Station quizzes — extraction review & answer key

20 units · 118 station pages · 472 problems · 133 auto-checked

Snippet edits applied overnight — see the "✂️ Snippet edits" tables per unit: every rewrite was generated from the guide's own words (no invented facts) and every drop is listed with its reason. Revert everything by deleting quizgen/snippet-overrides.json and rebuilding.

Every item below was pulled out of kbase/mom/guides/unit-*-teaching.html by extract_items.py and this page is the audit trail: prompt, key, level, source row, and exactly what the auto-checker will accept. Auto items grade themselves in the student quiz; talk items are discussion/graph prompts where the group types nothing useful and the key sits behind one tap. Station numbers show which sections made the 7-question group-work cut (greyed rows were dropped by the cap). Use this page as the answer key when you walk the room.

01 · Logic & Reasoning [LR1] 01-logic-reasoning-lr18 stations built / 8 sections in the guide11 auto of 32 items

Jump to: 1.1 · 1.1.5 · 1.2 · 1.2.5 · 1.3 · 1.4 · 1.5 · | 🖨 QR sheet for this unit ↗

1.1 — Vocabulary Station 1 QR sheet ↗ student page ↗

🔥 Think first: If you had to explain what a line is to an alien who has never seen one, how would you (without using the word "line")?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	True or false: a plane extends infinitely in all directions. ★	True —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Use the figure: name the plane a common way (e.g. three non-collinear points).	Plane ABC (or any three non-collinear points on it). —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Can two different planes intersect at a single point? Explain.	No — two planes intersect in a line (or not at all), never in a single point. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	How many planes can contain a given line? Explain.	Infinitely many — the line can be rotated like an axis to sweep out infinitely many planes. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

1.1.5 — Angles Station 2 QR sheet ↗ student page ↗

🔥 Think first: Without a protractor, estimate: is 55° closer to 45° or to 90°? How confident are you?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Classify a 95° angle.	Obtuse —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Measure an angle with a protractor and write its measure in degrees. ★	e.g. 72° (varies) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Two angles are supplementary. One is 3 times the other. Find both. ★	x + 3x = 180 → x = 45°, other = 135°. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	List 3 ways to name the angle with vertex B between two rays from A and C. ★	∠ABC, ∠CBA, ∠B. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

1.2 — Segment Addition Postulate Station 3 QR sheet ↗ student page ↗

🔥 Think first: Five points are spaced out evenly on a ruler. Add the two gaps to get the total — done. Now WHY does the whole length always equal the sum of the parts?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner twosided	A, B, C collinear, AB = 9, BC = 7. Find AC. ★	16 16	auto	`#` Just the number — no units, no x=.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Medium Intermediate twosided	A, B, C collinear, AC = 40, AB = 24. Find BC. ★	16 16	auto	`#` Just the number — no units, no x=.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Hot Advanced twosided	B between A,C; AB = 3x, BC = x+6, AC = 30. Find x. ★	3x + x+6 = 30 → 4x = 24 → x = 6. 6, x=6	auto	`x=#` Just the value, or write it as x=# — both count.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Spicy Challenge twosided	A, B, C, D collinear in order. AB = BC = CD and AD = 27. Find BD.	BD = BC+CD = 9+9 = 18 (AD/3 = 9 each). 18	auto	`#` Just the number — no units, no x=.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.

1.2.5 — Overlapping Segments Conjecture Station 4 QR sheet ↗ student page ↗

🔥 Think first: Two rulers overlap by a middle part. If both ends stick out the same amount, what is true about the overlapping middle?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner twosided	A, B, C, D collinear, AB = CD = 4, AD = 18. Find BC. ★	BC = 18 − 4 − 4 = 10. 10	auto	`#` Just the number — no units, no x=.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Medium Intermediate twosided	A, B, C, D collinear in order, AB = CD. If AD = 22 and BC = 8, find AB. ★	AB=(AD−BC)/2 = (22−8)/2 = 7. 7	auto	`#` Just the number — no units, no x=.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Hot Advanced twosided	A, B, C, D collinear, AB = 3y−1, CD = 2y+3, and AB = CD. Find y. ★	3y−1 = 2y+3 → y = 4. 4, y=4	auto	`y=#` Just the value, or write it as y=# — both count.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Spicy Challenge discuss	Why does AB = CD force AD to be symmetric around the midpoint of BC?	With AB and CD equal, the distance from A to the midpoint equals that from D behind it — reflect across the midpoint of BC and AB maps to CD. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

1.3 — Truth Tables Station 5 QR sheet ↗ student page ↗

🔥 Think first: If I tell you "the lights are out AND the door is locked," and the lights are out but the door is open — is my statement true?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	p=T,q=T. p AND q = ? ★	True —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	p=T,q=F. p OR q = ? ★	True —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	p=F,q=T. ¬p AND q = ? ★	¬p=T, q=T → True. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge arith	Two statements. How many rows in the truth table, and why?	4 rows because each of 2 statements has 2 possible values: 2×2=4. 4	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.

1.4 — Inductive & Deductive Reasoning Station 6 QR sheet ↗ student page ↗

🔥 Think first: "Every crow I have seen is black" — is that enough to say all crows are black? What single thing could change your mind?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Inductive or deductive: "Every triangle I tested has 3 sides, so all triangles have 3 sides." ★	Inductive —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Find a counterexample to "All prime numbers are odd." ★	2 is prime and even. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Classify: "If a figure is a square then it is a rhombus. Y is a square, so Y is a rhombus." ★	Deductive, valid. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Can inductive reasoning ever prove a statement? Explain. ★	No — it can strongly suggest a conjecture, but only one counterexample (or a proof/deductive argument) settles truth. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

1.5 — Handshake Formula Station 7 QR sheet ↗ student page ↗

🔥 Think first: Ten people all shake once. Guess the answer, then derive it — why is it NOT 10×9?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner counting	4 people — total handshakes. ★	4·3/2 = 6. 6	auto	`#` Just the number — no units, no x=.	Count the small case first — work out the answer for 3 or 4 by hand and look for the pattern. Each new item pairs with every one already there, so the total grows by a predictable amount.
Medium Intermediate discuss	A 10-team league, everyone twice? (round-robin, pairs once) ★	10·9/2 = 45 games. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced counting	A choir of 20 members — total unique pairings. ★	20·19/2 = 190. 190	auto	`#` Just the number — no units, no x=.	Count the small case first — work out the answer for 3 or 4 by hand and look for the pattern. Each new item pairs with every one already there, so the total grows by a predictable amount.
Spicy Challenge discuss	Show the handshake formula has the same meaning as choosing 2 from n (n choose 2).	n choose 2 = n!/(2!·(n−2)!) = n(n−1)/2, exactly the handshake count. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T1), Note Making & Rubric Station 8 QR sheet ↗ student page ↗

🔥 Think first: Boil the whole unit to ONE line that would survive on a desert island — what is it?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Classify: inductive/deductive "numbers 2,4,6,8 so next is 10." ★	Inductive —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate twosided	A, B, C collinear, AC=50, AB=3x, BC=2x. Find x. ★	5x=50 → x=10. 10, x=10	auto	`x=#` Just the value, or write it as x=# — both count.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Hot Advanced discuss	Find a counterexample to "all right angles are generated by congruent legs." ★	Any two different right angles are both 90° but need not come from congruent figures — e.g. a 3-4-5 right triangle. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Prove that if two angles are vertical, they are congruent, citing the linear-pair postulate.	See seminar A — let ∠1 and ∠3 be vertical; each is supplementary to ∠2 so ∠1=∠3. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

02 · Logic & Reasoning 2 [LR2] 02-logic-reasoning-lr27 stations built / 7 sections in the guide2 auto of 28 items

Jump to: 2.1 · 2.3 · 2.3.5 · 2.4 · 2.5 · 2.6 · | 🖨 QR sheet for this unit ↗

2.1 — 2.2 Constructions (NOT tested) Station 1 QR sheet ↗ student page ↗

🔥 Think first: Using only an unmarked straightedge and a compass, can you place a point exactly halfway along a line segment? Try to convince a skeptic it is exact.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	List the two allowed construction tools.	straightedge + compass —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	To copy a segment AB onto a ray, what compass setting do you use?	Set the compass to length AB and mark that distance from the ray endpoint. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Describe the construction of a perpendicular bisector of segment PQ.	Arcs of equal radius from P and Q; the two intersections define a line that cuts PQ at its midpoint at 90°. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Justify: the constructed midpoint is exactly halfway.	Both arc-intersection points are equidistant from P and Q (same radius), so the line they define is the set of points equidistant — hence it contains and bisects PQ at its midpoint. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

2.3 — Special Angle Pairs Station 2 QR sheet ↗ student page ↗

🔥 Think first: Two lines cross. If one angle is 30°, can you find all four WITHOUT measuring?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Two angles are if their measures sum to 90°. ★	complementary —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	m∠X = 117°. Find its supplement. ★	63° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Vertical angles: one is (5x−3)°, the other (2x+9)°. Find the angle. ★	5x−3 = 2x+9 → x=4 → angle = 17°. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	An angle and its supplement are equal. Find it, then explain. ★	x = 180−x → x=90°; a right angle is its own supplement. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

2.3.5 — Transversals & Parallel Lines Station 3 QR sheet ↗ student page ↗

🔥 Think first: Two parallel rails are "cut" by a crossing tie. Are the two acute angles on opposite sides equal? Why?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Corresponding angles on parallel lines are ______. ★	equal —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Parallel lines, transversal: an alternate interior angle is 73°. Find its consecutive interior neighbor. SolutionConsecutive interior = 180 − 73 = 107°.	— —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Consecutive interior: (4x−20)° and (x+5)°. Find each. ★	4x−20+x+5=180 → 5x=195 → x=39 → 136° and 44°. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	If two lines are cut by a transversal and a pair of corresponding angles is NOT equal, what must be true? ★	The lines are not parallel. Equal corresponding angles is the test for parallelism. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

2.4 — Triangle Angle Sum Conjecture Station 4 QR sheet ↗ student page ↗

🔥 Think first: Fold the three corners of a paper triangle together at one point — what do they form? What does that prove?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Angles 58°, 67°, and a missing angle sum to 180°. Find it. ★	55° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	A 90° and a 41° angle — find the third. ★	49° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Angles (x+30)°, x°, (3x−10)°. Find each. ★	x+30+x+3x−10=180 → 5x=160 → x=32 → 62°,32°,86°. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Can a triangle have angles 50°, 60°, and 70°? Justify. ★	Yes — they sum to 180°. Every triangle whose angles sum to 180° exists. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

