Discrete Math Adventures · Teacher Guide

Grade 2 enrichment · 36 weekly assignments · one school year

Discrete Math Adventures

A full year in the mathematics of structure — sorting, counting, graphs, and algorithms — sequenced so that each week’s puzzle lands near the standard the class is already working on. Every answer, teaching note, and extension is here.

Cadence: One assignment a week, two pages
Time: 30–40 minutes, or split in two
Setting: In class or take-home
Prep: None beyond photocopying

How to run it

One assignment a week, in order. Page 1 opens with a line naming what the class is doing that week, then a two-minute grade-level warm-up, then the main activity. Page 2 has a practice set, a Talk About It prompt, a hint, and the ★ Challenge Zone.

The pages are written to the student, clearly enough to go home in a backpack. Everything a teacher needs to launch one — what to say, what to watch for, what the answers are — lives here rather than on the page, so the sheet works either way.

The order matters more than it looks. Week 3’s parity is the tool that cracks Week 26. Week 7’s sorting method returns in Week 14 on lengths. Weeks 8 and 30 are the same strategy on numbers and then on shapes. Where a week leans on an earlier one, the page says so by name.

Four things worth protecting

Let the stuck moments happen. Weeks 18, 26, and 27 are built around tasks that cannot be done. Rescuing a student early costs them the discovery.

Ask “how do you know?” more than “what’s the answer?” Most weeks have short answers and long reasons; the reasons are the content.

Accept multiple correct answers. Map colorings, robot programs, schedules, and Venn placements often have many. Have students check against the rule, not against a neighbor.

Skipping a week is fine. Nothing except the named callbacks depends on continuity, and the review weeks (13 and 25) absorb a missed week without damage.

Why the year is ordered this way

The sequence follows a typical California Grade 2 pacing guide rather than the internal logic of discrete mathematics. Money week lands on coin counting. Time week lands on clock arithmetic. Geometry weeks land on map coloring and partitioning. The arrays that lead toward multiplication sit in the last month, where the class meets them.

That costs a little elegance — graph theory is split across Weeks 23, 26, 27, and 28 instead of running together — and buys something more useful: the enrichment sheet almost always reinforces what was taught on Tuesday.

If your district’s pacing differs, the year still works in order. The tie-ins are a bonus, not a dependency.

The strongest curriculum ties

Week 3 is 2.OA.3 outright — odd and even developed by pairing, and even numbers written as two equal addends.

Week 32 is 2.G.2 outright, and can substitute for a regular lesson on rows and columns.

Weeks 33–35 are 2.OA.4 three times over: combinations as an array, choice trees, and loops as repeated addition. Students who have stalled on arrays sometimes arrive through the loop.

Weeks 9, 12, 16, and 28 carry heavy grade-level addition — good weeks to schedule when the computation practice needs to be visible.

Weeks 1–13

Trimester 1

Sorting, number structure, and the beginnings of proof.

The class is on fluency within 20, place value, and adding within 100. These thirteen weeks shadow that: parity by pairing, binary as a second base, sorting and searching as ways of comparing numbers, and magic squares and balances as missing-addend practice with a puzzle wrapped around it.

Week 01

What’s My Rule?

Classification by attribute, and inferring a rule from examples and non-examples.

In class this weekAdding & subtracting within 20

2.OA.32.G.1MP7

Running itVague rules (“they’re boxy”) get sharpened by holding up a new shape and asking “would this one count?”

Answer key

Shapes: the shape has 4 sides. The no box holds triangles, circles, pentagons, hexagons.
A: even numbers. B: multiples of 5 (ends in 5 or 0). C: both digits are the same.
For A, some students say “add 4 each time” — also consistent with the four examples shown. Both are defensible given this data, and noticing that is worth more than the answer.

★ Challenge Zone

3, 13, 23, 30, 31: the number contains the digit 3.
Student-made rules: check that the no box actually rules things out. A random no-box means a weak rule.

Week 02

Two Circles, One Middle

Two-attribute sorting, the overlap, and the often-forgotten region outside both circles.

In class this weekSorting; even and odd

2.OA.32.G.12.MD.10

Running itThe classic error is treating the overlap as a third category. Ask: “is C red? is C a star?” Two yeses is what the middle means.

Answer key

Red only: A, E. Stars only: B, F. Both: C, G (so 2).
Stars in all: B, C, F, G = 4. Answering 2 means the middle was forgotten.
Outside both: D (green circle) and H (blue square).
Numbers — middle: 12, 20, 18. Outside both: 7, 3. They are odd and 10 or less.

★ Challenge Zone

Squares and circles: nothing can be both, so the middle is empty. An empty region is a legitimate answer.
No. Outside both means odd and not bigger than 10. A student answering “yes, 101” dropped one condition.

Week 03

Pair Them Up

Parity built from pairing rather than memorized last digits — which is what makes odd + odd explainable.

In class this weekOdd and even numbers (2.OA.3)

2.OA.3

Running itCounters help enormously. Two odd groups each have one lonely counter; the two lonely ones pair with each other.

