Teacher Guide · Find the Rule, Grade 6

Week 1 Patterns & Rules

A Letter For Any Number

One letter says what a hundred examples cannot.

In class this week: Writing and reading expressions (6.EE.2, 6.EE.6)

Teaching noteA letter is not an unknown to be found. It is every number at once, and that sentence is the whole year in one line. Look at the difference between rows: a constant difference is the number that multiplies n.

Question	Answer
1 sticks for 1	4
1 sticks for 2	7
1 sticks for 3	10
1 sticks for 4	13
1 sticks for 5	16
1 sticks for 10	31
1a each new square adds	3
1b the rule	3n + 1
1c at n = 3	10
1c at n = 10	31
2 3n + 1 at n = 5	16
2 2n - 3 at n = 5	7
2 n(n + 1) at n = 5	30
2 (n + 1) / 2 at n = 5	3
3 2(n + 3) at n = 1	8
3 2n + 6 at n = 1	8
3 2(n + 3) at n = 4	14
3 2n + 6 at n = 4	14
3 2(n + 3) at n = 10	26
3 2n + 6 at n = 10	26
3 same every time	yes
challenge triangles rule	2n + 1
challenge 7 12 17 22 rule	5n + 2
challenge at n = 100	502

Week 2 Patterns & Rules

A Pattern That Grows

Draw the shape, count the dots, and let the picture hand you the formula.

In class this week: Expressions and equivalent expressions (6.EE.2, 6.EE.4)

Teaching noteThe arrangement is the argument. The oblong splitting into two staircases is next week's page, so leave it visible.

Question	Answer
1 square 1	1
1 square 2	4
1 square 3	9
1 square 4	16
1 square 5	25
1 the rule	n squared
2a oblong at n = 4	20
2b the rule	n(n + 1)
3 each staircase	10
challenge pentagonal differences	4, 7, 10
challenge second differences	3, 3
challenge n rows of n plus 2 at n = 3	15

Week 3 Patterns & Rules

Triangular Numbers

1, 3, 6, 10, 15. You have met them three times already. Now you get the formula.

In class this week: Expressions and equivalent expressions (6.EE.2, 6.EE.3)

Teaching noteTwo triangles slotted together make an oblong, and that picture gives the formula without any algebra. Grade 3 met 1, 3, 6, 10, 15 in Pascal's triangle without knowing why.

Question	Answer
1a the shape	an oblong
1b oblong is	5 by 6
1b dots	30
1c one triangle	15
2a the rule	n(n + 1) / 2
2 triangular 1	1
2 triangular 2	3
2 triangular 3	6
2 triangular 4	10
2 triangular 5	15
2 triangular 6	21
2 triangular 10	55
2 triangular 100	5050
challenge hundredth	5050
challenge two in a row	a square number

Week 4 Ways of Counting

The Handshake Formula, Proved

Grade 2 counted them. Grade 3 found the rule. This week you prove it.

In class this week: Expressions and equivalent expressions (6.EE.2, 6.EE.4)

Teaching noteThis closes a thread open since Grade 2 week 12. Two proofs, the double count and the staircase, and both are worth doing: the second is the one that still works when some pairs refuse to shake.

Question	Answer
1a each person shakes	n minus 1
1a shakes counted	n(n minus 1)
1b counted by	2
1c the real number	n(n minus 1) / 2
2 handshakes for 6	15
2 handshakes for 10	45
2 handshakes for 20	190
2 handshakes for 100	4950
2b triangular for	n minus 1
2 formula agrees everywhere	yes
challenge twelve teams	66
challenge 66 handshakes	12

Week 5 Patterns & Rules

Adding One to n

Pair the ends. The whole sum falls out in one line.

In class this week: Expressions; the distributive property (6.EE.2, 6.EE.3)

Teaching noteThe same formula as week 3, from a completely different picture. Ask the children whether two proofs make it more believable, and let them argue.

Question	Answer
1a each column	9
1b columns	8
1b both rows total	72
1c counted	2
1c the sum	36
2a each column for n	n + 1
2a columns for n	n
2b the formula	n(n + 1) / 2
3 sum to 20	210
3 sum to 100	5050
3 sum to 1000	500500
challenge evens to 100	2550
challenge 100 minus 49	3825

Week 6 Patterns & Rules

The Odd Numbers Make a Square

1, then 1 + 3, then 1 + 3 + 5. The answers are square, and the picture says why.

