A great way to teach adding and subtracting integers is to start off with visual aids. Find small objects that children can easily move around to help them solve the problems. The next step is to move from 3-D objects to 2-D visualization. Draw a number line, and have the children count forward for addition and backwards for subtraction. Lastly, give the children numerical equations, and encourage them to draw visual representations to solve the problems.
An example of the objects method: for "7 plus 2," you would give the child a pile of seven M&Ms and have him or her move two M&Ms to that pile. Then, ask the child to count the total number of M&Ms.
By using a number line, you can introduce the concept of subtracting to get negative integers. For example, for "2 minus 7," have the child start at two and count backwards seven spots. Then, ask the child what number he or she has reached on the line.
While independently solving numerical equations, the child could draw shapes to visually see how many objects should be added or taken away, or the child could draw a number line. Eventually, the child should be comfortable enough to add and subtract integers confidently without visual aid.
While carrying out these methods, you should provide multiple examples and eventually decrease the amount of aid you provide the child per method. To avoid confusion for the child, you want to make sure that she can independently solve the problems using a particular method before moving on to the next step.