Answer key

7 is odd. 8 is even.
even + even = even (10) · even + odd = odd (11) · odd + odd = even (12)
Equal parts: 8 = 4 + 4, 12 = 6 + 6, 16 = 8 + 8, 20 = 10 + 10.
9 cannot split into two equal parts — the closest is 4 + 5, and they are not equal. That is what odd means.
Even numbers circled: 24, 46, 68, 70.

★ Challenge Zone

15 cookies: not fair between 2 (odd); fair between 3 (5 each); fair between 5 (3 each). Parity settles the 2 case only — worth naming.
7 + 7 + 7 is odd (21). odd + odd = even, then even + odd = odd. Ask for the reasoning before the arithmetic.

Week 04

Always, Sometimes, Never

The difference between an example and a reason. “Sometimes” requires producing a case both ways.

In class this weekFluency within 20

2.OA.32.G.1MP3

Running itFor every “always”, ask “could you find one that breaks it?” Failing to find a counterexample is how children first feel what proof is for.

Answer key

Page 1: 1 Always · 2 Sometimes · 3 Never · 4 Always · 5 Always · 6 Sometimes.
Item 6 (a rectangle is a square) will cause argument. It is true only when all four sides match. This is the most valuable item on the page.
Page 2: 1 Never (odd + odd is even) · 2 Sometimes · 3 Always · 4 Sometimes.
Item 4 catches almost everyone: 100 is bigger than 10 and has three digits.

★ Challenge Zone

Why never: an even number splits into pairs with nothing left over; a number ending in 5 always leaves one behind. Testing ten examples is not this reason — say that difference out loud.
Student statements: the real test is whether their “sometimes” genuinely has both cases.

Week 05

How Computers Count

Binary as place value with a different multiplier. A genuine second angle on 2.NBT.1.

In class this weekPlace value to 1000 (2.NBT.1)

2.NBT.12.NBT.3

Running itPhysical cards with dots on one side work best — flipping a card face down is writing a zero. Four students each holding a card is better still.

Answer key

5 = 0101 · 9 = 1001 · 12 = 1100 · 15 = 1111
3 = 0011 · 6 = 0110 · 10 = 1010 · 13 = 1101 · 14 = 1110
Biggest with four cards: 15. 0 0 0 0 is zero.
Normal numbers multiply by 10 each place; computer numbers multiply by 2. That one sentence is the whole lesson.

★ Challenge Zone

Counting 0–15, the last column reads 0, 1, 0, 1… and every even number has 0 there. Week 3’s parity, hiding in binary.
With a 16 card the biggest is 31; the next card is worth 32. Each new card doubles — and is one more than everything before it combined.

Week 06

Number Machines

Function rules: input, rule, output. And the discovery that order of operations matters.

In class this weekSkip counting by 5s, 10s, 100s (2.NBT.2)

2.NBT.22.OA.2MP7

Running itPush past the first plausible rule. Ask “does your rule work for every example, or just the first one?”

Answer key

+ 10 machine: 12 → 22, 40 → 50, and 28 → 38.
Secret machine: double. 10 → 20, 20 → 40.
Machine C: subtract 5. 60 → 55. Machine D: count by 5s (times 5). 6 → 30.
Machine E: the number stays the same (add 0). 99 → 99. Students often resist calling this a rule — it is one.
Counting by 5s: 5, 10, 15, 20, 25, 30, 35, 40.
Talk: + 10 and double agree at 10 (and only there).

★ Challenge Zone

3 → +10 → 13 → double → 26.
Other order: 3 → double → 6 → +10 → 16. Different answer — order matters. This is a genuinely surprising result at this age and worth dwelling on.

Week 07

Put Them in Order

Two sorting algorithms on the same data, and comparing methods by effort rather than by answer.

In class this weekComparing numbers (2.NBT.4)

2.NBT.4MP8

Running itNumber cards on the desk beat pencil work — the swapping should be physical.

Answer key

Sorted: 8, 12, 27, 34, 50.
Method 2 passes from 34, 12, 50, 27, 8:
pass 1 → 12, 34, 27, 8, 50 · pass 2 → 12, 27, 8, 34, 50
pass 3 → 12, 8, 27, 34, 50 (not sorted yet) · pass 4 → 8, 12, 27, 34, 50 ✓
Three-digit cards: 204, 240, 418, 481.

★ Challenge Zone

At most 4 passes for 5 cards — one fewer than the number of cards. So 10 cards need at most 9.
By digit sum: 12 (3), 50 (5), 34 (7), 8 (8), 27 (9). A different order — same algorithm, different comparison.

Week 08

Twenty Questions

Binary search. Students discover that halving beats guessing one at a time, and by how much.

In class this weekGreater than / less than (2.NBT.4)

2.NBT.4MP7MP8

Running itPlay it aloud as a class first, with you picking the number. Let someone try the one-at-a-time method and feel it fail.

