Teacher Guide · It Cannot Be Done, Grade 4

Week 1 Logic & Proof

The Thing That Never Changes

Find what a move cannot touch, and you can rule out the future.

In class this week: Number and shape patterns (4.OA.5)

Teaching noteThe year opens on its hardest idea, so do not rush it. A child who finds a clever sequence of moves has missed the point; the answer is that no sequence exists, and the reason is a number no move can touch. The challenge genuinely reverses when 2 becomes 3, and it takes three moves, so let them find that rather than telling them.

Question	Answer
1a start	7
1b two down turned up	down by 2
1b one of each	no change
1b two up turned down	up by 2
1c all three changes	even
2a goal count	0
2a odd or even	even
2b can you finish	no
2c moves tried	0
3a six cups possible	yes
3b moves needed	3
challenge seven cups flip three	yes
challenge fewest moves	3

Week 2 Number Structure

Odd, Even and Impossible

Odd and even is not a label. It is a tool that rules things out.

In class this week: Number patterns; place value (4.OA.5, 4.NBT.4)

Teaching noteParity as a tool rather than a label. The 1-to-10 challenge is the best question in the trimester: the total is 55, and flipping any sign changes it by an even amount, so it can never land on 0.

Question	Answer
table even plus even	even
table even plus odd	odd
table odd plus even	odd
table odd plus odd	even
1a four evens	even
1b four odds	even
1c five odds	odd
1d rule	odd when n is odd, even when n is even
2 fifteen from four evens	no
2 twentyone from six odds	no
2 twentyone from five odds	yes
2 thirty from three odds	no
challenge can reach zero	no
challenge total of 1 to 10	55

Week 3 Number Structure

Factor Rectangles

Every rectangle you can build from a number is a fact about that number.

In class this week: Factors, multiples, prime and composite (4.OA.4)

Teaching noteGrade 3 met factor rectangles. Here they are equipment. The odd factor count of a square was left hanging last year and gets its reason this week.

Question	Answer
1a pairs of 24	1 x 24, 2 x 12, 3 x 8, 4 x 6
1b factors of 24	1, 2, 3, 4, 6, 8, 12, 24
1c how many	8
2 factors of 36	9
2 factors of 48	10
2 factors of 13	2
3a the square	36
3b its factor count	9
challenge most factors below 50	48
challenge that count	10
challenge exactly three factors	4, 9, 25, 49

Week 4 Number Structure

Primes and Composites

Cross out every number that is a multiple of something. Look at what survives.

In class this week: Prime and composite numbers (4.OA.4)

Teaching noteThe sieve never asks whether anything is prime; it only crosses out. Stopping at 7 because 11 x 11 is past 100 is the part worth insisting on.

Question	Answer
1a how many primes	25
1b largest	97
1c eleven squared	121
2 is 91 prime	no
2 91 factor pair	7 x 13
2 is 97 prime	yes
2 is 87 prime	no
2 87 factor pair	3 x 29
3a is one prime	no
3b is two prime	yes
challenge twin pairs below 50	3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43
challenge how many twin pairs	6
challenge longest composite run	7

Week 5 Number Structure

Building From Primes

Break a number apart however you like. The bottom of the tree is always the same.

In class this week: Factors; multi-digit multiplication (4.OA.4, 4.NBT.5)

Teaching noteTwo different trees for 72 give the same primes. That is the whole page. Do not state it, have them build both.

Question	Answer
1a primes of 72	2, 2, 2, 3, 3
1b multiplied back	72
2 same primes either way	yes
3 primes of 60	2 x 2 x 3 x 5
3 primes of 100	2 x 2 x 5 x 5
3 primes of 45	3 x 3 x 5
challenge the number	120
challenge smallest four different primes	210

Week 6 Number Structure

Multiples That Meet

Two rhythms started together. When do they line up again?

In class this week: Factors and multiples (4.OA.4)

Teaching noteLeast common multiple by listing. The pair 4 and 12 behaves differently because 4 already divides 12, and noticing that is worth more than the answers.

