Subtracting a Negative - The Third Umpire

My Lovely Year 7s are starting their topic on directed numbers/integers at the moment. Over the years we have had a lot of discussions about how to get them to really understand about adding a negative and especially subtracting a negative.

With some help from my lovely husband, it seems that sport or some kind of game is the way to go. He played a version of cricket where each batsman stays in for a fixed number of overs, but gets a negative score added if they get "out". I thought it was a great idea, and had lots of potential for negative numbers - what if the third umpire overturns an out?

Here's the game we played today:

Students came up in pairs, one pair at a time, to play. They had 30 seconds to throw a bean bag between them (from behind marked lines) as many times as they could.

Scoring:
1 point for each successful catch.
-1 points for each time the bean bag hit the ground but was picked up again in less than 2 seconds.
-5 points for each time the bean bag stays on the ground for 2 seconds or more.

Also I stopped them mid-game so we could discuss the scores on the board.

I considered the penalties for dropping as negative numbers, rather than subtractions, so that we could write our number statements that way. That also allowed us to subtract a negative score if a penalty was overturned.

For example:

A pair of students catch the bean bag 22 times, then drop it. I rule that it is on the ground for more than 2 seconds.

So I write 22 + (-5) =

Here we can talk about the use of brackets, how they may or may not appear and why we might use them. We can also talk about how we would say it. Do we say "22 plus minus 5" or "22 plus negative 5" or can we say "22 plus take-away 5"? Which ones make sense? Which is the most clear?

Now they know that the pair have lost points, because they did something wrong, so 22 + (-5) = 17.

But they did appeal that it was less than 2 seconds. We took a vote. My decision was overruled.

So I write 17 - (-5) =

I'm removing the negative score. They all know already that the result should be back at 22. So we can now discuss why that is.

In summary at the end of the games we talked about adding and subtracting positive numbers, and adding and subtracting negative numbers, since we had done a bit of all of these.

After we played that version a few times, we switched to a system where the scores started at -20. This allowed us to do the same types of calculations, but in the negative numbers and crossing zero. I have a number line across the front of my room which helps a lot in these situations.