Every expression can be rewritten as a sum of terms

0:00Last time, we settled an expression by putting a bracket into it. Thirty, plus, open bracket, five times four, close bracket. Fifty. But now look at the line you actually meet most of the time. Thirty plus five times four. No brackets anywhere on it. So does a line like that just mean nothing? It cannot mean nothing. People write lines like that constantly, and they get answers out of them.

0:31There has to be a settled way to read it. There is. And here is the good part. It is not a rule you have to be told and then remember. It comes out of one move, and you can do that move to absolutely any expression. One move, and everything else follows from it. Here it is. Start with the easiest expression anybody has ever written. Twelve plus seven.

0:58That falls into two pieces. Twelve, and seven. Those pieces have a name. They are the terms of the expression. And the thing that cuts an expression into terms is the plus sign. Only the plus sign. Twelve is one term, seven is the other, and the plus is the join between them. While we are learning this, it helps to draw a loop around each term. That is a learner's scaffold, by the way, and not real notation. Nobody draws those loops on finished work.

1:33But for the next few minutes they earn their place, because this whole idea is about which bits stay together. So an expression built only out of plus signs falls straight into its terms. Which leaves the obvious problem. What about all the ones that are not? Eighty three minus fourteen. There is no plus sign in that anywhere. So we make one, and this is the single move the whole video rests on.

2:02Taking fourteen away is exactly the same as adding negative fourteen. So we rewrite it. Eighty three, plus, negative fourteen. And now there is a plus sign, so now it has terms. Eighty three, and negative fourteen. Look hard at that second term. It is negative fourteen. The minus sign came with it. That is the part people get wrong. The sign belongs to its term, and it travels with it.

2:33Why is the rewriting allowed at all? Picture six counters on a table, and four of them taken away. Now picture six counters, and four negative counters put down instead, each one cancelling a counter. You are left with the same two either way. Taking away is adding the opposite, and that is why the move is free. Now one with a multiplication in it. Six times five, plus three.

3:04How many terms is that? There are three numbers written down, so the tempting answer is three. It is two. Six times five is one single piece, and three is the other. Because the cut is the plus sign, and there is only one plus sign there. A multiplication sign does not cut anything. It holds its two numbers together. Here is another. Two, minus ten, plus four times six.

3:34Rewrite the subtraction first. Two, plus negative ten, plus four times six. Four numbers, and three terms. Two, negative ten, and four times six. And a division behaves the same way. Something divided by something is one piece as well. Let us run a few, because the counting is the actual skill here. Thirteen minus two, plus six. Rewritten, that is thirteen plus negative two plus six. Three terms. Five plus six times three.

4:11Two terms. Five, and six times three, because the multiplication holds together. Four plus fifteen minus nine. Three terms. Four, fifteen, and negative nine. Now a harder one, and it catches almost everybody. Twenty three, minus two times four, plus sixteen. The middle term is not two times four. It is negative two times four. The minus was sitting in front of the whole product, so the whole product is negative. Negative eight.

4:46Three terms, then. Twenty three, negative eight, and sixteen. Which come to thirty one. Now we can go back and answer the question we opened with. Thirty plus five times four, with nothing bracketed. Its terms are thirty, and five times four. So settle each term on its own. Thirty is already thirty. Five times four is twenty. Then add the terms up. Thirty and twenty. Fifty. Which is exactly what the bracket forced last time.

5:21So that bracket was never really needed. Reading by terms arrives at the same number without it. And there is the rule, in one sentence, and you built it rather than being handed it. Settle each term, then add the terms. One more thing, and it is the useful part. Once they are terms, you may add them in whatever order you like. Here is a longer one. Five times, bracket, three plus two, close bracket, plus seven times eight, plus three.

5:56That looks like a lot. It is three terms. Find the cuts. The plus signs that are not inside the bracket. There are two of those, so there are three terms. The first term is five times a bracket, and that bracket lives inside the term. Settle it. Three plus two is five, and five times five is twenty five. Second term, seven times eight. Fifty six. Third term is just three. Nothing to settle.

6:27Now add them. Twenty five, fifty six, and three. Eighty four. And notice that the bracket did not make a new term. It sat inside one. Where this really pays off is turning a situation into a line you can trust. Four pancakes at twenty three each, and a tip of five for the waiter. Four times twenty three, plus five. Two terms. Ninety two, and five. Ninety seven altogether.

6:58And notice the tip is a term of its own. It is not five for each pancake. Next one. Thirty three children are playing, and the caller shouts five. They scramble into groups of five, and three children are left standing. Six groups of five, and three left over. Six times five, plus three. Two terms, thirty three children. And one more. A hundred kilograms of rice, packed into two kilogram bags, on top of four bags already stacked.

7:33Four, plus a hundred divided by two. Two terms. Four, and fifty. Fifty four bags. One last situation, and this one is a puzzle rather than an exercise. You have to pay four hundred and thirty two. You may use coins worth one and worth five, and notes worth ten, twenty, fifty and a hundred. Here is one way to do it. Four hundreds, one twenty, one ten, and two ones.

8:04Written out, that is four times a hundred, plus one times twenty, plus one times ten, plus two times one. Four terms. And they add to four hundred and thirty two, which is the point. Here is another way. Eight fifties, one ten, four fives, and two ones. Also four terms, also correct, and a completely different expression. How many ways are there in total? Sixty three thousand, nine hundred.

8:34The first one uses eight pieces of money, and nothing at all does it in fewer than eight. Last thing, and it is a warning, because this does not cover everything. Two arrangements of blocks. On the left, five columns with two blocks in each, and then one column of three. That is five times two, plus three. Two terms. Thirteen blocks. On the right, two columns, and each column is five blocks sitting above three blocks.

9:07Now try writing that as a sum of terms with those same numbers. Two times five, plus three. That comes to thirteen. But count the picture and there are sixteen. Because the five and the three both get doubled, and the only way to write that is with a bracket. Two times, bracket, five plus three. So both tools are real. Most lines are a sum of terms, and some situations still need a bracket to say what is grouped.

9:39But once an expression is a sum of terms, the pieces are yours to settle and to add in any order at all, and that freedom is worth a proper look.