Teacher Guide · The Best Way, Grade 5

Week 1 Steps & Cost

What Is a Step?

Before you can say one method is better, you have to say what you are counting.

In class this week: Multi-digit multiplication and division (5.NBT.5, 5.NBT.6)

Teaching noteThe year opens with a method and a lower bound on the same page, which is the shape it returns to in weeks 10, 27 and 30. Children find the loser argument slippery: every card except the champion must lose at least once, and one comparison makes exactly one loser.

Question	Answer
1a cards left	7
1b cost	7
2a losers	7
2b losers per comparison	1
2c cannot beat	7
3 8 cards	7
3 20 cards	19
3 100 cards	99
3 n cards	n minus 1
challenge lazy way	13
challenge tournament	9

Week 2 Steps & Cost

Two Sorts, Counted

Two methods, the same cards, the same finish, and very different bills.

In class this week: Multi-digit operations; expressions (5.NBT.5, 5.OA.1)

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.

Question	Answer
1a selection run	28
1b depends on start	no
2 card 3	1
2 card 9	1
2 card 1	3
2 card 5	3
2 card 8	2
2 card 2	6
2 card 6	4
2 total	20
3 already sorted	7
3 backwards	28
3 our hand	20
3 which ties selection	exactly backwards
challenge twenty cards	190
challenge hundred cards	4950
challenge the rule	n times n minus 1, over 2

Week 3 Steps & Cost

Sorting by Splitting

Cut the pile in half, sort both halves, then zip them together.

In class this week: Problem solving with whole numbers (5.OA.1, 5.NBT.5)

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.

Question	Answer
1a merge two ones	1
1a four such merges	4
1b merge two twos	3
1b two such merges	6
1c merge two fours	7
1c total	17
3 8 selection	28
3 8 merge	17
3 16 selection	120
3 16 merge	49
3 64 selection	2016
3 64 merge	321
challenge 1024 merge	9217
challenge 1024 layers	10

Week 4 Steps & Cost

Searching a Sorted List

Halving is so much better than looking that the numbers stop feeling real.

In class this week: Place value and powers of ten (5.NBT.1, 5.NBT.2)

Teaching notePlay this out loud before writing. The million-name row is the moment: twenty looks.

Question	Answer
1 32 looks	5
1 1000 looks	10
1 1000000 looks	20
1 how much longer	only 10 more looks
3a needs to be	sorted
3b searches to pay off	1000
challenge billion	30
challenge real factor	100

Week 5 Number Structure

Doubling and Halving

The numbers behind last week. They get very big very fast.

In class this week: Powers of ten and place value (5.NBT.2)

Teaching noteThe doubling table is the reference page for the rest of the year. Twenty doublings passing a million surprises adults too.

Question	Answer
1 two to the 1	2
1 two to the 2	4
1 two to the 3	8
1 two to the 4	16
1 two to the 5	32
1 two to the 6	64
1 two to the 7	128
1 two to the 8	256
1 two to the 9	512
1 two to the 10	1024
1a just over	one thousand
1b halvings of 1024	10
2 two to the 16	65536
2 two to the 20	1048576
2 two to the 30	1073741824
2 compare with a million	just over a million
3a ten to the sixth	1000000
3b doublings past a million	20
challenge penny day	27
challenge that amount	1342177.28

Week 6 Steps & Cost

The Greedy Rule

Take the biggest that fits, then do it again. Simple, fast, and usually right.

In class this week: Decimal operations with money (5.NBT.7)

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.

Question	Answer
1 coins for 68	7
2 41 greedy	4
2 99 greedy	9
2 87 greedy	6
3 can you beat 99	no
challenge 63	6
challenge 64	7
challenge without the nickel	11

Week 7 Steps & Cost

When Greedy Fails

Change three coins and the simple rule starts giving wrong answers.

In class this week: Decimal and whole number operations (5.NBT.7, 5.OA.1)

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.

