Subtracting with tokens by first putting zero pairs in

0:00Last time, tokens were for adding. Green for a press of plus, red for a press of minus, and a pair of them worth nothing at all. Now for taking away. And taking away with tokens is exactly what it sounds like. You lay the first number out on the table, and then you lift the second number off it. Here is plus five, minus plus four. Lay out plus five. Five green tokens.

0:29Now take out plus four. That means lift four green tokens off the table. One green is left. So the answer is plus one. Nothing clever happened there at all. It is worth asking what made that one so easy, because it is going to stop being easy in a moment, and it helps to know what we are about to lose. It was easy because everything the question asked for was already sitting on the table.

1:00It wanted four greens taken off the table. There were five greens sitting there. So there was nothing to arrange first, and nothing to think about. Try another one of the same kind. Minus seven, minus minus five. Lay out minus seven. Seven red tokens. Now take out minus five, which means lift five reds off. Two reds are left on the table. So minus seven, take away minus five, is minus two.

1:34Now look at what just happened, because it is the thing people find hardest about negative numbers. We started at minus seven. We took something away. And we ended up at minus two, which is higher up than where we started. Taking away made the number bigger. With a rule about signs, that is just something you have to accept. With tokens, you watched it happen. There were seven reds pulling the total down. Five of them left the table. Only two are still pulling.

2:08Take negatives out of a pile, and you take away some of what was making it negative. And here is a question worth sitting with for a moment. Minus seven, take away minus five, came to minus two. What about minus seven, plus plus five? Seven reds, and five greens arriving. Five pairs cancel. Two reds are left. That is minus two as well. Same answer. Taking five reds away, and adding five greens, did the same thing to the pile.

2:43Hold that thought. It is going to come back at the end of this video, and by then we will be able to say why it has to be true. Now the case where all of that falls over, and it does not take much to break it. Plus five, minus plus six. Lay out plus five. Five green tokens on the table. Now the instruction says take out plus six.

3:10Lift off one green, two, three, four, five. And now we are stuck. There is no sixth green to lift. That is a real problem, not a made-up one. The instruction asks for six greens and the table has five. Notice exactly what is missing, because this number is about to do all the work. One green. We are short by exactly one. So here is the move that fixes it, and it looks like cheating for about ten seconds.

3:43Put a zero pair onto the table. One green, and one red, together. Watch the value while it happens. Five, and after the pair goes down, still five. Of course it is. A green and a red together are worth nothing at all, and adding nothing to something is the one move in mathematics guaranteed to leave it exactly as it was. But look at the picture now. There are six greens on that table, and one red.

4:14It is the same amount of money, in different change. And now the instruction can be carried out. Take out plus six. Six greens, and there are six greens. Off they go. One, two, three, four, five, and the sixth one, the one that was not there a moment ago. And what is left on the table is a single red token. So plus five, minus plus six, is minus one.

4:44And that answer is the answer to the original question, not to some new one. The pair we put in was worth nothing, so the table never stopped being worth plus five. We did not change the number. We changed how it was written down. Here are twelve to run, and something is hiding in them. The first six. Plus ten minus plus seven is plus three. Minus eight minus minus four is minus four. Minus nine minus minus four is minus five.

5:20Plus nine minus plus twelve is minus three. Minus five minus minus seven is plus two. Minus two minus minus six is plus four. The second six. Minus five minus minus seven, plus two. Plus ten minus plus thirteen, minus three. Minus seven minus minus nine, plus two. Plus three minus plus eight, minus five. Minus two minus minus seven, plus five. Plus three minus plus fifteen, minus twelve. And there is the thing that was hiding. Minus five minus minus seven is in both lists. The same question, asked twice, in two different sets of instructions.

6:09Now one that has nothing at all to work with. Plus four, minus minus six. Lay out plus four. Four green tokens. Now take out minus six, which means lift six reds off the table. There are no reds on the table. Not one. We are short by all six of them. So six zero pairs go in. Six greens and six reds arrive together, and the table is still worth plus four.

6:40Now there are ten greens and six reds down. Take the six reds out. Ten greens are left. Plus four, minus minus six, is plus ten. Taking away made it bigger by six. So the whole procedure turns on one number, and you can read it straight off the question before you touch a single token. How many do you want? How many are there? The difference between those two counts is how many pairs to put in.

7:11Try it. Minus three, minus plus five. How many greens are wanted? Five. How many greens are on the table? None, because minus three is three reds. So five pairs. And once they are in, the answer falls out. Three reds, plus five pairs, is five greens and eight reds. Take the five greens out. Eight reds. Minus eight. You can put in more pairs than that if you like, and you will still get the right answer. Exactly enough is just what makes it finish in one step.

7:49Now the important habit. Do not trust it. Check it. Take plus three, minus plus eight. On tokens, that is three greens down, eight greens wanted, so five pairs in. Eight greens and five reds. Take out the eight greens. Five reds are left. Minus five. Now the same question on the line. Start at plus eight. Finish at plus three. What movement takes you there? Five places to the left. Minus five.

8:21Two completely different procedures, sharing nothing but the question. They land on the same number, and they will always land on the same number. One last look at the table, because there is a pattern in it now. Every single time, taking something out ended with putting its opposite in. To take six reds out, six greens came in with them. To take six greens out, six reds came in.

8:51And that is the whole of it. Taking a number away is the same as putting its partner down. Which is exactly the question left hanging earlier. Minus seven take away minus five, and minus seven plus plus five, had to agree, because they are the same physical act described two ways. The tokens were never a slower method. They were the reason the rule is true.