Question	Answer
1a first shared	12
1b next two	24 and 36
1c what they all are	multiples of 12
2 lcm 6 and 8	24
2 lcm 5 and 7	35
2 lcm 12 and 18	36
2 lcm 4 and 12	12
2 the odd pair	4 and 12, because 4 already divides 12
challenge two buses	9:36
challenge three buses	60

Week 7 Number Structure

The Biggest Shared Factor

The other end of last week. Not where they meet, but what they share.

In class this week: Factors and multiples (4.OA.4)

Teaching noteThe mirror of last week. Children confuse the two all year, so the bag problem is there to give greatest common factor something concrete to be for.

Question	Answer
1a common factors	1, 2, 3, 4, 6, 12
1b greatest	12
2 gcf 18 and 30	6
2 gcf 15 and 28	1
2 gcf 48 and 60	12
2 the coprime pair	15 and 28
3 largest number of bags	12
3 in each bag	2 cookies and 3 brownies
challenge bunches	6
challenge in each bunch	5 roses and 7 tulips

Week 8 Logic & Proof

Coloring Proves It

Color the board first. The coloring does the arguing.

In class this week: Multi-step problems and reasoning (4.OA.3)

Teaching noteThe coloring is added by the solver, not given by the puzzle, and children find that suspicious. It is worth saying out loud that you are allowed to add anything you like to a problem as long as it is true.

Question	Answer
1a shaded per domino	1
1a white per domino	1
1c twelve dominoes shaded	12
1c twelve dominoes white	12
2 3 by 3 shaded	5
2 3 by 3 white	4
2 3 by 3 possible	no
2 6 by 4 shaded	12
2 6 by 4 white	12
2 6 by 4 possible	yes
2 5 by 5 shaded	13
2 5 by 5 white	12
2 5 by 5 possible	no
2 6 by 4 less two white shaded	12
2 6 by 4 less two white white	10
2 6 by 4 less two white possible	no
challenge five by five middle shaded	12
challenge five by five middle white	12
challenge five by five middle tileable	yes

Week 9 Logic & Proof

The Domino Problem

Sixty-two squares, thirty-one dominoes, and it cannot be done.

In class this week: Multi-step problems and reasoning (4.OA.3)

Teaching noteThe best page in the first trimester. Four lines: the claim, the coloring, the two counts, the contradiction. The second board is not ruled out by the coloring AND is genuinely possible, which is the distinction the whole year turns on.

Question	Answer
1a corner color	shaded
1b shaded left	30
1b white left	32
1c dominoes cover of each	31
1d contradiction	31 dominoes need 31 of each color, but there are 30 and 32
2a two top corners colors	shaded and white
2b ruled out by color	no
2b actually possible	yes
3 the rule	whenever the two removed squares are the same color

Week 10 Dots & Lines

Degree Tells You

Add up every degree in any drawing at all. The answer is never odd.

In class this week: Multi-step problems and reasoning (4.OA.3)

Teaching noteTwo theorems, one useful. The second, that odd-degree dots always come in pairs, gets used from here to Grade 6. Grade 3 asked children to notice it; this is where they prove it.

Question	Answer
1a sum of degrees	14
1b lines	7
1b doubled	14
1c theorem	twice the number of lines
2a odd dots here	2
2c always	even
2d odd ones come in	pairs
3 seven houses degree three	impossible
3 six houses degree three	not ruled out
3 nine people five hands	impossible
challenge five dots four threes one two	not ruled out

Week 11 Dots & Lines

Trace It or Prove You Cannot

The old bridge puzzle, settled in one line, without walking anywhere.

In class this week: Multi-step problems and reasoning (4.OA.3)

Teaching noteKonigsberg in three lines. The degrees are 5, 3, 3, 3, which is four odd dots, and the rule allows only 0 or 2. Children who want to try walking it should be allowed to, once.

Question	Answer
1a degrees	3, 5, 3, 3
1b how many odd	4
1c verdict	impossible
2a lines used	2
2b middle dot degree	even
3 a square odd dots	0
3 a square traceable	yes
3 a square with one diagonal odd dots	2
3 a square with one diagonal traceable	yes
3 a square with both diagonals odd dots	4
3 a square with both diagonals traceable	no
3 an envelope odd dots	2
3 an envelope traceable	yes

Week 12 Logic & Proof

Impossible Fair

Six claims. Prove each one, or knock it down. You may not shrug.