Question	Answer
1a biggest that fits	4
1a left to make	2
1b greedy total	3
1c best	2
2 6 greedy	3
2 6 best	2
2 8 greedy	2
2 8 best	2
2 9 greedy	3
2 9 best	3
2 11 greedy	3
2 11 best	3
2 12 greedy	3
2 12 best	3
challenge amounts greedy loses below 20	6, 10, 14, 18

Week 8 Dots & Lines

Shortest Path With Costs

Not the fewest roads. The cheapest total, in dollars and cents.

In class this week: Adding and subtracting decimals (5.NBT.7)

Teaching noteDecimals in the edge weights are deliberate: 5.NBT.7 is the spine of Grade 5. The cheapest route uses four roads and the fewest-roads route uses three and costs more, which is the whole point of the page.

Question	Answer
1 cheapest to A	0.00
1 cheapest to B	2.50
1 cheapest to C	1.80
1 cheapest to D	5.50
1 cheapest to E	4.40
1 cheapest to F	6.90
2a cheapest to F	6.90
2b the towns	A to C to E to D to F
2c roads on it	4
2c fewest roads exists	yes
2c what it costs	7.10

Week 9 Dots & Lines

The Cheapest Network

Two completely different methods. Same answer, every time.

In class this week: Adding decimals; problem solving (5.NBT.7, 5.OA.1)

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

Question	Answer
1 kruskal total	7.80
1 roads taken	A to C, B to C, C to E, D to F, E to D
2 prim total	7.80
2 same roads	yes
2 same total	yes
3a roads	5
3a towns	6
3b roads to sort	9
challenge with BD at fifty cents	5.70

Week 10 Steps & Cost

Packing the Boxes

Same parcels, same boxes. Change the order they arrive in and you need a whole extra box.

In class this week: Volume and measurement (5.MD.3, 5.MD.5)

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.

Question	Answer
1a boxes first fit	4
1b total size	26
1c lower bound	3
2 boxes sorted first	3
2 matches lower bound	yes
2 true best	3
challenge lower bound	3
challenge reachable	3

Week 11 Steps & Cost

Scheduling the Jobs

Six jobs, thirteen days of work, and it can all be done in ten.

In class this week: Problem solving; measurement (5.OA.1, 5.MD.1)

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

Question	Answer
1a chain days	10
1b every job added	13
2 finishes on day	10
3a which job to shorten	any job on the critical path
3b speeding up plant	nothing, it is not on the critical path
challenge frame at six	12
challenge with a roof	12

Week 12 Steps & Cost

Method Fair

One problem, four methods, one bill each.

In class this week: Problem solving (5.OA.1)

Teaching noteThe sort-once-or-search-once question is the trimester in one line. There is no single right answer and children should be allowed to argue about the switching point.

Question	Answer
selection on 64	2016
merge on 64	321
look one at a time on 64	64
binary search on 64	6
1a the fair pair	the two sorts, or the two searches
2c searches to pay off	53

Week 13 Logic & Proof

Case Files

Everything from the first twelve weeks, priced.

In class this week: Review & reasoning

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

Question	Answer
1 thirty cards	29
2 selection on twelve	66
2 insertion worst on twelve	66
3 greedy eight	2
3 best eight	2
4 lower bound	3
4 true best	3
5 finishes on day	12

Week 14 Ways of Counting

Counting on a Grid

Streets on a coordinate grid, and only two directions allowed.

In class this week: The coordinate plane (5.G.1, 5.G.2)

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.

Question	Answer
1b routes to END	35
2 routes to (2, 1)	3
2 routes to (3, 1)	4
2 routes to (2, 2)	6
2 routes to (4, 3)	35
3a steps right	4
3a steps up	3
3b every route length	7
challenge five by three	56
challenge four by four	70

Week 15 Ways of Counting

Paths That Avoid a Block

Count everything, then subtract the ones you did not want.

In class this week: The coordinate plane; problem solving (5.G.1, 5.OA.1)

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

Question	Answer
1a routes to the block	3
1b routes on from it	6
1c bad routes	18
2a all routes	35
2b surviving	17
3 the slow way agrees	yes
challenge block one one	15
challenge block two corners	7

Week 16 Ways of Counting

Two Circles, Then Three

Add the three clubs and you have counted some children twice and some three times.