2.5 — Isosceles Triangles Station 5 QR sheet ↗ student page ↗

🔥 Think first: If two sides of a triangle are equal, must the two base angles be equal too? Build one and look.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Isosceles vertex 70° — each base angle? ★	55° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Base angles each 48° — the vertex angle? ★	84° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Base angle = vertex angle − 30°. Find all angles. ★	Let base=b, vertex=b+30: 2b+(b+30)=180 → 3b=150 → b=50°, vertex 80°. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Why is an equilateral triangle also equiangular? Find its angle. ★	All three sides equal → all three angles equal → each is 60°. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

2.6 — Triangle Inequality Theorem Station 6 QR sheet ↗ student page ↗

🔥 Think first: Can a 2-inch, a 3-inch, and a 9-inch stick form a triangle? Predict, then try to connect them.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Can 3, 7, 4 form a triangle? ★	No — 3+4=7, not > 7. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Can 6, 6, 6? ★	Yes — all equal, equilateral. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced countask	Two sides 8 and 3 — range of the third side? ★	5 < x < 11. 5<x<11, x<11andx>5, x>5andx<11	auto	`#<x<#` Put x in the middle: #<x<# — or type both parts joined with and.	You are told a property of the shape and asked for its sides — set up the one formula that links the two. Use the sum formula for the interior angles, or the 'exterior angles add to 360' fact, then solve it for n.
Spicy Challenge discuss	Sides a, b, a+b — possible? ★	No — a+b is not less than a+b (needs strictly greater). —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T2) Station 7 QR sheet ↗ student page ↗

🔥 Think first: Boil the whole unit to ONE sentence that survives on a desert island.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Vertical angles sum to? (each pair) ★	They are equal, not a fixed sum. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Complement of 34°? ★	56° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced simplify	Transversal: corresponding (5x)° and (x+80)°. Find x. ★	5x = x+80 → x=20. 20, x=20	auto	`#` Just the number — no units, no x=.	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.
Spicy Challenge discuss	Is 5,5,10 a triangle? ★	No — 5+5=10, not > 10. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

03 · Quadrilaterals [CP1] 03-quadrilaterals-cp19 stations built / 9 sections in the guide9 auto of 36 items

Jump to: 3.1 · 3.2 · 3.2.5 · 3.3 · 3.4 · 3.5 · 3.6 · 3.7 · | 🖨 QR sheet for this unit ↗

3.1 — Polygon Classification Station 1 QR sheet ↗ student page ↗

🔥 Think first: Name a shape with 3, 5, and 7 sides in three seconds each.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	A 5-sided polygon. ★	pentagon —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Which is NOT a polygon: circle, triangle, square? ★	a circle —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced countask	Each exterior 45° — sides? ★	8 8	auto	`#` Just the number — no units, no x=.	You are told a property of the shape and asked for its sides — set up the one formula that links the two. Use the sum formula for the interior angles, or the 'exterior angles add to 360' fact, then solve it for n.
Spicy Challenge discuss	Why is a non-square rectangle not regular? ★	Not all sides equal. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.2 — Interior Angles of Polygons Station 2 QR sheet ↗ student page ↗

🔥 Think first: A hexagon — can you find its angle sum by breaking it into triangles?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Heptagon interior sum. ★	900° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Regular hexagon one interior. ★	120° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced countask	Interior sum 1800 — sides? ★	12 12	auto	`#` Just the number — no units, no x=.	You are told a property of the shape and asked for its sides — set up the one formula that links the two. Use the sum formula for the interior angles, or the 'exterior angles add to 360' fact, then solve it for n.
Spicy Challenge discuss	Why does (n−2)·180 work? ★	From one vertex, n−3 diagonals split the polygon into n−2 triangles, each 180°. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.2.5 — Exterior Angle Sum Station 3 QR sheet ↗ student page ↗

🔥 Think first: Walk any closed hallway, turning only at corners. How much do you turn in total?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Exterior sum of any polygon. ★	360° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Regular 15-gon one exterior. ★	24° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced countask	Each exterior 30 — sides? ★	12 12	auto	`#` Just the number — no units, no x=.	You are told a property of the shape and asked for its sides — set up the one formula that links the two. Use the sum formula for the interior angles, or the 'exterior angles add to 360' fact, then solve it for n.
Spicy Challenge discuss	Why always 360°? ★	Turning once fully around a polygon is a full 360° turn. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.3 — Classifying Triangles & Quadrilaterals Station 4 QR sheet ↗ student page ↗

🔥 Think first: Sort a handful of shapes into the fewest bins. What rules do you invent?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	No equal sides — triangle type. ★	scalene —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	One pair of parallel sides — quadrilateral. ★	trapezoid —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Which is always true: square is a rhombus, or rhombus is a square? ★	A square is a rhombus. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	A rectangle but not a square. ★	e.g. a 2x5 rectangle. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.4 — Parallelogram Properties Station 5 QR sheet ↗ student page ↗

🔥 Think first: Push a rectangle top to the side. It is still a parallelogram. What stays equal?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Consecutive angles of a parallelogram. ★	supplementary —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	A 40° angle — the others. ★	40°, 140°, 140° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Half-diagonals (4x−3),(3x+2). ★	x = 5 → 17 each —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	A parallelogram with a right angle. ★	a rectangle. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.5 — Rhombus, Rectangle & Square Properties Station 6 QR sheet ↗ student page ↗

🔥 Think first: If a rhombus has a right angle, what does it become?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Rhombus diagonals are ______. ★	perpendicular —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate simplify	Rectangle diagonals 8 and (3x−1). ★	x = 3 3	auto	`#` Just the number — no units, no x=.	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.
Hot Advanced discuss	Square side 7 diagonal. ★	7√2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Why is a square all of rhombus+rectangle+parallelogram? ★	Meets every definition. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.6 — Kite & Trapezoid Properties Station 7 QR sheet ↗ student page ↗

🔥 Think first: A trapezoid and a kite — what makes each special? Half plus half?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	A trapezoid is isosceles when its legs are ______. ★	congruent —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate arith	Bases 12 and 18 — midsegment. ★	15 15	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced measure	Kite with perpendicular diagonals 6 and 10 — area. ★	30 30	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Why is the midsegment parallel to the bases? ★	It connects the leg midpoints and averages the base length. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.7 — Midsegment Properties Station 8 QR sheet ↗ student page ↗

🔥 Think first: —

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	A triangle midsegment parallel to a side of length 16 — its length. ★	8 8	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate arith	Two midpoints on sides, third side 13 — the midsegment. ★	6.5 6.5	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced discuss	Midsegment (3x) and third side (4x+2). Find each. ★	2·3x = 4x+2 → 2x = 2 → x = 1 → 3 and 6. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Why is the midsegment parallel? ★	It joins midpoints of two sides; by the triangle-midsegment theorem the slope matches the third side. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T3) Station 9 QR sheet ↗ student page ↗

🔥 Think first: Boil this unit to ONE sentence that survives a desert island.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Decagon interior sum. ★	1440° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	A parallelogram 120° — the others. ★	60°,120°,60° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced measure	Kite diagonals 7 and 4 — area. ★	14 14	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Can a rhombus be a rectangle? ★	Yes — if it is a square. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

04 · Area of Polygons [AV1] 04-area-polygons-av14 stations built / 4 sections in the guide12 auto of 16 items

Jump to: 4.1 · 4.2 · 4.3 · | 🖨 QR sheet for this unit ↗

4.1 — Rectangles & Parallelograms Station 1 QR sheet ↗ student page ↗

🔥 Think first: A 3×4 parallelogram is leaning — can Houdini-ish you rearrange it into a rectangle without changing the area?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner measure	Rectangle 8 by 5 — area. ★	40 40	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Medium Intermediate measure	Parallelogram base 12, height 4 — area. ★	48 48	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced measure	Area 90, base 15 — the height. ★	6 6	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Why does A=b·h for a parallelogram? ★	Slice the "leaning" piece off and slide it to make a rectangle of width b and height h. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

4.2 — Triangles, Trapezoids & Kites Station 2 QR sheet ↗ student page ↗

🔥 Think first: A triangle is half of what simple shape? Use it to guess the area formula.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner measure	Triangle base 6, height 8 — area. ★	24 24	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Medium Intermediate measure	Trapezoid bases 9 and 5, height 4 — area. ★	28 28	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced measure	Kite diagonals 8 and 13 — area. ★	52 52	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Why is the triangle area ½·b·h but the rectangle b·h? ★	A triangle is half a parallelogram/rectangle of the same base and height. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

4.3 — Area of Regular Polygons Station 3 QR sheet ↗ student page ↗

🔥 Think first: A regular hexagon — can you see 6 identical triangles in it? Their total is the area.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner measure	Regular polygon apothem 3, perimeter 30 — area. ★	45 45	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Medium Intermediate discuss	Hexagon apothem √3, perimeter 24 — area. ★	½·√3·24 = 12√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced measure	Apothem 7, area 140 — perimeter. ★	P = 2·140/7 = 40 40	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Show that ½·a·P gives the sum of 6 triangles for a hexagon. ★	Each of the 6 triangles has base = side, height = apothem; total = ½·a·(6·side) = ½·a·P. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T4) Station 4 QR sheet ↗ student page ↗

🔥 Think first: Boil this unit to ONE sentence.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner measure	Rectangle 9 by 6 — area. ★	54 54	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Medium Intermediate measure	Triangle base 13, height 4 — area. ★	26 26	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced measure	Trapezoid bases 7 and 11, height 6 — area. ★	54 54	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge measure	Composite: a 10×6 rectangle plus a triangle height 4 on its 6-side — total area. ★	60 + ½·6·4 = 60 + 12 = 72. 72	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.

