Practice Worksheets

Student copy — no answers. One per concept, session-labeled (W#D#). The teacher copy carries the answer key.

Day 0 · before the unit

User Interview: Motion Matters

This is your Day 0 prework — do it before the unit starts. Your job is to find one real person and listen like a designer: what makes their task frustrating, tiring, or error-prone? What do they wish could be different?
Name
Section
Date
Before the interview:Reflect:Turn the motion into two points
The arm's base is at (0, 0). x = how far across, y = how high — both in cm. So (10, 8) means 10 cm across and 8 cm up. Keep each point within the arm's reach: 8–40 cm from the base. (Closer than about 8 cm and the elbow can't fold tight enough to get there.)
Start (x, y)
End (x, y)
W1D2 · Day 2

Criteria vs. Constraints

Tell a measurable success test from a fixed design limit — and sharpen vague wishes.
Name
Section
Date
Example: “Within 1 cm of the target — yes/no” is a criterion. “The tray rail is 8 cm high” is a constraint.
  1. Classify: “The arm must fit on a 30 cm desk.”
  2. Classify: “The end-effector lands within 1.5 cm of the target.”
  3. Classify: “The motion completes in under 3 seconds.”
  4. Classify: “The tray rail is 8 cm tall.”
  5. Classify: “The arm should feel smooth.”
  6. Sharpen “the arm is fast” into a criterion.
  7. Sharpen “lightweight” into a criterion.
  8. “Easy to use” is an opinion. Make it measurable with a User-Defined Scale.
  9. In one sentence: how is a constraint different from a criterion?
⭐ Stretch: Write one “criterion” a user could argue is BOTH met and not met, depending on how you measure it. How would you rewrite it so there's no argument?
W1D3 · Day 3

Degrees ↔ Radians

Convert an angle between degrees and radians.
Name
Section
Date
radians = degrees × π/180
degrees = radians × 180/π
Example: 90° × π/180 = π/2 ≈ 1.5708 rad
  1. Convert 30° to radians (decimal).
  2. Convert 45° to radians.
  3. Convert 60° to radians.
  4. Convert 120° to radians.
  5. Convert 270° to radians.
  6. Convert π/3 rad to degrees.
  7. Convert 2π/3 rad to degrees.
  8. Convert 0.785 rad to degrees.
⭐ Stretch: Convert 200° to radians, then convert your answer back to degrees. Do you land on 200° again? Explain why you must.
W1D4 · Day 4

Arc Length: s = r · θ

How far the arm's tip travels when it sweeps an angle.
Name
Section
Date
s = r · θ (θ must be in radians)
Example: r = 18 cm, 45° = 0.785 rad → s = 18 × 0.785 ≈ 14.1 cm
  1. r = 18 cm, sweep 90° (= 1.571 rad). Find s.
  2. r = 22 cm, sweep 60° (= 1.047 rad). Find s.
  3. r = 18 cm, sweep 30° (= 0.524 rad). Find s.
  4. r = 40 cm (max reach), sweep 90°. Find s.
  5. r = 22 cm, sweep 120° (= 2.094 rad). Find s.
  6. If you double the sweep angle, what happens to the arc length?
⭐ Stretch: Two arms sweep the same 60°, but one has r = 18 and the other r = 36. Predict the ratio of their arc lengths before you compute — then check.
W2D1 · Day 6

Sine & Cosine on the Unit Circle

Read sine and cosine straight off the coordinates of a point.
Name
Section
Date
cos θ = x
sin θ = y
(the point at angle θ)
Example: The point at 60° is (0.5, 0.866) → cos 60° = 0.5, sin 60° = 0.866
  1. cos 0° = ?
  2. sin 90° = ?
  3. cos 60° = ?
  4. sin 30° = ?
  5. cos 45° = ?
  6. sin 120° = ?
  7. cos 180° = ?
  8. The point at 30° is (0.866, 0.5). What is sin 30°?
  9. cos 150° = ? (reference angle 180° − 150° = 30°)
  10. sin 135° = ? (reference angle 45°)
  11. The shoulder is at 140°. Give its reference angle, and say whether cos 140° is positive or negative.
⭐ Stretch: Without a calculator, explain why sin 150° = sin 30° using the unit circle. Which coordinate is doing the work?
W2D2 · Day 7

Tangent & Inverse Tangent

Find an angle from the two legs of a right triangle — no hypotenuse needed.
Name
Section
Date
tan θ = opposite / adjacent
θ = arctan(opp / adj)
Example: opposite 8, adjacent 10 → θ = arctan(8/10) = arctan(0.8) ≈ 38.7°
  1. tan θ = opposite over what?
  2. A target is 8 cm up, 10 cm across. θ = arctan(8/10) = ?
  3. Legs 15 up, 20 across. θ = arctan(15/20) = ?
  4. Legs 6 and 8. θ = arctan(6/8) = ?
  5. A target 24 up, 10 across. θ = arctan(24/10) = ?
  6. Legs 5 and 12. θ = arctan(5/12) = ?
  7. You know both legs but not the hypotenuse. Which ratio finds the angle?
⭐ Stretch: For what angle is tan θ = 1? What does that tell you about the two legs of the triangle?
W2D3 · Day 8

SOHCAHTOA — Choose & Solve

Pick the ratio that uses the two sides you have, then solve for the angle.
Name
Section
Date
SOH sin θ = O / H
CAH cos θ = A / H
TOA tan θ = O / A
Example: Have opposite & hypotenuse, need the angle → use sine.
  1. You have opposite and hypotenuse, need the angle. Which ratio?
  2. You have adjacent and hypotenuse. Which ratio?
  3. You have opposite and adjacent. Which ratio?
  4. sin θ = 10/26. Find θ.
  5. cos θ = 20/25. Find θ.
  6. tan θ = 8/10. Find θ.
  7. Fill in the stem: “I have ___ and ___. I need ___. So I use ___.”
⭐ Stretch: A target sits straight up at (0, 15). Which ratio breaks, and why? What is the angle — and how do you know without a formula?
W2D4 · Day 9