Answer key

Worked example (11): more than 8? YES → 9–16. More than 12? NO → 9–12. More than 10? YES → 11–12. More than 11? NO → 11. Four questions.
Most questions ever needed for 1–16: 4.
Halving: 16 → 8 → 4 → 2 → 1.

★ Challenge Zone

1–32 needs 5 questions. 1–64 needs 6. Every time the list doubles, you need exactly one more question.
Same idea as Week 5: each yes/no answer is one bit. Four cards make 16 numbers; four questions find one of 16.

Week 09

The Shortest Way to the Gold

Weighted shortest path, and the discovery that fewer stops does not mean fewer steps.

In class this weekAdding within 100 (2.NBT.5)

2.NBT.52.OA.1

Running itThis page carries the most grade-level arithmetic in the trimester — four sums within 100 on page 1 alone.

Answer key

HOME → BARN → GOLD = 80 · HOME → BARN → CAVE → GOLD = 80
HOME → POND → CAVE → GOLD = 65 ◀ shortest · HOME → POND → GOLD = 100
Second map: START → MILL → END = 70; START → FARM → END = 70; START → FARM → MILL → END = 65 ◀ shortest.
In both maps the shortest route has the most stops. That is the point of the page.

★ Challenge Zone

Pond–Cave floods: the shortest becomes 80, and two routes tie for it. Ties surprise students who expect one answer.
Longest: HOME → POND → GOLD at 100 — 35 steps more than the shortest.

Week 10

Magic Squares

Missing addends inside a structure, and a first taste of proving a number has to be what it is.

In class this weekMissing addends (2.OA.1)

2.OA.12.NBT.5MP7

Running itInsist on finding a line that is missing only one number rather than guessing. That habit is the transferable part.

Answer key

Square 1: 8 1 6 / 3 5 7 / 4 9 2 (every line makes 15)
Square 2: 9 2 7 / 4 6 8 / 5 10 3 (every line makes 18)
Square 3: the total is 15 (from the full middle row). Completed: 4 9 2 / 3 5 7 / 8 1 6.
Talk: the middle number is always the magic total divided by 3 (5, 6, 5).

★ Challenge Zone

1 to 9 add to 45, and the three rows cover every number exactly once, so the three row-totals add to 45. Three equal totals making 45 must each be 15. This is a real proof and second graders can follow it.
Add 10 to every number: still magic. Each line gains 30, so the new total is 45.

Week 11

Balance Puzzles

Equations as balance rather than as “the answer goes here”. Substitution, one step at a time.

In class this weekEquations and unknowns (2.OA.1)

2.OA.12.OA.2

Running itA real balance scale with cubes is worth the setup time. Students who see it physically stop treating = as “makes”.

Answer key

▲ = 5, ■ = 7, ● = 8 → ▲ + ■ + ● = 20
★ = 4, ♥ = 7, ◆ = 9 → ★ + ♥ + ◆ = 20
Which balance: ★+★=8 YES · ♥+★=◆+3 NO (11 vs 12) · ◆+◆=20 NO (18) · ♥+♥=★+10 YES (14 = 14)

★ Challenge Zone

▲ = 4 and ■ = 8. Since ■ is 4 more than ▲, the puzzle becomes ▲ + ▲ + ▲ + 4 = 16. Guess-and-check is a perfectly good route here and should be praised, not corrected.
Student-made puzzles: check theirs has exactly one answer. Many will accidentally write one with several.

Week 12

The Handshake Problem

Counting pairs, and the double-counting correction. The triangular numbers arrive.

In class this weekAdding within 100 (2.NBT.5)

2.OA.12.NBT.5MP8

Running itDo it physically with five volunteers before anyone draws anything. The double-counting becomes obvious in the body first.

Answer key

10 handshakes. A shakes 4 hands, B shakes 4 hands.
Why not 20: each handshake involves two people, so counting “4 hands each” counts every handshake twice.
Table: 2 → 1, 3 → 3, 4 → 6, 5 → 10, 6 → 15, 7 → 21.
Each new friend adds one more handshake than the last one did.
Sums: 4+3+2+1 = 10 · 5+4+3+2+1 = 15 · 6+5+4+3+2+1 = 21.

★ Challenge Zone

10 students → 45 handshakes. Build it from the table rather than drawing.
Waves: 90 — exactly double, because a wave only goes one way, so nothing is double-counted.

Week 13

Detective Week

Trimester 1 review, run as a case file. Deduction by elimination on a 4×4 grid.

In class this weekReview & reasoning

MP1MP3

Running itInsist on the grid rather than mental solving. Marking an ✗ is the visible act of reasoning; the ✓ comes for free after.

Answer key

Ana — cat · Ben — dog · Cruz — fish · Dee — rabbit.
Case 1: the rule is multiples of 10 (ends in 0).
Case 2: 23 + 23 is even — odd + odd. (It is 46.)
Case 3: the rule is half. 40 → 20.
Case 4: a good first question is “is it more than 8?”