In class this week: Multi-step problems and reasoning (4.OA.3)

Teaching noteThree true and three false, and the asymmetry is the lesson: one example kills a claim, but no number of examples holds one up. Claim 5 fails on a count, not a coloring.

Question	Answer
1 three odds always odd	true
2 odd factor count means square	true
3 seven people three hands	false
4 every prime is odd	false
5 four by six less one	false
6 by two and three means by six	true
how many true	3

Week 13 Logic & Proof

Case Files

Everything from the first twelve weeks, in one folder.

In class this week: Review & reasoning

Teaching noteReview. Four of the five cases are settled without arranging anything, which is worth pointing out explicitly.

Question	Answer
1 nine cups	impossible
2 gcf	14
2 lcm	210
3 eleven houses degree five	impossible
4 shaded and white	16 and 18
4 possible	no
5 primes of 84	2 x 2 x 3 x 7
5 factors of 84	12

Week 14 Number Structure

Divisible by 2, 5 and 10

Why does only the last digit matter? There is a reason, and it is short.

In class this week: Place value to one million (4.NBT.1, 4.NBT.2)

Teaching noteA rule with its reason attached. Children already know the last-digit tests; the work here is being able to say why, from place value.

Question	Answer
1 3000 by 10	yes
1 400 by 10	yes
1 70 by 10	yes
1 0 by 10	yes
1a ten is two times	2
1a and	5
1b decided by	the ones digit
2 3470 by 2	yes
2 3470 by 5	yes
2 3470 by 10	yes
2 3470 by 4	no
2 8216 by 2	yes
2 8216 by 5	no
2 8216 by 10	no
2 8216 by 4	yes
2 9005 by 2	no
2 9005 by 5	yes
2 9005 by 10	no
2 9005 by 4	no
2 12348 by 2	yes
2 12348 by 5	no
2 12348 by 10	no
2 12348 by 4	yes
3a hundred by four	yes
3a thousand by four	yes
3b digits needed for four	2
challenge digits needed for eight	3
challenge 1048 by eight	yes
challenge 3036 by eight	no
challenge last two digits for twenty	00, 20, 40, 60, 80

Week 15 Number Structure

Divisible by 3 and 9

Add the digits. It is not a trick, and here is why it works.

In class this week: Place value and multi-digit arithmetic (4.NBT.1, 4.NBT.4)

Teaching noteThe 9 + 1 split is the reason the digit-sum test works, and it is the page next week depends on. Do not let them skip to the rule.

Question	Answer
2 4275 digit sum	18
2 4275 by 3	yes
2 4275 by 9	yes
2 1236 digit sum	12
2 1236 by 3	yes
2 1236 by 9	no
2 8001 digit sum	9
2 8001 by 3	yes
2 8001 by 9	yes
2 5555 digit sum	20
2 5555 by 3	no
2 5555 by 9	no
1a the leftovers are	the digits of the number
1b divisible when	the digit sum is
2 reverse true	no
3 six means	2
3 and	3
3 1236 by six	yes
3 8001 by six	no
challenge 123456789 by nine	yes
challenge smallest over 1000 by 8 and 9	1008

Week 16 Number Structure

Casting Out Nines

A real check on multiplication, from the rule you proved last week.

In class this week: Multi-digit multiplication (4.NBT.5)

Teaching noteA real check with a real limitation. It catches the wrong answer in part 2 and misses the transposed digits in part 3, and being honest about that is the point of the page.

Question	Answer
1 first factor cast	9
1 second factor cast	7
1 those multiplied cast	9
1 answer cast	9
1a match	yes
1b if not	the answer is definitely wrong
2 43 by 26 left cast	2
2 43 by 26 answer cast	2
2 43 by 26 wrong	no
2 58 by 37 left cast	4
2 58 by 37 answer cast	5
2 58 by 37 wrong	yes
2 64 by 45 left cast	9
2 64 by 45 answer cast	9
2 64 by 45 wrong	no
2 the wrong one	58 x 37
2 the right answer	2146
3 918 cast	9
3 981 cast	9
3 does it spot it	no

Week 17 Number Structure

Wrapping Around, Again

Casting out nines was not a trick. It was arithmetic on a wheel.

