One of the simplest ways to help students really understand the concept of negative integers is debt. It is difficult for many students to visualize what "negative" means. Many things that are tangible do not have a clear negative. But students often understand what it means to have less than zero dollars. Explain to them that if they have $5, but then borrow $10 from a relative to buy something, they are now in debt. A simple economic simulation can be started at the bringing of math class to drive this point home. Students can get "paid" with classroom dollars for attending class, completing homework, being on task, etc. However, they can pay taxes and have to "purchase" things, such as pencils and bathroom and water privileges. They can "rent" their desk and chair. Set up your classroom economy so that most students will start the year in debt, and this will be an excellent real-life application of negative integers.
A number line is a wonderful visual for students to see that numbers that are less than zero have a negative value, and a reminder that numbers that are farther away from zero have less value, even though their digit is higher. For example, the idea that -4 is greater than -6 is a concept that takes some getting used to. A fun number line game is to make a simple printed sign for every student in the class to hold, with a positive integer, a negative integer or zero printed on it. Assign each child a different integer. Have them quickly arrange themselves into a proper number line. When they are done, have them switch numbers with a classmate and quickly try it again.
Many middle school students are familiar with "The 24 Game" and have probably played it in math class since the third grade. "The 24 Game" is a card game in which the object is to use four numbers to come up with an answer of 24. Operations including addition, subtraction, multiplication and division are used to make the four digits on the card equal 24. For example, if a card had the numbers 1, 5, 4 and 1, a student could use the numbers to state that (5 +1) x 4 x 1 = 24. An advanced version of the game for older students can include negative numbers as well.
"Integer War" is played like a regular game of "War." Students divide up a regular deck of cards, with each student turning over one card at a time in order for the two cards to go to "war" with each other. The player who throws the card with the highest value wins both cards, and two more cards are thrown until one student ends up with all the cards. When two cards with the same value are thrown at the same time, the students turn three cards face down and then a fourth card on top of those that they flip over to show its value. Whoever throws the fourth card with the highest value wins all 10 cards. In the integer variation, all red cards are negative and black cards are positive. Jokers can equal zero and face cards can equal 10 (if they are black) and -10 (if they are red).