★ Challenge Zone

No clue can be removed. Drop clue 1 and Ana/Ben can swap cat and dog; drop clue 2 or 3 and two arrangements survive. Every clue is load-bearing — a satisfying thing to verify.
Student-written puzzles: the usual failure is too few clues, giving several answers. Have them solve their own before trading.

Weeks 14–25

Trimester 2

Measurement, money, time, and data — each with a discrete idea attached.

This is the stretch where the regular curriculum leaves pure number, and the pairings get more interesting: sorting applied to lengths, tilings as repeated units, coin counting as exhaustive listing, greedy algorithms as making change, modular arithmetic as a clock face, and graphs as a bar-graph data source.

Week 14

Sorting by Length

Week 7’s algorithm applied to a non-numeric attribute — the first explicit transfer of a method.

In class this weekMeasuring length (2.MD.1, 2.MD.4)

2.MD.12.MD.42.NBT.5

Running itPaper strips cut to these lengths make the comparison-without-a-ruler challenge real rather than imagined.

Answer key

Lengths: A = 6, B = 9, C = 3, D = 7, E = 5.
Sorted shortest to longest: C, E, A, D, B.
B − C = 6 · D − E = 2 · C + E = 8
Which two equal B + C (12)? D and E (7 + 5).

★ Challenge Zone

Sorting by comparison alone works fine. Method 1 needs 4 + 3 + 2 + 1 = 10 comparisons for five strips — the handshake number from Week 12, which is a nice thing to point out.
Two equal strips are allowed. The sorted list is then not unique, which is worth noticing but not a problem.

Week 15

Covering a Strip

Tilings, organized listing, and the Fibonacci recurrence discovered from the student’s own data.

In class this weekLength as repeated units (2.MD.1)

2.MD.12.OA.4MP8

Running itThe organizing move is: first every way that starts with a short tile, then every way that starts with a long one. Model it for length 3.

Answer key

Length 1 → 1 way. Length 2 → 2. Length 3 → 3. Length 4 → 5.
Length 5 → 8. Length 6 → 13.
The pattern: each number is the sum of the two before it (1, 2, 3, 5, 8, 13 — the Fibonacci numbers).
Why: a strip of length 5 either starts with a short tile (leaving a strip of 4) or a long tile (leaving a strip of 3). So the ways for 5 are the ways for 4 plus the ways for 3.

★ Challenge Zone

Length 7 → 21 (13 + 8).
With a 3-unit tile added: 1, 2, 4, 7, 13… Now you add the three numbers before, for the same reason — three choices of first tile.

Week 16

Two Ways Across the Park

Adding lengths to compare whole routes; the same structure as Week 9 with measurement units.

In class this weekAdding lengths; number line (2.MD.5, 2.MD.6)

2.MD.52.MD.62.NBT.5

Running itThe number line is where the comparison becomes visible. Have them use two colors for the two paths.

Answer key

Path 1 (through OAK): 24 + 18 = 42 cm
Path 2 (through POND): 30 + 22 = 52 cm
Path 1 is shorter, by 10 cm.
More lengths: 45 − 28 = 17 · 34 + 29 = 63 · 80 − 55 = 25

★ Challenge Zone

24 + 18 + 30 = 72. That is the only set of the four that works.
Shortening a part of Path 2 is the only thing that could change the answer, and only if it drops Path 2 below 42 — a good discussion about which improvements actually matter.

Week 17

Color the Kingdom

Graph coloring. Finding a constraint that makes fewer colors impossible, not just hard.

In class this weekShapes and attributes (2.G.1)

2.G.1MP1

Running itCrayons needed. Enforce the shared-border rule: touching at one corner does not count. Accept any correct coloring and have students check borders rather than compare with a neighbor.

Answer key

Map A — one valid 3-coloring: 1 red, 2 blue, 3 green, 4 green, 5 red, 6 blue. Many others work.
Two colors fail because regions 2, 3, and 5 all touch one another.
Map B (ring of five): 3 colors. An odd ring can never be done in two.
Map C (two rows of three): 2 colors — a checkerboard. This is the contrast: same number of regions, different answer.

★ Challenge Zone

Regions 1 and 4 must be different in any 3-coloring, even though they never touch. If both were red, then 2, 3, and 5 would need three colors from the two remaining. A genuinely surprising result.
A ring of six needs only 2 colors. Odd rings need 3; even rings need 2. That is the whole idea, and students can find it.

Week 18

When Three Colors Is Not Enough

A map that provably needs four colors, and the story of the Four Color Theorem.

In class this weekAttributes; impossibility (2.G.1)

2.G.1MP3

Running itLet them fail with three colors first. The stuck moment is the lesson — do not rescue it early.

Answer key

Three colors do not work. Wedges A, B, and C all touch each other, so they use all three; then D in the middle touches all three.
D is the region that gets stuck.
Four is the fewest that works.
Any student map with four mutually touching regions is correct — the wedge-and-centre shape is the simplest.