In class this week: Problem solving with whole numbers (5.OA.1)

Teaching noteFill the middle first and work outward. Every number given already includes what is inside it, which is the source of nearly every error on this page.

Question	Answer
1 region ABC	3
1 region AB	4
1 region AC	5
1 region BC	2
1 region A	6
1 region B	3
1 region C	4
2a seven regions add to	27
2b the short way	27
2c same answer	yes
2c no club	3
3 counted in the total	1
challenge neither of two	9
challenge nothing of three	2

Week 17 Ways of Counting

Counting the Leftovers

Sometimes the fastest way to count what you want is to count what you do not.

In class this week: Place value and multi-digit numbers (5.NBT.1, 5.OA.1)

Teaching noteThe words <b>at least one</b> should now trigger a reflex: count them all, count the ones with none, subtract. The first digit having 8 choices rather than 9 is the usual slip.

Question	Answer
1a three digit numbers	900
1b first digit choices	8
1b second digit choices	9
1b third digit choices	9
2a no seven	648
2b at least one seven	252
3 at least one five	252
3 four digit at least one zero	2439
challenge at least two sevens	27
challenge no digit over five	1080

Week 18 Fractions

Sharing Fairly

Dividing by a fraction is a counting question in disguise.

In class this week: Dividing with unit fractions (5.NF.7)

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.

Question	Answer
1a pieces from one	4
1b from three	12
1c as a division	12
2a shared between three	one twelfth
3 four by a half	8
3 two by a sixth	12
3 a third by two	1/6
3 a half by four	1/8
challenge rope pieces	12

Week 19 Fractions

A Fraction of a Fraction

Cut a square one way, then the other. The overlap is the answer.

In class this week: Multiplying fractions as area (5.NF.4, 5.NF.6)

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

Question	Answer
1a shaded	6
1b in all	12
1c product	6 over 12
1c simplified	1/2
2a shaded across	3
2a shaded down	2
2b grid across	4
2b grid down	3
3 1 over 2 of 2 over 5 shaded	2
3 1 over 2 of 2 over 5 in all	10
3 1 over 2 of 2 over 5 answer	1/5
3 3 over 4 of 2 over 3 shaded	6
3 3 over 4 of 2 over 3 in all	12
3 3 over 4 of 2 over 3 answer	1/2
3 2 over 5 of 3 over 4 shaded	6
3 2 over 5 of 3 over 4 in all	20
3 2 over 5 of 3 over 4 answer	3/10
challenge four fifths of five eighths	1/2
challenge half of half of half	1/8

Week 20 Number Structure

Powers and Place Value

Every column is ten times the one on its right, and that is a rule you can push around.

In class this week: Powers of ten and place value (5.NBT.2)

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 round prevents a lot of later trouble.

Question	Answer
1 2.5 times ten cubed	2500.0
1 340 by ten squared	3.4
1 0.06 times ten to the fourth	600.0
1 9100 by ten cubed	9.1
2a columns moved	3
2a direction	left
2b columns moved	2
2b direction	right
challenge which is bigger	they are equal
challenge 47000	4.7 times ten to the fourth

Week 21 Number Structure

Base Five

A whole number system with only five digits. Not a code, a way of counting.

In class this week: Place value and powers (5.NBT.1, 5.NBT.2)

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.

Question	Answer
1a first column	125
1a second column	25
1a third column	5
1a fourth column	1
1b in base ten	266
2 2031 base five	266
2 400 base five	100
2 1234 base five	194
2 63 in base five	223
2 100 in base five	400
3 counting to twelve	10, 11, 12, 13, 14, 20, 21, 22
challenge 200 in base five	1300
challenge 500 in base five	4000
challenge which base gives 100 as 25	5

Week 22 Number Structure

Carrying in Base Five

Column addition works in any base. Only the moment you carry changes.

In class this week: Adding within a place value system (5.NBT.7, 5.NBT.1)

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.

