in real world perspective helped a lot (and tamed my credit impulses as an adult!)
"The greatest sign of success for a teacher... is to be able to say, "The children are now working as if I did not exist."
 M. Montessori
Proud nonmember of the HSLDA
 M. Montessori
Proud nonmember of the HSLDA
operations with integers (positive and negative numbers)
Two huge helps are to clarify what "add" and "subtract" mean. "Add" means "combine with" or "join with" or "put together with" or even "gain" (like in football yardage). "Subtract" means "remove" or "lose."
Secondly, also clarify that "positive" means "have" or "having" and "negative" means "owing". So +5 means "having 5" and 3 means "owing 3."
Using these synonyms consistently helps bring the ideas of addition and subtraction out of the automaticorobscure realm into the realm of meaningfulness.
So, for 3 + 5, focus first on the MEANING: 3 + 5 means "having 3 joined with owing 5, PERIOD." Focus on the meaning first, NOT THE ANSWER yet. Getting the meaning first will lead to the answer later! (sometimes moments later). But ALWAYS focus first on meaning. In fact, PRACTICE verbalizing the meaning of a problem like "3 + 5."
Next step: now go for simplifying the meaning. So if "I have 3 and I owe 5, then at the end of the day, I still owe 2"  and "owe 2" means "negative 2" which is written 2.
Another example 2: 3  5. The meaning is "I have 3 and I remove [or lose] owing you 5." This doth require wonderful concentration of the mind. Here's the best realworld connection for this.
Imagine your bank had charged you a $5 fee for your checking account, so they removed $5 from your balance and your balance went down by 5 to make your balance now be $3. After they did this, then they contacted you again, saying, "Oops  our mistake. That $5 fee you owed us?  remove that $5 fee." What happens to your balance?  it goes up by $5, right? Removing a $5 charge is the same as adding a positive 5, right?
So here are the math symbols for what just happened: 3  5 = 3 + 5 = 8.
Ted Pride's advice elsewhere on this thread is also good: you don't need to know WHY something works in order to know HOW to DO it  and his rule summary works, yes.
My explanation above focuses on the why; but you also should focus on the how, by practice, practice, practice. My booklet "Math Games to Supplement ANY Math Curriculum" has several math games that a student can play that involve the necessary practicepracticepractice of adding and subtracting positive and negative numbers.
Hope this helps!
Bob Hazen
Secondly, also clarify that "positive" means "have" or "having" and "negative" means "owing". So +5 means "having 5" and 3 means "owing 3."
Using these synonyms consistently helps bring the ideas of addition and subtraction out of the automaticorobscure realm into the realm of meaningfulness.
So, for 3 + 5, focus first on the MEANING: 3 + 5 means "having 3 joined with owing 5, PERIOD." Focus on the meaning first, NOT THE ANSWER yet. Getting the meaning first will lead to the answer later! (sometimes moments later). But ALWAYS focus first on meaning. In fact, PRACTICE verbalizing the meaning of a problem like "3 + 5."
Next step: now go for simplifying the meaning. So if "I have 3 and I owe 5, then at the end of the day, I still owe 2"  and "owe 2" means "negative 2" which is written 2.
Another example 2: 3  5. The meaning is "I have 3 and I remove [or lose] owing you 5." This doth require wonderful concentration of the mind. Here's the best realworld connection for this.
Imagine your bank had charged you a $5 fee for your checking account, so they removed $5 from your balance and your balance went down by 5 to make your balance now be $3. After they did this, then they contacted you again, saying, "Oops  our mistake. That $5 fee you owed us?  remove that $5 fee." What happens to your balance?  it goes up by $5, right? Removing a $5 charge is the same as adding a positive 5, right?
So here are the math symbols for what just happened: 3  5 = 3 + 5 = 8.
Ted Pride's advice elsewhere on this thread is also good: you don't need to know WHY something works in order to know HOW to DO it  and his rule summary works, yes.
My explanation above focuses on the why; but you also should focus on the how, by practice, practice, practice. My booklet "Math Games to Supplement ANY Math Curriculum" has several math games that a student can play that involve the necessary practicepracticepractice of adding and subtracting positive and negative numbers.
Hope this helps!
Bob Hazen
Last edited by Bob Hazen on Wed Dec 26, 2007 6:31 am, edited 1 time in total.

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Some good reallife MODELS for integers are:
 temperature in a thermometer
 altitude vs. sea depth
 earning money vs. being in depth.
When first teaching integer operations, tie them in with one of these models.
I'll take for example the temperature.
Assuming n is a positive integer, the simple rules governing this situation are:
* x + n means the temperature is xÂ° and RISES by n degrees.
* x âˆ’ n means the temperature is xÂ° and DROPS by n degrees.
It's all about MOVEMENT â€” moving either "up" or "down" the thermometer n degrees.
For example:
* 6 âˆ’ 7 means: temperature is first 6Â° and drops 7 degrees.
* (6) âˆ’ 7 means: temperature is first 6Â° and drops 7 degrees (it's even colder!).
* (2) + 5 means: temperature is first 2Â° and rises 5 degrees.
* 4 + 5 means: temperature is first 4Â° and rises 5 degrees.
These simple situations handle adding or subtracting a positive integer. Practice those first, until kids are familiar with these cases.
The remaining cases to handle ar adding or subtracting a negative integer:
* (2) + (5) would mean: temperature is first 2Â° and you "add" more negatives so it gets even colder.