05 · Prisms and Pyramids [AV2] 05-prisms-pyramids-av25 stations built / 5 sections in the guide11 auto of 20 items

Jump to: 5.1 · 5.2 · 5.3 · 5.4 · | 🖨 QR sheet for this unit ↗

5.1 — Vocabulary & Nets Station 1 QR sheet ↗ student page ↗

🔥 Think first: Fold a paper triangle-prism net in your head. How many faces, edges, vertices?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	A solid with 2 circular bases is a ______. ★	cylinder —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	A solid with one circular base and a vertex is a ______. ★	cone —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	How many vertices on a triangular prism? ★	6 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	What net folds into a square pyramid? ★	One square + four triangles. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

5.2 — Prism & Pyramid Volume Station 2 QR sheet ↗ student page ↗

🔥 Think first: A box 2×3×4 — can you guess its volume without counting cubes?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner measure	Rectangular prism 3,4,5 — volume. ★	60 60	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Medium Intermediate measure	Pyramid, base area 18, height 5 — volume. ★	30 30	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced measure	Prism, base area 27, height 9 — volume. ★	243 243	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Why is a pyramid exactly ⅓ of a prism of the same base and height? ★	Cavaleri/Cavalieri-style: the pyramid fills ⅓; a cube is 3 pyramids of equal volume. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

5.3 — Surface Area Station 3 QR sheet ↗ student page ↗

🔥 Think first: Wrap a 4×5×3 box in paper — what is the least paper?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	Cube side 3 — SA. ★	54 54	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate arith	Rectangular prism 4,4,5 — SA. ★	2(16)+2(20)+2(20) = 112 112	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced arith	Square pyramid base 8, slant 6 — SA. ★	64 + 4·½·8·6 = 64+96 = 160 160	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Spicy Challenge discuss	When does box wrapping need SA vs volume? ★	Wrapping (material covering the outside) needs SA; filling needs volume. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

5.4 — Applications Station 4 QR sheet ↗ student page ↗

🔥 Think first: Are you filling the box or wrapping it? Two different numbers.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	Compute the water in a 6×3×2 prism tank. ★	36 36	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate arith	Fabric for a 5×3×2 box lid-less (top open). ★	SA minus top = 2(15)+2(10)+(3·2)? Actually sides+base = 30+20+6=56 56	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced measure	A pyramid heap base 30, height 6 — volume. ★	⅓·30·6 = 60 60	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Why does a cone/pyramid hold a third of a prism of the same base/height? ★	Cavalieri principle of equal stacked layers — a ⅓ factor. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T5) Station 5 QR sheet ↗ student page ↗

🔥 Think first: Boil this unit to ONE sentence.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner measure	Pyramid base area 9, height 6 — volume. ★	18 18	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Medium Intermediate measure	Rectangular prism 2,3,4 — volume. ★	24 24	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced discuss	Cone r=2,h=9 — volume in π. ★	12π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Sphere r=1 — volume in π. ★	4/3π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

06 · Circles [CP2] 06-circles-cp26 stations built / 6 sections in the guide3 auto of 24 items

Jump to: 6.1 · 6.2 · 6.3 · 6.4 · 6.5 · | 🖨 QR sheet for this unit ↗

6.1 — Circle Vocabulary Station 1 QR sheet ↗ student page ↗

🔥 Think first: Trace a jar lid. Now describe every line you could draw that touches the circle just once.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	A line that touches a circle at exactly one point is a ______. ★	tangent —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate arith	Radius 8 — diameter. ★	16 16	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced discuss	A chord that does not pass through the center relates to a diameter how? ★	It is shorter than the diameter. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Name the region bounded by an arc and a chord.	a segment (of a circle). —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

6.2 — Central, Inscribed Angles & Chords Station 2 QR sheet ↗ student page ↗

🔥 Think first: An inscribed angle opens wide or narrow — why does it only depend on the arc it sees?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Inscribed angle 25° — its arc. ★	50° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Central angle 110° — arc. ★	110° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Inscribed angle halves an 84° arc — the angle. ★	42° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	An angle inscribed in a semicircle is ____ — why? ★	90°; it intercepts a 180° arc, half is 90°. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

6.3 — Clock Angles Station 3 QR sheet ↗ student page ↗

🔥 Think first: At exactly 3:00, how full of the circle is between the hands?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Angle at 1:00. ★	30° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Angle at 5:00. ★	150° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Which time gives 180°? ★	6:00 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Why is each hour mark 30°? ★	360°/12 hours = 30° per hour. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

6.4 — Tangent Properties Station 4 QR sheet ↗ student page ↗

🔥 Think first: A tangent just kisses the circle — at what angle does it meet the radius?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Tangent ⊥ radius — angle at tangency. ★	90° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate simplify	Tangents from P: 5 and (x+1). Find x. ★	x = 4 4	auto	`#` Just the number — no units, no x=.	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.
Hot Advanced arith	A tangent and radius form a right triangle; radius 3, tangent 4 — distance from center to the external point. ★	√(9+16) = 5 5	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Spicy Challenge discuss	Prove the two tangents from an external point are equal. ★	The two right triangles have equal hypotenuse (center-to-P) and equal leg (radius), so congruent by HL → tangents equal. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

6.5 — Cyclic Quadrilaterals Station 5 QR sheet ↗ student page ↗

🔥 Think first: Stretch a rubber-band square until all four corners touch a circle — what stays true about opposite corners?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Cyclic quadrilateral opposite angles. ★	supplementary —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	One angle 58°, its opposite. ★	122° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Opposite angles (4x)° and (5x)° — find each. ★	4x+5x=180 → x=20 → 80°,100° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Can a parallelogram inscribed in a circle be anything besides a rectangle? ★	Only a rectangle — both pairs of opposite angles are 90° (supplementary and equal). —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T6) Station 6 QR sheet ↗ student page ↗

🔥 Think first: Boil this unit to ONE sentence.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Central angle 140° — arc. ★	140° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Inscribed angle on a 100° arc. ★	50° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Radius 5 — circumference. ★	10π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Angle at 2:00. ★	60° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

Jump to: 7.1 · 7.2 · 7.3 · | 🖨 QR sheet for this unit ↗

7.1 — Circumference & Area of a Circle Station 1 QR sheet ↗ student page ↗

🔥 Think first: Trace a jar lid. One string measures the rim. Is that string longer or shorter than the straight flashlight-beam across? About how many times?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Circumference of a circle with radius 12 cm. ★	24π cm —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate measure	Area of a circle with radius 6 cm. ★	36π cm² 36picm^2	auto	`#picm^#` Type the expression with no spaces; use ^ for exponents and / for fractions.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced discuss	Circumference of a circle with diameter 16 cm. ★	16π cm —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	A circle has circumference 20π. Find its area. ★	100π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

7.2 — Sector, Segment & Annulus Areas Station 2 QR sheet ↗ student page ↗

🔥 Think first: A pizza is cut into 8 equal slices. What fraction of the pizza is one slice — and how does that match (θ/360)·πr²?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner measure	Area of a sector with radius 4 m and central angle 45°. ★	2π m² 2pim^2	auto	`#pim^#` Type the expression with no spaces; use ^ for exponents and / for fractions.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Medium Intermediate twosided	Annuulus: r = 3 cm, R = 6 cm. ★	27π cm² 27picm^2	auto	`#picm^#` Type the expression with no spaces; use ^ for exponents and / for fractions.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Hot Advanced discuss	Segment area: r = 8, central angle 90°. ★	16π − 32 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	A sector of radius 12 has area 24π. Find its central angle. ★	(θ/360)·144π = 24π → θ = 60° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

7.3 — Surface Area of Cones & Cylinders Station 3 QR sheet ↗ student page ↗

🔥 Think first: Wrap a sheet of paper around a soup can. The paper is a rectangle: one side is the height, the other is the circumference. Surface area = the paper + two lids.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Surface area of a cylinder: r = 3, h = 10. ★	78π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Surface area of a cone: r = 4, slant height 9. ★	52π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced arith	A cone has SA 96π and radius 6. Find the slant height. ★	10 10	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Spicy Challenge arith	A cylinder has SA 112π and radius 4. Find the height. ★	10 10	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.

— Review (T7) Station 4 QR sheet ↗ student page ↗

🔥 Think first: Boil this unit down to ONE sentence you can say in 10 seconds.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Circumference of a circle with radius 12. ★	24π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate measure	Sector of radius 4 m with angle 45° — area. ★	2π m² 2pim^2	auto	`#pim^#` Type the expression with no spaces; use ^ for exponents and / for fractions.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced discuss	Cylinder r = 3, h = 10 — surface area. ★	78π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Segment r = 8, angle 90° — area. ★	16π − 32 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

08 · Right Triangles [RT1] 08-right-triangles-rt15 stations built / 5 sections in the guide10 auto of 20 items

Jump to: 8.1 · 8.2 · 8.3 · 8.4 · | 🖨 QR sheet for this unit ↗

8.1 — Pythagorean Theorem Station 1 QR sheet ↗ student page ↗

🔥 Think first: Three squares of sides 3, 4, 5 — do the two small squares tile the big one exactly? Tape it and see.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	Legs 6 and 8 — hypotenuse. ★	10 10	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate arith	Legs 9 and 12 — hypotenuse. ★	15 15	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced arith	Leg 30, hypotenuse 50 — other leg. ★	40 40	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Spicy Challenge discuss	Both legs are x, hypotenuse is 10 — find x. ★	x² + x² = 100 → x = 5√2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

8.2 — Pythagorean Triples & Converse Station 2 QR sheet ↗ student page ↗

🔥 Think first: Can you find a right triangle whose sides are all whole numbers and the hypotenuse is 13? (Answer: 5-12-13.)

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Is (3, 4, 5) a Pythagorean triple? ★	yes (9 + 16 = 25) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Do sides 6, 8, 10 make a right triangle? ★	yes —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Do sides 25, 70, 74 make a right triangle? ★	no —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge simplify	A triple is (11, 60, c). Find c. ★	121 + 3600 = 3721 = 61² → c = 61 61, c=61	auto	`#` Just the number — no units, no x=.	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.

8.3 — Applications Station 3 QR sheet ↗ student page ↗

🔥 Think first: A 10-ft ladder with its foot 6 ft out — how high? Guess first, then check with 100 = 36 + 64.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner scale	A 13 ft ladder, foot 5 ft from the wall — height reached. ★	12 ft 12, 12ft	auto	`#ft` Type the expression with no spaces; use ^ for exponents and / for fractions.	Write the scale as a ratio with matching units (cm with cm, km with km). Multiply by the ratio that cancels the unit you are leaving.
Medium Intermediate measure	20-inch diagonal, 12-inch-high monitor — area. ★	192 in² 192in^2	auto	`#in^#` Type the expression with no spaces; use ^ for exponents and / for fractions.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced discuss	Square with diagonal 22 — perimeter. ★	44√2 ≈ 62.2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Square with diagonal 14 — side and area. ★	side 7√2; area 98 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

8.4 — 8.5 Distance Formula & Circle Equations Station 4 QR sheet ↗ student page ↗

🔥 Think first: Stand at (0,0). Who is farther: the point 3 right and 4 up, or 2 right and 5 up? Count the diagonal distance, not the straight-line guess.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	Distance between (0, 0) and (3, 4). ★	5 5	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate simplify	Equation of the circle with center (4, −14), radius 2. ★	(x−4)² + (y+14)² = 4 4	auto	`#` Just the number — no units, no x=.	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.
Hot Advanced discuss	Center and radius of (x+5)² + (y+6)² = 36. ★	center (−5,−6), radius 6 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Classify triangle A(−2,1), B(3,11), C(7,9). ★	scalene (11.2, 4.5, 12.0) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T8) Station 5 QR sheet ↗ student page ↗

🔥 Think first: Boil this unit to one sentence in 10 seconds.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	Legs 5 and 12 — hypotenuse. ★	13 13	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate scale	Monitor: 20-inch diagonal, 12 high — width. ★	16 in 16, 16in	auto	`#in` Type the expression with no spaces; use ^ for exponents and / for fractions.	Write the scale as a ratio with matching units (cm with cm, km with km). Multiply by the ratio that cancels the unit you are leaving.
Hot Advanced discuss	Distance (−2,1) to (3,11). ★	5√5 ≈ 11.2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Center and radius of (x+5)² + (y+6)² = 36. ★	(−5,−6), 6 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

09 · Special Right Triangles [RT2] 09-special-right-triangles-rt26 stations built / 6 sections in the guide5 auto of 24 items

Jump to: 9.1 · 9.2 · 9.3 · 9.4 · 9.5 · | 🖨 QR sheet for this unit ↗

9.1 — Simplifying Radicals Station 1 QR sheet ↗ student page ↗

🔥 Think first: Build √72 with square tiles: how many 6×6 squares hide inside 72? The leftover is the odd bit inside the root.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Simplify √12. ★	2√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Simplify √512. ★	16√2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Simplify √72. ★	6√2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge rationalexp	Simplify (3√7)². ★	63 63	auto	`#` Just the number — no units, no x=.	Rewrite the fractional exponent as a root: x^(1/n) is the nth root. Evaluate the root before you do any multiplication.

