Teacher Guide · Count Every Way, Grade 3

Week 1 Ways of Counting

In Order, Every Time

There is a way to list things so that nothing gets skipped.

In class this week: Multiplication as equal groups (3.OA.1)

Teaching noteThe whole year starts here and the point is not the number 6. It is that a child who lists at random cannot tell you whether they are finished. Insist on the fixed order even from children who get 6 straight away. Common wrong answer on part 3: five, because 55 gets left out.

Question	Answer
1c lunches	6
2a with pear	8
3 two-digit from 3,5,7	9
3 list	33, 35, 37, 53, 55, 57, 73, 75, 77
challenge ABC	6
challenge ABCD	24

Week 2 Logic & Proof

Did I Miss Any?

A list is not finished just because you stopped writing.

In class this week: Multiplication and division (3.OA.3)

Teaching noteThree questions, and children will want to skip the third. The 10-cent challenge is the first time they have to argue that a list is complete rather than check it, so give it more time than it looks like it needs.

Question	Answer
1a duplicate	penny and dime
1b missing	penny and quarter
1c should be	6
2b Sam's list	11, 12, 21, 22
3 two boxes	4
3 three boxes	8
challenge 10 cents	4

Week 3 Ways of Counting

Two Stages

When two choices come one after the other, you multiply.

In class this week: Equal groups (3.OA.1, 3.OA.3)

Teaching noteIf a child says the answer is 7, they added instead of multiplying. Send them back to the lines in the picture and ask how many come out of one shirt. The shop question in part 3 has equal answers on purpose.

Question	Answer
1a lines	12
1b per shirt	4
1b total	12
2 sandwiches	15
2 hats	12
2 colors	28
2 books	30
3a shop one	12
3a shop two	12
challenge lock	50
challenge cone types	7

Week 4 Ways of Counting

Rows and Columns Count

An array is not a picture of a count. It is the count.

In class this week: Arrays and area models (3.OA.3)

Teaching noteThe array is the bridge between last week's picture and the multiplication they are doing in class. Part 2 is the commutative property arrived at physically rather than stated.

Question	Answer
1a rows	24
1b columns	24
1c multiply	24
2a turned	24
3 rows drawn	3
3 per row	4
challenge seats	180
challenge working	172
challenge arrays of 12	6

Week 5 Ways of Counting

Three Stages, One Tree

Two stages you can hold in your head. Three, you draw.

In class this week: Properties of multiplication (3.OA.5)

Teaching noteWatch for children who count the tree's branches instead of its endings. The tree gets unwieldy at three stages, which is the argument for trusting the rule.

Question	Answer
1a paths	8
1b multiply	8
2 three breads	12
2 four sauces	24
2 five shirts	30
2 four colors	32
3 endings	12
challenge repeats allowed	27
challenge no repeats	6

Week 6 Ways of Counting

Order Matters

Gold, silver and bronze are not the same as just three winners.

In class this week: Multiplying three factors (3.OA.5)

Teaching noteThe slot method, not listing. A child who lists six finishes will not be able to do four runners. PIP in the challenge has 3 arrangements, not 6, because the two P's are identical, and nobody gets this first time.

Question	Answer
1a gold	3
1b silver	2
1c bronze	1
1d multiply	6
3a gold of four	4
3a silver of four	3
3a bronze of four	2
3b results	24
challenge four books	24
challenge five books	120
challenge PIP	3

Week 7 Ways of Counting

Order Does Not Matter

Picking a team is not the same as handing out medals.

In class this week: The unknown factor (3.OA.6)

Teaching noteThe grid does the work. Do not let a child divide by 2 because you told them to: they should be able to point at two squares in the grid that are the same trip. Part 3 is the assessment for the whole week.

Question	Answer
1a squares	25
1b crossed	5
1c after crossing	20
1d halved	10
2 four friends	6
2 five friends	10
2 six friends	15
2 eight friends	28
3a bench	T
3b captain	M
3c toppings	T
challenge ten friends	45
challenge arranging two of five	20

Week 8 Ways of Counting

Handshakes, Bigger

Everybody shakes hands with everybody, once. Now do it for a hundred people.

