Integer rules don't have to be confusing

Integer rules don't have to be confusing. When kids need help with integers, and you need help teaching integers, these techniques may be just what you need.

When I first began teaching Algebra I in eighth grade many years ago, I found that students often confused the rules of operations with integers. I was determined to develop a way to teach them so that they could remember without getting the rules for addition and subtraction confused with those of multiplication and division.

The strategies I developed during those years proved to be successful for us, and I hope you will find that they help your students, too.

RULE FOR ADDITION

The rule for adding integers states that when the signs are the same, you add (find the sum) of the "raw numbers" (absolute values) and use the same sign for your answer. If the signs are different, you find the difference, which is the answer in subtraction. and use the sign of the one which has the most (the bigger absolute value.)

After spending time on the number line illustrating the concept and working with manipulatives, I taught them the following cheer:

ADDITION OF INTEGERS CHEER

Me: I say "same," you say "sum;" I say "same," you say "sum." SAME!

Them: SUM!

Me: SAME!

Them: SUM!

Me: SAME!

Them: SUM!

Me: SAME!

Them: SUM!

Me: I say "different," you say "difference;" I say "different," you say "difference" -- DIFFERENT!

Them: DIFFERENCE!

Me: DIFFERENT!

Them: DIFFERENCE!

Me: DIFFERENT!

Them: DIFFERENCE!

Me: DIFFERENT!

Them: DIFFERENCE!





RULE FOR SUBTRACTION

Subtraction of integers is, by definition, "adding the opposite." After explaining this to students, I used storytelling as a vehicle for memory

"The long subtraction symbol is really a dangerous sheet of ice. If you aren't careful, you could fall down and break something important. You need to use a ski pole to stick down into the ice which changes what come next (you don't fall down anymore.)"

So at every "ice sheet" we used a "ski pole" to change the subtraction to addition. Then we changed the sign of the integer that came next. At that point we proceeded to use the rules for adding integers that we learned in the cheer.

Honestly, this strategy for remembering integer rules is one of the best I ever developed. It reduced the number of sign errors tremendously so give it a try and see if it works for your kids, too!



RULE FOR MULTIPLICATION AND DIVISION

The rule is the same for both operations - when the signs are different, the quotient/product is negative, and when the signs are the same it is positive.

I knew I needed to come up with something entirely different from the rules for adding and subtracting - but what? It needed to have the words "multiply" and "divide," and it needed to be something in the middle school student's

context That meant it had to be something about his or her social life! Since a cell phone looks like a negative when turned on its side, I went with this story:

"You have decided that you want your number of boyfriends/girlfriends to multiply until you have to divide your time among them. So when you go to the movies you absolutely must NOT forget your cell phone because you need to be able to text your other girlfriends/boyfriends without the one you're with knowing what you are doing."

This part of the story reminded them of the integer rule that when multiplying/dividing a negative and a positive, the answer is negative and that they must not leave the negative sign behind.

The second part of the story served as a reminder that the answer is positive when multiplying/dividing with two negative numbers. Years ago, I used a beeper and a cell phone for the story, but I have updated it now!

"If you have a cell phone and an Ipad, that is a really positive thing because you can text AND leave messages on Facebook at the same time."



These techniques work really well to help kids remember integer rules. They can also provide opportunities for integrating art into math instruction, for example, by drawing a winter picture with an ice sheet.

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LET THE FORCE BE WITH YOU






Integer Game You Can Make

Have students make game boards from posters with spaces on which to move game pieces. Kids love to create their own boards, and it is a great way to integrate art into your instruction.

Place one black and one white die with each game board along with two tokens, one for each player.


Integer game you can make.



Integer rules are easy to learn with this board game.



RULES:

Numbers on the black die represent negative numbers and numbers on the white die represent positive numbers. Players take turns rolling the dice and adding their results together using the rules for addition of integers.

Since the signs of the two numbers are always different, the "raw numbers" (absolute values) are subtracted and the sign of the answer is the same as the on which has "leftovers" - the number with the bigger absolute value.

My children never got tired of this game, and they loved playing on their own boards. It is a great way to reinforce integer rules.

If the answer is positive, the players must move their tokens forward the same number of spaces. If it is negative, the players must move their tokens backward that number of spaces.

The player whose token has advanced the farthest when time is called is the winner.



Memory Help for Absolute Value

Absolute value is distance from zero. One problem my students had was keeping the symbol after finding this value. So - you guessed it - I made up a story.

"The absolute value symbol - | | - is really a double-bladed razor. If there is a 'hair' (a negative) it shaves it off. You can't keep an old nasty, gunked up razor so you throw it away. If there is NO hair, you don't need the razor so you put it away for later."

As an additional reminder, I began putting a single hair on all my smiley faces - and I still do!