Forward Kinematics

Given both joint angles, predict where the arm's tip lands.
Name
Section
Date
one link: x = L · cos θ, y = L · sin θ
both links: x = L₁·cos θ₁ + L₂·cos(θ₁ + θ₂)
y = L₁·sin θ₁ + L₂·sin(θ₁ + θ₂)
θ₂ is measured from LINK 1, so link 2 points at θ₁ + θ₂ — not at θ₂
(L₁ = 18 cm, L₂ = 22 cm)
Example: θ₁ = 60°, θ₂ = 30° → link 2 points at 90°; x = 9.0 + 0 = 9.0 cm, y = 15.6 + 22 = 37.6 cm
  1. θ₁ = 60°. Find x₁.
  2. θ₁ = 60°. Find y₁.
  3. θ₁ = 90°. Find x₁ and y₁.
  4. θ₁ = 60°, θ₂ = 30°. Which direction does link 2 point, measured from the ground?
  5. Link 2 is 22 cm and points at 90°. Find x₂ and y₂.
  6. Now add both links for θ₁ = 60°, θ₂ = 30°. Where does the tip land?
  7. θ₁ = 45°, θ₂ = 45°. Find the tip position.
  8. A student computes link 2 for θ₁ = 60°, θ₂ = 30° as 22 · cos 30° = 19.1. What did they forget?
  9. Forward kinematics goes from ___ to ___.
  10. A shoulder has sin θ₁ = 0.6, and the tip is LEFT of the base. Find cos θ₁.
  11. A student reports cos θ = 0.6 and sin θ = 0.9 for the same angle. Can both be right?
⭐ Stretch: Find joint angles that put the tip as FAR from the base as possible. What's the max reach, and why is it L₁ + L₂?
W3D2 · Day 12

Law of Cosines: the Tool

Run the Law of Cosines in both directions on ordinary triangles — no arm, no θ₂′.
Name
Section
Date
c² = a² + b² − 2ab · cos γ (side you want, from two sides + the angle between)
cos γ = (a² + b² − c²) / (2ab) (angle you want, from all three sides)
γ is the angle BETWEEN a and b, and it sits opposite c.
Example: Sides a = 7, b = 9, γ = 50°: c² = 49 + 81 − 126(0.643) ≈ 49, so c ≈ 7.
  1. a = 5, b = 8, γ = 60°. Find c². (cos 60° = 0.5)
  2. Sides 6, 8 and 11. Find the angle opposite the side of length 11.
  3. Sides 9, 12 and 15. Find the angle opposite 15. What does your answer tell you about this triangle?
  4. Sides 10, 10 and 14. Find the angle opposite 14.
  5. You have a right triangle, its hypotenuse, and one acute angle, and you want the opposite side. Law of Cosines or SOHCAHTOA? Why?
⭐ Stretch: Put γ = 90° into c² = a² + b² − 2ab · cos γ and simplify. Which theorem falls out, and what does that say about how the two rules are related?
W3D3 · Day 13

Law of Cosines: the Elbow Angle θ₂′

Solve the elbow angle for a target — the arm's triangle isn't a right triangle.
Name
Section
Date
cos(θ₂′) = (r² − L₁² − L₂²) / (2 · L₁ · L₂)
L₁ = 18, L₂ = 22 (L₁² = 324, L₂² = 484, 2·L₁·L₂ = 792)
Example: Target (10, 8): r² = 164 → cos(θ₂′) = (164 − 324 − 484)/792 = −0.813 → θ₂′ = 144.4°
  1. Target (10, 8). Find r² = x² + y².
  2. Continue: cos(θ₂′) = (164 − 324 − 484) / 792 = ?
  3. θ₂′ = arccos(−0.813) = ?
  4. Target (18, 22). Find r², then cos(θ₂′), then θ₂′.
  5. Target (20, 10): r² = 500. cos(θ₂′) = (500 − 808)/792 = ? (808 = L₁² + L₂² = 324 + 484.) Then θ₂′ = ?
  6. Target (30, 0): r² = 900. cos(θ₂′) = (900 − 808)/792 = ? Then θ₂′ = ?
  7. Your formula gives cos(θ₂′) = 1.2. What does that mean?
  8. Now the other end: what would cos(θ₂′) = −1.2 mean?
  9. A target gives cos(θ₂′) = −0.98, so an angle does exist. Can the real arm reach it?
  10. A triangle has sides 7, 9 and 12 — no right angle anywhere. Find the angle opposite the 12 side.
  11. Two sides are 6 and 8 with a RIGHT angle between them. Use the Law of Cosines to find the third side. What happens to the last term?
  12. You have a right triangle and know two legs, and you want the angle. Which tool, and why not the other one?
  13. Target (24, 18). r² = 900. Find cos(θ₂′), then θ₂′.
⭐ Stretch: Find a target where θ₂′ = 90° exactly (a right-angle elbow). What has to be true about r² for that to happen?
W3D4 · Day 14

The Shoulder Angle θ₁ = α + β

Aim the whole arm at the target with the aim angle α and the lean β — and compute BOTH yourself.
Name
Section
Date
d = √(x² + y²)
α = arctan(y / x), then CHECK THE QUADRANT — if x is negative, add 180°
cos β = (L₁² + d² − L₂²) / (2·L₁·d)
θ₁ = α + β
Example: Target (20, 15): d = 25 · α = arctan(15/20) = 36.9° · cos β = (324 + 625 − 484)/(2·18·25) = 0.517 → β = 58.9° · θ₁ = 95.8°
  1. α = arctan(y/x) for (10, 8) = ?
  2. α for (18, 22) = arctan(22/18) = ?
  3. α for (14, 20) = arctan(20/14) = ?
  4. α for (24, 6) = arctan(6/24) = ?
  5. Find β yourself for (10, 8). First d = √(10² + 8²) = 12.8, then cos β = (L₁² + d² − L₂²) / (2·L₁·d) with L₁ = 18, L₂ = 22.
  6. Now θ₁ for (10, 8): θ₁ = α + β = ?
  7. Find β yourself for (24, 10). (d = 26)
  8. Find β yourself for (18, 22), then θ₁. (d = 28.4, α = 50.7°)
  9. For (24, 10) one student correctly gets β = 56.5°. Another mistakenly uses its supplement, 123.5°. Carry both through to θ₁ (α = 22.6°), then check each against the shoulder's 20°–150° travel. Why is that check not enough to catch the mistake?
  10. A target sits at (−10, 8) — left of the base. The calculator gives arctan(8 / −10) = −38.7°. Is −38.7° the aim angle α? Explain, and correct it if not.
  11. θ₁ is built from which two pieces — and is β the same for every target?
⭐ Stretch: Two targets share the same α but have different r. Which piece of θ₁ changes — α or β? Reason it out, then test with numbers.
W3D5 · Day 15

