Teacher Daily Cards · Week 2
How can triangles predict movement? — one page per session, generated from the current curriculum.
How to read these cards. Scan the header icons:
🔧 hardware-dependent ·
★ Gate/Milestone. In the run of show, each teacher cue is its own line;
Say: /
Ask: mark exact moves.
▽ below / ◆ at / △ above are the three differentiation tracks, and
⏱ If behind today tells you what to cut when the clock is short.
When it goes sideways, work down this ladder. You do not have to invent a fix mid-period. 1. The day's ⏱ If behind today line on the card — it names exactly what to cut, and on gate days what must NOT be cut. 2. The week's If it goes sideways card at the back of this pack, when the problem is bigger than the clock. 3. The Boost Card on the table, when it is one student stuck rather than the room. 4. Message Liz after class, not during. If your situation is not named anywhere, every recovery move here follows one principle: protect the thing being assessed, and let everything else move.
Week 2 · Day 6Sin, cos & the unit circle
Read sine and cosine straight off the unit circle.
Competencies this lesson builds
- ◆ REPRESENT (Convert between forms · Computational Thinking) — cos θ is the x and sin θ is the y — the unit circle and the right triangle are the same fact in two costumes.
- ◆ PATTERN (Notice what repeats · Computational Thinking) — sin 30° = sin 150°. Values repeat around the circle, and that repetition is exactly the ambiguity IK will have to resolve.
Warm-up · Daily Driver (on the board): Your Brief gave the motion a start and an end, both measured in cm from the same place. What sits at (0, 0)? (base or hand)
Note: Warm recall pulls from earlier days by design (spaced retrieval). If many students missed the recalled day (whole-class absence, fire drill), swap the matching Boost Card refresher in for the recall block that day only, then resume the pattern. Applies to every recall day from here on.
Main work block · run of show (45 min) — one cue per line
- 0–5 · Entry Task
- The base.
- Every coordinate in their Brief is measured from where the arm is bolted down — which is the hook into “a point at an angle has coordinates.”
- 5–20 · Teacher-led · x = cos, y = sin
- Drop a point on the unit circle.
- Its x-coordinate IS cos θ; its y IS sin θ.
- That’s the whole idea — sine and cosine are just coordinates.
- Read a few off the diagram.
- 20–38 · Guided practice
- Sin & Cosine worksheet.
- Have them read values off the labeled circle before reaching for the calculator.
- 38–45 · Exit
- cos 0° = ?
- Math Move (SOHCAHTOA).
- Preview Day 7: “tangent — when you don’t know the hypotenuse.”
Students produce → worked problems on “Sine & Cosine on the Unit Circle”
Three tracks — below / at / above- ▽ Below — Read values straight off the labeled circle — no calculator — for the four quadrantal angles, then the 30/45/60 set.
- ◆ At — Read the labeled values, then check two of them on the calculator and reconcile any disagreement.
- △ Above — Predict the SIGN of cos θ and sin θ in each quadrant before computing, and say which quadrant an obtuse elbow angle lands in. (Sets up Day 12.)
⏱ If behind today (35 min, or yesterday ran over): Read four values off the labeled circle together, then send the Sin & Cosine worksheet home. Keep the exit ticket — cos 0° is the check that x = cos actually landed. Day 9 rebuilds forward kinematics directly on this idea, so if it did not stick, spend Day 7's warm-up here rather than pushing on.
Students will ask · what to sayAsk · “Which one is x again — sin or cos?”
Say: Cos = x (both start with the ‘co’ sound of ‘across’). Sin = y (up).
Watch for- Swapping sin and cos. Anchor: cosine is the horizontal (x), sine is the vertical (y).
Boost Cards: BC1, BC2, BC3, BC4
Substitute today? Run it as a paper day — warm-up, work block, and exit ticket exactly as written above.
There is no arm today, so nothing to defer.
Week 2 · Day 7Tangent & the missing arm length
Use tangent when you don’t know the hypotenuse.
Competencies this lesson builds
- ◆ CHOOSE (Pick the right tool · Computational Thinking) — Tangent is the tool you reach for when the hypotenuse is unknown — students pick it for a reason, not by habit.
- ◆ REPRESENT (Convert between forms · Computational Thinking) — tan θ = y/x rearranges to θ = arctan(y/x) — the same relationship solved for the thing you need.
Warm-up · Daily Driver (on the board): What is sin 90°? (the y-coordinate at the top of the circle)
Main work block · run of show (45 min) — one cue per line
- 0–5 · Entry Task
- sin 90° = 1.
- Bridge from last class.
- 5–20 · Teacher-led · tangent
- tan θ = opposite / adjacent — never needs the hypotenuse.