In class this week: Fluency within 100 (3.OA.7)

Teaching noteTwo separate ideas: one fewer, because nobody shakes their own hand, and halve, because each handshake sits in the grid twice. Children who lose one of the two usually lose the halving.

Question	Answer
1 lines in circle	15
2 eight people	28
2 ten people	45
2 twelve people	66
2 twenty people	190
3 rule first blank	one fewer
3 rule second blank	2
challenge 21 handshakes	7
challenge hundred people	4950

Week 9 Dots & Lines

Dots, Lines and Degree

How many lines meet at a dot, and what that number gives away.

In class this week: Fluency within 100 (3.OA.7)

Teaching noteDegree is new vocabulary and it is worth being fussy about. The sum being twice the number of lines is the first real theorem in the book, and week 35 pays it off, so make sure part 1c gets said out loud.

Question	Answer
1a sum of degrees	12
1b lines	6
2a sum of degrees	10
2b lines	5
3a four dots degree three	6
3b ten lines	20
challenge five dots degree three	impossible
challenge seven people degree four	14

Week 10 Logic & Proof

Pigeons and Holes

Sometimes you can be sure without looking.

In class this week: Two-step word problems (3.OA.8)

Teaching noteThe uncomfortable part is proving something about boxes nobody has looked in. Children want to name the full box. They cannot, and saying so is the lesson. The hairs challenge is a genuine argument, not a trick.

Question	Answer
1a one each	4
1b at least	2
2 socks	2
2 children	3
2 letters	2
2 pencils	5
challenge must two	yes
challenge must three	yes
challenge must four	no
challenge hairs	yes

Week 11 Steps & Rules

Sorting Your Hand

Two people can both sort correctly and one can still do far less work.

In class this week: Adding and subtracting within 1000 (3.NBT.2)

Teaching noteCount comparisons on real cards, physically. The best and worst cases are the point: the same method, the same answer, very different work. This is the first week of the Steps and Rules strand and Grade 5 builds a whole year on it.

Question	Answer
1a putting 3 in	1
1b putting 1 in	3
1c total	8
2 four eight two six	5
3 already sorted	3
3 backwards	6
challenge five sorted	4
challenge five backwards	10

Week 12 Steps & Rules

Splitting the Search

Guessing one at a time is the slowest possible plan.

In class this week: Rounding to the nearest 10 or 100 (3.NBT.1)

Teaching notePlay this out loud as a game before anybody writes. The 1000 row is the moment worth waiting for: ten questions against a thousand guesses.

Question	Answer
1b questions for 32	5
2 sixteen	4
2 thirty-two	5
2 sixty-four	6
2 one hundred	7
2 one thousand	10
3a slow to 32	32
3b slow to 1000	1000

Week 13 Logic & Proof

Case Files

One case file. Everything from the first twelve weeks is in here somewhere.

In class this week: Review & reasoning

Teaching noteA review week with no new rules. Case 5 is deliberately open: four comparisons is achievable, and a child who says it is not has usually only tried one hand.

Question	Answer
1 lunch cart	24
2 complete	no
2 missing	CD
3 six dots degree two	6
4 tallest stack	4
5 four comparisons possible	yes

Week 14 Shape & Space

Cutting a Rectangle

Area is an array that forgot it was an array.

In class this week: Area by tiling (3.MD.6, 3.MD.7)

Teaching noteArea arrives as an array, which they have had since week 4. Splitting a rectangle is the distributive property in a form a third grader can see.

Question	Answer
1a repeated	24
1b multiply	24
1c two halves	24
2a seven threes	21
2a seven twos	14
2b added	35
2c direct	35
challenge nine eights	72

Week 15 Shape & Space

Same Area, Different Shape

Twelve squares can be laid out several ways, and the fence changes every time.

In class this week: Area and perimeter (3.MD.7, 3.MD.8)

Teaching noteSame area, different perimeter. The square being cheapest is a result they can discover here and will prove properly much later.

