Teacher Guide

The second amount is half the first. 25 percent of $80 and 50 percent of $40 are both $20, and the good answers state the relationship rather than only producing a pair.

10 percent is $8, 5 percent is $4, half of 5 percent is $2, so $14. Checking it as 17.5 percent of $80 directly should give the same, and a student who checks has done the more valuable half.

$27 is 60 percent of the original, so the original is $45. The step most people get wrong is dividing $27 by 0.4 instead of 0.6, and it is worth asking which percentage the $27 IS.

$50.00. The wrong answer subtracts 8 percent of $54; the right one divides $54 by 1.08. Students who notice the two differ by 32 cents and want to know why have found the whole idea.

$50 up 20 percent is $60, and 20 percent off $60 is $48. It is not $50 because the second percent works on the larger number, which is week 11 arriving early and is worth saying so.

'Up to 90 percent off, on at least one item' is honest and nearly useless. 'Everything at least 15 percent off' is shorter, promises less and helps more.

Per day is almost always the smallest-sounding frame and per year the largest. Students who notice that the choice of frame is itself information have got it.

Surround it with more expensive things, or show a bigger crossed-out number, to make it feel cheap. To make it feel expensive, put it beside cheaper ones or remove the crossed-out number entirely.

At most a dollar per item, so at most $10 for ten items, and usually far less. That makes it safe for a small basket and worth doing exactly when you cannot add the real prices in your head.

The prices have to differ by more than the percents do, in the opposite direction. 10 percent of $100 against 40 percent of $10 is $10 against $4, which is more than double.

30 then 30 is 51 percent off. Three in a row is about 65.7 percent, and the good answers notice it approaches 100 without ever reaching it.

A sign is unfair when the crossed-out price was never charged to anybody. A fair version states how long the old price was charged, which is the one fact a customer would need to judge it.

-$15 is worse to owe; -$5 is the larger number. Both are true, they answer different questions, and a student who says so has understood negative numbers better than one who picks a side.

-$40, then -$28, -$16, -$4, +$8. It stops being negative in week 4, and students who say week 3 have counted the balances rather than the weeks.

A percentage fee scales, so it falls hardest on nobody: 50 percent of a $2 shortfall is $1, and of a $200 one is $100. What it costs the people charging it is the money a flat fee collected from the small shortfalls, and a good answer says so.

$10, $10, $10, $40 and $10, $20, $30, $40 have the same range and nothing else in common. The range ignores everything between the two ends, which is most of the information.

One very large value drags the mean and leaves the median. $1, $1, $1, $1, $96 has a median of $1 and a mean of $20. The reverse, a median more than double the mean, never happens with five prices: the middle price and the two above it are each at least the median, so the mean is at least three fifths of it. A very small value can pull the mean below the median, just never that far.

How many were surveyed, and what the spread was. A refusal does not prove anything, but a seller with a large sample and a narrow spread has every reason to say so, and a good answer notices that.

Same unit price and still better: it comes with something else, or it is the amount you actually need, or the smaller one will not go stale before you finish it.

It could raise the small pack's price, which makes the big one look better without changing it. That is legal, common, and worth a class discussion about whether it is dishonest.

Buy 5 get 1 free is about 17 percent off, because 1 of the 6 you take home is free. Buy 1 get 2 free is about 67 percent off, because 2 of the 3 are. The difference is the share of what you take home that is free: more than half of it free means more than 50 percent off.

'The lowest price of any shop on this street today' is checkable and much weaker as an advertisement, which is the trade the week is about.

The most useful true thing is usually a question back: what will you use it for. That is honest, it helps, and it is what a good seller does anyway.

An advertisement of only checkable claims may be worse at selling and better at being trusted. Whether that is a good trade is a real question, and it is worth letting the class disagree about it.

Rule e is a flat price however many times you go, which is the best deal for somebody who goes often and the worst for somebody who goes twice. Students who connect it to week 29 of the Grade 5 book are right to.

Two rules with the same coefficient of n never meet. 3n + 12 and 3n + 20 stay eight dollars apart forever, and the cheaper one is cheaper at every n, which is the only case where the joining fee alone decides.

6n + 5 at $50 gives 7; 7n + 5 gives 6. One fewer. A $5 rise in the joining fee gives 6n + 10 at $50, which is also 6, so at this budget the two changes cost the same and at a larger budget they do not.

A short list usually means a narrow claim. 'Free delivery' has almost nothing behind it to check, which makes it honest and small.

'This costs $2.40 for 200 g' is complete and tells a customer almost nothing about whether to buy it. Completeness and usefulness are different properties.

A friend with nothing riding on it is a fair person to ask the two judgment questions, b and d. The three fact questions, a, c and e, still go to the seller or the label, because a friend may simply not know the unit price, the return policy or the running cost.

The honest version of 'premium quality' is a list of what it is made of and how long it is guaranteed. It sells worse and it is what a customer would actually want.

Any rule based on the size of the decision, whether it can be undone, and whether it repeats. The usual failure is a rule that only looks at the price, which gets row c wrong.

The question people did not ask is often about the running cost or the return policy, and a common reason is embarrassment rather than not knowing to ask.

A refusal to answer is not always disqualifying: they may not know, or it may not be their job. The case where it is not a red flag is worth finding rather than being told.

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

Common answers are interest, insurance and anything involving a contract. All three are later years in the pathway and a student naming one has read the arc rather than the book.