IK Solve Card — Full Pipeline

Solve a target end to end: r → θ₂′ → α → θ₁, and check it's reachable.
Name
Section
Date
Step 0 · reachability: r = √(x² + y²), and 8 ≤ r ≤ 40 — check BEFORE you solve
1. cos(θ₂′) = (r² − L₁² − L₂²) / (2·L₁·L₂)
2. θ₂′ = arccos(that)
3. α = arctan(y/x), then check the quadrant — if x is negative, add 180°
4. cos β = (L₁² + r² − L₂²) / (2·L₁·r)
5. β = arccos(that)
6. θ₁ = α + β
Example: Target (10, 8) → r² = 164, θ₂′ = 144.4°, α = 38.7°, θ₁ = 128.2°
  1. Target (18, 22): find r².
  2. Target (18, 22): find θ₂′.
  3. Target (18, 22): find α, then β yourself (d = 28.4), then θ₁.
  4. Target (14, 20): find r², then cos(θ₂′), then θ₂′.
  5. Target (14, 20): find α, then β yourself (d = 24.4), then θ₁.
  6. Is the target (50, 0) reachable? (max reach 40 cm)
  7. Reachability rule: a target is reachable only if ___ ≤ r ≤ ___.
⭐ Stretch: Design a back-and-forth motion whose two endpoints need the SAME θ₂′ but different θ₁. What does that look like on the arm?
W5D1 · Day 21

Period, Midline, Amplitude

Read the three numbers of a repeating motion off a table or a description.
Name
Section
Date
period = time for one full round trip (out AND back)
midline = (max + min) ÷ 2
amplitude = (max − min) ÷ 2
Example: Angles run 101.4° to 128.2°: midline = 114.8°, amplitude = 13.4°
  1. A motion takes 7 s to go out and 7 s to come back. What is its period?
  2. An arm completes 5 full cycles in 30 s. What is its period?
  3. A motion runs between 60° and 96°. Find the midline.
  4. Same motion, 60° to 96°. Find the amplitude.
  5. As the arm carries the cup, the end-effector's distance from the base runs between 12 cm and 28 cm. Find the midline and the amplitude.
  6. A motion runs between −25° and −5°. Find the midline and the amplitude.
  7. A motion has midline 40° and amplitude 8°. What are its highest and lowest values?
  8. An arm reaches its far point 9 s after leaving the start. What is its period?
⭐ Stretch: Two motions have the same period and the same amplitude but different midlines. Sketch both on one pair of axes. What is the only thing that looks different — and what does that tell you about which of the three numbers controls height?
W5D1 · Day 21

Three Numbers, Other People's Motions

Read period, midline and amplitude out of a context that has no robot in it.
Name
Section
Date
period = time for one FULL cycle, out and back
midline = (max + min) ÷ 2
amplitude = (max − min) ÷ 2
Example: A swing from 12° to 48°: midline 30°, amplitude 18°
  1. A playground swing takes 1.6 s to travel from the front of its arc to the back. What is its period?
  2. A piston completes 150 cycles in one minute. What is its period, in seconds?
  3. A weather station logs a low of 41°F at 5 a.m. and a high of 67°F at 3 p.m. Find the midline and the amplitude.
  4. A Ferris wheel is 24 m across and its lowest seat sits 1.5 m above the ground. Find the highest point, then the midline and the amplitude.
  5. A tide gauge 3.4 miles offshore records a high of 8.2 ft at 04:10 and a low of 1.6 ft at 10:22. Find the midline and the amplitude. One fact here is not needed — say which.
  6. A cooling fan's blade tip swings between −22 cm and +6 cm of its rest position. Find the midline and the amplitude.
  7. A heart-rate trace has a period of 0.75 s. How long is it from one peak to the next trough?
  8. A motion has midline 30 cm and amplitude 11 cm, and takes 9 s to get from its lowest point to its highest. Give its period, its highest value and its lowest value.
⭐ Stretch: Two students time the same swing. One times 10 full swings and divides by 10; the other times a single swing. Which method gives the better period, and why? What has to be true about the swing for the extra trouble to be worth it?
W5D2 · Day 22

Reading a Motion Graph

Find period, midline and amplitude on a picture instead of in a list.
Name
Section
Date
period = width of one repeat, along the t axis
amplitude = height from the midline up to the peak
midline = (peak + trough) ÷ 2
Example: Peak 128.2°, midline 114.8° → amplitude = 128.2 − 114.8 = 13.4°
  1. A graph peaks at 48° with its midline drawn at 40°. Find the amplitude.
  2. A graph has midline 17 m and amplitude 15 m. How high is its peak?
  3. Same graph. How low is its trough?
  4. A graph peaks at 7 and bottoms at 1. Find its midline and amplitude.
  5. A curve leaves 101.4° at t = 0 and is next back at 101.4° heading the same way at t = 8 s. What is the period?
  6. Two students graph the SAME motion. One uses a vertical axis of 95°–135°, the other 0°–140°. Whose amplitude is larger?
  7. On a graph the curve sits exactly on its midline at t = 2 s and again at t = 6 s. What does that tell you?
  8. A graph's period is 8 s. How long does it take to get from the start to the far point?
⭐ Stretch: Draw the SAME motion twice: once on a vertical axis that fits it snugly, once on an axis running from zero. Measure the amplitude off both drawings. Explain to someone who has not done this why the two drawings cannot disagree.
W5D2 · Day 22

Reading a Graph Someone Else Drew

Find the three numbers on a graph or a log that was not drawn for this lesson.
Name
Section
Date
period = one full repeat along the horizontal axis
midline = (peak + trough) ÷ 2
amplitude = (peak − trough) ÷ 2
Example: A tide graph peaking at 8 ft and troughing at 2 ft: midline 5 ft, amplitude 3 ft
  1. A depth-against-time graph crosses its midline going UP at t = 3 s, going DOWN at t = 9 s, and going UP again at t = 15 s. Find the period.
  2. The same graph peaks once between 3 s and 9 s. At what time?
  3. A temperature graph runs from a low of 8°C to a high of 24°C. A student writes "amplitude = 16°C." Name the error and give the amplitude.
  4. A pump's outlet pressure is logged every 2 s — t=0: 30, t=2: 44, t=4: 50, t=6: 44, t=8: 30, t=10: 16, t=12: 10, t=14: 16, t=16: 30. Find the period, the midline and the amplitude.
  5. A graph of a buoy's height shows a peak of 9.4 ft at 06:00 and the next peak at 18:24. The buoy is painted red and weighs 180 lb. Find the period, and say which facts you did not need.
  6. A graph has its midline at 5 and peaks at 12. What is its lowest value?
  7. Two troughs in a row sit at t = 2.5 s and t = 6.1 s. Find the period, and say where the peak between them is.
  8. Using your answers to question 7: that motion has a midline of 40 and a peak of 46. What is its value at t = 2.5 s?
⭐ Stretch: The same motion is graphed twice: once with the vertical axis running 0 to 50, once running 28 to 34. Which of the three numbers looks different on the two drawings, and which cannot? What does that tell you about reading amplitude by eye?
W5D3 · Day 23