In class this week: Expressions and equivalent expressions (6.EE.2)

Teaching noteA picture that settles every n at once. The gnomon going from side k to side k + 1 adds 2k + 1, which is why the strips are the odd numbers.

Question	Answer
1a the strips	1, 3, 5, 7, 9
1b how many strips	5
1b side of the square	5
1c the rule	n squared
1c check at five	25
2 a column of	k
2 a row of	k
2 that is	2k + 1
3 sum to 19	100
3 first 25 odd	625
3 how many make 400	20
challenge terms to 99	50
challenge sum to 99	2500
challenge evens to 100	2550

Week 7 Patterns & Rules

Looking Back or Jumping Ahead

Two kinds of rule. One needs the answers before it, and one does not.

In class this week: Expressions; dependent and independent quantities (6.EE.2, 6.EE.9)

Teaching noteTwo kinds of rule, priced by how many steps each takes. This is the Grade 5 habit applied to formulas. The strip coverings have a closed form and it is horrible, which is a legitimate reason to prefer looking back.

Question	Answer
1a looking back steps to twenty	19
1b jumping ahead steps	1
2 coverings of 1	1
2 coverings of 2	2
2 coverings of 3	3
2 coverings of 4	5
2 coverings of 5	8
2 coverings of 6	13
2 coverings of 7	21
2 coverings of 8	34
3a better for the first ten	looking back
3b better for the thousandth	jumping ahead
challenge doubling plus one	1, 3, 7, 15, 31
challenge its closed form	2 to the n, minus 1

Week 8 Logic & Proof

When a Pattern Lies

1, 2, 4, 8, 16. Everybody says 32. Everybody is wrong.

In class this week: Expressions and reasoning (6.EE.2, 6.EE.9)

Teaching noteThe most important page in the book. 1, 2, 4, 8, 16 and then 31. Everybody says 32, including adults. Count the six-point picture twice, because two chords nearly meet and that near-miss is exactly where the missing region hides.

Question	Answer
1 regions with 1 points	1
1 regions with 2 points	2
1 regions with 3 points	4
1 regions with 4 points	8
1 regions with 5 points	16
1 regions with 6 points	31
2a cases that fitted doubling	5
2b is that a lot	no
3a n squared plus n plus 41 at 41	1763
3a is it prime	no
3a its factors	41 x 43

Week 9 Logic & Proof

Proof by Picture

Some arrangements settle a question before a single symbol is written.

In class this week: Expressions and equivalent expressions (6.EE.3, 6.EE.4)

Teaching noteA picture drawn for n = 4 proves nothing unless the child can say how to draw it for any n. That sentence is the test, and it is what separates this page from week 8.

Question	Answer
1 the fact	3 times 5 equals 5 times 3
2 sixth plus tenth	16
2 the general claim	two triangular numbers in a row make a square
3a nine as a difference	5 squared minus 4 squared
3b in general	(k + 1) squared minus k squared
challenge first n evens	n(n + 1)
challenge powers of two sum	one less than the next power of 2

Week 10 Logic & Proof

Proof by Cases

Split the question into a few kinds, settle each one, and you have settled them all.

In class this week: Reasoning about expressions (6.EE.2, 6.EE.5)

Teaching noteTwo cases, both settled, nothing left out. Ask every time whether the cases really do cover everything, because a proof by cases with a gap proves nothing at all.

Question	Answer
1 n even so n plus 1 is	odd
1 n odd so n plus 1 is	even
1a in both cases	even
1b other cases	no
1c proved for	all
1 checked by search	yes
3b cases needed for three in a row	3
3b checked by search	yes

Week 11 Ways of Counting

Counting the Same Thing Twice

Count one thing two ways. The two answers must be equal, so you have a rule for free.

In class this week: Expressions and equivalent expressions (6.EE.3, 6.EE.4)

Teaching noteCount one collection two ways and a rule appears without anything being solved. The three places in the table are all things this line of books already did without naming it.

Question	Answer
1a by people	60
1b by kinds	60
1c so	the two counts are equal
3 row six total	64
3 come or stay	64
challenge 45 games	10
challenge club size	12

Week 12 Patterns & Rules

Rule Fair

Six patterns. Find each rule, and say how sure you are.