In class this week: Multi-digit arithmetic and patterns (4.NBT.4, 4.OA.5)

Teaching noteCasting out nines and the remainder wheel turn out to be the same thing. If that lands, the child has understood something most adults have not.

Question	Answer
1 38 mod 9	2
1 38 mod 5	3
1 100 mod 9	1
1 100 mod 5	0
1 918 mod 9	0
1 918 mod 5	3
1 918 digit sum matches	yes
3 last digits of three	3, 9, 7, 1, 3, 9
3 cycle length	4
challenge seven to the thirty last digit	9
challenge which land on zero	the multiples of 9

Week 18 Number Structure

The Day of the Week

A whole year on a seven-wheel, and no counting on fingers.

In class this week: Multi-step problems; measurement (4.OA.3, 4.MD.2)

Teaching noteThe offsets are the real cumulative day counts, and verify.py checks them against the month lengths. October matching January surprises people every time.

Question	Answer
January leaves	0
January starts on	Monday
February leaves	3
February starts on	Thursday
March leaves	3
March starts on	Thursday
April leaves	6
April starts on	Sunday
May leaves	1
May starts on	Tuesday
June leaves	4
June starts on	Friday
July leaves	6
July starts on	Sunday
August leaves	2
August starts on	Wednesday
September leaves	5
September starts on	Saturday
October leaves	0
October starts on	Monday
November leaves	3
November starts on	Thursday
December leaves	5
December starts on	Saturday
1a march divided	8
1a march remainder	3
3a same as january	October
3b other shared groups	February, March, November; April, July; September, December

Week 19 Shape & Space

Lines of Symmetry

Fold it. If the two halves land on each other, you found a line.

In class this week: Lines of symmetry (4.G.3)

Teaching noteFold and check, physically. A line that nearly works does not count, and cutting the shape out settles arguments faster than talking about them.

Question	Answer
2 square	4
2 rectangle that is not a square	2
2 equilateral triangle	3
2 isosceles triangle	1
2 regular hexagon	6
2 regular pentagon	5
2 parallelogram that is not a rectangle	0
2 scalene triangle	0
3 the two zeroes	the parallelogram and the scalene triangle
challenge seven sides	7
challenge twelve sides	12

Week 20 Shape & Space

Turning a Shape

Fold lines are not the only kind of symmetry. Some shapes only turn.

In class this week: Lines of symmetry; angle measure (4.G.3, 4.MD.5)

Teaching noteThe parallelogram is the row that matters: turn symmetry with no fold lines at all. It shows the two kinds of symmetry are independent.

Question	Answer
2 square folds	4
2 square turn order	4
2 rectangle, not a square folds	2
2 rectangle, not a square turn order	2
2 equilateral triangle folds	3
2 equilateral triangle turn order	3
2 regular hexagon folds	6
2 regular hexagon turn order	6
2 parallelogram, not a rectangle folds	0
2 parallelogram, not a rectangle turn order	2
2 isosceles triangle folds	1
2 isosceles triangle turn order	1
1 square order	4
3a turns but no folds	parallelogram, not a rectangle
3b folds but no turns	isosceles triangle

Week 21 Shape & Space

The Eight Ways to Move a Square

Every move that leaves a square looking the same. There are exactly eight.

In class this week: Angle measure and reasoning (4.MD.5, 4.OA.3)

Teaching noteThe first time a child sees a set of moves as an object with its own size. That doing any two in a row always lands on another of the eight is the genuine surprise.

Question	Answer
1a turns	4
1a flips	4
1a total	8
2 turn90 then turn90	turn 180
2 turn90 then turn270	stay
2 flip across then flip down	turn 180
2 flip across twice	stay
3a back after doing twice	flip across, flip down, flip on one slant, flip on the other, turn 180
3b undoes turn 90	turn 270
challenge triangle moves	6
challenge rectangle moves	4

Week 22 Shape & Space

Angles That Must Add Up

Tear the three corners off any triangle at all. They make a straight line.

In class this week: Angle measure and adding angles (4.MD.5, 4.MD.7)

Teaching noteTearing the corners is not a proof and it is worth saying so, while also letting them do it. The splitting-into-triangles trick is what generalizes.