Writing the Equation

Turn three numbers into y = A·cos(Bx) + D, and back again.
Name
Section
Date
B = 2π ÷ period period = 2π ÷ B
D = midline |A| = amplitude A is negative when the motion starts at its LOWEST point
Example: period 8 s → B = 2π ÷ 8 = π/4 ≈ 0.785
  1. Find B for a period of 12 s. Give the exact form and the decimal.
  2. Find B for a period of 4 s.
  3. Find B for a period of 20 s.
  4. Find the period when B = π/3.
  5. Find the period when B = π/5.
  6. In y = −13.4·cos((π/4)t) + 114.8, what is the amplitude?
  7. Evaluate θ₁(3) for θ₁(t) = 114.8 − 13.4·cos((π/4)·t). Radian mode.
  8. Evaluate the same equation at t = 1.
⭐ Stretch: Your equation has B in it. Predict, without computing, what happens to the graph if you double B — and what happens to the period. Then check with B = π/4 against B = π/2.
W5D3 · Day 23

Building the Equation for Someone Else's Motion

Turn three numbers into y = A·cos(Bx) + D when the motion is not an arm, and read them back out.
Name
Section
Date
B = 2π ÷ period
A = amplitude, made NEGATIVE if the motion starts at its lowest point
D = midline
Example: period 8, midline 30, amplitude 5, starting at the top: y = 5·cos((π/4)x) + 30
  1. A wheel turns once every 10 s. Find B. Exact form.
  2. A buoy rides between 2 ft and 14 ft with a period of 6 s, starting at the TOP. Write its equation.
  3. The same buoy, but it starts at the BOTTOM. What changes, and what is the equation?
  4. Read the three numbers back out of y = 12·cos((π/9)t) + 45.
  5. A ventilator cycles 15 times a minute between 4 and 20 cm of water pressure, starting at the low end. The machine costs $18,000. Write the equation, and say which fact you did not use.
  6. A student writes B = period ÷ 2π. Pick a period, compute their B and the correct B, and say which is right and how you know.
  7. Using your equation from question 5, find the pressure at t = 1 s.
  8. A motion has B = π/12. What is its period?
⭐ Stretch: Two motions have B = π/6 and B = π/3. One is faster. Say which BEFORE computing anything and explain how you knew, then find both periods and check.
W5D4 · Day 24

Graph ↔ Equation

Read an equation off a graph, and a graph off an equation.
Name
Section
Date
From a graph: midline = (peak + trough) ÷ 2, amplitude = peak − midline, B = 2π ÷ period
Sign of A: + if the motion starts at its top, − if it starts at its bottom
Example: Top 48°, bottom 32°, repeats every 12 s, starts at the top → y = 8·cos((π/6)x) + 40
  1. A graph tops at 48°, bottoms at 32°, repeats every 12 s and starts at its top. Write its equation.
  2. A motion starts at its LOWEST point, 18°, rises to 42°, repeats every 10 s. Write its equation.
  3. For y = 5·cos((π/2)x) + 30, find the period.
  4. For the same equation, find the highest and lowest values.
  5. Evaluate y = 8·cos((π/6)x) + 40 at x = 0 and at x = 6.
  6. A graph repeats every 6 s. A classmate proposes y = 12·cos((π/6)x) + 100. Which of the three numbers is wrong, and what should it be?
  7. Every slot of a proposed equation checks out except that the motion starts at its bottom and the equation starts at its top. What single change fixes it?
  8. Sketch one full repeat of y = 5·cos((π/2)x) + 30. List the five anchor points.
⭐ Stretch: Write an equation whose graph starts at its midline rather than at a top or a bottom. Can you do it with cosine alone and no sliding left or right? Explain what that tells you about the form y = A·cos(Bx) + D.
W5D4 · Day 24

Graph to Equation, Equation to Graph

Go both directions on a motion that is not an arm: a picture in and an equation out, then back again.
Name
Section
Date
From a graph: D is the midline, A is peak − midline, B is 2π ÷ period
From an equation: read the same three off the slots
Example: A wave peaking at 9, troughing at 1, repeating every 4: y = 4·cos((π/2)x) + 5
  1. A graph peaks at 9 and troughs at 1, one full repeat takes 4 units, and it starts at a peak. Write the equation.
  2. Sketch two full cycles of y = 3·cos((π/5)x) + 7. Label the highest value, the lowest value, and where the second cycle ends.
  3. What window would you set to see two clear cycles of y = 20·cos((π/30)x) + 65? Give an x range and a y range.
  4. A graph of y = A·cos(Bx) + D has D = 0. Where is its midline, and what does that mean for the graph?
  5. A wave is drawn on graph paper with 2 mm squares. It peaks at 14, troughs at −6, and repeats every 8 units. Write the equation, and say which fact you did not need.
  6. Two equations: y = 5·cos((π/4)x) + 10 and y = 5·cos((π/2)x) + 10. Without drawing either, say every way the graphs differ.
  7. Using question 6: at what x does each graph first return to its peak?
  8. A student graphs y = −4·cos((π/6)x) + 9 and draws it starting at 13. What did they get wrong?
⭐ Stretch: Draw y = 5·cos((π/4)x) + 30 and y = −5·cos((π/4)x) + 30 on one pair of axes. Find every x between 0 and 8 where they meet, and say what is true about both motions at those points.
W5D5 · Day 25

Radians and Co-terminal Angles

Trade degrees for radians in both directions, and find the angle that points the same way.
Name
Section
Date
radians = degrees × π/180 degrees = radians × 180/π
180° = π radians one full turn = 360° = 2π radians
co-terminal: add or subtract a full turn
Example: 45° × π/180 = π/4 ≈ 0.7854 rad
  1. Convert 120° to radians. Exact form and decimal.
  2. Convert 240° to radians.
  3. Convert 135° to radians.
  4. Convert 1 radian to degrees.
  5. Convert 2 radians to degrees.
  6. A joint spins to 488.2°. Find the co-terminal angle between 0° and 360°.
  7. Find the co-terminal angle between 0° and 360° for −40°.
  8. Find the co-terminal angle between 0° and 360° for 750°.
⭐ Stretch: A circle of radius 5 and a circle of radius 500 both have an angle of 1 radian marked on them. The arcs are wildly different lengths. Explain why the ANGLE is the same anyway.
W5D5 · Day 25