Question	Answer
1 twelve by one	26
1 six by two	16
1 four by three	14
1 least fence	4 x 3
2 sixteen by one	34
2 eight by two	20
2 four by four	16
2 the square	4 x 4
challenge smallest of 36	6 x 6
challenge its perimeter	24

Week 16 Shape & Space

Same Perimeter, Different Area

Now hold the fence still and see what happens to the ground.

In class this week: Perimeter (3.MD.8)

Teaching noteThe mirror of last week and worth teaching straight after it. The challenge about the wall changes the answer, which surprises adults too.

Question	Answer
1 seven by one	7
1 six by two	12
1 five by three	15
1 four by four	16
1 most	4 x 4
2 one by nine	9
2 two by eight	16
2 three by seven	21
2 four by six	24
2 five by five	25
2 biggest	25
challenge 24 fence	36
challenge against a wall	72

Week 17 Ways of Counting

Rectangles Inside a Grid

How many rectangles are hiding in a grid? More than you think.

In class this week: Multiplication problems (3.OA.3)

Teaching noteCounting by size is week 1 again, in a place where children do not expect it. Almost everybody forgets the whole grid counts as a rectangle.

Question	Answer
1 one by one	6
1 two by one	4
1 three by one	2
1 one by two	3
1 two by two	2
1 three by two	1
1 total	18
2 total	9
3 total	30
challenge three by three	36
challenge five by two	45

Week 18 Ways of Counting

Factor Pairs

Every way to build a rectangle out of 12 squares is a fact about the number 12.

In class this week: Unknown factors (3.OA.6, 3.OA.4)

Teaching noteFactor pairs as rectangle shapes. The square number having an odd factor count is left hanging on purpose: it is a Grade 4 week.

Question	Answer
1a pairs of 12	1 x 12, 2 x 6, 3 x 4
1b factors of 12	1, 2, 3, 4, 6, 12
1c how many	6
2 factors of 18	6
2 factors of 24	8
2 factors of 7	2
2 factors of 16	5
3a only two	7
3b square	16
challenge exactly two factors	2, 3, 5, 7, 11, 13, 17, 19
challenge odd number of factors	1, 4, 9, 16

Week 19 Ways of Counting

Only Two Rectangles

Some numbers can only be built one way, and that turns out to matter.

In class this week: Multiplication fluency (3.OA.7)

Teaching notePrimes defined by shape rather than announced. Be firm that 1 is neither prime nor composite, and be ready for the argument about 2.

Question	Answer
1a rectangles of 7	1
1a factors of 7	2
1b rectangles of 9	2
1b factors of 9	3
1c prime	7
2 primes to 30	2, 3, 5, 7, 11, 13, 17, 19, 23, 29
2 how many	10
3 is one prime	no
challenge 51	3 x 17
challenge primes adding to 20	3 + 17, 7 + 13

Week 20 Ways of Counting

Sharing Without Remainder

Division asks two different questions, and they are not the same question.

In class this week: Division as equal groups (3.OA.2)

Teaching noteTwo meanings of division, which most curricula run together. A child who draws the same picture for both parts has not seen the difference yet.

Question	Answer
1a groups of six	4
1b shared among four	6
2 thirty-six by four	9
2 thirty-six by six	6
2 thirty-six by nine	4
2 exact count	3
3 the name	factors
challenge bag counts	7
challenge under fifty	12, 24, 36, 48

Week 21 Number Structure

What Is Left Over

The leftovers are not a mistake. They repeat, and the repeating is useful.

In class this week: Interpreting remainders (3.OA.8)

Teaching noteRemainders as a repeating cycle rather than as a failure. The cycle is what makes next week possible.

Question	Answer
1a full groups	4
1a left over	3
2 seventeen by five	3 r 2
2 twenty-nine by four	7 r 1
2 thirty by seven	4 r 2
2 hundred by nine	11 r 1
3 pattern	1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0
3 length	4
challenge hundred by three	1
challenge thousand by three	1

Week 22 Number Structure

The Days Wrap Around

Thirty days from Monday. You can work it out without counting thirty days.

