Turning a situation into an expression: terms, variables, coefficients

0:00Somebody is at a shop counter, buying boxes. Every red box holds four pens. Every blue box holds five pencils. Both kinds are sealed, so you cannot look inside and count. The question is how many pens and pencils go home in the bag. And you cannot answer it, because nobody has said how many boxes get bought. Nothing about the arithmetic is hard here. The difficulty is that one of the numbers has not turned up yet.

0:33That is not a fault in the question. It is the whole reason algebra exists. So write a letter for the thing that has not been decided yet. Let x be the number of red boxes, and y the number of blue ones. Now, x is not a number somebody has hidden from you and you are meant to find. There is nothing here to find. It is a declaration that this quantity is still free to move. Any number of boxes is allowed, and the letter stands for all of them at once.

1:09That distinction is worth holding on to, because almost every difficulty people have with algebra starts with treating a free quantity as a hidden one. And the letter does not have to settle down later, either. In this situation it never does. Now count. Each red box brings four pens with it, and there are x of them, so the pens come to four times x. We write that as four x.

1:40Not four, and then x. Four multiplied by x. Read it aloud as four pens for every red box, and there are x red boxes. Same on the other side. Five pencils a box, y boxes, five y. And notice what that four actually is. It is not a quantity of pens. It is a rate: four pens for every one box. You can test that without any algebra at all. Buy one more red box and the total goes up by four. Always by four, whatever you already had.

2:18One more thing happens at the counter. The shopkeeper hands over three loose pens, free. Those three do not multiply anything. They arrive once, and they stay. So the total is four x, plus five y, plus three. Now set both letters to zero. Buy nothing at all. What is left is three. That is what a constant term is for. It is the part of the total that survives when everything free has been switched off.

2:49Free things do not scale with what you buy, so those three cannot be folded into either of the other terms. Look at what we have built, because every piece of it has a name and a job. The plus signs cut the expression into terms. There are three of them here: four x, five y, and three. The letters are the variables. Two of those. The number in front of each letter is its coefficient. Four, and five.

3:20And the term with no letter in it at all is the constant. Try the whole thing. Two red boxes and three blue: eight, plus fifteen, plus three. Twenty six items in the bag. The names are worth having because they let you ask precise questions about an expression you have never seen before. Second situation, and a differently shaped answer comes out of it. A rectangular garden, l metres long and w metres wide.

3:53Nothing in the setup says which of those two is bigger, so read the edges by where they are rather than by how long they look. Wire fencing runs along the length at a hundred a metre. Wooden fencing runs along the width at eighty a metre. Here is where the factor of two gets lost. Count the edges. A rectangle has four of them, not two. Two edges of length l, and two of width w. You are fencing both of each pair.

4:25So the wire is two l metres at a hundred, which is two hundred l. And the wood is two w at eighty, which is a hundred and sixty w. Then the garden gets planted. Seed over the whole area, at fifty a square metre. The area is l times w, so that cost is fifty l w. And that term is a different animal from the other two. Two hundred l only listens to the length. A hundred and sixty w only listens to the width. Fifty l w listens to both of them.

5:05Watch what that does. At ten metres by four, the three costs are two thousand, six hundred and forty, and two thousand. Four thousand six hundred and forty altogether. Now double the length. Four thousand, six hundred and forty, four thousand. Eight thousand six hundred and forty, which is not double, because one of the three parts never moved. So doubling one measurement does not double the bill. It doubles two of the three parts and leaves the third exactly where it was.

5:43Third situation. A twenty centimetre wire, bent into a rectangle. Call the length x. What is the width? A very common answer is twenty minus x, and it is wrong. The twenty is the whole way round, not half of it. So do it properly. Twice the length plus twice the width is twenty. Halve everything. Length plus width is ten. Which makes the width ten minus x. That halving is the only step where twenty turns into ten, and it is exactly the step the wrong answer skips.

6:21Ten is half of twenty because the boundary of a rectangle contains each of the two measurements twice over. Now this does not describe one rectangle. It describes a whole family of them. Seven by three: seven and three make ten. Five and a half by four and a half: also ten. Every choice of x gives a different rectangle, and every single one uses exactly twenty centimetres of wire.

6:53But not every x. A side has to be a positive length. So x has to be more than zero, and ten minus x has to be more than zero as well, which puts x strictly between zero and ten. Nothing in the setup says that out loud. It comes from insisting that the shape actually exists. The area is length times width. So it is x times ten minus x.

7:22Multiply that out. x times ten is ten x. x times minus x is minus x squared. Ten x minus x squared. And there is a square, arriving without anybody asking for one. Nothing in the story mentioned a squared quantity. It came from x multiplying itself. That one change is what makes this expression a different kind of object from the first one we built. While we are here: the biggest area in that family is at x equals five. A five by five square, twenty five square centimetres. Every other choice gives you less.

8:07Three situations, three expressions. Put them side by side. The boxes: two letters, and no letter ever multiplies another one. The garden: also two letters, but one of its terms multiplies them together. The wire's area: only one letter, and that letter multiplies itself. So counting letters and reading powers are two different measurements, and they do not track each other. Two letters can give a highest power of one. One letter can give a highest power of two.

8:42Those are the two things that get confused with each other, and telling them apart is what everything after this depends on. So. A letter is a quantity that has not been pinned down. A coefficient is a rate: how much the total moves for one more of whatever the letter counts. A constant is what is left standing when every letter goes to zero. And a term is one contribution to one total.

9:13None of that is notation for its own sake. Every piece of it points at something in the story it came from. That is the habit worth carrying forward. For every symbol on the page, be able to point at what it is measuring.