- When you know two legs and want the angle, tangent is the tool.
- Work θ = arctan(8/10) = 38.7°.
- 20–38 · Guided practice
- Tangent & inverse-tangent worksheet.
- Students solve for the angle in their own scenario’s legs.
- 38–45 · Exit
- Two legs, no hypotenuse — which ratio?
- (tangent).
- Math Move.
- Preview Day 8: “choosing the right ratio.”
Students produce → worked problems on “Tangent & Inverse Tangent”
Three tracks — below / at / above- ▽ Below — Right triangle drawn and labeled; they choose the ratio and solve one side.
- ◆ At — Set up and solve for the missing length with tangent on their own triangle.
- △ Above — Try a target that forces an obtuse angle — does SOHCAHTOA still reach it? Why not? (Bridge to Day 12.)
⏱ If behind today (35 min, or yesterday ran over): One worked tangent problem + one solo; worksheet to homework.
Students will ask · what to sayAsk · “When do I use tangent vs sine/cosine?”
Say: Tangent when you have the two legs (opp & adj) and no hypotenuse. Next class we make the choice explicit.
Ask · “arctan of what — 8/10 or 10/8?”
Say: Opposite over adjacent. If the angle is at the base, opposite is the height (y), adjacent is the across (x).
Watch for- Flipping opp/adj. Draw the triangle and label relative to θ every time.
Boost Cards: BC1, BC2, BC3, BC4
Substitute today? Run it as a paper day — warm-up, work block, and exit ticket exactly as written above.
There is no arm today, so nothing to defer.
Week 2 · Day 8Choose the right tool
Pick sin, cos, or tan — and justify it before you compute.
Competencies this lesson builds
- ◆ CHOOSE (Pick the right tool · Computational Thinking) — The whole lesson: name the tool (sin, cos, tan, Law of Cosines) and say why it fits before computing anything.
- ★ JUSTIFY (Explain with evidence · Problem Solving) — Students write the one-sentence reason out loud — “tan, because I have opposite and adjacent and no hypotenuse.”
Warm-up · Daily Driver (on the board): tan θ = opposite over what? (one word)
Main work block · run of show (45 min) — one cue per line
- 0–5 · Entry Task
- adjacent.
- Prime the ratio choice.
- 5–18 · Teacher-led · choose first
- Give the sentence stem: “I have ___ and ___.
- I need ___.
- So I use ___.” Half of trig is picking the ratio that uses the two sides you HAVE.
- Model 2–3.
- 18–38 · Guided practice
- SOHCAHTOA choose-&-solve worksheet.
- Insist they fill the stem before computing.
- Distribute BC2/BC3.
- 38–45 · Exit
- Have opp & hyp — which ratio?
- (sine).
- Math Move.
- Preview Day 9: “forward kinematics — angles to position.”
Students produce → worked problems on “SOHCAHTOA — Choose & Solve”
Three tracks — below / at / above- ▽ Below — A two-column match — known/unknown → tool. They match, not derive.
- ◆ At — Pick the tool for each scenario and justify the choice in a phrase.
- △ Above — Write a scenario that needs TWO tools in sequence, then solve it.
⏱ If behind today (35 min, or yesterday ran over): Run the tool chooser as a whole-class discussion; skip the individual scenarios.
Students will ask · what to sayAsk · “I picked a ratio but it uses a side I don’t have.”
Say: Then it’s the wrong ratio. The right one uses ONLY the two sides you already know.
Watch for- Jumping to compute before choosing. Enforce the sentence stem — it’s the whole lesson.
Boost Cards: BC1, BC2, BC3, BC4
Substitute today? Run it as a paper day — warm-up, work block, and exit ticket exactly as written above.
There is no arm today, so nothing to defer.
Week 2 · Day 9 🔧Forward kinematics on paper
Given the joint angles, predict where the tip lands.
Competencies this lesson builds
- ○ MODEL (Name the quantities · Mathematical Modeling) — Forward kinematics IS a model: feed it joint angles, it predicts where the real end-effector lands.
- ◆ REPRESENT (Convert between forms · Computational Thinking) — Each link is the hypotenuse of its own right triangle, so students move between the arm diagram, x = L·cos θ, and a coordinate pair.
Warm-up · Daily Driver (on the board): If you have the adjacent leg and the hypotenuse, which ratio? (sin/cos/tan)
Main work block · run of show (45 min) — one cue per line
- 0–5 · Entry Task
- cosine (adj/hyp).
- Warm-up.
- 5–20 · Teacher-led · FK
- Say it the way the screen says it: forward kinematics tells you where the arm will land before you move it.
- Start with the two angles, do the math, get the (x, y) the end-effector reaches.
