I am teaching negative numbers to my 6th grade class. Most of them have either never worked with operations on signed numbers or have only worked with them briefly. As soon as I brought it up, many students groaned. When I inquired about the groans they told me, “Oh – I HATE those!” and “I just don’t GET them.” A few students however were happy to offer up the “rules” that they had memorized. Unfortunately, the pretest I had given showed that even those students were not computing the signed numbers correctly.

I basically had a blank slate. So, I decided to start from scratch and introduce signed numbers using red and yellow integer circles. I made a worksheet where they progressed through the following problems.

- addition of numbers
- addition of negative numbers
- SUMMARY
- subtraction of numbers
- zero pairs
- subtraction of negative numbers
- SUMMARY

I had them draw the circles on their papers and even encouraged them to use red and yellow markers. The loved it and understood it all! Until we came to subtraction of negative numbers like 4 – (-3). For some reason, this concept really gets kids. Even with the circles in front of them, they want to make this problem 4 + (-3). It happens over and over again. In my past experience it is misunderstood by so many that I honestly thought about teaching multiplication and division of signed numbers first and then just saying, “You are just distributing a -1!” But, my conscience would not let me. I wanted them to really SEE it! So, I plugged on, showing them how if you have four yellow circle (+) and you put three red circle (-) down to cancel and get 1 circle then you DON’T have 4 – ( -3), you have 4 + (-3). At this point, I heard many “lightbulb” moments throughout the room.

We also went outside and drew number lines with chalk on the sidewalk and then “walked” the problems. They actually walked out how a subtraction of a negative ends up adding. They really seemed to get it.

Before class was over I did a three question “quick check” and did not let them use the circles (but they could draw them). The questions were…

1. -3 + (-4)

2. 5 – 7

3. 5 – (-6)

Everyone in the entire class got number one right. Only two kids missed number two. But, over half of the class missed number 3! The most common incorrect answer was -11 !!

In summary, I think that the chips were almost a success as they now get everything but “subtraction” of negative numbers. Almost isn’t good enough however. And, I don’t want to just tell them the rule and have them compute it. I want them to understand it and to know it. But, I feel like they have done so much work with this (two days) and are still lost on this concept.

Maybe I am expecting too much too soon. I have not taught this grade level before. Is this such a new and foreign concept to them that it just takes a while to absorb and a lot of practice to be able to calculate this sort of problem? Will they eventually get it or must I insist on them getting it now.

UPDATE: FOR NEXT YEAR:

I don’t know about you guys, but my brains works fiendishly at night while I sleep. I wake up in the morning refreshed and with new ideas churning (especially if I am thinking of an issue (or blog about it) right before I go to bed. So, here is what I am going to try next year.

Day One – Addition and subtraction of numbers, excluding subtraction of negative numbers. I am going to use three models.

- circles
- money (thanks Cathy!)
- “walking” the numberline

I want them to be very secure in this concept and to UNDERSTAND that 3 + (-4) is the SAME thing as 3 – 4. To strengthen this I am going to draw the circles on the board and have THEM make up the equations, then have different people write their equations on the board. The kids will see how the different equations mean the same thing. I may have them make up money stories as well for this.

DAY TWO: Subtraction of Negatives

For the warm-up I will review yesterday and introduce zero pairs. Then, I am going to START with the number line (or the money), or both. I will then move to modeling with the circles. THEN I am going to have them write the equations for the ones that I model. Finally, I am going to insist that they re-write 8 – (-2) = 8 + 2, over and over again. I want to spend an entire day on this because they seem to really get everything else, but this throws them!

If you have any comments, suggestions, or great ways you conveyed this to students I would really appreciate it!

I feel your pain! I have had success as you have using tiles to represent integers. I have also used number lines but the approach that worked the best was connecting integers to money.

Basically we talk about gains and losses. Then, when we add integers we are actually combining the gains and losses to see whether they have a profit or a loss. Kids get it. Even when you use long strings like this, 4 + (-5) + 2 + 3 + (-6), they figured out to combine all their losses to one number, combine all their gains to one number and they end up with (-11) + 9. Then we say if you lose $11 and gain $9, what is the overall result? They easily say that there is a loss of $2 so the answer is -2.

Then when we move to subtraction it gets a bit trickier but we talk it through. I will try to show and explain below with some examples.

5 – 2 is a gain of $5 taking away a gain of $2 which is a profit of $3 = 3

5 – (-2) is a gain of $5 taking away a loss of $2. If someone takes away a loss (or a debt) you no longer have a loss so it is a gain so it ends up being a profit of $7 = 7

(-5) – 2 is a loss of $5 then taking away a gain of $2 so overall it is a loss of $7 = -7

(-5) – (-2) is a loss of $5 then taking away a loss of $2 which means gaining $2 so the result is an overall loss of $3 = 3

After lots of fun with this then the kids can generally figure out the ‘rules’ of subtraction. Acting it out with money really helps too. When they act it out lots of light bulbs go on!

I hope that this makes sense. Good luck with it 🙂

Cathy,

Thank you so much for this logical answer! Yesterday when going through the warm-up I first talked about money, THEN drew the integer circles. They totally got the money thing. I did not talk about money on the subtraction of a negative however and that was the only one they didn’t get. After your comment I am thinking there is definitely a correlation there! Thanks so much for the tip. I am going to Walmart to buy fake money today! lol!

I’ve done walking the number line, with negation being turning around (180 degrees), and negation of negation turning around 360. Moving their bodies around on a number line helped them understand addition and subtraction.

But this was with 1st graders at the top of their class, not 6th graders. They all got it, but I don’t know if they retained it through the next 5 years. The first graders at the bottom of the class were still working on counting and relating numbers to digits, so would not have benefited from the negative number exercises. (Luckily I was just a parent doing a pullout math circle, so did not have to figure out how to remediate, only how to enrich.)

I would teach subtraction as addition combined with negation, and work on negation of negation of negation … until the kids could spin around.

The money idea and saying you take away a loss of $2 for a problem like 5 – (-2) is brilliant! This was perfect timing too since I will teach my Jr. High kids this concept starting Monday! I have tried the counters and the number lines, but the were always vague when it came to that one concept.

There are two interpretations of subtraction: “take away” and “how many more”. It is hard to grasp the idea of taking away a negative, and examples such as canceling a debt are somewhat removed from daily experience.

I had better luck with the second interpretation. If you mark 6 and -5 on a number line, then it is easy to see that they differ by 11, just by counting the spaces between them. This suggests a practical example. On a winter day, the temperature rose from -5F to 6F. What was the change in temperature?

David,

Thanks! I definitely want to add more number line work next year. I think if I could use several models (the counters, number line, and debt problems) I can reach different kids. A student also showed me a neat thing her brother taught her. She put all of the negatives in one column, positives in another, added up the columns and then added the final number. It was an extra step but really made it easy to add problems with multiple numbers.

A teacher I work with uses Keep Change Change, or you can use Freeze Change Change. You keep the first number, change the subtraction to addition, and change the last number to the opposite. The kids often write KCC over the problem. Of course, some students still forget, and some want to apply KCC to any problem we do. Keep up the good work!

Nicole,

Thank you so much for the comments! I do not teach changing all of the signs as I saw so much confusion with it at the high school level (like you mentioned in your comment). Instead, I teach that the sign

goes with the numberand the only time we change signs is with the subtraction of a negative. The 4th and 5th grade math teacher as well as the 8th grade teacher at my school all teach it that way so we are consistent from 4th on. Hopefully it will make it easier for them but it is such a bear of a concept!