Radians When the Problem Never Says So

Trade degrees for radians both ways, and use radians for arc length, on problems with no arm in them.
Name
Section
Date
degrees → radians: × π/180
radians → degrees: × 180/π
arc length s = r · θ, with θ in RADIANS
Example: 150° = 150 × π/180 = 5π/6 ≈ 2.618 rad
  1. Convert 225° to radians. Exact form.
  2. Convert 7π/6 to degrees.
  3. A bike wheel of radius 34 cm turns through 2.5 radians. How far does it roll?
  4. The same wheel turns through 90°. How far now?
  5. A clock's minute hand is 12 cm long and the face is 30 cm across. How far does the tip travel in 20 minutes?
  6. An angle of 1 radian: is it bigger or smaller than 60°, and by how much?
  7. A satellite dish of radius 1.8 m sweeps an arc 2.7 m long. Through what angle did it turn — in radians, then in degrees?
  8. Why does s = r·θ fail when θ is in degrees? Answer with a number, not a rule.
⭐ Stretch: Your calculator shows sin(30). In DEG mode the answer is 0.5. In RAD mode it is about −0.988. Explain why neither calculator is wrong — and say what information is missing from “30”.
W6D1 · Day 26

Exact Values on the Unit Circle

Write sine and cosine exactly, and get the sign from the quadrant.
Name
Section
Date
the point at angle θ is (cos θ, sin θ)
Q I: both + Q II: sin +, cos − Q III: both − Q IV: sin −, cos +
for 90°–180°, the matching first-quarter angle is 180° − θ
Example: cos 120° = −cos 60° = −1/2
  1. sin 60° and cos 60°, exact form.
  2. sin 45° and cos 45°, exact form.
  3. cos 120°, exact form.
  4. sin 135°, exact form.
  5. cos 150°, exact form.
  6. sin 180° and cos 180°.
  7. sin 240° and cos 240°, exact form.
  8. The class motion runs 101.4° to 128.2°. Which quadrant is it in, and what are the signs of sine and cosine throughout?
⭐ Stretch: Without a calculator, explain why sin 150° = sin 30° but cos 150° = −cos 30°. Which coordinate is doing the work in each case?
W6D1 · Day 26

Exact Values Away From the First Quadrant

Write sine and cosine exactly when the angle is past 180° or handed to you in radians.
Name
Section
Date
30°: 1/2 and √3/2 · 45°: √2/2 twice · 60°: √3/2 and 1/2
The quadrant fixes the sign: II sin +, III both −, IV cos +
Example: sin 210° = −1/2: reference angle 30°, and quadrant III makes sine negative
  1. sin 210°, exact form.
  2. cos 300°, exact form.
  3. sin(5π/4), exact form.
  4. cos(11π/6), exact form.
  5. A wind turbine blade is 54 m long, and its tip sits at 240° measured from the horizontal. Give the exact cosine of that angle, and say which fact you did not need.
  6. A point on the unit circle is (−1/2, −√3/2). Which angle is it, between 0° and 360°?
  7. Which of these cannot be a point on the unit circle: (0.6, 0.8), (√3/2, 1/2), (0.5, 0.9)?
  8. Using question 6: give the sine and cosine of the angle 180° larger than your answer.
⭐ Stretch: Every exact value on the unit circle uses only three numbers: 1/2, √2/2 and √3/2. Put them in order, then explain why the largest goes with the angle nearest the horizontal axis for cosine, and nearest the vertical axis for sine.
W6D2 · Day 27

Reference Angles and Two Solutions

Find every angle with a given sine or cosine, not just the one the calculator gives.
Name
Section
Date
reference angle α (always positive and acute): Q I θ · Q II 180° − θ · Q III θ − 180° · Q IV 360° − θ
sine POSITIVE → α and 180° − α sine NEGATIVE → 180° + α and 360° − α
cosine POSITIVE → α and 360° − α cosine NEGATIVE → 180° − α and 180° + α
arcsin always answers between −90° and 90°; arccos between 0° and 180°
Example: sin θ = 0.5 → θ = 30° and 150°
  1. Find the reference angle of 150°.
  2. Find the reference angle of 240°.
  3. Find the reference angle of 300°.
  4. Find the reference angle of 128.2°.
  5. sin θ = 0.866. Find BOTH angles between 0° and 360°.
  6. cos θ = −0.5. Find BOTH angles between 0° and 360°.
  7. cos θ = 0.866. Find BOTH angles between 0° and 360°.
  8. sin θ₁ = 0.866 and the shoulder only travels between 101.4° and 128.2°. What is θ₁, and why is the calculator's 60° not the answer?
⭐ Stretch: arcsin only ever answers between −90° and 90°. Work out what range arccos answers in, by thinking about which angles have each cosine exactly once. Check three values on a calculator.
W6D2 · Day 27

Every Angle, Not Just the One the Calculator Gives

Find all the angles in one turn with a given sine or cosine, on problems with no arm in them.
Name
Section
Date
Find the reference angle α first — always positive and acute — then let the sign pick the quadrants.
sine POSITIVE → α and 180° − α sine NEGATIVE → 180° + α and 360° − α
cosine POSITIVE → α and 360° − α cosine NEGATIVE → 180° − α and 180° + α
Add or subtract whole turns of 360° for more
Example: sin θ = 0.5 gives 30° and 150° in one turn
  1. sin θ = 0.766. Find both angles between 0° and 360°, to one decimal place.
  2. cos θ = 0.5. Find both angles between 0° and 360°.
  3. cos θ = −0.342. Find both angles between 0° and 360°, to one decimal place.
  4. A calculator returns arcsin(−0.5) = −30°. Give two angles between 0° and 360° with that sine.
  5. A crane's cable makes an angle whose sine is 0.6, and the cable is rated to 4 tons. Give every angle between 0° and 360° with that sine, to one decimal place.
  6. Why does a calculator hand back only one answer?
  7. sin θ = 0.9, and θ is known to be obtuse. Which angle is it?
  8. Using question 7: give a third angle with the same sine, from outside 0° to 360°.
⭐ Stretch: arccos never returns an angle in quadrant III or IV, and arcsin never returns one in II or III. Between them they cover only half the circle each. Work out which half each covers, then explain how you still reach every angle.
W6D3 · Day 28