Question	Answer
1a corners make	a straight line
1b add to	180
2 60 and 70 third	50
2 60 and 70 possible	yes
2 90 and 45 third	45
2 90 and 45 possible	yes
2 30 and 30 third	120
2 30 and 30 possible	yes
2 90 and 95 third	-5
2 90 and 95 possible	no
2 the impossible row	90 and 95
3a quadrilateral triangles	2
3a quadrilateral total	360
3b pentagon triangles	3
3b pentagon total	540
challenge hexagon total	720
challenge ten sided total	1440

Week 23 Shape & Space

Tiling the Floor

Only three regular shapes tile a floor. The reason is one division.

In class this week: Angle measure (4.MD.5, 4.MD.7)

Teaching noteThree shapes tile, and the argument for why nothing with seven or more sides can is short enough for a fourth grader to hold. Two octagons leave exactly 90 degrees, which is a square, and that is a real bathroom floor.

Question	Answer
2 triangle corner angle	60
2 triangle 360 divided	6
2 triangle tiles	yes
2 square corner angle	90
2 square 360 divided	4
2 square tiles	yes
2 pentagon corner angle	108
2 pentagon 360 divided	3.33
2 pentagon tiles	no
2 hexagon corner angle	120
2 hexagon 360 divided	3
2 hexagon tiles	yes
2 octagon corner angle	135
2 octagon 360 divided	2.67
2 octagon tiles	no
1 triangles at a point	6
1 squares at a point	4
1 pentagon degrees left	36.0
3a which tile	triangle, square, hexagon
3b lands between	2
3b and	3
challenge two octagons leave	90
challenge which shape fits	square

Week 24 Shape & Space

Impossible Tilings

Two boards, one corner different, and the answers are not the same.

In class this week: Multi-step problems and reasoning (4.OA.3)

Teaching noteThe sharpest page of the year. Same board, same tiles, two different corners, and one is settled while the other is not. Resist the urge to tell them whether the second is possible, because the honest answer is that this argument does not say.

Question	Answer
1 5 by 5 squares	25
1 5 by 5 divides	no
1 5 by 5 ruled out by counting	yes
1 4 by 4 squares	16
1 4 by 4 divides	no
1 4 by 4 ruled out by counting	yes
1 8 by 8 squares	64
1 8 by 8 divides	no
1 8 by 8 ruled out by counting	yes
1 6 by 4 squares	24
1 6 by 4 divides	yes
1 6 by 4 ruled out by counting	no
2a the three counts	21, 22, 21
2a equal	no
3a top left counts	20, 22, 21
3a top left equal	no
3a top left verdict	impossible
3b top right counts	21, 21, 21
3b top right equal	yes
3b top right verdict	not ruled out
challenge five by five remove which color	shaded
challenge nine by nine counts	27, 27, 27

Week 25 Logic & Proof

Puzzle Fair

A little of everything from Weeks 14 to 24.

In class this week: Review & reasoning

Teaching noteReview. Puzzle 6 needs a count and a coloring together, and comes out possible.

Question	Answer
1 71325 by 3	yes
1 71325 by 9	yes
1 71325 by 5	yes
2 is 1232 right	no
2 the right answer	1242
3 six fold lines shape	a regular hexagon
3 its turn order	6
4 nine sided corner angle	140
4 tiles	no
5 december starts on	Saturday
6 shaded and white	24 and 24
6 possible	yes

Week 26 Ways of Counting

Choosing a Team

Line them up, then divide out the orderings you did not want.

In class this week: Multi-step problems (4.OA.3)

Teaching noteOver-count on purpose, then divide by exactly how many times each team was counted. Children want to subtract, and asking them why dividing is right usually fixes it.

Question	Answer
1a first pick	5
1a second	4
1a third	3
1a multiplied	60
1b orders of one team	6
1c teams	10
2 choose 2 from 6 lineups	30
2 choose 2 from 6 orders	2
2 choose 2 from 6 teams	15
2 choose 3 from 6 lineups	120
2 choose 3 from 6 orders	6
2 choose 3 from 6 teams	20
2 choose 4 from 6 lineups	360
2 choose 4 from 6 orders	24
2 choose 4 from 6 teams	15
2 choose 3 from 7 lineups	210
2 choose 3 from 7 orders	6
2 choose 3 from 7 teams	35
3 why they match	choosing 2 to go is the same as choosing 4 to stay behind
challenge 3 from 8	56
challenge 5 from 8	56
challenge pizzas	35

Week 27 Ways of Counting

The Triangle Grows

Every number is the two above it added together, and that rule builds the whole thing.