Question	Answer
1a three plus four	7
1a is it five or more	yes
1b write	2
1b carry	1
1c fives column	4
1c answer	42
2a 23 in base ten	13
2a 14 in base ten	9
2b added	22
2b converted back	22
3 12 plus 13	30
3 12 plus 13 in base ten	15
3 34 plus 22	111
3 34 plus 22 in base ten	31
3 444 plus 1	1000
3 444 plus 1 in base ten	125
challenge 1234 plus 4321	11110

Week 23 Steps & Cost

How a Machine Adds

Two digits, one carry, and that is the entire arithmetic of a computer.

In class this week: Place value in another base (5.NBT.1)

Teaching noteFour cases is the entire arithmetic of a machine. That is a genuinely startling fact and worth pausing on.

Question	Answer
1 0 and 0 write	0
1 0 and 0 carry	0
1 0 and 1 write	1
1 0 and 1 carry	0
1 1 and 0 write	1
1 1 and 0 carry	0
1 1 and 1 write	0
1 1 and 1 carry	1
2a sum in base two	10001
2b bits in the answer	5
3a 1011 in base ten	11
3a 0110 in base ten	6
3b added	17
challenge 1111 plus 0001	10000
challenge bits for 100	7
challenge bits for 1000	10

Week 24 Steps & Cost

Logic Gates

Three tiny machines. Wire them together and you get last week’s adding.

In class this week: Reasoning and patterns (5.OA.1, 5.OA.3)

Teaching noteCarry is AND and write is exclusive or. Two gates make an adder, and the child has already built the truth table for it the week before without knowing.

Question	Answer
1a AND ones	1
1b OR ones	3
2a carry gate	AND
2b the row where OR differs	1 and 1
3 NAND outputs	1, 1, 1, 0
challenge both zero gate	OR then NOT, which is NOR

Week 25 Logic & Proof

Puzzle Fair

A little of everything from Weeks 14 to 24.

In class this week: Review & reasoning

Teaching noteReview. Puzzle 5 leaves a third of a meter over, which several children will round away.

Question	Answer
1 blocked five by two	9
2 none of three	6
3 no nine at all	648
4 302 base five in ten	77
4 88 in base five	323
5 whole pieces	7
5 left over	1/3
6 NOR outputs	1, 0, 0, 0

Week 26 Dots & Lines

Flow Through a Network

Water, not routes. How much can you push from one end to the other?

In class this week: Problem solving; volume (5.OA.1, 5.MD.5)

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.

Question	Answer
1 S to A to T	5
1 S to B to T	4
1 S to A to B to T	1
1 total sent	10
2a leaving S	10
2b arriving at T	12
2c which matched	the pipes leaving S
challenge S to B at eight	12
challenge A to T at nine	10

Week 27 Dots & Lines

The Bottleneck

The answer is never about the big pipes. It is about the narrow place.

In class this week: Problem solving (5.OA.1)

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.

Question	Answer
1 cut at S	18
1 cut after A and B	5
1 cut after C	9
1 smallest cut	5
2a flow reached	5
2b compare	they are equal
3a can never be more than	the smallest cut
3b the bottleneck pipes	A to C and B to C
challenge A to C at six	9
challenge new cut	9

Week 28 Dots & Lines

Matching People to Jobs

Greedy gets stuck. Then you fix it without starting over.

In class this week: Problem solving (5.OA.1)

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.

Question	Answer
1 greedy matched	3
1 left out	B
1 greedy gives A	1
1 greedy gives B	nothing
1 greedy gives C	2
1 greedy gives D	3
2a who has job one	A
2c after the swap	4
3 A ends with	2
3 B ends with	1
3 C ends with	3
3 D ends with	4
challenge without C to three	3

Week 29 Dots & Lines

Coloring, and How Good Is Greedy?

Six clubs, two slots are enough, and greedy uses three. The order is to blame.

In class this week: Problem solving (5.OA.1)

Teaching noteThe order defeats greedy. Two slots are enough, the graph has no triangle at all, and greedy in the wrong order uses three. Dijkstra is also greedy and never wrong, and the difference between those two facts is worth discussing.