- Each link is the hypotenuse of its own right triangle: x₁ = L₁·cos θ₁, y₁ = L₁·sin θ₁.
- Add link 2.
- Work an example.
- 20–35 · Guided practice
- Forward Kinematics worksheet.
- Keep L₁=18, L₂=22 on the board.
- 35–42 · Arm Check
- Enter θ₁=60°, θ₂=30°, run it, measure where the end-effector actually lands, compare to the prediction.
- The gap is the lesson.
- 42–45 · Exit + preview
- FK goes from angles → position.
- Preview Day 10: “predict, then test — the FK Prediction Card.”
Students produce → worked problems on “Forward Kinematics”
Three tracks — below / at / above- ▽ Below — One link at a time — compute L₁'s tip, then add L₂, with the formula split into two boxes.
- ◆ At — Full forward kinematics for their angle pair, algebra shown.
- △ Above — Predict, then find a DIFFERENT angle pair that lands the tip near the same point (preview of two solutions).
⏱ If behind today (35 min, or yesterday ran over): Paper FK only; defer the arm measurement to Day 10's station rotation.
Students will ask · what to sayAsk · “Why doesn’t the arm land exactly where I predicted?”
Say: The model assumes perfect rigid links and a perfect pivot. Real slack/flex shifts it a few mm. That gap is a BUILD error, not a math error — and it’s the point of the next card.
Watch for- Forgetting the second link (only computing link 1). The tip is link 1 + link 2.
Boost Cards: BC1, BC2, BC3, BC4 · arm setup (BC7)
Substitute today? Run it as a paper day — warm-up, work block, and exit ticket exactly as written above.
No arm needed: defer the Arm Check to the next arm day, or have students measure from the reference video (same motion, ruler in frame) — not yet filmed, so until it is, demo the motion by hand with a ruler or use the Day-11 aimer in the app, which simulates the same landing. The prediction and the error reasoning are unchanged — only where the measured number comes from.
Week 2 · Day 10 · Gate/Milestone ★ 🔧The FK Prediction Card
Milestone 2 — predict a tip position, then test it on the arm.
Competencies this lesson builds
- ★ DESIGN (Create for user needs · Problem Solving) — The user check-in: half the unit has passed since the Brief, so students take the card back to the person it's for and record what they actually said.
- ◆ REPRESENT (Convert between forms · Computational Thinking) — Gate/Milestone 2 — the predicted tip is computed from angles and expressed as coordinates.
- ◆ CHOOSE (Pick the right tool · Computational Thinking) — Students pick the right ratio for each link without being told which.
- ○ TEST (Prediction vs. reality · Mathematical Modeling) — The prediction is checked against where the arm actually lands. The gap is the lesson.
- ◆ DEBUG (Find & fix errors · Computational Thinking) — Predicted ≠ measured — students have to say where the error came from.
Warm-up · Daily Driver (on the board): Link-1 x at θ₁ = 60° with L₁ = 18? (cos 60° = 0.5)
Note: STATION PROTOCOL — ONE measured run per crew, not per student. Every crew of four has its own arm, so all crews run at the same time; there is no rotation and no posted order. The reason it is one run per crew is pedagogical, not logistical: the crew shares a target, so a single landing is the common evidence every member measures their own prediction against. Each crew agrees on ONE angle pair; every student predicts it INDEPENDENTLY on their own card (that is the individual math the gate assesses); the crew's OPERATOR drives the single run at their own arm; the whole crew measures that one landing and each student explains their own gap. A crew whose arm will not home or connect moves to the front arm rather than losing the block. Two-part day: anyone who needs it drafts the card the night before (Day 9 FK refresher). Class time is the run, the comparison, and tagging — not the full solve.
Main work block · run of show (45 min) — one cue per line
- 0–5 · Entry Task
- 5–12 · Frame the Gate
- Say: “Your crew picks ONE angle pair.
- Each of you predicts it alone — your own algebra.
- Your OPERATOR runs it once; you all measure the same landing.” Have each crew commit its pair on a sticky note before anyone computes — that stops mid-run changes.
- The GAP between predicted and measured is the whole card.
- Show the W2D5 exemplars.
- 12–30 · Predict (individually, at tables)
- Everyone computes their own predicted (x, y) at their table.
- No one is at the arm yet.
- Circulate for the link-2 angle (θ₁ + θ₂, not θ₂) and calculator mode.
- Push them to show the algebra, not just a number.
- 30–38 · Arm stations — one run per crew
- All crews run at once, each at their own arm.
- The OPERATOR enters the crew's pair and runs it; the RECORDER reads the landing in cm; everyone in the crew writes down that same measured (x, y).