9.2 — Add, Subtract & Multiply Radicals Station 2 QR sheet ↗ student page ↗

🔥 Think first: How many √7s are in 12√7 + 3√7? Count the coefficients: 15 altogether.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner rationalexp	Simplify √8 · √2. ★	4 4	auto	`#` Just the number — no units, no x=.	Rewrite the fractional exponent as a root: x^(1/n) is the nth root. Evaluate the root before you do any multiplication.
Medium Intermediate discuss	Simplify 3√112 + 3√7. ★	15√7 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Simplify √50 − √18. ★	2√2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Rationalize 3/√7. ★	3√7/7 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

9.3 — Special Right Triangles Station 3 QR sheet ↗ student page ↗

🔥 Think first: Fold a square corner to corner: the diagonal is longer than the side — but by exactly √2, every time, no matter the square size.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	45-45-90 with legs 11 — hypotenuse. ★	11√2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	30-60-90, short leg 5 — hypotenuse and long leg. ★	10 and 5√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	30-60-90 with hypotenuse 14 — the legs. ★	7 and 7√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	30-60-90 with long leg 9 — the hypotenuse. ★	6√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

9.4 — Areas from Special Triangles Station 4 QR sheet ↗ student page ↗

🔥 Think first: An equilateral triangle of side 10 — guess its height: 5? 8? 9? (It is 5√3 ≈ 8.66.)

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Area of an equilateral triangle with side 6. ★	9√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Area of an equilateral triangle with side 10. ★	25√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced measure	Area of a right isosceles triangle with hypotenuse 14. ★	49 49	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge measure	Area of a right isosceles triangle with hypotenuse 10. ★	25 25	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.

9.5 — Circle Applications Station 5 QR sheet ↗ student page ↗

🔥 Think first: A square just fits inside a circle. Which is bigger: the circle or the square? By how much in radius 5?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Square inscribed in circle radius 5 — side. ★	5√2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Square inscribed in circle radius 5 — shaded area (circle minus square). ★	25π − 50 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Regular hexagon inscribed in circle radius 8 — area. ★	96√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Regular hexagon with side 16 — area. ★	384√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T9) Station 6 QR sheet ↗ student page ↗

🔥 Think first: Boil this unit to one sentence in 10 seconds.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Simplify √48. ★	4√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	30-60-90 with hypotenuse 14 — legs. ★	7 and 7√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced measure	Right isosceles with hypotenuse 14 — area. ★	49 49	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Regular hexagon with side 16 — area. ★	384√3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

10 · Similarity & Congruence [PSC1] 10-similarity-congruence-psc17 stations built / 7 sections in the guide17 auto of 28 items

Jump to: 10.1 · 10.2 · 10.3 · 10.4 · 10.5 · 10.6 · | 🖨 QR sheet for this unit ↗

10.1 — Proportions, Ratios & Similarity Station 1 QR sheet ↗ student page ↗

🔥 Think first: A photo printed 2 cm wide and 4 cm wide: does the face stay the same shape? Shrink a 4-by-6 photo to width 3 — what happens to the height?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner literal	Solve for x: x/6 = 4/3. ★	x = 8 8, x=8	auto	`x=#` Just the value, or write it as x=# — both count.	Treat every other letter as a plain number — x is the only unknown. Undo what is attached to x: addition/subtraction first, then multiplication/division.
Medium Intermediate arith	A 4-by-6 rectangle is similar to one of width 10 — its height? ★	15 15	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced discuss	A 5-12-13 triangle is scaled by 2.5 — the three new sides? ★	12.5, 30, 32.5 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge measure	The perimeters of two similar hexagons are in the ratio 7:11. The smaller perimeter is 49 — the larger? ★	49/7·11 = 77 77	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.

10.2 — Triangle Similarity Shortcuts (SSS, SAS, AA) Station 2 QR sheet ↗ student page ↗

🔥 Think first: Two triangles, one with a 50° and a 70° angle, one with a 70° and a 50° angle — same or different? Guess before you compute the third angle.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Triangles with angle pairs 45° & 60° and 45° & 60° — similar? ★	yes (AA) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Sides 6, 8, 10 vs 9, 12, 15 — similar? ★	yes (SSS, k = 2/3) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced simplify	Sides 5, 9, 12 are similar to 15, 27, x — find x. ★	36 36, x=36	auto	`#` Just the number — no units, no x=.	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.
Spicy Challenge discuss	Why must the angle in SAS be the INCLUDED angle? ★	Two sides and a non-included angle do not lock the shape — two different triangles can share them (the ambiguous case). —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

10.3 — Parallel Lines in Triangles Station 3 QR sheet ↗ student page ↗

🔥 Think first: Draw any triangle, draw one line parallel to a side, and measure the four pieces. Does AD/DB always equal AE/EC? Try two different triangles and compare.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	AD = 8, DB = 4, AE = 10 — find EC. ★	5 5	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate arith	AD = 6, DB = 2, AE = 9 — find EC. ★	3 3	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced arith	AD = 4, AB = 10, AE = 6 — find AC. ★	15 15	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Spicy Challenge discuss	AD = 5, DB = 10, AE = 7, EC = 14 — is DE ∥ BC? ★	yes (5/10 = 7/14) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

10.4 — Indirect Measurement Station 4 QR sheet ↗ student page ↗

🔥 Think first: At noon your shadow is half your height. Without measuring anything else, how tall is the tree whose shadow is 15 ft?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner scale	6-ft person, 4-ft shadow; tree 20-ft shadow — tree height? ★	30 ft 30, 30ft	auto	`#ft` Type the expression with no spaces; use ^ for exponents and / for fractions.	Write the scale as a ratio with matching units (cm with cm, km with km). Multiply by the ratio that cancels the unit you are leaving.
Medium Intermediate scale	4-ft stick casts 6 ft; pole casts 27 ft — pole height? ★	18 ft 18, 18ft	auto	`#ft` Type the expression with no spaces; use ^ for exponents and / for fractions.	Write the scale as a ratio with matching units (cm with cm, km with km). Multiply by the ratio that cancels the unit you are leaving.
Hot Advanced scale	5.5-ft student, mirror 6 ft away, treetop 18 ft from mirror — tree height? ★	16.5 ft 16.5, 16.5ft	auto	`#ft` Type the expression with no spaces; use ^ for exponents and / for fractions.	Write the scale as a ratio with matching units (cm with cm, km with km). Multiply by the ratio that cancels the unit you are leaving.
Spicy Challenge simplify	Flagpole shadow 36 m while a 2-m sign casts 3 m — pole height? ★	24 m 24m	auto	`#m` Type the expression with no spaces; use ^ for exponents and / for fractions.	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.

10.5 — Corresponding Parts of Similar Triangles Station 5 QR sheet ↗ student page ↗

🔥 Think first: Two similar triangles overlap. If the small one has sides 3, 4, 5 and the big one starts at 6, what are the other two sides — without computing?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	ABC ~ DEF, AB = 9, DE = 6, BC = 12 — find EF. ★	8 8	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate arith	ABC ~ DEF, AB = 10, AC = 14, DF = 21 — find DE. ★	15 15	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced discuss	8-15-17 triangle scaled by 3 — new sides? ★	24, 45, 51 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge measure	ABC ~ DEF, perimeter of ABC = 36, k = 2/3 — perimeter of DEF? ★	24 24, k=24	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.

10.6 — Similar Areas & Volumes Station 6 QR sheet ↗ student page ↗

🔥 Think first: A 1-inch cube and a 2-inch cube: one holds how many of the other in volume? Guess, then count the little cubes.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Similar triangles, scale factor 3:4 — ratio of areas? ★	9:16 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate measure	Figure with area 12 scaled by 2 — new area? ★	48 48	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced discuss	Similar figures with areas 4 and 9 — ratio of sides? ★	2:3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Similar solids with volumes 8 and 27 — ratio of surface areas? ★	4:9 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T10) Station 7 QR sheet ↗ student page ↗

🔥 Think first: Boil the whole unit to one sentence in 10 seconds.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner fraction	Solve x/14 = 3/7. ★	x = 6 6, x=6	auto	`x=#` Just the value, or write it as x=# — both count.	Clear the fraction by multiplying both sides by the denominator. Simplify before you type — reduced form is what gets marked right.
Medium Intermediate arith	ABC ~ DEF, AB = 8, DE = 12, BC = 10 — EF? ★	15 15	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced discuss	Similar figures with areas 25 and 49 — side ratio? ★	5:7 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Similar cones, linear scale 3:1 — ratio of volumes? ★	27:1 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

11 · Volume of Cones & Cylinders 11-volume-cones-cylinders5 stations built / 5 sections in the guide4 auto of 20 items

Jump to: 11.1 · 11.2 · 11.3 · 11.4 · | 🖨 QR sheet for this unit ↗

11.1 — Volume of Cylinders & Cones Station 1 QR sheet ↗ student page ↗

🔥 Think first: Three identical paper cones fill one same-base, same-height cylinder with sand or rice — pour it and watch the ratio land on exactly three.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Cylinder with radius 3, height 5 — volume? ★	45π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Cone with radius 4, height 9 — volume? ★	48π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced measure	Cylinder with volume 200π and height 8 — radius? ★	5 5	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Cylinder with diameter 10 and height 8 — volume? ★	200π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

11.2 — Volume of Spheres & Hemispheres Station 2 QR sheet ↗ student page ↗

🔥 Think first: A sphere of radius r fits snugly inside a cylinder of radius r and height 2r. What fraction of the cylinder does the sphere fill? Guess before computing.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Sphere with radius 3 — volume? ★	36π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Hemisphere with radius 9 — volume? ★	486π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced measure	Sphere with volume 36π — radius? ★	3 3	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Hemisphere with radius 6 — volume? ★	144π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