One Ratio From Another

Use sin²θ + cos²θ = 1 with a quadrant to find the ratio you were not given.
Name
Section
Date
sin²θ + cos²θ = 1
cos θ = ±√(1 − sin²θ) sin θ = ±√(1 − cos²θ)
the square root gives the SIZE. the quadrant gives the SIGN.
Example: sin θ = 3/5 in quadrant II → cos θ = −4/5
  1. cos θ = 0.6, quadrant I. Find sin θ.
  2. sin θ = 3/5, quadrant II. Find cos θ, exact form.
  3. cos θ = −5/13, quadrant III. Find sin θ, exact form.
  4. sin θ = −8/17, quadrant IV. Find cos θ, exact form.
  5. cos θ = −3/5, quadrant III. Find sin θ, exact form.
  6. sin θ = 12/13, quadrant II. Find cos θ, exact form.
  7. A student computes √(1 − 3/5) = √(2/5) ≈ 0.632. What did they do wrong?
  8. A student gets the size 4/5 correct for an angle in quadrant III and answers +4/5. What is missing?
⭐ Stretch: Pick any angle, get its sine and cosine on a calculator, square both and add. Do it for four angles in four different quadrants. Why does the answer not care which quadrant you chose?
W6D3 · Day 28

One Ratio From the Other, With the Quadrant

Use sin²θ + cos²θ = 1 to get the ratio you were not given, then let the quadrant choose the sign.
Name
Section
Date
cos θ = ±√(1 − sin²θ) sin θ = ±√(1 − cos²θ)
The root gives the size. The quadrant gives the sign.
Example: sin θ = 3/5 in quadrant II gives cos θ = −4/5
  1. sin θ = 8/17, quadrant II. Find cos θ, exact form.
  2. cos θ = −7/25, quadrant III. Find sin θ, exact form.
  3. sin θ = −0.28, quadrant IV. Find cos θ to three decimal places.
  4. Given sin θ = 0.6 with θ in quadrant I, a student computes √(1 − 0.6) = 0.632 for cos θ. Name the error and give the right answer.
  5. A ramp rises at an angle θ with sin θ = 0.44. The ramp is 9.5 m long and made of steel. Find cos θ to three decimal places, and say which facts you did not need.
  6. Can sin θ = 1.2 for any angle? Answer using the identity.
  7. Using question 1: find tan θ, exact form.
  8. θ is in quadrant III and sin θ = −3/5. Find cos θ, then say what sin²θ + cos²θ equals — without computing it.
⭐ Stretch: Take sin θ = 3/5 in quadrant II, find cos θ, then find sin θ again FROM your cosine. You should land back on 3/5. Do it — then explain what would have gone wrong if you had skipped the quadrant on the way back.
W6D4 · Day 29

Periodic Word Problems

Pull period, midline and amplitude out of sentences, then answer what the sentences cannot.
Name
Section
Date
read the period from "one full turn / cycle takes …"
midline = (highest + lowest) ÷ 2 amplitude = (highest − lowest) ÷ 2
boards at the bottom → A negative
Example: Wheel 30 m across, lowest seat 2 m, 40 s per turn → h(t) = 17 − 15·cos((π/20)t)
  1. A Ferris wheel is 30 m across with its lowest seat 2 m above the ground. How high is the highest seat?
  2. Same wheel. Find the midline and the amplitude.
  3. Same wheel, one turn every 40 s. Find B, exact form.
  4. Write the equation for a rider boarding at the lowest point.
  5. Using that equation, how high is the rider 5 s after boarding? Radian mode.
  6. The water at a dock is 4 m deep at low tide and 10 m at high tide, one low tide to the next taking 12 hours. Find all three numbers.
  7. For h = 25 − 20·cos((π/6)·t), find the maximum and minimum height.
  8. For that same equation, find the period.
⭐ Stretch: A rider boards a Ferris wheel at the SIDE rather than the bottom — at midline height, going up. The three numbers are unchanged. Explain why y = A·cos(Bt) + D can no longer describe them, and what you would need instead.
W6D4 · Day 29

Three Numbers Out of a Paragraph

Pull period, midline and amplitude out of prose, build the equation, then answer what the prose never said.
Name
Section
Date
B = 2π ÷ period A = amplitude (negative if it starts low) D = midline
Then substitute the time you were asked about.
Example: Lowest seat 2 m, wheel 30 m across, one turn in 40 s: h = 17 − 15·cos((π/20)t)
  1. A Ferris wheel 40 m across has its lowest seat 2 m up and turns once every 90 s. Find the midline, the amplitude, and B in exact form.
  2. Write the equation for that wheel, for a rider boarding at the bottom.
  3. Using question 2: how high is a rider 30 s after boarding?
  4. A city's daylight runs from 9.1 hours on December 21 to 15.3 hours on June 21. Find the midline and the amplitude.
  5. A tide runs from 1.2 ft to 7.8 ft, with 12.4 hours between successive high tides. The dock is 240 ft long. Find the midline, the amplitude and B to four decimal places, and say which fact was not needed.
  6. A paragraph says a motion "reaches its maximum twice a day." What is its period, in hours?
  7. Why can y = A·cos(Bt) + D not describe a rider who boards at the SIDE of the wheel, going up?
  8. Using question 4, with t in months counted from December 21: write an equation for daylight, then estimate the daylight on March 21.
⭐ Stretch: A paragraph gives a high, a low, and the time between two consecutive HIGHS. A classmate uses the time from a high to the next LOW as the period. Work out what their B would be against the correct one, and what their graph would look like.
W6D5 · Day 30

Mixed Review — Exam Format

Every skill from Days 21-29, in the shape an exam asks for it.
Name
Section
Date
no new formulas — see the sheets for Days 21 to 29
Example: Weighted to the six exams counted: the identity-plus-quadrant item appears most.
  1. sin θ = 3/5, quadrant II. Find cos θ, exact form.
  2. cos θ = 5/13, quadrant IV. Find sin θ, exact form.
  3. Find the period of y = 4·sin((π/3)x) + 2.
  4. A wave has midline 4, peak 7 and repeats every 10 units. Find B, exact form.
  5. Convert 240° to radians, exact form.
  6. Find cos(5π/6), exact form.
  7. A seat's height is h = 25 − 20·cos((π/6)t). Find its maximum height.
  8. A wave has period 4. Find B, exact form.
⭐ Stretch: Look back at which questions you missed. Write one sentence for each naming the SPECIFIC step that went wrong — not "I forgot" but "I took the square root and stopped before the quadrant." That sentence is what you review.
W6D5 · Day 30