In class this week: Number and shape patterns (4.OA.5)

Teaching noteThe triangle rebuilt from its rule. The row totals doubling and the third diagonal being last year's handshake numbers are both left for them to find.

Question	Answer
1a row five	1, 5, 10, 10, 5, 1
2a row totals	1, 2, 4, 8, 16, 32, 64, 128
2b each total is	double the one before it
2c third number down	1, 3, 6, 10, 15, 21
3 the missing one counts as	nothing, so the edge stays 1
challenge row eight	1, 8, 28, 56, 70, 56, 28, 8, 1
challenge row nine	1, 9, 36, 84, 126, 126, 84, 36, 9, 1
challenge row ten	1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1
challenge biggest in row ten	252
challenge row ten total	1024

Week 28 Ways of Counting

The Triangle and the Teams

Last week's triangle and the week before's teams are the same numbers.

In class this week: Number and shape patterns (4.OA.5)

Teaching noteWhere the triangle and the teams turn out to be the same numbers. The come-or-stay-home argument for the row total is the cleanest thing in the book.

Question	Answer
1 teams from six	1, 6, 15, 20, 15, 6, 1
1a where is fifteen	row 6, position 2
1b choosing none fits	yes
2 choose 3 from 6	20
2 choose 2 from 7	21
2 choose 3 from 7	35
2 choose 4 from 8	70
3a row six total	64
3b two to the six	64
challenge 3 from 8	56
challenge 5 from 8	56
challenge which row totals 1024	10

Week 29 Dots & Lines

Trees

Joined up, with no way round. Count the lines and you will find a rule.

In class this week: Multi-step problems and reasoning (4.OA.3)

Teaching noteAnother invariant wearing different clothes: add one dot, add exactly one line, so the gap never changes. Worth connecting back to Week 1 out loud.

Question	Answer
1a the cycle	P, Q, T, S and back to P
1b tree dots	5
1b tree lines	4
1c other dots	5
1c other lines	5
2 tree with 3 dots	2
2 tree with 5 dots	4
2 tree with 8 dots	7
2 tree with 12 dots	11
2 the rule	n minus 1
challenge ten dots eight lines connected	no
challenge ten dots twelve lines cycles	3

Week 30 Dots & Lines

The Cheapest Network

Join every town for the least money. The answer is always a tree.

In class this week: Multi-step problems (4.OA.3)

Teaching noteTake the cheapest road that does not close a loop, and it works. Children distrust this because it seems too easy, and the cycle argument is why it is not luck.

Question	Answer
2a roads built	4
2a towns	5
2b tree rule	yes
2c total cost	14
1 roads taken	A to C, B to C, B to D, D to E
challenge with BD at one	10
challenge all equal how many	21

Week 31 Steps & Rules

Two Ways to Multiply

Two methods, the same answer, and a fair way to compare the work.

In class this week: Multiplying multi-digit numbers (4.NBT.5)

Teaching noteTwo methods, identical work. The count of times-table facts does not depend on the method, which is a genuinely useful disappointment.

Question	Answer
1a boxes	4
1a facts	4
1c answer	918
2c standard facts	4
3 46 by 23 boxes	4
3 46 by 23 facts	4
3 7 by 385 boxes	3
3 7 by 385 facts	3
3 124 by 56 boxes	6
3 124 by 56 facts	6
challenge 385 by 47	18095
challenge cast check	yes
challenge three by three facts	9
challenge the rule	a times b

Week 32 Steps & Rules

The Euclid Machine

Take the smaller from the bigger, over and over. It always stops, and where it stops matters.

In class this week: Multi-step problems; factors (4.OA.3, 4.OA.4)

Teaching noteThe machine finds the greatest common factor without ever looking for a factor. Why it has to stop is the same shape of argument as Week 1. The worst pair below 30 is 29 and 28, not anything exotic, because subtracting 1 repeatedly is the slowest thing it can do.