- Budget ~60–75 seconds per crew — six crews fit comfortably.
- Crews waiting keep working the algebra or start the Day-11 preview; crews finished compare predicted vs measured at their table.
- 38–45 · Submit + close
- Each student states their own error in cm and says whether the gap is a math error or a build error — that is the class work, and it needs every one of these minutes.
- Then submit.
- Tags against REPRESENT, CHOOSE, DEBUG, TEST, DESIGN (pass = all 6 FK checklist must-haves — the sixth is the user check-in, which is what DESIGN tags).
- Preview Day 11: “the reverse problem — going backward.”
Students produce → FK Prediction Card (Gate/Milestone — pass 6/6 must-haves)
Three tracks — below / at / above- ▽ Below — FK skeleton with the link-1 row already computed (x₁ = 18·cos θ₁, y₁ = 18·sin θ₁ filled in); they do link 2 and the sum. Still their own prediction, still their own algebra.
- ◆ At — Full independent prediction for the crew's pair, algebra shown, then the measured gap and its source.
- △ Above — After predicting the crew's pair, find a DIFFERENT pair that lands near the same point — the two-solution preview Day 11 opens with.
⏱ If behind today (35 min, or yesterday ran over): The predict block is the time sink, not the station — crews run in parallel, so the arm costs about two minutes. Have every crew's angle pair committed on a sticky note before the bell so no one is deciding at the machine. Still short? Run the comparison as a whole-class debrief instead of table-by-table, and let the math-error-or-build-error call move to the Day 11 warm-up. Never cut the individual predict block — it is the only part the card assesses, and never let a crew skip its run, because the measured gap is must-have #3. If the shortfall is unfinished cards rather than the clock, see the week's Contingencies card.
Students will ask · what to sayAsk · “What exactly goes on the board?”
Say: Both links in full, and the reason for the second: x = L₁·cos θ₁ + L₂·cos(θ₁ + θ₂), y = L₁·sin θ₁ + L₂·sin(θ₁ + θ₂). Say the reason out loud — link 2 points at θ₁ + θ₂ because θ₂ is measured from LINK 1, not from the ground. A student who treats θ₂ as an absolute angle gets a believable wrong answer every time, and this is where that gets prevented.
Ask · “My prediction is way off, not a little.”
Say: Start with the angle you used for link 2 — it should be θ₁ + θ₂ — and your calculator mode. A large gap points at the math, the units, the mode and the angle convention first; a small one can be arm slack, calibration or measurement. Do not diagnose from size alone — check the suspect before acting on it, which is the habit Day 17 builds on.
Ask · “Why don't I get my own run on the arm?”
Say: Because the MATH is what's being assessed, and that's yours alone — you predicted it independently. The arm just tells the crew what reality did. Everyone in your crew predicted the same pair, so you can compare whose algebra was closest to the same measurement — that comparison is more useful than four separate runs at the same machine.
Watch for- No algebra shown (just an answer). The card is about the reasoning, not the number.
- A crew that changes its angle pair AFTER the run so the prediction matches. The pair is committed before anyone computes — the gap is the point, not a clean score.
- Students copying a crewmate's prediction instead of computing their own. The measured run is shared; the prediction is not.
Boost Cards: BC1, BC2, BC3, BC4 · arm setup (BC7)
Substitute today? Run it as a paper day — warm-up, work block, and exit ticket exactly as written above.
No arm needed: defer the Arm Check to the next arm day, or have students measure from the reference video (same motion, ruler in frame) — not yet filmed, so until it is, demo the motion by hand with a ruler or use the Day-11 aimer in the app, which simulates the same landing. The prediction and the error reasoning are unchanged — only where the measured number comes from.
Week 2 · ContingenciesIf it goes sideways
Three recovery moves for this week — preserve rigor, ease the crunch. Reach for this card when the problem is bigger than the clock; if it is only the clock, the day's ⏱ If behind today line is the faster fix.
- Arms or laptops are down — FK is a paper prediction first — run all the Day 9–10 math on paper. The 'measured' box becomes a single teacher demo of one pose, or defer measurement to a later station day.
- Half haven't finished the FK Card by Day 10 — A pass is one predicted (x, y) with the algebra shown + a measured value + the gap explained. One angle pair is enough; a second is stretch.
- A drill kills warm recall — Skip the Daily Driver; open with the worked FK example on the board and go straight to guided practice.
- The app or the internet is down — Days 6–9 print in full with answer keys, and the arm is on its own wi-fi network so the Day 9 and Day 10 runs still happen. On Day 10 hand out the printed FK Prediction Card and its checklist: they predict on paper, run the arm, record the measurement, and you tag the paper.
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