Mixed Set, Second Pass

Every skill from Days 21-29 again, in exam wording, with nothing telling you which day a question came from.
Name
Section
Date
No formula sheet. Deciding WHICH tool a question wants is half of what is being tested.
Example: "Find the period of y = 4·cos((π/6)x) + 1" is a Day 23 question wearing no label
  1. Find the period of y = 6·cos((π/8)x) + 3.
  2. sin(4π/3), exact form.
  3. cos θ = 24/25, quadrant IV. Find sin θ, exact form.
  4. cos θ = −0.6. Find both angles between 0° and 360°, to one decimal place.
  5. Convert 5π/3 to degrees.
  6. A paddle wheel 12 ft across sits with its lowest point 1 ft above the water and turns once every 8 s. Write the equation for a paddle's height, starting at the bottom.
  7. A pendulum 1.2 m long swings between −14° and +14°, completing 25 swings in 60 s. The bob weighs 300 g. Find the period and the amplitude, and say which facts were not needed.
  8. Asked for cos 150° in exact form, a student writes √3/2. The size is right. Name the step they skipped, and give the answer.
⭐ Stretch: Sort your mistakes on this sheet into two piles: arithmetic wrong, and method wrong. Count each. Write one sentence saying which pile worries you more and why.
W1D5 · Day 5 · Gate/Milestone

Project Brief

Your Brief names the real person you're solving for, the exact motion the arm must make, and how you'll know it worked. Everything you build over the rest of the unit points back to this page.
Name
Section
Date
User
The real, specific person you're solving for — a name, not “someone.”
I am solving for ___, my ___ (relationship), who struggles to ___.
The motion (start → end, in cm)
Name the task as a movement between two points, measured in cm from the arm's base (0, 0).
The arm carries ___ from the start point (__, __) to the end point (__, __), then back, so that ___.
Success criterion 1
A number + a yes/no test — accuracy.
The end-effector lands within ___ cm of each target — yes / no.
Success criterion 2
A number + a yes/no test — time or speed.
The full motion completes in under ___ seconds — yes / no.
Success criterion 3
A number + a yes/no test, built FROM the physical constraint. Name the constraint, then write the test: the rail is 8 cm (constraint) → the arm clears it by ≥ 2 cm (criterion).
The ___ (obstacle) is ___ cm, so the arm clears it by at least ___ cm — yes / no.
Before you submit
W2D5 · Day 10 · Gate/Milestone

FK Prediction Card

Forward kinematics predicts where the arm's tip lands from the joint angles. YOUR CREW agrees on ONE angle pair; you each predict it on paper independently, your OPERATOR runs it ONCE on the arm, and you all measure the same landing. The prediction and the explanation are yours, not a team effort; the measured run is the crew's. Predict it on paper, run the arm, and explain the gap between prediction and reality.
Name
Section
Date
Your crew's angle pair (θ₁, θ₂)
You CHOOSE this pair, you do not solve for it — today runs angles → position, not the other way round. Each joint has its own travel: θ₁ from 20° to 150°, θ₂ from −160° to 160°. Those are the two joints taken separately, not a promise about the pair — about a fifth of the combinations inside both ranges put the tip below the base line, and the arm refuses those. So agree on a pair as a crew, everyone predicts where it lands independently, and check the predicted y is positive before the OPERATOR runs anything. Negative y means pick again.
θ₁ = ___°, θ₂ = ___°
Link 1 — from the base to the elbow
Link 1 is 18 cm and it leaves the base at θ₁. On its own it is a right triangle: how far across, how far up.
x₁ = 18·cos(θ₁) = ___ y₁ = 18·sin(θ₁) = ___
Link 2 — from the elbow to the tip
Link 2 is 22 cm, and it points at θ₁ + θ₂ — it starts wherever link 1 stopped turning, so the angles add.
θ₁ + θ₂ = ___° x₂ = 22·cos(θ₁+θ₂) = ___ y₂ = 22·sin(θ₁+θ₂) = ___
Show one substitution
The nine boxes above take results. This one takes the working: write ONE of the four link formulas with your own angle substituted in, before you evaluate it. This is the line the checklist means by "shows the algebra, not just an answer".
x₁ = 18 · cos ___° = ___
Predicted tip (x, y) — add the two links
Nothing new to work out. Add what you already have: across plus across, up plus up.
x = x₁ + x₂ = ___ y = y₁ + y₂ = ___ → (___, ___)
Measured tip on the arm (x, y)
OPERATOR: enter θ₁ in the shoulder field. Add 90° to your signed θ₂ and enter that result in the elbow field. Keep the original θ₂ in your worksheet calculations. Check that both field values are within the installed arm's limits before running — shoulder J₁ 20° to 150°, elbow J₂ −70° to 250°. Then drive ONE run for the crew, and everyone measures where the end-effector actually lands.
(__, __)
Why predicted ≠ measured
Say whether the gap is a math error or a build error, and how big it is.
The gap is about ___ cm; it's a ___ error because ___.
👤 User check-in — their exact words
Show them your sketch from the Brief and remind them of the motion you promised. Then be straight about what you tested today: not their motion, a pair your crew picked to test the method, because aiming at their target is next week's work. What you CAN report is that you can now predict where the tip lands, and how far off you were, in cm. Give them that number, not the angles — θ₁ = 62° means nothing to them. Then ask three things. One: “I was off by ___ cm. For the motion I promised you, is that close enough?” Two: “What would get in the way of using this?” Three: “What would you want checked first, and why?” If they only say “looks good”, ask “which part would help you, and what still needs checking?” Write their answer in THEIR words, not your summary of it.
___ said: “___”
What their words tell us to keep, change, or investigate
A quote on its own proves nothing. Say what it means for your design — one line. Rosa saying “a centimeter off would still catch the thread” is not a compliment or a complaint, it is a number: your accuracy criterion is looser than what she needs. This is the line the Final scores in Section 3, where the ask is what you chose or changed BECAUSE of what they said.
Their words mean ___, so I need to ___.
Our next check or test
One thing you will actually do, small enough to do next class. “Tighten the criterion, then find where the error comes from before Day 15” is a next check. “Make it better” is not.
Next we will ___.
Before you submit
W4D1 · Day 16 · Gate/Milestone