11.3 — Surface Area of Spheres & Hemispheres Station 3 QR sheet ↗ student page ↗

🔥 Think first: Peel an orange, flatten the peel over a paper circle with the same radius — the peel covers about four such circles. Try it and count.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Sphere with radius 5 — surface area? ★	100π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Hemisphere with radius 3 — surface area? ★	27π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced measure	Sphere with surface area 144π — radius? ★	6 6	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Hemisphere with radius 7 — surface area? ★	147π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

11.4 — Applications (Composite Figures) Station 4 QR sheet ↗ student page ↗

🔥 Think first: A cylinder with a dome on top: which parts do you add, and which hidden face do you NOT count in surface area? Point at the shared circle.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Cylinder r = 3, h = 10 with hemisphere cap r = 3 — total volume? ★	108π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Cylinder r = 2, h = 8 with hemisphere cap — total volume? ★	112π/3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Sphere with surface area 144π — its volume? ★	288π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Composite SA: dome 18π + wall 60π + base 9π — total? ★	87π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T11) Station 5 QR sheet ↗ student page ↗

🔥 Think first: Boil the unit to one sentence in 10 seconds.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Cylinder r = 5, h = 12 — volume? ★	300π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Cone r = 6, h = 9 — volume? ★	108π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced measure	Sphere with surface area 100π — radius? ★	5 5	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Spicy Challenge discuss	Hemisphere with radius 9 — volume? ★	486π —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

12 · Right Triangles [RT3] 12-right-triangles-trig-rt34 stations built / 4 sections in the guide9 auto of 16 items

Jump to: 12.1 · 12.2 · 12.3 · | 🖨 QR sheet for this unit ↗

12.1 — Trig Definitions — SOH-CAH-TOA Station 1 QR sheet ↗ student page ↗

🔥 Think first: Could you measure the height of the flagpole by only knowing one angle and one length you can reach? Sketch the right triangle before anyone says trigonometry.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	3-4-5 triangle: sin of the angle opposite the 3-leg? ★	3/5 3/5	auto	`#/#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate arith	5-12-13 triangle: cos of the angle opposite the 5-leg? ★	12/13 12/13	auto	`#/#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced ratio	tan 34° = x/12 — find x (nearest tenth). ★	8.1 8.1, x=8.1	auto	`#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Spicy Challenge ratio	sin 41° = x/20 — find x (nearest tenth). ★	13.1 13.1, x=13.1	auto	`#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.

12.2 — Inverse Trig Functions (Missing Angles) Station 2 QR sheet ↗ student page ↗

🔥 Think first: If sin A = 1/2, you already know A without a calculator. What angle has sine exactly 1/2? Now what if sin A = 12/13?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	tan x = 3/4 — find x (nearest tenth). ★	36.9° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	cos x = 8/17 — find x (nearest tenth). ★	61.9° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	sin A = 12/13 — find A (nearest tenth). ★	67.4° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Both acute angles of a 9-40-41 triangle. ★	12.7° and 77.3° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

12.3 — Applications (Angles of Elevation & Depression) Station 3 QR sheet ↗ student page ↗

🔥 Think first: You are in a rowboat and the lighthouse is 500 ft tall. If you guess the angle of elevation, how far out are you? Try 40° and 10° — which is farther?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner scale	Lighthouse 500 ft, elevation 40° — distance to boat (whole ft)? ★	596 ft 596, 596ft	auto	`#ft` Type the expression with no spaces; use ^ for exponents and / for fractions.	Write the scale as a ratio with matching units (cm with cm, km with km). Multiply by the ratio that cancels the unit you are leaving.
Medium Intermediate scale	90-ft kite string at 65° — horizontal distance? ★	38.0 ft 38.0, 38.0ft	auto	`#ft` Type the expression with no spaces; use ^ for exponents and / for fractions.	Write the scale as a ratio with matching units (cm with cm, km with km). Multiply by the ratio that cancels the unit you are leaving.
Hot Advanced simplify	35-m lighthouse, 10° depression — distance? ★	199 m 199m	auto	`#m` Type the expression with no spaces; use ^ for exponents and / for fractions.	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.
Spicy Challenge discuss	200-ft tower, 400-ft cable — angle with the ground? ★	30° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T12) Station 4 QR sheet ↗ student page ↗

🔥 Think first: Boil the unit to one sentence in 10 seconds.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	5-12-13 triangle: tan of the angle opposite the 5-leg? ★	5/12 5/12	auto	`#/#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate discuss	sin A = 24/25 — angle A? ★	73.7° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced scale	40-ft rope at 35° — height of attachment? ★	22.9 ft 22.9, 22.9ft	auto	`#ft` Type the expression with no spaces; use ^ for exponents and / for fractions.	Write the scale as a ratio with matching units (cm with cm, km with km). Multiply by the ratio that cancels the unit you are leaving.
Spicy Challenge discuss	cos x = 15/17 — angle x? ★	28.1° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

13 · Triangle Proofs [LR3] 13-triangle-proofs-lr38 stations built / 8 sections in the guide2 auto of 32 items

Jump to: 13.1 · 13.2 · 13.3 · 13.4 · 13.5 · 13.6 · 13.7 · | 🖨 QR sheet for this unit ↗

13.1 — Conditional Statements Station 1 QR sheet ↗ student page ↗

🔥 Think first: Is the statement — if it is Halloween, then it is October 31 — still true if it is NOT Halloween? When does a conditional lie?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Hypothesis and conclusion of — if a shape is a triangle, it has 3 sides. ★	hyp: triangle; concl: has 3 sides —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Converse of — if it is Halloween, then it is October 31 — and its truth value? ★	If Oct 31 then Halloween — true —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Contrapositive of — if it rains, the game is cancelled. ★	If not cancelled, then no rain —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Is — if x² = 9 then x = 3 — true? Correct it if false. ★	False — biconditional needs x = 3 or −3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

13.2 — Shortcuts for Triangle Congruence Station 2 QR sheet ↗ student page ↗

🔥 Think first: You have two triangles and can only check THREE pairs of parts. Which three pairs lock the shape completely — and which three fail?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Two sides + included angle match — which shortcut? ★	SAS —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Two angles + the side between them match — which? ★	ASA —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Right triangles with a congruent leg and hypotenuse — which? ★	HL —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Why does AAA fail for congruence? ★	Same shape but any size — similar, not congruent —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

13.3 — Two-Column Proofs with Algebra Station 3 QR sheet ↗ student page ↗

🔥 Think first: Solve 2x + 6 = 14. Now write WHY each step is legal — props to: given, addition, division.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Given 2x + 6 = 14 — first step to prove x = 4? ★	Subtract 6 from both sides —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Given 3(x − 2) = 12, prove x = 6 — first step? ★	Divide both sides by 3 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced twosided	Given 5x + 9 = 29 — prove x. ★	x = 4 4, x=4	auto	`x=#` Just the value, or write it as x=# — both count.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Spicy Challenge discuss	Given 2(x + 7) = 30, prove x = 8 — which two properties? ★	Division then Subtraction —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

13.4 — Flowchart Proofs with Triangles Station 4 QR sheet ↗ student page ↗

🔥 Think first: Can you order a recipe with only arrows? Draw how three facts combine into one conclusion without writing a single sentence.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	First box in any flowchart proof? ★	The givens —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Shared side AC in a triangle proof comes from which reason? ★	Reflexive property —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Two lines cross — the pair of angles at the crossing are... ★	Vertical angles, congruent —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Given midpoint M of AB — what does the definition give you? ★	AM ≅ MB —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

13.5 — Two-Column Proofs with Triangles Station 5 QR sheet ↗ student page ↗

🔥 Think first: Given two parallel lines and a shared diagonal, you already have two pairs of congruent angles for FREE. Can you spot them without measuring?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	AB ∥ CD with diagonal AC — which angle pair is created? ★	Alternate interior angles, congruent —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Shared segment in two triangles — reason? ★	Reflexive property —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	∠A ≅ ∠D, AB ≅ DE, ∠B ≅ ∠E — shortcut to prove the triangles congruent? ★	ASA —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	If you prove ΔABC ≅ ΔCDA, what does CPCTC then give you that is NOT a side of the shortcut? ★	The remaining corresponding parts (e.g. BC ≅ DA) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

13.6 — Proofs with Corresponding Parts (CPCTC) Station 6 QR sheet ↗ student page ↗

🔥 Think first: You proved two triangles congruent. How many pairs of parts are now free to claim? Count them.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	CPCTC can only be used after...? ★	The triangles are proven congruent —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	After SSS proves two triangles congruent, CPCTC lets you claim...? ★	Any remaining corresponding part —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	To prove M is a midpoint via CPCTC — what do you show? ★	AM ≅ MB, then definition of midpoint —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	CPCTC gives you a pair of alternate interior angles — what follows? ★	The lines are parallel (converse) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

13.7 — Proofs of Definitions Station 7 QR sheet ↗ student page ↗

🔥 Think first: What would you have to measure to prove someone is at the exact middle of a wall? That is a definition proof in real life.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	To prove CM bisects ∠ACB, you show...? ★	∠ACM ≅ ∠MCB —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	To prove CM perpendicular AB, you show...? ★	The angles are right angles —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	M is midpoint of AB — the definition statement? ★	AM ≅ MB —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Recipe order for a definition proof. ★	Congruence → CPCTC → definition —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T13) Station 8 QR sheet ↗ student page ↗

🔥 Think first: Boil the unit to one sentence in 10 seconds.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Two angles + the side BETWEEN them — shortcut? ★	ASA —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate twosided	Given 5x + 9 = 29 — prove x. ★	x = 4 (subtract 9, divide by 5) 4, x=4	auto	`x=#` Just the value, or write it as x=# — both count.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Hot Advanced discuss	CPCTC after congruence gives...? ★	Any remaining corresponding part —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	To prove AB ∥ CD via CPCTC — which angles do you target? ★	A pair of alternate interior angles —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

14 · Right Triangles [RT4] 14-law-of-sines-cosines-rt45 stations built / 5 sections in the guide8 auto of 20 items

Jump to: 14.1 · 14.2 · 14.3 · 14.4 · | 🖨 QR sheet for this unit ↗

14.1 — SAS Triangle Area Station 1 QR sheet ↗ student page ↗

🔥 Think first: Draw ANY triangle on a grid, pick two sides and the angle between them. You can compute its area without ever drawing a height. How?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner measure	Sides 6 and 9, included angle 30° — area? ★	13.5 13.5	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Medium Intermediate measure	Sides 8 and 11, included 40° — area (nearest tenth)? ★	28.3 28.3	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Hot Advanced discuss	Sides 5 and 7, included 60° — exact area? ★	35√3/4 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge countask	Area 20, side b = 8, included 30° — the other side? ★	10 10, b=10	auto	`#` Just the number — no units, no x=.	You are told a property of the shape and asked for its sides — set up the one formula that links the two. Use the sum formula for the interior angles, or the 'exterior angles add to 360' fact, then solve it for n.