In class this week: Writing and evaluating expressions (6.EE.2, 6.EE.6)

Teaching noteThe L, C and P marks are the assessment. An honest C is worth more than a dishonest P, and children need telling that explicitly.

Question	Answer
1 pattern A rule	4n - 1
1 pattern B rule	n squared
1 pattern C rule	n(n + 1)
1 pattern D rule	2 to the n, minus 1
1 pattern E rule	n(n + 1)(2n + 1) / 6
1 pattern F rule	the week 8 one
1a which is the week 8 one	F
1b pattern E is	the running total of the square numbers

Week 13 Logic & Proof

Case Files

Everything from the first twelve weeks, in one folder.

In class this week: Review & reasoning

Teaching noteReview. Case 4 is the whole first trimester in one question and the right answer is that four cases is not a proof.

Question	Answer
1 hexagons rule	5n + 1
2 fifteen teams	105
3 terms to 41	21
3 total to 41	441
4 what to say	four cases is not a proof
5 three in a row	yes

Week 14 Ratio & Rate

Ratio as a Rule

Three cups of flour to two of water. That is a rule with a letter hiding in it.

In class this week: Ratio concepts and reasoning (6.RP.1, 6.RP.3)

Teaching noteA ratio is a rule, and plotting the pairs makes that visible. The line goes through the origin because zero flour needs zero water.

Question	Answer
1 water for 3 flour	2
1 water for 6 flour	4
1 water for 9 flour	6
1 water for 12 flour	8
1 flour for 20 water	30
1a water is	two thirds
1b the rule	2n / 3
3 6 to 4 same as 3 to 2	yes
3 9 to 5 same as 3 to 2	no
3 15 to 10 same as 3 to 2	yes
challenge white for 35 blue	21
challenge 4 to 6 and 6 to 9	yes

Week 15 Ratio & Rate

One of Them

Reduce everything to one, and any two things become comparable.

In class this week: Unit rate and rate reasoning (6.RP.2, 6.RP.3)

Teaching noteBoth unit rates rank the shops identically, and it is worth asking why they must. The cheapest shop is neither the one with the lowest shelf price nor the biggest pack.

Question	Answer
1 A price for one	0.25
1 B price for one	0.23
1 C price for one	0.26
1 D price for one	0.25
1a cheapest per pencil	B
1b lowest shelf price	C
1b biggest pack	D
2b same winner	yes
3 miles per hour	60
3 dollars per hour	9
3 words per minute	60
challenge better car	the first
challenge by how much per gallon	0.5

Week 16 Ratio & Rate

Out of a Hundred

Percent is a ratio wearing a uniform. Everything is out of a hundred.

In class this week: Percent as a rate per 100 (6.RP.3c)

Teaching notePercent exists so that differently sized results can be compared without thinking. The up 20 then down 20 challenge catches nearly everybody.

Question	Answer
1a as a fraction	37 over 100
1b as a decimal	0.37
1c as a percent	37 percent
2 18 out of 25 percent	72
2 21 out of 30 percent	70
2 34 out of 40 percent	85
2 7 out of 8 percent	87.5
2 best result	7 out of 8
3a 25 percent of 60	15
3b 10 percent of 250	25
3c discount	6
3c you pay	34
challenge better score	43 out of 50
challenge up then down	96

Week 17 Dots & Lines

Scaling a Network

Every road doubles in price. What happens to the cheapest route?

In class this week: Ratio and rate reasoning (6.RP.3, 6.EE.2)

Teaching noteThe plus-10 row is the page. Multiplying every road keeps the cheapest route because it scales all routes equally; adding charges a three-road route three times and a two-road route twice, so it can flip the answer. At plus 9 the two tie.

Question	Answer
1a cheapest route	Depot to North to South to East
1a cheapest cost	10
1b roads used	3
2 as printed cost	10
2 as printed route	Depot to North to South to East
2 doubled cost	20
2 doubled route	Depot to North to South to East
2 multiplied by 10 cost	100
2 multiplied by 10 route	Depot to North to South to East
2 plus 10 each cost	39
2 plus 10 each route	Depot to North to East
2 which changed	plus 10 each
challenge halved cost	5.0
challenge smallest that changes it	10

Week 18 Dots & Lines

Everyone Plays Everyone

A round robin, counted with the handshake formula and read as ratios.