In class this week: Elapsed time (3.OA.8, 3.MD.1)

Teaching noteThe first genuinely useful piece of modular arithmetic. Children who count 30 days on a calendar are not wrong, only slow, so let them do it once and then show the division.

Question	Answer
1a whole weeks	4
1a left over	2
1c answer	Wednesday
2 tuesday plus ten	Friday
2 sunday plus hundred	Tuesday
2 thursday plus 365	Friday
2 friday plus 49	Friday
3 does not move	Friday plus 49
challenge 365 from saturday	Sunday
challenge 366 from saturday	Monday

Week 23 Data

Reading a Scaled Graph

One square does not always mean one thing, and that is where mistakes live.

In class this week: Scaled bar graphs (3.MD.3)

Teaching noteRead the key first. That single habit is the whole week. Two graphs of the same numbers looking different is a good conversation and not a trick question.

Question	Answer
1a room one	35
1b four minus two	20
1c all four	125
2 room one squares	3.5
2 room two squares	2
2 room three squares	3
2 room four squares	4
2 not whole	Room 1
3 squares at one each	40
challenge scale of five shows 34	no
challenge a scale that works	5

Week 24 Data

Counting From a Picture

Half a symbol is where nearly everybody gets it wrong.

In class this week: Scaled picture graphs (3.MD.3)

Teaching noteHalf symbols are where pictographs go wrong. The rule for choosing a key, that it should be a factor of every number, ties straight back to week 18.

Question	Answer
1a monday	35
1b thursday	25
1c most	Friday
1d all week	170
2a can it show 27	no
2b closest	25
2c a key that works	3
3 my key	6
challenge two and a half at six	15
challenge exactly twenty	no

Week 25 Logic & Proof

Puzzle Fair

A little of everything from Weeks 14 to 24. No new rules this week.

In class this week: Review & reasoning

Teaching noteReview of weeks 14 to 24. Puzzle 3 needs a real test, not a feeling: 91 is 7 times 13 and looks prime to nearly everyone.

Question	Answer
1 least fence	6 x 4
2 rectangles in three by three	36
3 is 91 prime	no
3 factor pair	7 x 13
4 wednesday plus hundred	Friday
5 key	7
6 full cups	14
6 left	2

Week 26 Fractions

Parts of a Whole

Equal parts. The word equal is doing all the work in that sentence.

In class this week: Fractions as equal parts (3.NF.1)

Teaching noteEqual is the load-bearing word. Children who shade a bigger piece and call it a fourth have missed the definition, not the drawing.

Question	Answer
1b fraction	1/4
1c three pieces	3/4
3 eight parts take three	3/8
3 six parts take one	1/6
3 three parts take two	2/3
3 ten parts take seven	7/10

Week 27 Ways of Counting

Making One Whole

Counting, but with fraction pieces. Same job: list them in order, miss none.

In class this week: Fractions on a strip (3.NF.1, 3.NF.3)

Teaching noteConverting everything to sixths turns a fraction problem into the counting problem they have been doing since week 1. That move is the lesson.

Question	Answer
1a sixths in a half	3
1a sixths in a third	2
1b whole in sixths	6
3a how many ways	7
challenge with eighths	9
challenge shorter list	the sixths list

Week 28 Fractions

Same Amount, Different Name

One half and three sixths are the same amount wearing different clothes.

In class this week: Equivalent fractions (3.NF.3)

Teaching noteThe strips must be the same length or nothing lines up. Same-top comparison is the one that catches children, because the bigger number makes the smaller fraction.

Question	Answer
1a half in fourths	2
1a half in sixths	3
1b two thirds in sixths	4
1c third in fourths	no
2 three eighths	<
2 four sixths	>
3 two thirds	>
3 three fourths	>
challenge names for a half	1/2, 2/4, 3/6
challenge new line-ups from twelfths	4

Week 29 Dots & Lines

Coloring a Schedule

Five clubs, and some of them cannot meet at the same time.

In class this week: Two-step problems (3.OA.8)

Teaching noteTwo halves to the answer: proving three slots are needed, then finding three that work. Children give one and stop. Ask for both every time.