★ Challenge Zone

Five is impossible on a flat map. That is exactly the Four Color Theorem — conjectured 1852, proved 1976 by Appel and Haken with computer assistance. Students cannot prove it, but they can feel why nobody could draw a counterexample.
On a doughnut the answer is 7. A doughnut has a hole, so regions can wrap around and touch in ways they cannot on a flat page.

Week 19

How Many Ways to Make 15¢?

Exhaustive counting with a systematic organizer — and the realization that one choice is forced.

In class this weekMoney (2.MD.8)

2.MD.8MP7

Running itReal coins, or coin manipulatives. The organizing move is to fix the number of dimes first and let the pennies fall out.

Answer key

15¢ — 6 ways: (1 dime, 1 nickel) · (1 dime, 5 pennies) · (3 nickels) · (2 nickels, 5 pennies) · (1 nickel, 10 pennies) · (15 pennies).
20¢ — 9 ways: 2 dimes · 1 dime + 2 nickels · 1 dime + 1 nickel + 5p · 1 dime + 10p · 4 nickels · 3 nickels + 5p · 2 nickels + 10p · 1 nickel + 15p · 20 pennies.
Why the organizer works: once dimes and nickels are chosen, the pennies are forced. So there are really only two choices to make, not three.

★ Challenge Zone

25¢ — 12 ways with pennies, nickels, and dimes (6 with no dimes, 4 with one dime, 2 with two dimes).
Allowing a quarter makes 13. The only new way is the single quarter itself.

Week 20

The Fewest Coins

A greedy algorithm — and, in the challenge, a case where greedy is provably wrong.

In class this weekMoney word problems (2.MD.8)

2.MD.82.NBT.5MP3

Running itThe challenge is the most important part of this week. Do not skip it for students who finish the front quickly.

Answer key

68¢ = 2 quarters, 1 dime, 1 nickel, 3 pennies = 7 coins
47¢ = 1 quarter, 2 dimes, 2 pennies = 5 coins
93¢ = 3 quarters, 1 dime, 1 nickel, 3 pennies = 8 coins
36¢ = 1 quarter, 1 dime, 1 penny = 3 coins
80¢ = 3 quarters, 1 nickel = 4 coins. Most coins needed: 93¢.

★ Challenge Zone

Greedy in the 1-7-10 country: 10 + 1 + 1 + 1 + 1 = 5 coins.
The real best is 7 + 7 = 2 coins. Greedy fails. It works with our coins because of how their values are built — not because “biggest first” is always right. This is a genuine algorithmic insight and second graders can hold it.

Week 21

Clock Arithmetic

Modular arithmetic on a familiar object. Wrapping is not an error — it is the system.

In class this weekTelling time to 5 minutes (2.MD.7)

2.MD.72.NBT.2MP2

Running itA geared demonstration clock earns its keep here. Have students physically walk the hand around past 12.

Answer key

From 9 o’clock: +5 → 2 · +8 → 5 · +12 → 9 · −3 → 6
Table: 11 + 4 → 3 · 7 + 8 → 3 · 10 + 5 → 3 · 6 + 12 → 6 · 2 + 11 → 1
Minutes: 10:40 + 25 = 11:05 · 3:50 + 15 = 4:05 · 8:05 − 10 = 7:55
Why +12 returns you home: twelve hours is exactly one trip around the whole clock.

★ Challenge Zone

100 hours after 3 o’clock is 7 o’clock. Skip-count by 12: 96 hours is eight full trips, leaving 4 more. 3 + 4 = 7.
Tuesday + 30 days is Thursday. 28 days is four full weeks, leaving 2.

Week 22

Getting Ready for the Party

Precedence, partial order, and the critical path — why more helpers stop helping.

In class this weekTime and duration (2.MD.7)

2.MD.7MP1

Running itSticky notes on the board that students physically reorder make the “many valid orders” point land instantly.

Answer key

Could go first: Bake cake, or Blow up balloons.
One valid order: Bake cake, Frost cake, Set table, Blow up balloons, Hang balloons. Many others work.
Could happen at the same time: any cake job with any balloon job (e.g. Bake cake and Blow up balloons).
Never at the same time: any two joined by an arrow — Bake cake and Frost cake, for instance.
Alone: 50 minutes. With a helper: 30 minutes — the cake chain is three jobs long and cannot be split, so no number of helpers beats 30.

★ Challenge Zone

10 orders. The three cake jobs stay in order and the two balloon jobs stay in order; you are choosing which two of the five slots the balloon jobs take.
Adding Buy candles makes it 40 minutes with one helper — six jobs is 60 minutes of work split two ways, and the schedule can actually reach it.

Week 23

Dots and Lines

Graphs as a model of relationships; degree; the handshake lemma; and a real bar graph of the data.

In class this weekBar graphs (2.MD.10)

2.MD.102.OA.2

Running itDraw the class’s own graph on the board first. Say explicitly that “graph” here does not mean bar graph — then have them build a bar graph from it anyway.

