The following article is all about how I help students to understand directed numbers. I would like to focus on the subtraction and addition of directed numbers.
Where I teach the first time students learn about directed numbers in a math class is when they enter secondary schools. I find they already have some knowledge of how to add and subtract numbers, as they learn this when at primary school. I find this is a good starting point, as students are aware that when you add to something it means get bigger, and when you subtract it gets smaller.
The greatest hurdle I find the students encounter is when they start to add or subtract negative numbers.
The first thing I do is have students construct a basic number line of range -10 through to +10. I then try the students out to see how capable they are of subtracting and adding small positive numbers. I leave the + sign out at this point.
I ask the students about the number their subtracting, in relation to if it's a positive or negative. Students realize it is positive. I also ask them what direction they travel along the number line, starting from their original number. Before long students realize that adding means one way and subtracting is the other.
The next step is to talk to students about real life examples of when things rise and fall. Some good examples the students provide are:
1. Temperature change.
2. Stock market.
3. Football scores.
4. Water levels.
5. Bank balances.
6. Time line.
My next step is to talk about what are the different types of numbers. Before long some student will list negative numbers. I ask them about when they added and subtracted numbers and if they were able to do this to positive numbers, could they do it to negative numbers too.
I regularly refer to a number line as I find it helps the students with the direction they need to take when adding and subtracting, and also where their starting point is.
In a basic sum, for example 6 + 4, I explain that 6 is their stating point and 4 is how far they need to travel along the number line.
I then ask them if they knew what the opposite to positive is, and they generally say negative. I then go on to the part where if I add a positive number, such as six, what would happen if I added negative six. Students soon realize that it must travel in the opposite direction, thus adding a negative simply means subtract.
Students form a an understanding that + - = -.
My next step is to go back to the 6 + 4 example, and ask them 'Instead of adding four, what about subtracting four. Soon students realize that - + and + - both mean to subtract.
Hence: odd signs mean to subtract.
The final part, that I find number lines are still essential to incorporate into is - - (subtracting a negative). I ask the students about how subtracting a positive and adding a negative make subtract. therefore, what would happen if instead of subtracting a positive, I now subtract a negative. This part kids find hard to comprehend. So I make reference to the direction of travel when I changed a positive to a negative. After awhile students start to form an understanding of how like signs add and unlike signs subtract.
+ + AND - - = +
- + AND + - = -
I find it essential for students to reinforce their understanding by completing a set amount of core work.
I also find it useful to incorporate several activities designed to engage the students, so they don't look at this topic as a bunch of number crunching math work. Some of the activities I use are:
1. Tug of war.
2. Die games, there are many out there, so I wont mention any.
3. Board games. I.e. move forward 3 places or move back 2 places.
4. Stock market projects. Where they're given a set amount to invest and they track it's progress over a set period of time.
5. Money games. Where the students buy and sell stuff.
This is a basic overview of what I do, and I'm sure other mathematicians or math teachers have great ideas. So with this I welcome any other article that may help others to understand more about directed numbers.