Writing an Expression or Formula from a Context

Picture a taxi that charges three pounds the moment you close the door, then two pounds for every mile you travel. You want the price of a four-mile trip, then an eleven-mile trip, then a trip across town you have not even taken yet. You could grind out each journey separately for the rest of your life, or you could write one single line that prices every journey there will ever be. Turning an ordinary English sentence into that line is the skill here, and it is more slippery than it looks.

First job: separate the part of that fare that changes from the part that never does.

Two pounds a mile depends entirely on how far you go, so it grows as the journey grows. The three pounds is charged once and then never again, no matter what happens after that. The word every is the giveaway. Two pounds for every mile means the two is multiplied by however many miles there are, so it has to touch the letter that counts them. A charge that happens once cannot be multiplied by anything, so it stands completely alone, with no letter anywhere near it. Call the miles m and the total cost C, and the whole sentence becomes C equals three plus two m. One line, and every possible journey is priced. Both of those letters were handed to you there. When a question does not hand them over, you choose them yourself, and you write down what each one stands for. Let m be the number of miles travelled. That takes about five seconds to write, and without it a letter is just a shape on the page that only you can read.

Try the same pattern on a different sentence. A ticket costs five pounds, plus one pound fifty for every extra guest, and the number of extra guests is g. Which of these three is the total cost in pounds? Five plus one point five g, or one point five plus five g, or six point five g? Take your pick. I'll wait. The answer is five plus one point five g. The five pounds is charged once, so it stands alone, and the one pound fifty arrives with every guest, so it multiplies g. Six point five g is the tempting one, because it looks tidier than the right answer. It glues the fixed five onto the rate, and then charges that five pounds all over again for every single guest.

The version that actually stops people is the one where a question hands you two relationships instead of one.

In a bag, the number of red counters is twice the number of blue counters, b. There are also five more green counters than red counters. Write an expression, in terms of b, for the total number of counters in the bag. Three colours in that bag, and only one letter allowed in the answer. The phrase in terms of b is an instruction: when you finish, b is the only letter still standing. So here is your handle for this whole video: one letter, one line each, one expression. One letter is already decided for you, because the question named b as the number of blue counters. Blue is simply b, and that is your first line written down. Now one line each. Red is twice blue, so red is two b, and that goes on a line of its own underneath the first. Green is five more than red, and red is the word that matters in that sentence. Green is not five more than b, it is five more than two b, so green is two b plus five. That is exactly where the marks leak away. The five gets added to whatever the question measured green against, and here green was measured against red, not against blue. Three colours named, three lines written, three ticks. Blue ticked, red ticked, green ticked, and nothing in the question left sitting there unused.

One expression left to go. The total in the bag is blue plus red plus green, so those three lines of yours need adding together. Take your time. I'll wait right here. The answer is five b plus five. b plus two b plus two b makes five b, and the five that came from the green counters stands on its own. That last squeeze, adding the like terms together, is tidying rather than translating, and the video on collecting like terms drills it properly. Here it is only the final click.

And that is the finish line. Nothing in this question asks what b actually is, so there is no equals sign with a number hiding at the end. Building the expression was the entire job. When a question does give you a total and ask you to find the value, that is the video called Setting Up and Solving an Equation from a Context. Same first move, one extra half.

Now the exam-room side of this, because that sticking point is documented rather than guessed.

Here is one line from an examiner report, about a question where two given ratios had to be combined into a single relationship. Many students struggled to start this problem. There were 2 main methods to solving this problem. The first method was to use the 2 ratios to form an equation, which could then be solved. This is the method that most tried to attempt but struggled to combine the different parts. Read that last sentence again, because it is precise. Those students chose a sensible method and they found the separate parts, and what defeated them was joining those parts together. That is why every relationship gets its own written line. A line you can see is a line you can add, and a relationship you only held in your head is the one that quietly goes missing. The wording carries an instruction too. In terms of b means your final answer contains b and nothing else, however tidy r for red and g for green looked on the way there. So an answer left as blue plus red plus green has not been finished, however correct every separate line above it happens to be.

Right, let me gather the method into three instructions small enough to carry into an exam. One letter: name the unknown quantity, and write down in words what that letter actually stands for. One line each: every relationship the question gives you gets a line of its own, including the ones measured against another part rather than against your letter. One expression: push those lines together into the single thing the question asked for, and then stop, because building it was the job. And keep hold of the two signals. Each, per and every mean multiply by the letter, while a charge that happens only once stands alone.

Next in the chapter: Setting Up and Solving an Equation from a Context.

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