Answer key

Friends: Mia 3, Leo 2, Ivy 2, Sam 3, Ann 2.
Most: Mia and Sam (tied). Fewest: Leo, Ivy, Ann (tied).
Lines: 6. Column total: 12.
Why double: every line has two ends, so each line gets counted once at each of its two dots.
Bar graph: equal bars for Leo, Ivy, Ann. Mia has 1 more than Leo.

★ Challenge Zone

Every dot with 2 lines: possible — a ring of five.
Every dot with 3 lines: impossible. The column total would be 15, but that total is always double the number of lines, so it must be even. A real impossibility proof, at seven years old.

Week 24

Who Wins the Tournament?

Counting by what is eliminated rather than by what is drawn — and logarithmic growth again.

In class this weekReading data (2.MD.10)

2.MD.102.NBT.5MP8

Running itLet them count boxes first and get 7 the hard way. The elimination argument lands better once they have the number.

Answer key

Round 1: 4 games. Round 2: 2. Round 3: 1. Total: 7.
Table: 2 teams → 1 game, 1 round · 4 → 3 games, 2 rounds · 8 → 7, 3 · 16 → 15, 4 · 32 → 31, 5
Teams knocked out to leave one champion: 7.
So games = teams − 1, always, for any knockout tournament.

★ Challenge Zone

100 teams → 99 games, no drawing required. Ninety-nine teams have to be eliminated and each game eliminates exactly one.
64 teams → 6 rounds; 128 → 7. Doubling the teams adds one round — the same halving pattern as Week 8.

Week 25

Puzzle Fair

Trimester 2 review, run as stations.

In class this weekReview & reasoning

MP1MP3

Running itRun it as actual stations if you can. The fifth station — writing a puzzle — tells you more about understanding than the other four combined.

Answer key

Station 1: 2 colors (the map is a checkerboard).
Station 2: 57¢ = 2 quarters, 1 nickel, 2 pennies = 5 coins.
Station 3: 8 + 7 → 3 o’clock; 8 − 10 → 10 o’clock.
Station 4: 16 teams → 15 games, 4 rounds. Known without drawing: every game eliminates one team.
Station 5: open. Check their stated answer actually solves their puzzle.

★ Challenge Zone

To force three colors on Station 1, add a region that touches three regions which already touch each other — or draw a diagonal border creating an odd ring.
The worst amounts under a dollar need 9 coins — 94¢ (3 quarters, 1 dime, 1 nickel, 4 pennies) and 99¢ (3 quarters, 2 dimes, 4 pennies).

Weeks 26–36

Trimester 3

Networks, geometry, and the arrays that lead toward multiplication.

The strongest curriculum overlap in the year sits at the end: Weeks 32 to 35 are all 2.OA.4 and 2.G.2 in disguise. Before that, Euler paths use odd and even as a working tool, and spanning trees and lattice paths carry the three-digit addition the class is doing.

Week 26

Trace It Without Lifting

Euler paths. Students derive a real theorem from data they generate themselves.

In class this weekOdd and even as a tool (2.OA.3)

2.OA.3MP8

Running itHave them count degrees and predict before tracing. The prediction is the mathematics; the tracing only checks it.

Answer key

Odd dots — A: 2 · B: 4 · C: 2 · D: 0 · E: 2 · F: 0
Traceable: A yes · B no · C yes · D yes · E yes · F yes
The rule: traceable with exactly 0 or 2 odd dots.
With 2 odd dots you must start at one of them and you will finish at the other. With 0 you finish where you started.
Why: passing through a dot uses two lines, one in and one out. So an odd dot can only be the start or the end.

★ Challenge Zone

Königsberg: impossible. All four landmasses are odd, and 4 is more than 2. Euler settled this in 1736 — the first result in graph theory.
Figure B: erase any one line. The two dots it touched drop from 3 to 2, leaving exactly 2 odd dots.

Week 27

The Bridges of the Old City

Applying last week’s rule to a famous real problem, and to maps the students design.

In class this weekAdding within 1000 (2.NBT.7)

2.OA.32.NBT.7MP3

Running itDouble bridges are easy to miscount. Have students trace each bridge with a finger while counting aloud.

Answer key

Bridge counts: North 3, South 3, Island 5, East 3.
All four are odd — four odd landmasses, and the rule allows at most two. The walk is impossible.
Map X-Y-Z: X 3, Y 3, Z 2 → 2 odd → possible, starting at X and finishing at Y (or the reverse).
Map P-Q-R-S: every island has 2 → 0 odd → possible, and you finish exactly where you started.

★ Challenge Zone

Start and finish in the same place: every island must have an even number of bridges.
Finish somewhere else: exactly two islands are odd, and those two are the start and the end.

Week 28

Connecting the Towns

A minimum spanning tree, built with a greedy rule that this time actually is optimal.

In class this weekAdding within 1000 (2.NBT.7)

2.NBT.72.NBT.5MP1

Running itCompare with Week 20: greedy failed there and works here. Students who ask why are asking the right question — it is fine to say that mathematicians proved this one works.

