Teacher Guide

The gap between 230 and 203 is 27 parts, which is 27 cents. Students who expected those two amounts to be close have found exactly the confusion week 2 is about.

There are 80 amounts strictly between $5.09 and $5.90. Students who answer 81 have counted one of the endpoints; students who answer 4 have listed four and stopped, which is what the question literally asked for and is worth accepting with a nudge.

$0.01 gets to $10.00. Any amount ending in a penny that carries the total to a whole dollar works, and the three cheapest are $0.01, $1.01 and $2.01 if the shelf allows it.

Four coins to $1.15: a dollar, a dime and a nickel is only three. With exactly four coins there are two answers: a dollar coin and three nickels, or two half dollars, a dime and a nickel. Accept either and press for the second one.

Any plan that survives a zero week has slack in it somewhere, and the slack was taken from something. The good answers name what they gave up. Answers that just spread $24 over four weeks have not noticed the question.

To the cent, the amount in cents has to divide by 4, 5 and 6, so by 60: any multiple of $0.60 works, and $0.60 itself gives 15, 12 and 10 cents. The smallest whole-dollar amount is $3.00, giving $0.75, $0.60 and $0.50. Students who look only for whole dollars each week land on $60.00, which breaks the page's under-$50 rule. Students who found it by listing multiples have done real work.

Neither set is wrong. A good set of categories covers everything the student spends on, has no overlaps that make a purchase ambiguous, and is short enough to keep. Students who reach the third of those without prompting have understood the week.

Many answers. Four different amounts totaling $12.00 with no repeats is easy; five is tighter and forces at least one amount under $2.00, which is a good moment to ask whether a $1.00 category is worth having.

Same price per one but a better buy: any pair where one comes with something else, or is the amount you actually need, or will not go stale. Students who say 'because you only wanted two' have jumped ahead to week 11, which is fine.

With $10.00, one 15-pack at $9.00 leaves $1.00 and one loose at $0.80, so 16 items and $0.20 left. Twelve loose at $0.80 is $9.60 for 12. The pack wins on count, which is the point.

The pack is $3.00 for any number up to 6. Loose beats it at 1, 2 and 3 ($0.90, $1.80, $2.70) and loses from 4 ($3.60). At 7 to 12 you need two packs, $6.00, against $6.30 to $10.80 loose. The crossover is 4 and students should say what happens at exactly 4.

A $12.00 plan with four categories and no whole dollars is easy to make and hard to make sensible. Ask why a real plan would rarely look like that.

Amounts ending in 25, 50 or 75 that total $7.50 exactly: $2.25, $2.50 and $2.75 is one; $1.75, $2.75 and $3.00 fails because $3.00 does not end in one of the three, which is a good argument to have.

Three rows over and the total under requires one row far enough under to outweigh them. Students who produce one have understood that a total is not a summary of the rows.

Five rows summing to zero with none matching: three over by $1, $2 and $3, two under by $4 and $2. Many answers; the constraint students miss is that no row may be exact.

Categories you cannot stop should be funded first and generously, because being over on them is not a decision you get to make. Students reaching that have found the idea week 28 formalizes.

Needs to divide by 3 and by 4, so by 12. $12.00 is the smallest, giving $4.00 and $3.00.

A quarter and two tenths both land on whole cents whenever the amount is a multiple of 20 cents, so $20, $25 and $30 all work. Students who test rather than assert have done the week's real work.

In tenths, with all three parts different and give the smallest, there are four sets: 7 + 2 + 1, 6 + 3 + 1, 5 + 4 + 1 and 5 + 3 + 2 tenths. If swapping spend and save counts as a different plan, there are eight. A common wrong one is 6/10 + 2/10 + 1/10, which adds to 9/10. Ask for the check every time.

$2.50 a week reaches $60 at exactly 24 weeks, which is equal, not more. It costs more than $60 from week 25. Students who answer 24 have stopped at equal; ask them to read 'more than' again.

$9.00 a month is $108.00. $2.00 a week is $104.00. The monthly one is more, by $4.00, which can surprise a student because $2.00 a week feels more frequent. Frequency is not size.

Three of the five rows come to exactly $100.00, which is worth pointing out. One pair that works: $0.50 five times a week, and $10.00 ten times in the 40 weeks. Both come to $100.00.

No right number. Answers that multiply a guessed frequency by a guessed size and divide by the weeks are doing exactly the right thing, and the arithmetic is checkable even though the guess is not.

A plan they are confident of keeping is usually one with more slack in it, and the slack came from saving. Naming that trade honestly is a better answer than a confident plan with nothing given up.

The lump sum can be spent or committed at once and has to last; the weekly one arrives whether or not you planned well and is harder to lose all at once. Both answers are good.

Average $6.00 failing three weeks out of four needs one large week: $15, $3, $3, $3 averages $6.00 and fails in three. Students who find one have understood what an average hides.

Depends on how long the job takes. $18.00 for the job beats $5.00 an hour if the job takes under 3.6 hours. Students who ask that question before answering have got it.

A tenth of $6, $4, $7, $4 is $0.60, $0.40, $0.70, $0.40, and the plan gets $5.40, $3.60, $6.30, $3.60. Plan on $3.60, the smallest, which joins this week to week 26.

$2.00 a week reaches $12.00 at week 6, drops to $2.00, then needs $26.00 more at $2.00 a week, so 13 more weeks: week 19 from the start.

Impossible. If every week is over, every actual exceeds its plan, and the totals must too. Students who say so and explain why have done something better than an example.

Neither is doing better without knowing what each of them wanted. A student who says the bars can only be judged against the plan, not against another person, has understood the whole book.

Over by $1, $2, $3, $4 is growing, so week 5 is about $5 over. Raising the amount does not fix a growing gap, it just moves when it starts. Students who notice that have found something that is easy to miss.

Options are: raise the money coming in, lower a different category, or accept snacks at $8. Doing nothing is a real option and saying so is a better answer than inventing money.

$150 at $10 a month is 15 months with nothing taken. Each $20 taken costs two more months, and over the time it takes there are two to four of them. If the $20 goes in months 6, 12 and 18, the total reaches $150 in month 21: $210 saved, $60 taken. Depending on when in the year the $20 goes, the answer runs from 19 to 23 months, so any clearly reasoned answer in that range is good; the reasoning is the marked part.

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

A yearly plan differs by more than its numbers: it cannot be held in the head, it has to survive events, and it needs review points built in. Students who name any of those three have gone past the year.