Using a vertical number line can help provide certainty.
Adding and subtracting positive and negative numbers can be confusing for students. You can either start with a positive or a negative number and you can combine with a positive or a negative number. That makes for four types or patterns of problems. Then when you consider both addition and subtraction the total is 8 problem types. Rocket Math has three learning tracks to help students learn how to deal with integers. Mixed Integers includes all eight types, whereas Learning to Add Integers and Learning to Subtract Integers each just deal with four types. [Mixed Integers may be too hard for some or all of your students–meaning they can’t pass levels in 6 tries. In that case put them through the Learning to Add Integers and Learning to Subtract Integers first.]
Part 1: Using the vertical number line to solve integers problems
The first issue for students is just to be certain of the answer. A vertical number line, where “up” is more and “down” is less helps provide certainty.
I have posted a series of free lessons online (links below) that use a vertical number line and a consistent procedure to take the confusion out of the process. All eight types of problems can be solved with the same process on the vertical number line. Using the vertical number line there are two rules to learn. Rule 1: When you add a positive or subtract a negative you go up on the number line. Rule 2: When you subtract a positive or add a negative you go down on the number line.
So first thing to figure out is what you are being asked to do (add or subtract a positive or a negative) and then use the rule to tell you whether whether you’re going up or down. Next step in the procedure is to circle the starting point on the number line. Once you circle the starting point, you show how far you’re being asked to go. You simply make the right number of “bumps” going either up or down from where you start. That gives you the answer without any uncertainty. These online lessons are quick (about 2 minutes) and identify a pattern of whether the answer is like the sum or the difference between the numbers. Once students can recognize the pattern they can begin to answer fluently and without a struggle.
(1) Mixed Integers Set A1 Positive add a positive
(2) Mixed Integers Set A2 Positive subtract a positive
(3) Mixed Integers Set D Negative add a positive
(4) Mixed Integers Set G Negative subtract a positive
(5) Mixed Integers Set J Negative subtract a negative
(6) Mixed Integers Set M Positive subtract a negative
(7) Mixed Integers Set P Positive add a negative
(8) Mixed Integers Set S Negative add a negative
Part 2: Using the Rocket Math Integers learning track(s) to develop fluency in recognizing the type of problem
Here is a part of a page from the Mixed Integers learning track. The paired practice part of the program helps students learn to quickly and easily recognize each pattern. First students use the vertical number line to work a problem. In this example: -6 minus (-4). Then they have a set of problems with the same pattern (a negative subtracting a negative) which they should be able to orally answer without having to use the number line. Each worksheet includes all the types learned so far in the learning track.
As with all Rocket Math programs there is a 2 to 3 minute practice session (at this level I’d recommend 3 minutes), with a partner. Then the two switch roles. The practice is followed by a one-minute test. If the student can answer the problems in the test fluently (essentially without hesitations) the level is passed. As always, the students goals are individually determined by a Writing Speed Tests. If a given level is still difficult the student stays with that level a bit longer.
When a new pattern or type of problem is first introduced the one-minute tests will have a whole row of problems that are the same pattern. When that level is passed the next test will have two types of problems in each row. The next level has 3 types in a row, culminating in the fifth level where the problem types are mixed. This way the student develops fluency in recognizing the type of problem and how to derive the answer quickly. The Learning to Add Integers and Learning to Subtract Integers learning tracks take more time to learn the patterns, while Mixed Integers moves more quickly.
Don’t forget that Rocket Math has a money-back guarantee. So if this doesn’t work for you and your students we will refund your subscription price.