Answer key

Cheapest set: B–C (15), A–B (20), C–D (25), D–E (30). Total 90, using 4 roads.
C–E (50) and B–D (40) and A–C (35) are all skipped — each would close a loop or cost more.
Smaller map: Q–R (12), P–Q (18), R–S (22). Total 52, using 3 roads.
Why 4 roads for 5 towns: 3 roads cannot reach every town; a 5th would close a loop and be wasted.

★ Challenge Zone

Losing D–E (30): the only other way to reach E is C–E (50), so the new total is 110.
Picking most-expensive-first does connect every town, but costs far more — on this map, 50 + 40 + 35 + 30 = 155. Being greedy for the wrong thing is still greedy.

Week 29

Counting Routes

Lattice paths built up corner by corner. This is Pascal’s triangle, arrived at honestly.

In class this weekAddition patterns (2.NBT.7)

2.NBT.72.OA.4MP7

Running itThe discipline is to finish each row before moving up. Students who jump ahead get lost; students who go in order cannot fail.

Answer key

2 across, 2 up: 6 routes.
3 across, 2 up: 10 routes.
3 across, 3 up: 20 routes.
Why adding works: the only way to arrive at a corner is from below it or from its left, so the ways to reach it are the ways to reach those two added together.
Why every route is 4 steps on the first grid: you must go right twice and up twice, in some order.

★ Challenge Zone

4 across, 4 up: 70 routes.
Blocking the middle corner leaves 2 routes. Four of the six went through the middle. Students can find this by writing 0 at the blocked corner and continuing the addition — a lovely moment.

Week 30

Guess My Shape

Week 8’s halving strategy applied to attributes instead of numbers.

In class this weekShape attributes (2.G.1)

2.G.1MP7

Running itRule out “is it the circle?” style questions early by asking what happens when the answer is no.

Answer key

Exactly 4 sides — YES: A square, B rectangle, G diamond, H trapezoid.
NO: C triangle, D circle, E pentagon, F hexagon.
Most questions ever needed: 3, if every question splits the group in half.
Other good splitting questions: “does it have more than 4 sides?” (2 vs 6), “are all its sides the same length?”
Why ‘is it the circle?’ is weak: a no leaves seven shapes; a good question leaves four.

★ Challenge Zone

3 questions for 8 shapes, against 4 for 16 numbers in Week 8. Halving is the same idea in both.
16 shapes → 4 questions. 32 → 5. Doubling the collection adds exactly one question.

Week 31

Cutting Fair Shares

Equal parts that are not congruent. This is where “same size” and “same shape” come apart.

In class this weekHalves, thirds, fourths (2.G.3)

2.G.3MP3

Running itPaper folding beats drawing. Have them cut the parts out and stack them to check equality.

Answer key

Four ways to quarter a square: four vertical strips · four horizontal strips · four small squares (quadrants) · four triangles from the centre. All are correct.
Yes — two fourths of the same square can be different shapes and still be the same amount. Cutting them out and comparing settles the argument.
Unequal pizza pieces are not fourths. Fourths means four equal parts, not four parts.
One half is bigger than one fourth of the same object — the classic reversal, because the bigger number names the smaller piece.

★ Challenge Zone

Lines from the centre to the four corners give four triangles — equal, and none is a square or rectangle. A pinwheel cut also works.
Four small squares, each a quarter-size copy of the original. Every square can be cut into four smaller squares like this, forever.

Week 32

Filling a Rectangle

Rows and columns of same-size squares; the commutative property made visible.

In class this weekRows and columns (2.G.2)

2.G.22.OA.4

Running itThis is the most directly on-standard week in Trimester 3 and can substitute for a regular 2.G.2 lesson.

Answer key

4 rows of 3: by rows 3 + 3 + 3 + 3 = 12; by columns 4 + 4 + 4 = 12.
2 rows of 5: 5 + 5 = 10 and 2 + 2 + 2 + 2 + 2 = 10.
3 rows of 3: 3 + 3 + 3 = 9 both ways.
8 squares: 1×8 and 2×4 (and their turns).
Why rows and columns agree: they are two ways of counting the same squares, and nothing was added or removed.

★ Challenge Zone

12 squares: 1×12, 2×6, and 3×4 (with their turns).
12, 18, and 20 tie for the most arrangements between 1 and 20 — six each. Prime numbers (2, 3, 5, 7, 11, 13, 17, 19) give only one rectangle. Students are meeting factors here without the word.

Week 33

The Ice Cream Shop

Systematic counting, and the array as a picture of the same count.

In class this weekArrays and repeated addition (2.OA.4)

2.OA.4

Running itWatch for random listing. The move to teach: hold the flavour fixed, run through every topping, then switch.

Answer key

6 bowls: Vanilla with Sprinkles / Fudge / Cherry, then Chocolate with Sprinkles / Fudge / Cherry.
As addition: 3 + 3 = 6 (by rows) or 2 + 2 + 2 = 6 (by columns).
3 flavours × 3 toppings = 9, as 3 + 3 + 3.
Adding a fourth topping does not mean starting over — each flavour simply gains one more bowl.

