Teacher Guide

Over the same period, the cheap one's running cost has to equal three times its own price plus twice the other one's running cost. Over a year: $10 with $4.50 a month comes to $64, and $20 with $1 a month comes to $32. The good answers notice that the running cost has to outweigh a price gap and then double it, and say so in a sentence rather than only producing an example.

Printers, game consoles and pets all fit. The good answers notice who benefits: something sold cheaply so that the expensive part is sold repeatedly afterwards is a deliberate arrangement, not an accident.

Needs the running costs to differ by enough that two more years of the difference doubles the total. A $60 price with $5 a month against a $120 price with nothing is one: $120 each over one year, then $240 against $120 over three years.

It has to last less time, so you buy it more than once. That is week 7 and a student who reaches it here has read ahead in the right direction.

The ratio of cost to uses has to be the same, so the expensive one is used ten times as often. Good answers say that in a sentence about the thing rather than only about the arithmetic.

$40 over 15 uses is $2.67 rather than $1.33. Whether that changes the answer is the student's to decide, and a good response says what the number would have to be for them to walk away, which is week 27.

There is no arithmetic answer: over six years they cost the same, $180 each. Good responses name something that is not money, which is week 10 arriving early, or point out that six years of a thing you outgrow is not the same as six years of a thing you keep.

The honest rules are about the thing rather than the answer: how long it will last, how long you will want it, or how long until you outgrow it. Any rule that mentions which option you prefer is the thing this week is warning about.

They never cross. If the monthly amounts match, the price gap never closes and the cheaper one is cheaper forever, which is the only case in this book where the price tag alone is the right thing to look at.

What is right: it makes the two options comparable and it is what adults do. What is wrong: the rate is invented, small changes in it flip the answer, and it hides a choice inside what looks like a calculation.

Real cases exist: a library book, a hand-me-down, something a friend is giving away today. What is unusual is nearly always that somebody else already paid, or that it is not yours to keep. Good answers say which of those it is.

Almost any small change flips it, which is the point: keep it 25 months ($109.50 against $108.50), or make the running cost $3.60 a month ($108.40 against $108.00). The decision was within noise and was made as though it were obvious.

Two honest answers: the other three cover the shortfall equally, or the person who pays less owns less, which is week 17. Both are fair and they are not the same agreement.

1/2 and 1/3 leave 1/6. 1/2 and 1/5 leave 3/10. Students who find the second one without a common denominator may have guessed; ask them to check it.

Any multiple of 6 works, so $6, $12, $18 and so on up to $96. The common feature is exactly that: sixths have to be whole, so the amount has to divide by 6.

You pay more than an even split would have cost, and you agreed to. The good answers separate being worse off from having been treated unfairly, which are not the same thing and get confused constantly.

Both lost their share of it, so in fractions they lost equally and in dollars they did not. Getting a student to say both sentences is the week.

Nothing needs to move: each person is owed and owes the same amount. With four it is the same. Students who see the circle cancel have found something genuinely surprising.

$50 at $8 is six weeks with $2 left, so seven payments with a last one of $2. At $9 it is five weeks and $5, so six with a last one of $5. The good answers notice the last payment is smaller, not larger.

The $2 sentence should be light and specific. The $50 sentence should be specific and unembarrassed. Look for the small one written too heavily or the large one too vaguely; noticing that difference is the lesson.

Reasonable both ways. Taking it can turn a favor into a transaction; refusing it can be its own kind of rudeness. What matters is that the student notices a small sum of money has changed what the relationship is.

What happens if one of us wants to sell it, what if a third person wants to use it, and what if one of us loses it are all good answers. Any question that names a real situation the five lines miss is a good answer.

In row c nobody broke it, so 'whoever broke it pays' is unavailable and one of the other two has to do the work. A rule that fails on the most ordinary case, a thing wearing out, is not a good rule.

A person joining halfway makes the shares depend on time as well as money, which is week 16's split-by-use idea arriving in a harder form. Any reasonable treatment is a good answer.

Row b is about when you have the money rather than how much, which is a genuinely different question and is week 28.

Two people give the same options different scores and get different totals, which is fine, except that the method presents the result as though it came from the options rather than from the scorer.

Sometimes it is the right answer, for something you would never do at any price. Usually it means the decision was not made on the reasons given, and saying which of the two it is takes honesty rather than arithmetic.

A useful note names the amount, the date it stops, and what it was for. Notes that only say 'you decided this' are the ones later you resents.

How long the bought one lasts, whether it gets replaced with a newer one, and whether you could actually put the price together at all. Any of the three is a good answer.

Sixteen days over two years against $80 means owning wins on money at $10 a day. Good answers also ask whether it will still be borrowable next year, which is a risk with no price.

If the winner changes with the period, the honest period is the one that matches how long they will really have it, which is a fact about them rather than about the options.

The interesting finding can be about a kind of decision rather than a thing: bought in a hurry, bought with somebody else, bought after waiting. That is week 34 and it is worth letting them get there early.

Neither is wrong. They agree on the number and disagree about the value, which is exactly the split this week is teaching, and a student who says so has understood it.

Because the rule is about their own habits and situation, not about money. A true rule about a person does not transfer, and noticing that is more useful than any particular rule.

What they would add is the most useful thing a teacher can send back about this year, and it is worth collecting.

The missing week might be about earning, about giving, or about somebody else deciding for you. Any of those is a real gap and the student naming one has read the year as a whole.