14.2 — Law of Sines Station 2 QR sheet ↗ student page ↗

🔥 Think first: A ranger spots a fire from two campsites. Each camp measures the angle to the fire and to the other camp. With NO right angle anywhere, can you still find the fire?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner twosided	A = 40°, B = 60°, a = 10 — find b (nearest tenth). ★	13.5 13.5, b=13.5	auto	`#` Just the number — no units, no x=.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Medium Intermediate discuss	a = 12, b = 16, A = 35° — find B (nearest tenth). ★	49.9° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced twosided	A = 50°, C = 70°, a = 9 — find c (nearest tenth). ★	11.1 11.1, c=11.1	auto	`#` Just the number — no units, no x=.	Collect the variable terms on one side and the plain numbers on the other. Divide both sides by the coefficient of the variable — check whether the sign flips.
Spicy Challenge discuss	a = 5, b = 20, A = 30° — how many triangles fit? ★	none — sin B > 1 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

14.3 — Law of Cosines Station 3 QR sheet ↗ student page ↗

🔥 Think first: You know two sides of a triangle and the angle between them. The third side is NOT the difference of the other two — so what is it, exactly?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	Sides 8 and 11, included 40° — third side (nearest tenth)? ★	7.1 7.1	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate discuss	Sides 6 and 9, included 60° — third side? ★	3√7 ≈ 7.9 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Sides 5, 6, 7 — angle opposite the 7? ★	78.5° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Sides 4, 5, 6 — angle opposite the 6? ★	82.8° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

14.4 — Decision Tree (Which Law?) Station 4 QR sheet ↗ student page ↗

🔥 Think first: Four situations land on your desk. You may use exactly one law each. Right triangle, SAS, AAS, SSS — sort them into the tools before computing anything.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Two sides + included angle — which law? ★	Cosines —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Two angles + one side — which law? ★	Sines —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Three sides — which law? ★	Cosines —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Right triangle with an angle and a side — which tool? ★	Neither — plain trig —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T14) Station 5 QR sheet ↗ student page ↗

🔥 Think first: Cover the T14 formula sheet. From memory, which tool answers: two sides + included angle? Two angles + a side? Three sides?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner measure	SAS area — sides 9 and 14, included 35° (nearest tenth)? ★	36.1 36.1	auto	`#` Just the number — no units, no x=.	Break the figure into the simple shapes you know, find each piece, then add them. Check the units — a length answer, a squared answer and a cubed answer are all different.
Medium Intermediate discuss	a = 10, b = 15, A = 45° — how many triangles? ★	none (sin B > 1) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Sides 7, 8, 9 — angle opposite the 9? ★	73.4° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge arith	Sides 122 and 215 with included 83° — third side (nearest tenth)? ★	233.9 233.9	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.

15 · Circles and Polygons [CP3] 15-circles-polygons-cp37 stations built / 7 sections in the guide8 auto of 28 items

Jump to: 15.1 · 15.2 · 15.3 · 15.4 · 15.5 · 15.6 · | 🖨 QR sheet for this unit ↗

15.1 — Midpoints, Slopes & Systems of Equations Station 1 QR sheet ↗ student page ↗

🔥 Think first: Two lines cross on a grid. Can you find the meeting point without drawing either line? All you need is the algebra.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Midpoint of A(4, −8) and B(−2, −8)? ★	(1, −8) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Slope of AC for A(4, −8), C(−2, 4)? ★	−2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Perpendicular bisector equation of AC (through its midpoint)? ★	y = ½x − 2.5 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge fraction	Intersection of y = (2/3)x + 1 and y = (−1/6)x + 6? ★	(6, 5) 6,5	auto	`#,#` Ordered pair, x first then y: (#,#).	Clear the fraction by multiplying both sides by the denominator. Simplify before you type — reduced form is what gets marked right.

15.2 — Points of Concurrency Station 2 QR sheet ↗ student page ↗

🔥 Think first: A triangular park needs one light pole equidistant from the three paths. Where does the pole go — and how do you find that spot with a compass?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Tree equidistant from the sides of a triangular lot — which point? ★	Incenter —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Streetlight equidistant from three path lines — which point? ★	Incenter —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Capital equidistant from three cities (points) — which point? ★	Circumcenter —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Center of gravity of a hang-glider — which point? ★	Centroid —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

15.3 — Centroids Station 3 QR sheet ↗ student page ↗

🔥 Think first: Balance a cardboard triangle on one fingertip. There is exactly one spot where it stays level. Can you find it with three midpoints and a ruler?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner solve	AG = 12 with G the centroid — find GD. ★	6 6, g=6	auto	`#` Just the number — no units, no x=.	Undo what is being done to the variable — start with the operation furthest from it. Do the same thing to both sides as you work down to the variable.
Medium Intermediate solve	FG = 5 — find FC (F the midpoint of AB). ★	15 15, f=15	auto	`#` Just the number — no units, no x=.	Undo what is being done to the variable — start with the operation furthest from it. Do the same thing to both sides as you work down to the variable.
Hot Advanced discuss	Centroid of A(4, −8), B(−2, −8), C(−2, 4)? ★	(0, −4) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge arith	AG = 16, FG = 7, BF = 9, BE = 18 — find FC, GD, AF, BG. ★	FC = 21, GD = 8, AF = 9, BG = 12 12	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.

15.4 — Arc Length Station 4 QR sheet ↗ student page ↗

🔥 Think first: A ferris wheel turns one quarter of a full circle. How far did your seat travel? Guess, then compute: fraction of the circle times the circumference.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	60° arc, radius 10 — arc length? ★	≈ 10.5 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	45° MINOR arc, radius 2.68 mi — arc length? ★	≈ 2.1 mi —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	315° MAJOR arc, radius 2.68 mi — arc length (nearest mile)? ★	≈ 15 mi —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	22 ft ladder at 85° falls backward — how far does the top travel? ★	≈ 33 ft —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

15.5 — Circumcenter Station 5 QR sheet ↗ student page ↗

🔥 Think first: Three towns want a cell tower equally far from all three. Draw the three towns, then find the one point equidistant from all — with only a compass and straightedge.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner writeeq	Slope of AB for A(12, 8), B(10, 4)? ★	2 2, b=2	auto	`#` Just the number — no units, no x=.	Find the slope first (rise over run, or two points), then substitute one point to get b. Write it as y = mx + b, then check that each given point fits your equation.
Medium Intermediate discuss	Perpendicular bisector equation of AB? ★	y = −½x + 11.5 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced simplify	Circumcenter of A(12, 8), B(10, 4), C(8, 10)? ★	(9, 7) 9,7	auto	`#,#` Ordered pair, x first then y: (#,#).	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.
Spicy Challenge discuss	Radius of the circumscribed circle for that triangle? ★	√10 ≈ 3.2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

15.6 — Constructions Station 6 QR sheet ↗ student page ↗

🔥 Think first: With only a compass and a straightedge, can you cut a segment exactly in half? No ruler markings allowed. Try it — the arcs do the measuring for you.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Three perpendicular bisectors construct which point? ★	Circumcenter —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Three angle bisectors construct which point? ★	Incenter —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	The incenter construction inscribes which circle? ★	The circle tangent to all three sides —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Desmos project: which three constructions build the circumcenter, incenter, and centroid? ★	Perp bisectors, angle bisectors, medians —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T15) Station 7 QR sheet ↗ student page ↗

🔥 Think first: Boil the unit to one sentence in ten seconds: points on a triangle that decide where to put things, and the arc a swinging thing traces.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Tree equidistant from the three SIDES — which point? ★	Incenter —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate solve	AG = 12, G the centroid — find GD. ★	6 6, g=6	auto	`#` Just the number — no units, no x=.	Undo what is being done to the variable — start with the operation furthest from it. Do the same thing to both sides as you work down to the variable.
Hot Advanced discuss	45° minor arc, radius 2.68 mi — arc length? ★	≈ 2.1 mi —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge simplify	Circumcenter of A(12, 8), B(10, 4), C(8, 10)? ★	(9, 7) 9,7	auto	`#,#` Ordered pair, x first then y: (#,#).	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.

16 · Transformations [PSC3] 16-transformations-psc35 stations built / 5 sections in the guide3 auto of 20 items

Jump to: 16.1 · 16.2 · 16.3 · 16.4 · | 🖨 QR sheet for this unit ↗

16.1 — Rigid Transformations & Symmetry Station 1 QR sheet ↗ student page ↗

🔥 Think first: Look around the room: find three pairs of identical shapes that are copies of each other. How would you move one copy onto the other — slide, flip, or spin?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Name the three rigid transformations. ★	Translation, reflection, rotation —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate arith	Lines of symmetry of a square? ★	4 4	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced discuss	Which is NOT rigid: translation, reflection, rotation, dilation? ★	Dilation —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Rotational symmetry of a regular hexagon — smallest angle that maps it onto itself? ★	60° (order 6) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

16.2 — Transformations on the Coordinate Plane Station 2 QR sheet ↗ student page ↗

🔥 Think first: You have a point (3, 5) and a rule like (x, y) → (−y, x). Before computing, ROTATE your head: where does the point land?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Translate (2, 3) by (x, y) → (x − 4, y + 5)? ★	(−2, 8) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Reflect (4, −2) over the y-axis? ★	(−4, −2) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Rotate (3, 1) 90° ccw about the origin? ★	(−1, 3) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Rotate (2, −5) 180° about the origin? ★	(−2, 5) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

16.3 — Compositions of Transformations Station 3 QR sheet ↗ student page ↗

🔥 Think first: Do the moves commute? Reflect a point over the y-axis, then translate it right 3. Now translate first, then reflect. Same answer? Try (1, 1).

