Thermometers, sea level, football fields, number lines, mineshafts, tides and tidewater marks, bank accounts, credit and debt, even lost sheep--they're the negatives--and sheep that are not lost. Visually, graduated cylinders or measuring cups of water, or koolaid or fruit juice for colour.

What I found most beneficial is: 1. simplify the number of rules, 2. get integer rules all in one place (i.e. a recipe card easily tucked in between pages), and 3. practice saying the rules being used out loud or explaining which rule used was chosen or how it was used.

We synthesized the rules down to five:

1. Zero is not negative or positive

2. Addition:

* If the signs are the same, that is the sign of the

answer. Then you really add.

* If the signs are different, the sign of the biggest

number is the sign of the answer. Then you really

subtract. This is where it begins to boggle their minds!

3. Subtraction: use which ever wording works:

-Add the opposite value (7 - (-4) becomes 7 + (+4))

Then follow the adding rules.

-Change the sign of the number being subtracted and the minus sign to a plus. Then follow the adding rules

Either way, the operation sign changes to addition and the sign of the subtrahend changes.

4. When multiplying or dividing two integers, if there is

only one negative sign, the answer is negative.

Otherwise, it's positive.

Don't teach the rules by symbols (i.e - + - = -, - x - = +) Students try to memorize symbols only, and I can guarantee confusion between addition and multiplication rules. Use words.

Teach guidelines:

1. Read the question--properly! -5 - (-7) should be read as negative 5 minus (or take away) negative 7, NOT minus 5 negative minus seven or some such combination. It really helps!

2. Determine the operation and sign symbols.

3. Determine the rule that applies.

4. Determine the answer's sign.

5. Calculate.

I never found a concrete example for division. After some research, I found a mathematician who stated that it doesn't make "practical sense", but works in a mathematical context. I use the fact that division is a reverse of multiplication. If -2 x 6 = -12, then when you divide -12 by 6, the answer has to be negative because 2 x 6 gives a positive 12. Or use the underline symbol for division and cancelling. To divide -12 by -6, write -12/-6 (vertically-- not a slash / ) and you can see negative signs cancel.

For most integer questions, it's easier to determine an answer's sign first. An exception is addition and subtraction of signed fractions. Then determine common denominators before determining the answer's sign.

Practice one set of rules until it is being used comfortably before you introduce a new one; then after practicing the new rule, combine the two. Sometimes they are OK until you mix them up, and that underlines the importance of discerning the operation to be done and which rule applies first.