In class this week: Ratio and rate reasoning (6.RP.3, 6.EE.2)

Teaching noteThe games formula is week 4 again. The check that the wins add up to the games is a double count, and worth naming as one.

Question	Answer
1a each team plays	4
1b five times that	20
1b counted	2
1c games	10
2 Ash games	4
2 Ash percent	100
2 Birch games	4
2 Birch percent	75
2 Cedar games	4
2 Cedar percent	50
2 Elm games	4
2 Elm percent	25
2 Fir games	4
2 Fir percent	0
2 wins add to	10
2 equals the games	yes
3a twelve teams games	66
3b eight of eleven percent	72.7
challenge 66 games	12
challenge all equal	no

Week 19 Number Structure

Factors, By Primes

Grade 4 found common factors by listing. Primes make it a rule instead.

In class this week: Greatest common factor and least common multiple (6.NS.4)

Teaching noteThe prime table makes both answers mechanical. The rule that the two answers multiply to the two starting numbers falls straight out of the columns.

Question	Answer
1 primes of 24	2 x 2 x 2 x 3
1 primes of 36	2 x 2 x 3 x 3
2 gcf 24 and 36	12
2 lcm 24 and 36	72
2 product 24 and 36	864
2 gcf 12 and 18	6
2 lcm 12 and 18	36
2 product 12 and 18	216
2 gcf 15 and 28	1
2 lcm 15 and 28	420
2 product 15 and 28	420
2 gcf 20 and 50	10
2 lcm 20 and 50	100
2 product 20 and 50	1000
3 the rule	yes
challenge gcf 84 and 126	42
challenge lcm 84 and 126	252
challenge lcm from product	300

Week 20 Number Structure

All Four Quadrants

The grid goes left and down as well, and nothing else changes.

In class this week: The coordinate plane and negative numbers (6.NS.6, 6.NS.8)

Teaching noteReading the across number first is the whole discipline here. Reflections change the sign of exactly one coordinate.

Question	Answer
1 A across	3
1 A up	2
1 A quadrant	first
1 B across	-2
1 B up	3
1 B quadrant	second
1 C across	-3
1 C up	-2
1 C quadrant	third
1 D across	2
1 D up	-3
1 D quadrant	fourth
2a reflect A across the up-down axis	(-3, 2)
2a which changed	the across one
2b reflect A across the left-right axis	(3, -2)
3a A to B across	5
3a A to B up	1

Week 21 Shape & Space

Distance on a Grid

A delivery van cannot go through buildings. Its distance is a different distance.

In class this week: The coordinate plane; absolute value (6.NS.8, 6.NS.7)

Teaching noteEvery street route between two points is the same length, and asking why is the useful question. Grade 5 counted these routes; this year measures them.

Question	Answer
1a across	3
1a up	3
1a total	6
2 the rule	the across difference plus the up difference
3 (0, 0) to (3, 4)	7
3 (2, 5) to (2, 1)	4
3 (-1, 2) to (3, -1)	7
3 (4, 4) to (1, 1)	6
challenge four blocks from origin	16
challenge when equal	when the two points share a row or a column

Week 22 Shape & Space

A Square Circle

A circle is every point the same distance from the middle. Change the distance, change the circle.

In class this week: The coordinate plane; absolute value (6.NS.8, 6.G.3)

Teaching noteThe definition of a circle never mentions round. Change what distance means and the same words draw a diamond. This is the best page in the geometry trimester.

Question	Answer
2 points at radius 1	4
2 points at radius 2	8
2 points at radius 3	12
2 points at radius 4	16
2 points at radius 10	40
1a points at three	12
1b the shape	a diamond
1c is it a circle	yes, by the definition
2 the rule	4r
challenge radius four	16

Week 23 Shape & Space

Nets and Solids

Flatten a solid, count its parts, and put it back together.

In class this week: Nets and surface area (6.G.4)

Teaching noteCount faces from the net and edges by halving the face sides, because every edge is shared by two faces. That halving is double counting again.