★ Challenge Zone

Cone or cup with 2 flavours and 3 toppings: 12. Every bowl already found splits in two.
3 flavours, 3 toppings, cone or cup: 18. Push for “9 doubled” rather than a list of 18.

Week 34

Choice Trees

The same count in a second representation, and the doubling that comes from repeated choices.

In class this weekArrays and repeated addition (2.OA.4)

2.OA.4MP7

Running itFill in the first shirt’s branches together, then let them finish. Emphasise that each path is one outfit, not each box.

Answer key

6 paths: striped with red / blue / green, plain with red / blue / green.
Same as Week 33 because both are two choices followed by three choices — identical structure, different story.
Two coin flips: 4 endings (HH, HT, TH, TT). Three flips: 8.
Table: 1 flip → 2, 2 → 4, 3 → 8, 4 → 16.

★ Challenge Zone

The ice cream problem draws as a tree with 2 first-level branches and 3 below each — the same tree with different labels.
5 flips → 32 endings, of which exactly 1 is all heads.

Week 35

Program the Robot

Algorithms, shortest programs, and loops as repeated addition.

In class this weekRows, columns, repeated addition (2.OA.4)

2.OA.42.G.22.OA.1

Running itHave one student read arrows aloud while another moves a counter, following exactly what is said. Mismatches teach precision faster than corrections.

Answer key

One 8-move solution: ↑ ↑ ↑ ↑ → → → →
Fewest possible: 8. The robot must gain 4 columns and 4 rows, and each move changes only one of those — so 8 is a floor regardless of walls.
Going right first fails: the wall at C5 blocks the bottom row.
Loops: REPEAT 4 [↑] = ↑↑↑↑ · REPEAT 3 [→ ↑] = →↑→↑→↑ · ↓↓↓↓↓ = REPEAT 5 [↓]
REPEAT 4 [3 moves] = 12, and a 4-by-3 rectangle holds 12 squares. Same number, two pictures.

★ Challenge Zone

Star route, still 8 moves: ↑ ↑ ↑ → → (on the star) → → ↑. It works because the star already sits on a shortest path.
Square: REPEAT 4 [FORWARD 3, TURN RIGHT].

Week 36

Secret Codes

A shift cipher and a binary decode — a rule applied systematically, which is where the year started.

In class this weekPlace value review (2.NBT.1)

2.NBT.1MP2

Running itEnding on decoding is deliberate: every child gets a moment of the message resolving. Leave real time for them to write one for a partner.

Answer key

Slide code: N B U I J T G V O → MATH IS FUN
Computer code: 0100 = 4, 0001 = 1, 0100 = 4 → 4, 1, 4 → D A D
Their own message: the best clues describe the method without giving away the key. That distinction is worth naming.

★ Challenge Zone

QWT EQFG slid back two is OUR CODE.
Z is 26, so four cards (max 15) cannot reach it — add a 16 card and you can write up to 31.
MATH with cards 16 8 4 2 1: M = 13 = 01101, A = 1 = 00001, T = 20 = 10100, H = 8 = 01000.

Standards this year touches

California adopted the Common Core State Standards for Mathematics, so these are the CA CCSSM Grade 2 codes. This is enrichment — it reinforces and extends these standards rather than delivering them, and no standard here is taught to mastery by this program alone.

Content standards

2.OA.1Use addition and subtraction within 100 to solve one- and two-step word problems
2.OA.2Fluently add and subtract within 20
2.OA.3Determine whether a group of objects is odd or even; write an even number as a sum of two equal addends
2.OA.4Use addition to find the total number of objects arranged in rectangular arrays
2.NBT.1Understand three-digit place value
2.NBT.2Count within 1000; skip-count by 5s, 10s, and 100s
2.NBT.3Read and write numbers to 1000 using base-ten numerals
2.NBT.4Compare two three-digit numbers using >, =, and <
2.NBT.5Fluently add and subtract within 100
2.NBT.7Add and subtract within 1000
2.MD.1Measure the length of an object using appropriate tools
2.MD.4Measure to determine how much longer one object is than another
2.MD.5Use addition and subtraction within 100 to solve problems involving lengths
2.MD.6Represent whole numbers as lengths on a number line
2.MD.7Tell and write time to the nearest five minutes
2.MD.8Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies
2.MD.10Draw a picture graph and a bar graph to represent a data set
2.G.1Recognize and draw shapes having specified attributes
2.G.2Partition a rectangle into rows and columns of same-size squares and count them
2.G.3Partition circles and rectangles into halves, thirds, and fourths

Standards for Mathematical Practice

MP1Make sense of problems and persevere in solving them
MP2Reason abstractly and quantitatively
MP3Construct viable arguments and critique the reasoning of others
MP7Look for and make use of structure
MP8Look for and express regularity in repeated reasoning

The practice standards are where this program earns its place in the week. Nearly every page asks students to look for structure, argue from a reason, or persevere past a first failed attempt — and several ask them to prove something is impossible, which the regular curriculum rarely does at this age.