First let's begin with integer addition. If the two numbers have the same sign, you just add them together and keep the sign. Say this sentence to yourself: "This is like adding five oranges to four oranges since the answer is just nine oranges." Along similar lines, 4 + 5 = 9 and -4 + -5 = -9.
If the two numbers have opposite signs, then we subtract the base numbers and keep the sign that went with the largest number. If you have an auditory learning style, then associating positive numbers with a gain and negative numbers as a loss can help you to understand this better. For example, if you had the problem 15 + -4, you can say to yourself, "A gain of fifteen with a loss of four gives how much?" For visual and kinesthetic learners, it might be easier to consider two sides having a war. If there were 15 positives fighting 4 negatives, then the positives have an advantage of 11, which is the answer.
Integer subtraction can be made into an addition problem, which skips the need to have a new set of rules to remember for subtraction. To change a subtraction problem into an addition problem, you have to change the sign of the second number. To give an example, 4 - 3 would become 4 + -3. For a more difficult example, 56 - -41 would become 56 + 41. This is an easy idea to grasp for students of all learning styles because it's so basic of an idea.
Integer multiplication and division is performed just like multiplication and division on whole numbers, except you have to count off how many negative signs are involved in the operation. If there is an even number of negatives, then the result is positive. However, if there is an odd number of negatives in the operation, then the end result is negative. For auditory learners, it's usually better to count out the number of negatives. For kinesthetic and visual learners, pairing the negatives off mentally tends to work better.