The Best Way · Teacher Guide

Teaching noteThe year opens with a method and a proof that nothing beats it on the same page, which is the shape it returns to in weeks 10, 27 and 30. The loser argument can feel slippery, so say it slowly: every card except the champion must lose at least once, and one comparison makes exactly one loser.

Teaching noteSelection sort costs the same on every hand and insertion sort does not. That difference is worth more than either number. The backwards hand ties selection at 28, and it is not a coincidence: both end up making every possible comparison.

Teaching noteMerge sort by hand is slow and fiddly and the count is the point, not the sorting. Work one layer at a time. The 64-card row is where the gap stops being arguable.

Teaching notePlay this out loud before writing. The million-name row is the moment: twenty looks. In 3b each search saves about 990 looks, so 10,000 comparisons of sorting pay for themselves after about 10 searches.

Teaching noteThe doubling table is the reference page for the rest of the year. Twenty doublings passing a million is worth pausing on.

Teaching noteGreedy is right for American coins and the page says honestly that this is luck rather than proof. Do not let a child conclude that greedy always works, because next week is built on the opposite. The nickel challenge previews it: without the 5 cent coin, greedy uses 11 coins for 68 cents and the best is 8 (a quarter, four dimes and three pennies).

Teaching noteOne counterexample demotes greedy from always to usually. Six is the smallest amount where the coins 1, 3 and 4 defeat it, and the amounts below 20 where it loses are 6, 10, 14 and 18, which is a pattern worth noticing.

Teaching noteDecimals in the edge weights are deliberate: 5.NBT.7 is the spine of Grade 5. The cheapest route uses four roads. Three different routes use only three roads, and even the cheapest of those, A to B to D to F, costs more, which is the whole point of the page.

Teaching noteTwo methods that look nothing alike, one total. Both refuse to close a loop, and that shared refusal is the reason they agree.

Teaching noteSame parcels, same method, different order, one extra box. The lower bound is what makes the sorted answer provably right rather than merely better.

Teaching noteThirteen days of work finishing in ten. It is tempting to speed up whichever job looks hardest; the page teaches them to speed up whichever job is on the chain.

Teaching noteThe sort-once-or-search-once question is the trimester in one line. On paper the break-even in 2c is about 6 searches (321 comparisons, and each search saves 64 minus 6 looks), but children should be allowed to argue about where they would switch in practice.

Teaching noteReview. Case 4 needs a lower bound and a packing, and the two agree at 3.

Teaching noteGrade 3 built these grids without coordinates. The new part is noticing that every route is seven steps and a route is just a choice of which three go up.

Teaching noteCounting the bad ones and subtracting. The multiplication of the two halves is the step most easily skipped; ask why it is not an addition.

Teaching noteFill the middle first and work outward. Every number given already includes what is inside it, which is the easiest thing to miss on this page. In question 3 the answer is 0: counted 3 times, taken away 3 times, which is exactly why the formula adds the middle back once.

Teaching noteThe words "at least one" should now trigger a reflex: count them all, count the ones with none, subtract. Watch the first digit: it has 8 choices, not 9.

Teaching noteTwo genuinely different questions wearing the same notation. A child who answers 3 divided by one fourth with one twelfth has answered the other one.

Teaching noteThe rule falls out of the picture rather than being announced. Do not let them multiply before they have shaded and counted.

Teaching noteDigits do not change when you multiply by a power of ten. Only the column they sit in changes, and saying it that way around prevents a lot of later trouble.

Teaching noteBase five is a number system, not a code. Converting into base five is the greedy rule from week 6, applied to place values, and saying so out loud helps.

Teaching noteColumn addition is identical in any base. Only the moment you carry moves. The row 444 + 1 is the one to look at twice.

Teaching noteWith nothing carried in, one column has only four cases. A column with a carry coming in adds three bits, which a machine does by using the same four-row rule twice. In 2a the third and fourth columns from the right each have a carry coming in, so say this plainly there.

Teaching noteCarry is AND and write is exclusive or. Those two gates add one column with nothing carried in, which engineers call a half adder. A full adder, for a column with a carry coming in, uses two half adders and one OR gate. The child built the truth table for the half adder the week before without knowing.

Teaching noteReview. Puzzle 5 leaves a third of a meter over, which is easy to round away.

Teaching noteSend water down one path at a time and reduce the pipes as you go. The A to B pipe carries only 2 and is the reason the total reaches 10 rather than 9.

Teaching noteThe bottleneck is in the middle, not at the ends, so widening the obvious pipes changes nothing. Flow equals the smallest cut here, as it always does.

Teaching noteGreedy strands B and the swap chain rescues it without starting over. That is the first time this year a method is repaired rather than replaced.

Teaching noteThe order defeats greedy. Two slots are enough, the graph has no triangle at all, and greedy in the wrong order uses three. The cheapest-route method from Week 8 (known as the Dijkstra method) is also greedy and never wrong, and the difference between those two facts is worth discussing. The first challenge has no order that works, by design: every club has exactly two neighbors, so greedy can never need a fourth slot.

Teaching noteA method and a lower bound, meeting. The table at the end connects this to weeks 1, 10 and 27, which is the argument that the year had a shape. In week 27 the cut is a ceiling rather than a floor, because flow is something you want more of, but the shape is the same: a method, and a reason nothing beats it.

Teaching noteEnds in a 1 or ends in a 2, with no third case. That is the whole proof of the recurrence, and it is short enough for a fifth grader to own.

Teaching noteWeek 1 used n for the size of a pile; this is the first rule built from a table of a pattern. Grade 6 is a whole year of this, so it is worth taking slowly.

Teaching noteChecking is not proving, and the false rule surviving three tests before dying on the fourth is the most useful thing on the page.

Teaching noteSix days against under three hours, for the same 1,024 cards. Growth, not speed, is what the year has really been measuring.

Teaching noteThe numbers are arranged so the answer flips. Sorting daily, merge sort pays back in 41 days and saves over three thousand dollars a year. Sorting monthly, twelve runs save about $120 and the $400 is not worth spending. Same methods, same list, different answer.

Teaching noteAssembly. A page without a cost on it is not finished, and that is the only test.