Question	Answer
1a three that clash	Art, Band, Chess
1b slots needed	3
2 Art	1
2 Band	2
2 Chess	3
2 Drama	1
2 Eco	2
2 slots used	3
3 share a slot	Art and Drama
challenge sixth club	4

Week 30 Dots & Lines

Can Everyone Be Paired?

Four students, four jobs. Sometimes it fits and sometimes nothing you try will.

In class this week: Two-step problems (3.OA.8)

Teaching noteThe failing case is the pigeonhole principle from week 10 wearing a different hat, and a child who spots that has understood both weeks.

Question	Answer
1a only one job	B
1b A	2
1b B	1
1b C	3
1b D	4
2a how many get one	2
2b the problem three	A, B, C
3 the rule	the pigeonhole principle

Week 31 Steps & Rules

Instructions With a Repeat

The same four sides, written once instead of four times.

In class this week: Arithmetic patterns (3.OA.9)

Teaching noteRepeat blocks. Written instructions against carried-out instructions is the distinction worth drawing out, and it is the same idea as counting comparisons in week 11.

Question	Answer
1a program	F F R F F R F F R F F R
1b instructions	12
2a written	3
2b carried out	12
3a program	REPEAT 2 TIMES [ F F F F R F F R ]
3b written	8
3b carried out	16

Week 32 Steps & Rules

Instructions With a Question

A program that looks at what is in front of it and decides.

In class this week: Two-step problems (3.OA.8)

Teaching noteBranching. The sorting program at the end connects to week 11 directly: a repeat with a decision inside it is what sorting a hand actually is.

Question	Answer
1a no shaded	6
1b all shaded	0
2 seven three swap	yes
2 seven three after	3, 7
2 two nine swap	no
2 two nine after	2, 9
2 five five swap	no
2 five five after	5, 5

Week 33 Ways of Counting

Counting Paths on a Grid

Every corner knows how many routes reach it, if you ask the two corners behind it.

In class this week: Arithmetic patterns (3.OA.9)

Teaching noteBuilding up from the corners rather than tracing routes. Children who try to draw every route will get 10 and be exhausted, which makes the method land better.

Question	Answer
1c routes to END	10
2 two by two	6
2 three by two	10
2 three by three	20
2 four by two	15
challenge four by three	35

Week 34 Ways of Counting

The Triangle of Numbers

The route numbers, tipped over. And the handshakes are hiding in it.

In class this week: Arithmetic patterns (3.OA.9)

Teaching noteThe best page of the year. Three unrelated questions from weeks 2, 8 and 33 turn out to share one triangle. Do not explain why. Grade 6 does that.

Question	Answer
1a row five	1, 5, 10, 10, 5, 1
1b row six	1, 6, 15, 20, 15, 6, 1
2a row sums	1, 2, 4, 8, 16, 32, 64
2c third diagonal	1, 3, 6, 10, 15
3 fifteen is in row	6
3 twenty-one would be in row	7
challenge row seven	1, 7, 21, 35, 35, 21, 7, 1
challenge row eight	1, 8, 28, 56, 70, 56, 28, 8, 1
challenge where is twenty	6

Week 35 Logic & Proof

Prove It With Pairing

Nine people, three handshakes each. It cannot happen, and here is why.

In class this week: Even and odd; reasoning (3.OA.9)

Teaching noteThe first proof. A child who says they tried and failed has not proved anything, and the difference between that and the parity argument is the whole point of the week.

Question	Answer
1a nine times three	27
1a is it even	no
2 nine by three total	27
2 nine by three possible	no
2 six by three total	18
2 six by three possible	yes
2 seven by three total	21
2 seven by three possible	no
2 five by four total	20
2 five by four possible	yes
2 eleven by five total	55
2 eleven by five possible	no
challenge odd shakers	always even

Week 36 Ways of Counting

The Counting Book, Finished

Thirty-six weeks of pages. This week you turn them into one book.

In class this week: Review of the year

Teaching noteAssembly week. The only test that matters is whether a child can explain a page out loud without reading it.

Question	Answer
2a cones	24
2b pairs from ten	45
2c friday plus fifty	Saturday