The last case is least intuitive one:
* 1 âˆ’ (5) or subtracting a negative integer. I personally just remember the little rule of "two negatives turns into a positive".
Some people explain it this way. In (7) âˆ’ (3) you can think that you have 7 negatives at first, and you "take away" three of those negatives, leaving 4.
This rule of "two negatives makes a positive" might seem counterintuitive at first, but it is needful so that many principles of mathematics can continue to apply (for example distribuitive property).
This is continued at:
http://homeschoolmath.blogspot.com/2007 ... egers.html
See also an excellent treatise of integers vs. submarine depth at Text Savvy:
http://www.textsavvyblog.net/2006/06/ad ... egers.html
And my article:
http://www.homeschoolmath.net/teaching/integers.php
Maria Miller
 temperature in a thermometer
 altitude vs. sea depth
 earning money vs. being in depth.
When first teaching integer operations, tie them in with one of these models.
I'll take for example the temperature.
Assuming n is a positive integer, the simple rules governing this situation are:
* x + n means the temperature is xÂ° and RISES by n degrees.
* x âˆ’ n means the temperature is xÂ° and DROPS by n degrees.
It's all about MOVEMENT â€” moving either "up" or "down" the thermometer n degrees.
For example:
* 6 âˆ’ 7 means: temperature is first 6Â° and drops 7 degrees.
* (6) âˆ’ 7 means: temperature is first 6Â° and drops 7 degrees (it's even colder!).
* (2) + 5 means: temperature is first 2Â° and rises 5 degrees.
* 4 + 5 means: temperature is first 4Â° and rises 5 degrees.
These simple situations handle adding or subtracting a positive integer. Practice those first, until kids are familiar with these cases.
The remaining cases to handle ar adding or subtracting a negative integer:
* (2) + (5) would mean: temperature is first 2Â° and you "add" more negatives so it gets even colder.
The last case is least intuitive one:
* 1 âˆ’ (5) or subtracting a negative integer. I personally just remember the little rule of "two negatives turns into a positive".
Some people explain it this way. In (7) âˆ’ (3) you can think that you have 7 negatives at first, and you "take away" three of those negatives, leaving 4.
This rule of "two negatives makes a positive" might seem counterintuitive at first, but it is needful so that many principles of mathematics can continue to apply (for example distribuitive property).
This is continued at:
http://homeschoolmath.blogspot.com/2007 ... egers.html
See also an excellent treatise of integers vs. submarine depth at Text Savvy:
http://www.textsavvyblog.net/2006/06/ad ... egers.html
And my article:
http://www.homeschoolmath.net/teaching/integers.php
Maria Miller

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 Posts: 27
 Joined: Fri May 25, 2007 2:32 pm
My son is learning about this right now. I just drew a simple number line on the board with the zero, and several places in both directions. Then I would pick a place, say 4, and ask him to add 2. Then I would move 2 in the direction of the positive numbers, winding up at 2. My son is a visual learner, and this worked well for him. Good luck!
clarification on "two negatives make a positive"
On an earlier post in this thread, MariaMiller wrote:
"1 âˆ’ (5) or subtracting a negative integer. I personally just remember the little rule of 'two negatives turns into a positive'.... This rule of 'two negatives makes a positive' might seem counterintuitive at first, but it is needful so that many principles of mathematics can continue to apply (for example distribuitive property)."
Remember that the context of Maria's remark is the subtraction of a negative number.
BE CAREFUL: what she wrote here does not apply to the addition of two negative numbers. For example, 13 +  5 = 18, yes. My concern here is that I've seen kids take the rule "two negatives make a positive" and apply it (against all common sense, of course) to the above problem, resulting (incorrectly) in, "13 + 5 is two negatives, and two negatives make a positive, so 13 + 5 is +18." The problme here is that Maria's rule doesn't apply here  but kids can easily make the mistake of thinking that it does!!!
What Maria's rule of "two negatives make a positive" applies to would be:
subtracting a negative number, where you end up writing the subtract/negative sign twice in a row: 3  5 = 3 + 5 = 8.
multiplying a negative times a negative: 3 x 5 = +15.
But it doesn't apply to adding two negative numbers: 3 + 5 is not equal to +8.
I know Maria would agree with this.
Hope this helps!
Bob Hazen
"1 âˆ’ (5) or subtracting a negative integer. I personally just remember the little rule of 'two negatives turns into a positive'.... This rule of 'two negatives makes a positive' might seem counterintuitive at first, but it is needful so that many principles of mathematics can continue to apply (for example distribuitive property)."
Remember that the context of Maria's remark is the subtraction of a negative number.
BE CAREFUL: what she wrote here does not apply to the addition of two negative numbers. For example, 13 +  5 = 18, yes. My concern here is that I've seen kids take the rule "two negatives make a positive" and apply it (against all common sense, of course) to the above problem, resulting (incorrectly) in, "13 + 5 is two negatives, and two negatives make a positive, so 13 + 5 is +18." The problme here is that Maria's rule doesn't apply here  but kids can easily make the mistake of thinking that it does!!!
What Maria's rule of "two negatives make a positive" applies to would be:
subtracting a negative number, where you end up writing the subtract/negative sign twice in a row: 3  5 = 3 + 5 = 8.
multiplying a negative times a negative: 3 x 5 = +15.
But it doesn't apply to adding two negative numbers: 3 + 5 is not equal to +8.
I know Maria would agree with this.
Hope this helps!
Bob Hazen
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