Question	Answer
1 cube faces	6
1 cube edges	12
1 cube corners	8
1 square pyramid faces	5
1 square pyramid edges	8
1 square pyramid corners	5
1 triangular prism faces	5
1 triangular prism edges	9
1 triangular prism corners	6
1 tetrahedron faces	4
1 tetrahedron edges	6
1 tetrahedron corners	4
1 octahedron faces	8
1 octahedron edges	12
1 octahedron corners	6
2a cube face sides	24
2b faces meet at an edge	2
2c cube edges	12
challenge cube nets	11
challenge hexagonal prism faces	8
challenge hexagonal prism edges	18
challenge hexagonal prism corners	12

Week 24 Patterns & Rules

The Formula That Always Holds

Faces, edges and corners. One little sum comes out the same for every solid you try.

In class this week: Expressions and reasoning (6.EE.2, 6.G.4)

Teaching noteFive cases fitted 1, 2, 4, 8, 16 too, so the honest note on this page matters. The formula is true, the proof is beyond this book, and saying so is the lesson.

Question	Answer
1 cube C minus E plus F	2
1 square pyramid C minus E plus F	2
1 triangular prism C minus E plus F	2
1 tetrahedron C minus E plus F	2
1 octahedron C minus E plus F	2
1a every row gives	2
1b is that a proof	no
1b which week said so	week 8
2 twelve faces thirty edges	20
2 eight corners twelve edges	6
2 twenty faces twelve corners	30
challenge pentagonal prism	2
challenge hexagonal pyramid	2
challenge football faces	32
challenge football edges	90
challenge football corners	60

Week 25 Logic & Proof

Puzzle Fair

A little of everything from Weeks 14 to 24.

In class this week: Review & reasoning

Teaching noteReview. Puzzles 5 and 6 are the same formula twice, once forwards and once backwards.

Question	Answer
1 white for 63 blue	36
2 fifteen for 4.50	0.3
2 twentytwo for 6.16	0.28
2 cheaper	22 for $6.16
3 gcf 45 and 60	15
3 lcm 45 and 60	180
4 taxicab	10
5 corners	9
6 teams	8
6 games each	7

Week 26 Ways of Counting

n Choose k

The counting you have done since Grade 3, finally written as a formula.

In class this week: Expressions and equivalent expressions (6.EE.2, 6.EE.3)

Teaching noteThe over-count-then-divide reason, written once with letters. The n choose 2 line closes the handshake thread for the last time in the line.

Question	Answer
1a lineups	60
1b orders of one team	6
1c five choose three	10
2 6 choose 2	15
2 7 choose 3	35
2 10 choose 2	45
2 8 choose 4	70
3a n choose 2	n(n minus 1) / 2
3a where seen	the handshakes
3b n choose 1	n
3b n choose 0	1
challenge 12 choose 3	220
challenge 12 choose 9	220
challenge pizzas	126
formula agrees everywhere	yes

Week 27 Ways of Counting

Patterns in the Triangle

Every row, every diagonal and every sum in Pascal's triangle means something.

In class this week: Expressions and patterns (6.EE.2, 6.EE.9)

Teaching noteEvery pattern in the triangle has a short reason in choosing language. The building rule proof, take the newest person or leave them, is the best paragraph in the book.

Question	Answer
1a symmetry says	n choose n minus k
1a checked	yes
1b row n adds to	2 to the n
1b checked	yes
2 take the newest	k minus 1
2 leave the newest	k
3a third diagonal	the triangular numbers
3b third entry of row n	n choose 2
challenge hockey stick	20
challenge where it lands	6 choose 3
challenge biggest in row ten	252

Week 28 Ways of Counting

At Least One

Three words that always mean the same instruction: count them all, count the ones with none, subtract.

In class this week: Expressions and reasoning (6.EE.2, 6.EE.9)

Teaching noteThe words at least one should now trigger a reflex. Counting none is one multiplication and counting at least one directly is a sum of several cases.

Question	Answer
1a codes altogether	10000
1b choices with no seven	9
1b codes with no seven	6561
1c at least one seven	3439
1c checked by counting	yes
2 all teams of four	210
2 neither	70
2 at least one	140
2 both	28
challenge at least one 7 and one 3	974
challenge exactly one of them	112

Week 29 Ways of Counting

Three Circles, In Symbols

Grade 5 filled the regions in. This week the picture becomes a formula.

In class this week: Expressions and equivalent expressions (6.EE.2, 6.EE.3)

Teaching noteGrade 5 filled the regions in; this is where it becomes algebra. The all-three student is counted three times and removed three times, leaving zero, which is why the last term adds them back.