Question	Answer
1a stops at	6
1b by listing	6
1c same	yes
2 35 and 14 subtractions	3
2 35 and 14 stops at	7
2 30 and 12 subtractions	3
2 30 and 12 stops at	6
2 21 and 13 subtractions	6
2 21 and 13 stops at	1
2 stopping at one means	they share no factor except 1
challenge 1071 and 462 subtractions	11
challenge 1071 and 462 answer	21
challenge worst pair below thirty	29 and 28
challenge its subtractions	28
challenge what is special about them	they are next to each other

Week 33 Fractions

The Same Point on the Line

Equivalent fractions are not two things that are equal. They are one point with two names.

In class this week: Equivalent fractions (4.NF.1, 4.NF.2)

Teaching noteOne point, several names. The Grade 3 strips are gone because stacked strips make equivalent fractions look like a coincidence of alignment rather than one point.

Question	Answer
1a half in halves	1
1a half in fourths	2
1a half in eighths	4
1b three fourths in eighths	6
1c third on this line	no
2 1 over 2 doubled	2 over 4
2 1 over 2 tripled	3 over 6
2 3 over 4 doubled	6 over 8
2 3 over 4 tripled	9 over 12
2 2 over 3 doubled	4 over 6
2 2 over 3 tripled	6 over 9
2 5 over 6 doubled	10 over 12
2 5 over 6 tripled	15 over 18
3 two thirds in twelfths	8 over 12
3 three fourths in twelfths	9 over 12
3 the bigger	3 over 4
challenge three fifths equivalents	6 over 10, 9 over 15, 12 over 20
challenge in order	5 over 8, 2 over 3, 3 over 4

Week 34 Fractions

Parts of a Hundred

A hundred squares, and two ways to say the same shading.

In class this week: Decimal notation for fractions (4.NF.5, 4.NF.6)

Teaching noteThe eighth is the interesting case: 12 and a half squares cannot be shaded, and the reason is which primes divide 100.

Question	Answer
2 37 fraction	37 over 100
2 37 tenths and hundredths	3 and 7
2 37 decimal	0.37
2 50 fraction	50 over 100
2 50 tenths and hundredths	5 and 0
2 50 decimal	0.50
2 8 fraction	8 over 100
2 8 tenths and hundredths	0 and 8
2 8 decimal	0.08
2 70 fraction	70 over 100
2 70 tenths and hundredths	7 and 0
2 70 decimal	0.70
2 25 fraction	25 over 100
2 25 tenths and hundredths	2 and 5
2 25 decimal	0.25
1c whole rows	3
1c spare squares	7
3a quarter squares	25
3a quarter decimal	0.25
3b eighth squares	12 and a half
3b can you shade whole squares	no
challenge which are exact	one half, one fifth, one twentieth
challenge what the bottoms share	their only prime factors are 2 and 5

Week 35 Logic & Proof

The Weighing Puzzles

How much can one weighing possibly tell you? Count before you weigh.

In class this week: Multi-step problems and reasoning (4.OA.3)

Teaching noteThe only week where counting says what is possible rather than what is not. Three outcomes per weighing, so k weighings separate 3 to the k cases, and splitting into three groups rather than two follows from that.

Question	Answer
1a outcomes	3
1a name them	left down, right down, or level
1b two weighings	9
2a left down means	coins 1, 2 and 3
2b balances means	coins 7, 8 and 9
2c down to	3
3 1 weighings cases	3
3 1 weighings most coins	3
3 2 weighings cases	9
3 2 weighings most coins	9
3 3 weighings cases	27
3 3 weighings most coins	27
3 4 weighings cases	81
3 4 weighings most coins	81
talk twelve coins two weighings	impossible
challenge first weighing of 27	9 against 9
challenge twelve unknown cases	24
challenge three weighings enough	yes

Week 36 Logic & Proof

The Proof File, Finished

Thirty-three pages of reasons. This week they become one folder.

In class this week: Review of the year

Teaching noteAssembly. The only real test is whether a page can be said out loud without reading it.

Question	Answer
2a eleven people three hands	false
2b 51 is prime	false
2c octagon tiles alone	false
2d tree with twenty dots	true