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner translate	Translate (1, 2) by (x + 3, y)? ★	(4, 2) 4,2	auto	`#,#` Ordered pair, x first then y: (#,#).	Decide what the letter stands for, then read the sentence one operation at a time. Write the expression before you try to compute anything.
Medium Intermediate discuss	Then reflect that over the x-axis? ★	(4, −2) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Reflect (2, 3) over the y-axis, then the x-axis? ★	(−2, −3) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Two reflections over parallel lines equal which single move? ★	A translation —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

16.4 — Geometry Dash Project (Desmos) Station 4 QR sheet ↗ student page ↗

🔥 Think first: Design one level of a platformer using only three move tools: slide, flip, spin. What is the hardest obstacle you can build from just those three?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Which Desmos tool slides a shape sideways? ★	Translate —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	What does the Rotate tool spin around? ★	A center point —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Why only rigid transformations in the project? ★	Shapes stay congruent as they move —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	A moving hazard that mirrors across a line uses which transform? ★	Reflection —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T16) Station 5 QR sheet ↗ student page ↗

🔥 Think first: Boil the unit to one sentence in ten seconds: moves that copy shapes exactly, written as point rules.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner simplify	Reflect (4, −2) over the x-axis? ★	(4, 2) 4,2	auto	`#,#` Ordered pair, x first then y: (#,#).	Only like terms can combine — same variable and same exponent. Watch the sign in front of each term as you distribute or subtract.
Medium Intermediate discuss	Rotate (3, 1) 90° ccw about the origin? ★	(−1, 3) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Reflect (2, 3) over the y-axis, then the x-axis? ★	(−2, −3) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Regular octagon — smallest rotation mapping it onto itself? ★	45° (order 8) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

17 · Right Triangles 5 [RT5] 17-degrees-radians-rt55 stations built / 5 sections in the guide4 auto of 20 items

Jump to: 17.1 · 17.2 · 17.3 · 17.4 · | 🖨 QR sheet for this unit ↗

17.1 — Degrees and Radians Station 1 QR sheet ↗ student page ↗

🔥 Think first: How many quarters of a circle is 180°? Now measure that same turn by wrapping the RADIUS along the edge — how many radius-lengths fit?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Convert 5 radians into degrees (nearest tenth). ★	≈ 286.5° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Convert 54° into radians. ★	3π/10 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Convert 90° into radians. ★	π/2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Convert 2π/3 radians into degrees. ★	120° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

17.2 — Reference Triangles Station 2 QR sheet ↗ student page ↗

🔥 Think first: An angle of 210° looks wild. But draw it — every point on it is a mirror copy of a 30° angle. Find the mirror line.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Reference angle of θ = 7π/6? ★	π/6 (30°) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Quadrant of θ = 7π/6? ★	Quadrant III —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Reference angle of θ = π/4? ★	π/4 (45°) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Reference angle of θ = 5π/4 and its quadrant? ★	π/4, Quadrant III —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

17.3 — Secant, Cosecant & Cotangent (Optional) Station 3 QR sheet ↗ student page ↗

🔥 Think first: If sin 30° = 1/2, how many 30°-sines stack to reach 1? That stack IS cosecant.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner ratio	csc 30° if sin 30° = 1/2? ★	2 2	auto	`#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Medium Intermediate ratio	sec 60° if cos 60° = 1/2? ★	2 2	auto	`#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Hot Advanced ratio	cot 45° if tan 45° = 1? ★	1 1	auto	`#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Spicy Challenge discuss	sec θ in Quadrant II — positive or negative? ★	Negative (cos is negative) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

17.4 — Evaluating Trig Functions (Optional) Station 4 QR sheet ↗ student page ↗

🔥 Think first: The unit circle point at 60° is (1/2, √3/2). Which coordinate is sine and which is cosine — and how can you never forget?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner arith	sin(π/6) from the unit circle? ★	1/2 1/2	auto	`#/#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Medium Intermediate discuss	cos(π/4)? ★	√2/2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	sin(π/3)? ★	√3/2 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Unit circle point at θ = π/6? ★	(√3/2, 1/2) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (T17) Station 5 QR sheet ↗ student page ↗

🔥 Think first: Boil the unit to one sentence in ten seconds: angles in two languages, reduced to small triangles, read off a circle.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Convert 5 radians to degrees (nearest tenth)? ★	≈ 286.5° —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Convert 54° to radians? ★	3π/10 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Reference angle of 7π/6? ★	π/6 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Reference angle of π/4? ★	π/4 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

18 · Statistical Investigations I [Stats I] 18-statistical-investigations-i5 stations built / 5 sections in the guide0 auto of 20 items

Jump to: 1.1 · 1.2 · 1.3 · 1.4 · | 🖨 QR sheet for this unit ↗

1.1 — Statistical Questions & Hypotheses Station 1 QR sheet ↗ student page ↗

🔥 Think first: Is "how tall is your brother?" a statistical question? What would make it one?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Statistical or not: "What is the average age of our teachers?" ★	statistical (a distribution expected) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	A good statistical question expects ______. ★	variability across a group —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Null vs alternative hypothesis — what is being contrasted? ★	No-effect vs. an effect; data decide which is more plausible. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Give a statistical question with a binary answer that still shows variability. ★	"What proportion of students prefer band to orchestra?" — proportions vary across samples even though each answer is yes/no. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

1.2 — Study Design: Surveys, Experiments, Observational Studies Station 2 QR sheet ↗ student page ↗

🔥 Think first: If you want to know whether a drug works, why not just give it to everyone?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	A study assigns treatments randomly — what kind? ★	experiment —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Watching behavior without changing anything. ★	observational study —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Only ______ can establish cause. ★	a randomized experiment —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Control group vs treatment group — purpose. ★	Control provides a baseline to compare against, isolating the treatment effect. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

1.3 — Data Collection, Quality & Bias Station 3 QR sheet ↗ student page ↗

🔥 Think first: Ask only gym-goers how active people are. What did you stack the deck toward?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Sampling only gym-goers about exercise — which bias? ★	selection/coverage —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	A leading question produces ______ bias. ★	wording/response —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Larger random samples → ______ sampling variability. ★	lower —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Design a low-bias way to survey the whole school in a hour. ★	Stratify by grade, random sample within each, keep it short + anonymous — balances access/cost with representativeness. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

1.4 — Correlation vs. Causation Station 4 QR sheet ↗ student page ↗

🔥 Think first: Can ice cream really cause drowning? What else rises with both?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	"Storks bring babies" — the classic error? ★	correlation mistaken for causation —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	The hidden third variable is a ______. ★	lurking/confounding variable —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	To establish cause, you need a ______. ★	randomized experiment (or a controlling design) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Same real data, two headlines — one causal, one careful. Write both. ★	"Sleep boosts scores" (causal overclaim) vs. "Sleep is associated with higher scores in this sample" (careful). —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (Stats I) Station 5 QR sheet ↗ student page ↗

🔥 Think first: Boil Stats I to ONE sentence.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	A question answered by a distribution. ★	statistical question —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Only experiments can show ______. ★	causation —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Hidden variable explaining a correlation. ★	confounder —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Improve "texting and crashes" bias: suggest a fair design. ★	Random sample, neutral wording, and compare a controlled experiment on reaction time — not a self-report of crashes. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

19 · Probability [Prob] 19-probability6 stations built / 6 sections in the guide9 auto of 24 items

Jump to: 2.1 · 2.2 · 2.3 · 2.4 · 2.5 · | 🖨 QR sheet for this unit ↗

2.1 — Sample Spaces & Counting Station 1 QR sheet ↗ student page ↗

🔥 Think first: How many outcomes for 3 coins? Count them before you multiply.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Sample space of one coin flip. ★	{H, T} —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate arith	Die + coin — how many outcomes? ★	12 12	auto	`#` Just the number — no units, no x=.	Work the grouping symbols first, then exponents, then × and ÷ left to right, then + and −. Show each step on the board so the order you used is visible.
Hot Advanced discuss	Counting principle for 3 shirts × 2 jeans. ★	6 outfits —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge ratio	Two dice — probability of sum 7. ★	6/36 = 1/6 1/6	auto	`#/#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.

2.2 — Events: Unions, Intersections & Complements Station 2 QR sheet ↗ student page ↗

🔥 Think first: Roll two dice: what is the chance of at least one six? Easier to count its opposite.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner ratio	P(rolling a 5 or 6 on one die). ★	2/6 = 1/3 1/3, p=1/3	auto	`#/#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Medium Intermediate ratio	P(not rolling a 6). ★	5/6 5/6, p=5/6	auto	`#/#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Hot Advanced ratio	Deck: P(heart or face card). ★	11/26 11/26, p=11/26	auto	`#/#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Spicy Challenge discuss	Why subtract P(A∩B) in P(A or B)? ★	Both groups were counted twice — the overlap is double-counted, so subtract it once. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

2.3 — Conditional Probability & Independence Station 3 QR sheet ↗ student page ↗

🔥 Think first: "Given it is a heart, what is the chance it is red?" — you already shrink the deck.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	P(A\|B) uses which reduced space? ★	the B outcomes —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Independent events: P(A∩B) = ? ★	P(A)·P(B) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced absvalue	Deck: P(red \| it is a heart). ★	1 (all hearts are red) 1, p=1	auto	`#` Just the number — no units, no x=.	Get the absolute-value bars alone on one side before anything else. Then split it into two cases: what is inside the bars = the amount, and = its negative.
Spicy Challenge discuss	Two-way table independence check in one step. ★	Compare P(A\|B) to P(A) — equal means independent. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

2.4 — Experimental vs. Theoretical Probability Station 4 QR sheet ↗ student page ↗

🔥 Think first: Flip a coin 20 times. Would you trust 6 heads? 1000 flips?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner ratio	60 rolls, 12 sixes — experimental P(6). ★	1/5 1/5, p=1/5	auto	`#/#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Medium Intermediate ratio	Theoretical P(6) on a fair die. ★	1/6 1/6, p=1/6	auto	`#/#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Hot Advanced discuss	Law of large numbers says ______. ★	experimental → theoretical as trials grow —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Simulation advantage over a real survey/experiment. ★	Cheap + fast + repeatable for many trials to estimate a probability. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

2.5 — Expected Value & Decisions Station 5 QR sheet ↗ student page ↗

🔥 Think first: A game pays $3 on a 6, costs $1 to play. Is it fair? Who wins long-run?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	E = Σ P·value — this is ______. ★	expected value —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	P(win)=0.1 pays $10, else $0. EV? ★	0.1·10 = $1 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Tickets: 200 sold, $300 prize, $5 ticket — EV to you. ★	(1/200)(295) + (199/200)(−5) = 1.475 − 4.975 = −$3.50 —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Decision rule with EVs. ★	Compare EVs; pick the option with the higher expected value. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (Probability) Station 6 QR sheet ↗ student page ↗

🔥 Think first: Boil Probability to ONE sentence.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Sample space of two coins. ★	4 outcomes —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate ratio	P(at least one head in 2 flips). ★	1 − 1/4 = 3/4 3/4, p=3/4	auto	`#/#` Just the number — no units, no x=.	Write the count that matches what you want over the count of everything possible. Simplify the fraction, and use the reciprocal identity if the question gives you sin or cos and asks for csc or sec.
Hot Advanced discuss	P(H\|S) uses which denominator? ★	total S outcomes only —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	EV of a fair $1 game that pays $2 on heads. ★	(1/2)(1) + (1/2)(−1) = $0 — fair —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

20 · Data Analysis & Statistical Modeling [D&A] 20-data-analysis-statistical-modeling7 stations built / 7 sections in the guide0 auto of 28 items