Question	Answer
1 region ABC	3
1 region AB	6
1 region AC	3
1 region BC	2
1 region A	10
1 region B	7
1 region C	7
2a regions add to	38
2b the formula	the three pairs, then the triple
2c by the formula	38
2c same	yes
2c no language	2
3 counted by the singles	3
3 removed by the pairs	3
3 leaving them counted	0
challenge none of three	10

Week 30 Ratio & Rate

How Likely Is It?

Chance is a ratio: the outcomes you want against all the outcomes there are.

In class this week: Ratio and rate reasoning (6.RP.1, 6.RP.3c)

Teaching noteChance done properly is a Grade 7 subject. This page does only the counting part, which is a ratio, and the phrase equally likely is doing real work in the definition.

Question	Answer
1 red out of	10
1 red percent	30
1 blue out of	10
1 blue percent	50
1 green out of	10
1 green percent	20
1 not red	7
1 not red percent	70
2a percents add to	100
2b red and not red	100
challenge total from six red	15
challenge better bag	the first

Week 31 Data

Data Has a Shape

Nine numbers on a line. The shape says more than any single number can.

In class this week: Statistical questions and distributions (6.SP.1, 6.SP.2)

Teaching noteBoth plots must be on the same scale or the comparison means nothing. That is the commonest way a data picture misleads.

Question	Answer
1a A bunched around	7
1b B spread from	1
1b B spread to	13
1 both mean	7

Week 32 Data

Center and Spread

One number for where the data sits, one for how far it wanders.

In class this week: Measures of center and variability (6.SP.3, 6.SP.5)

Teaching noteChange one value from 9 to 90 and the mean moves nine points while the median does not move at all. That experiment is the page.

Question	Answer
2 Class A total	63
2 Class A mean	7
2 Class A median	7
2 Class A range	4
2 Class B total	63
2 Class B mean	7
2 Class B median	7
2 Class B range	12
2 which match	the mean and the median
2 which does not	the range
3 new mean	16
3 new median	7
3 median moved	no

Week 33 Data

Same Average, Different Story

Two classes, one average, and you would not want to swap them.

In class this week: Summarizing data in context (6.SP.5, 6.SP.5c)

Teaching noteBoth classes average 7 and you would not want to swap them. A report needs what was measured, how many, a center and a spread; a center alone is incomplete rather than wrong.

Question	Answer
1a both mean	7
1a both median	7
1b range A	4
1b range B	12
3 is the headline true	true but incomplete

Week 34 Data

Finding the Middle

The median needs the data sorted, and sorting has a price you already know.

In class this week: Measures of center; problem solving (6.SP.3, 6.EE.2)

Teaching notePricing statistics is the Grade 5 habit applied somewhere new. The mean is cheap and fragile, the median is dear and steady, and that trade is worth naming.

Question	Answer
ledger selection	4851
ledger merge	566
ledger biggest only	98
1a what else you get	the whole order
1b should the middle be cheaper	yes
2 cheapest to find	the mean
2 dearest to find	the median
challenge mean of 1000	999
challenge median of 1000	8977
challenge range of 1000	1997

Week 35 Logic & Proof

Pigeonhole, Grown Up

Grade 3 put marbles in boxes. Now the boxes are a city.

In class this week: Reasoning and expressions (6.EE.2, 6.EE.9)

Teaching noteKnowing the principle is easy and finding the boxes is not. The upper bound on hairs is doing all the work, and without it there is no argument at all.

Question	Answer
1c boxes	200000
1d conclusion	two people share a hair count
1d forced	5
2a some box holds	5
2b the general rule	T divided by B, rounded up
3 four hundred people birthdays	2
3 five numbers remainders after four	2
challenge twentythree people months	2

Week 36 Patterns & Rules

The Rule Book, Finished

Thirty-three pages of rules. This week they become one book, and the line ends.

In class this week: Review of the year

Teaching noteAssembly, and the end of the line. The last challenge asks the children to trace the handshake problem through all five books, which is worth doing with them.

Question	Answer
2a handshakes rule	n(n minus 1) / 2
2b sum to n	n(n + 1) / 2
2c first n odd numbers	n squared
2d regions rule	not 2 to the n minus 1