Question	Answer
1 bad order Art	1
1 bad order Drama	1
1 bad order Band	2
1 bad order Eco	2
1 bad order Chess	3
1 bad order Film	3
1 slots in the bad order	3
2 good order Art	1
2 good order Band	1
2 good order Chess	1
2 good order Drama	2
2 good order Eco	2
2 good order Film	2
2 slots in the good order	2
3a cannot be fewer than	2
3b true answer	2
3b triangle in the graph	no

Week 30 Logic & Proof

A Lower Bound

Two halves make an answer: a method that works, and a proof nothing beats it.

In class this week: Problem solving and reasoning (5.OA.1)

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.

Question	Answer
1a three that clash	P, Q, R
1b lower bound	3
2 slots used	3
2 matches	yes
3 week 1 lower bound	every loser needs a comparison
3 week 10 lower bound	total size divided by box size, rounded up
3 week 27 lower bound	the smallest cut
challenge sixth room	4

Week 31 Ways of Counting

Rules That Look Back

To count a strip of ten, count a strip of nine and a strip of eight.

In class this week: Patterns and rules (5.OA.3)

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.

Question	Answer
1 strip of 1	1
1 strip of 2	2
1 strip of 3	3
1 strip of 4	5
1 strip of 5	8
1 strip of 6	13
1 each is	the two before it added
2b ends in a one	5
2c ends in a two	4
3 strip of twelve	233
challenge with threes too	1, 2, 4, 7, 13, 24, 44, 81
challenge ones and threes	1, 1, 2, 3, 4, 6, 9, 13

Week 32 Ways of Counting

A Rule From a Table

Two patterns started together. One is always double the other, and you can see why.

In class this week: Generating patterns and graphing pairs (5.OA.3, 5.G.1)

Teaching noteThe first page in the book with a letter standing for a number. Grade 6 is a whole year of this, so it is worth taking slowly.

Question	Answer
1 pattern A	0, 3, 6, 9, 12, 15, 18
1 pattern B	0, 6, 12, 18, 24, 30, 36
1a B is	double
2 the points make	a straight line
3a A at step n	3 times n
3b B at step n	6 times n
challenge C double anything	no

Week 33 Logic & Proof

Two Rules, One Answer

A formula and a count that disagree cannot both be right, so check.

In class this week: Reasoning about patterns (5.OA.3, 5.OA.1)

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.

Question	Answer
1 biggest of six rule	5
1 biggest of six counted	5
1 selection of five rule	10
1 selection of five counted	10
1 tree of seven rule	6
1 tree of seven counted	6
2a n equals one rule	1
2a n equals one truth	1
2b n equals two rule	2
2b n equals two truth	2
2c n equals three rule	3
2c n equals three truth	3
2d n equals four rule	4
2d n equals four truth	5
2d first fails at	4
3 strip of seven	21
challenge smallest graph over n lines	3

Week 34 Steps & Cost

Sorting Really Big Lists

At eight cards it hardly matters. At a thousand it decides whether the job is possible.

In class this week: Multi-digit operations; patterns (5.NBT.5, 5.OA.3)

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

Question	Answer
1 8 selection	28
1 8 merge	17
1 64 selection	2016
1 64 merge	321
1 1024 selection	523776
1 1024 merge	9217
1a list grew by	8
1a selection grew by	72
1b merge grew by	18
3a selection seconds	523776
3a in days	6.1
3b merge seconds	9217
3b in hours	2.6
challenge million selection	499999500000
challenge million merge	18951425

Week 35 Steps & Cost

Which Method Would You Pay For?

The last real week. A decision, with money on it.

In class this week: Decimal operations in context (5.NBT.7, 5.OA.1)

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.

Question	Answer
table selection comparisons	49995000
table merge comparisons	123617
1a selection a morning	10.00
1b merge a morning	0.02
1c saved in a year	3640.61
2a days to pay back	41
2b cheaper over a year	merge sort
2b by how much	3240.61
3 twelve sorts a year saves	119.69
3 worth the 400 dollars	no
challenge at a tenth the price still worth it	no

Week 36 Steps & Cost

The Method Book, Finished

Thirty-three pages of methods, each with its price. This week it becomes one book.

In class this week: Review of the year

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

Question	Answer
2a largest of 500	499
2b find one in a sorted 500	9
2c search once	do not sort, just look
2d search a thousand times	sort with merge sort, then binary search