Jump to: 3.1 · 3.2 · 3.3 · 3.4 · 3.5 · 3.6 · | 🖨 QR sheet for this unit ↗

3.1 — Shape, Center & Spread Station 1 QR sheet ↗ student page ↗

🔥 Think first: A neighborhood has 9 houses worth $100k and one mansion worth $10M. Is the mean or median "typical" price?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Resistant-to-outliers center measure. ★	median —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Spread of the middle 50%. ★	IQR —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Right-skewed: mean vs median. ★	mean > median —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Shape drives which center/spread you report — explain. ★	Symmetric → mean+SD; skewed/outliers → median+IQR; shape is the key context. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.2 — Data Displays (Technology) Station 2 QR sheet ↗ student page ↗

🔥 Think first: Can you see a trend faster in a list of 200 numbers or their scatter plot?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Display for two quantitative variables. ★	scatter plot —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Histogram shows ______ of one variable. ★	the shape/distribution —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Boxplot summarizes ______. ★	five-number summary (center, spread, outliers) —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Read trend off a scatter: uphill points mean ______. ★	a positive association —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.3 — Normal Distribution, Mean & Standard Deviation Station 3 QR sheet ↗ student page ↗

🔥 Think first: Heights average 170 cm, SD 5. Is 185 cm common or rare?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	% within 1 SD (normal). ★	≈68% —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	% within 2 SD. ★	≈95% —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	SD measures ______. ★	spread from the mean —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	z-score purpose. ★	Compare values across distributions; locate a value in SD units to find normal percentages. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.4 — Inference from Random Samples Station 4 QR sheet ↗ student page ↗

🔥 Think first: Poll 30 students and find 60% favor pizza. Would a different 30 give exactly 60%?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Different random samples give slightly different results — this is ______. ★	sampling variation —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Larger random samples → ______ variation. ★	less —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Margin of error reflects ______. ★	sampling variation for the sample size —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Use multiple samples to judge an estimate. ★	If estimates cluster tightly, trust a single one; if they swing wide, report uncertainty. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.5 — Linear & Exponential Models, Correlation & Fit Station 5 QR sheet ↗ student page ↗

🔥 Think first: A scatter slopes up tightly; add a point that breaks it. What does the line do?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	r measures ______ of a linear association. ★	strength and direction —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	A residual is actual − ______. ★	predicted —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Constant-multiplier growth → model. ★	exponential —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Curved residual pattern means ______. ★	the linear model is wrong (try another form). —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

3.6 — Evaluating Reports & Misleading Data Station 6 QR sheet ↗ student page ↗

🔥 Think first: A Math bar of 5 and English of 7 is drawn with y starting at 5. Is English really "twice" math?

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Y-axis cut to exaggerate = ______. ★	truncated-axis distortion —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Same trend, two spins = ______. ★	competing explanations —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	Best first step to evaluate a stat claim. ★	Check the axis/scale and the sample. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	Fix a misleading chart. ★	Start axes at true zero (or show the cut), add full context, label units — so the eye reads the real magnitude. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

— Review (Data Analysis) Station 7 QR sheet ↗ student page ↗

🔥 Think first: Boil Data Analysis to ONE sentence.

Flavor	Problem	Key & accepted spellings	Grading	Format shown to students	Hints (gated 30s / 60s)
Mild Beginner discuss	Robust-to-outliers center/spread. ★	median + IQR —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Medium Intermediate discuss	Display for two quantitative variables. ★	scatter plot —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Hot Advanced discuss	A residual is actual − ______. ★	predicted —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.
Spicy Challenge discuss	The honest first check on any chart. ★	The axis scale/sample — before trusting the message. —	talk	discussion prompt	Say your idea out loud first, then write the sentence you would defend. Test your rule on a small example (easy numbers) before you commit.

Station quizzes — extraction review & answer key

01 · Logic & Reasoning [LR1] 01-logic-reasoning-lr18 stations built / 8 sections in the guide11 auto of 32 items

1.1 — Vocabulary Station 1 QR sheet ↗ student page ↗

1.1.5 — Angles Station 2 QR sheet ↗ student page ↗

1.2 — Segment Addition Postulate Station 3 QR sheet ↗ student page ↗

1.2.5 — Overlapping Segments Conjecture Station 4 QR sheet ↗ student page ↗

1.3 — Truth Tables Station 5 QR sheet ↗ student page ↗

1.4 — Inductive & Deductive Reasoning Station 6 QR sheet ↗ student page ↗

1.5 — Handshake Formula Station 7 QR sheet ↗ student page ↗

— Review (T1), Note Making & Rubric Station 8 QR sheet ↗ student page ↗

02 · Logic & Reasoning 2 [LR2] 02-logic-reasoning-lr27 stations built / 7 sections in the guide2 auto of 28 items

2.1 — 2.2 Constructions (NOT tested) Station 1 QR sheet ↗ student page ↗

2.3 — Special Angle Pairs Station 2 QR sheet ↗ student page ↗

2.3.5 — Transversals & Parallel Lines Station 3 QR sheet ↗ student page ↗

2.4 — Triangle Angle Sum Conjecture Station 4 QR sheet ↗ student page ↗

2.5 — Isosceles Triangles Station 5 QR sheet ↗ student page ↗

2.6 — Triangle Inequality Theorem Station 6 QR sheet ↗ student page ↗

— Review (T2) Station 7 QR sheet ↗ student page ↗

03 · Quadrilaterals [CP1] 03-quadrilaterals-cp19 stations built / 9 sections in the guide9 auto of 36 items

3.1 — Polygon Classification Station 1 QR sheet ↗ student page ↗

3.2 — Interior Angles of Polygons Station 2 QR sheet ↗ student page ↗

3.2.5 — Exterior Angle Sum Station 3 QR sheet ↗ student page ↗

3.3 — Classifying Triangles & Quadrilaterals Station 4 QR sheet ↗ student page ↗

3.4 — Parallelogram Properties Station 5 QR sheet ↗ student page ↗

3.5 — Rhombus, Rectangle & Square Properties Station 6 QR sheet ↗ student page ↗

3.6 — Kite & Trapezoid Properties Station 7 QR sheet ↗ student page ↗

3.7 — Midsegment Properties Station 8 QR sheet ↗ student page ↗

— Review (T3) Station 9 QR sheet ↗ student page ↗

04 · Area of Polygons [AV1] 04-area-polygons-av14 stations built / 4 sections in the guide12 auto of 16 items

4.1 — Rectangles & Parallelograms Station 1 QR sheet ↗ student page ↗

4.2 — Triangles, Trapezoids & Kites Station 2 QR sheet ↗ student page ↗

4.3 — Area of Regular Polygons Station 3 QR sheet ↗ student page ↗

— Review (T4) Station 4 QR sheet ↗ student page ↗

05 · Prisms and Pyramids [AV2] 05-prisms-pyramids-av25 stations built / 5 sections in the guide11 auto of 20 items

5.1 — Vocabulary & Nets Station 1 QR sheet ↗ student page ↗

5.2 — Prism & Pyramid Volume Station 2 QR sheet ↗ student page ↗

5.3 — Surface Area Station 3 QR sheet ↗ student page ↗

5.4 — Applications Station 4 QR sheet ↗ student page ↗

— Review (T5) Station 5 QR sheet ↗ student page ↗

06 · Circles [CP2] 06-circles-cp26 stations built / 6 sections in the guide3 auto of 24 items

6.1 — Circle Vocabulary Station 1 QR sheet ↗ student page ↗

6.2 — Central, Inscribed Angles & Chords Station 2 QR sheet ↗ student page ↗

6.3 — Clock Angles Station 3 QR sheet ↗ student page ↗

6.4 — Tangent Properties Station 4 QR sheet ↗ student page ↗

6.5 — Cyclic Quadrilaterals Station 5 QR sheet ↗ student page ↗

— Review (T6) Station 6 QR sheet ↗ student page ↗

07 · Circle Related Areas [AV3] 07-circle-related-areas-av34 stations built / 4 sections in the guide6 auto of 16 items

7.1 — Circumference & Area of a Circle Station 1 QR sheet ↗ student page ↗

7.2 — Sector, Segment & Annulus Areas Station 2 QR sheet ↗ student page ↗

7.3 — Surface Area of Cones & Cylinders Station 3 QR sheet ↗ student page ↗

— Review (T7) Station 4 QR sheet ↗ student page ↗

08 · Right Triangles [RT1] 08-right-triangles-rt15 stations built / 5 sections in the guide10 auto of 20 items

8.1 — Pythagorean Theorem Station 1 QR sheet ↗ student page ↗

8.2 — Pythagorean Triples & Converse Station 2 QR sheet ↗ student page ↗

8.3 — Applications Station 3 QR sheet ↗ student page ↗

8.4 — 8.5 Distance Formula & Circle Equations Station 4 QR sheet ↗ student page ↗

— Review (T8) Station 5 QR sheet ↗ student page ↗

09 · Special Right Triangles [RT2] 09-special-right-triangles-rt26 stations built / 6 sections in the guide5 auto of 24 items

9.1 — Simplifying Radicals Station 1 QR sheet ↗ student page ↗

9.2 — Add, Subtract & Multiply Radicals Station 2 QR sheet ↗ student page ↗

9.3 — Special Right Triangles Station 3 QR sheet ↗ student page ↗

9.4 — Areas from Special Triangles Station 4 QR sheet ↗ student page ↗

9.5 — Circle Applications Station 5 QR sheet ↗ student page ↗

— Review (T9) Station 6 QR sheet ↗ student page ↗

10 · Similarity & Congruence [PSC1] 10-similarity-congruence-psc17 stations built / 7 sections in the guide17 auto of 28 items

10.1 — Proportions, Ratios & Similarity Station 1 QR sheet ↗ student page ↗

10.2 — Triangle Similarity Shortcuts (SSS, SAS, AA) Station 2 QR sheet ↗ student page ↗

10.3 — Parallel Lines in Triangles Station 3 QR sheet ↗ student page ↗

10.4 — Indirect Measurement Station 4 QR sheet ↗ student page ↗

10.5 — Corresponding Parts of Similar Triangles Station 5 QR sheet ↗ student page ↗

10.6 — Similar Areas & Volumes Station 6 QR sheet ↗ student page ↗

— Review (T10) Station 7 QR sheet ↗ student page ↗

11 · Volume of Cones & Cylinders 11-volume-cones-cylinders5 stations built / 5 sections in the guide4 auto of 20 items

11.1 — Volume of Cylinders & Cones Station 1 QR sheet ↗ student page ↗

11.2 — Volume of Spheres & Hemispheres Station 2 QR sheet ↗ student page ↗

11.3 — Surface Area of Spheres & Hemispheres Station 3 QR sheet ↗ student page ↗

11.4 — Applications (Composite Figures) Station 4 QR sheet ↗ student page ↗

— Review (T11) Station 5 QR sheet ↗ student page ↗

12 · Right Triangles [RT3] 12-right-triangles-trig-rt34 stations built / 4 sections in the guide9 auto of 16 items

12.1 — Trig Definitions — SOH-CAH-TOA Station 1 QR sheet ↗ student page ↗