IK Solve Card

Inverse kinematics goes the other way — from a target point to the joint angles. Solve BOTH endpoints of your motion, state how the joints sweep between them, and verify on the arm.
Name
Section
Date
Step 0 · Reachability — is 8 ≤ r ≤ 40 at each endpoint?
Before the pipeline: compute r = √(x²+y²) at both points and confirm each is in the arm's practical 8–40 cm range (the elbow bends only to 160°, so 8 cm — not the geometric 4 cm — is the real floor). Out of range → fix the point now, not at arccos.
start (18, 22): r = √808 = 28.4 ✓ · end (10, 8): r = √164 = 12.8 ✓ — both in range
Start point (x, y) → r², cos(θ₂′), θ₂′, θ₁
Run the full IK pipeline for the start of the motion.
(18, 22): r² = 808, cos(θ₂′) = 0, θ₂′ = 90°, θ₁ = 101.4°
End point (x, y) → r², cos(θ₂′), θ₂′, θ₁
Same recipe for the end.
(10, 8): r² = 164, cos(θ₂′) = −0.813, θ₂′ = 144.4°, θ₁ = 128.2°
Path plan — how the joints sweep start → end
How each joint moves between the two poses.
θ₁: ___° → ___°, θ₂′: ___° → ___°, then reverse
Measured accuracy at both ends (cm)
Run the arm; measure the miss at each end.
start within ___ cm, end within ___ cm
Before you submit
W4D5 · Day 20 · Gate/Milestone

Final Presentation

Your record of the whole journey — from your first interview to your final calculations — presented to your user. Your goal: “How did my work make my user’s life easier, safer, or better?” Eight sections, all scored. Back every claim with a sketch, a number, or a calculation. Score with the 8-section rubric; use the leveled exemplars to calibrate what each level looks like.
Name
Section
Date
1. Your Design
What did you build, and what one movement does it make? How does that movement help your user? Keep this short. Sections 2 to 7 are where you show how you got there.
Written fresh — but it is a summary of everything below, so write it last.
2. Your User and the Problem
Who is your user, and what is hard for them to do? Write the movement as two points, in cm — for example: start (18, 22), end (10, 8). This section is your CLAIM. Section 3 is the proof.
Already written: your Brief, Day 5. Lift the user and the problem straight from it.
3. What You Learned From Your User
Write down something your user actually said — their words, not your summary. Say what you changed or chose because of it. List the limits you had to work inside. Give a number where there is one (the rail is 8 cm); some limits are not numbers, and that is fine. This section is the EVIDENCE for Section 2.
Already written: your interview notes from the first week.
4. Your Design Goals
Write three goals. For each one: what it is, why it matters to your user, and how you would test it. A test is something two people would agree on — “within 1.5 cm”, “under 3 seconds”, “lifts 500 g”, or a rating scale you define. Mark each goal CRITICAL (it fails without this) or DESIRED (it would be better with this). At least one goal should come from a physical limit. Write the limit and the test separately: the rail is 8 cm (the limit) → the arm clears it by at least 2 cm (the test).
Already written: your Brief's three criteria, as you sharpened them on Day 13.
5. Your Diagram
Draw the arm on a coordinate grid. Label L₁, L₂, the start point, the end point, and the angles. Write the units on it — cm and degrees. It does not need to look good. It needs to be right and easy to read.
Already drawn: the sketch from your Brief — start point, end point, the path between.
6. Show the Math
Solve for the joint angles at BOTH the start point and the end point. For each one show all four steps: find r, then cos(θ₂′), then θ₂′, then θ₁. Use your own numbers, and do not skip from the target to the answer. Say where you used the Law of Cosines and why you could use it there. Say what angle unit you worked in. If you converted between degrees and radians anywhere, show where and why.
Already solved: your FK Prediction Card (Day 10) and your IK Solve Card (Days 15–16). Copy the real numbers.
7. Did It Work?
For each endpoint, give four things: where your math said the arm would land, where it actually landed, the goal you set, and whether it passed. Then show one change you made: run 1 → what you found → what you changed → run 2. Did the number get better? What did your user or your classmates say, and what did you do about it?
Already measured: your two calibration runs — run 1, the cause you found, run 2.
8. What Did You Learn?
What surprised you about designing for a real person? How did trigonometry help you build or test the design? Where else could turning a position into joint angles be useful? With more time, what would you improve — and what in your results makes you say that?
Written fresh. Nobody can write this one for you, and it is the section that reads as yours.
Before you submit
W4D5 · Final Presentation

8-section scoring rubric

Goal: How did my work make my user’s life easier, safer, or better? Answer that clearly — with measured evidence, clear diagrams and accurate math — and you will have shown your audience a solution that is thought through, tested, and mathematically justified.

Be specific about the motion — Your arm solves ONE movement. “Helps with daily tasks” is a wish. “Moves the end-effector from (18, 22) to (10, 8) cm to bring a cup to the tray” is the work you did. Name it that clearly — two points, in cm.

Show criteria as measurable targets — Every criterion needs a measurable test — one anyone could run and get the same verdict. That can be a threshold (“no more than 1.5 cm”, “under 3 seconds”, “at least 500 g”), a pass/fail condition, or a defined rating scale for something that is genuinely a judgment. Not “lightweight”, not “fast”, not “accurate”. Sharpen anything vague from your Brief.

Put real numbers in the math — Don’t just show formulas. Use your own L₁ = 18, L₂ = 22 and your own target; solve θ₂′ then θ₁ at each endpoint and show every step. Say where you used the Law of Cosines and why it worked there. Say what angle unit you used — and if you converted between degrees and radians anywhere, show that conversion.

SectionMasteryConsistentEmerging
1. Your DesignNames the design, the motion as two points in cm, and the benefit to the user, in a few sentences that do not repeat later sections. The solution fits the user's need — simplicity counts in its favour, not against it.Names the design and the motion clearly; the benefit to the user is stated.A solution is present but the motion or the benefit is unclear or unstated.
2. Your User and the ProblemNames a specific person, the specific task they cannot do, and states the problem as a motion between two points. The problem is one this arm can actually solve.User and problem are clearly stated and connected.User or problem described in general terms; the problem is not tied to a motion.
3. What You Learned From Your UserQuotes the user's own words and shows which design decision each quote drove. Constraints are stated specifically, with a number wherever the constraint has one, and any relevant existing solution or background is identified where the problem called for it.Interview evidence is present and connected to at least one design choice; constraints are listed.Interview or research is summarised rather than evidenced; constraints missing or unnumbered.
4. Your Design GoalsEvery criterion carries a measurable test — a threshold, a pass/fail condition, or a defined rating scale — is marked CRITICAL or DESIRED as Section 4 asks, and is traced to something the user said. At least one is derived from a physical constraint, with the constraint and the test stated separately.Criteria are measurable and linked to the research; the constraint behind at least one is identifiable.Some criteria present; one or more are not yet measurable, or a constraint is offered in place of a test.
5. Your DiagramLabeled diagram showing start and end positions on the coordinate plane, with L₁, L₂, the target and the angles marked and the units given. Accurate and readable — a plain, correct diagram earns this.Diagram accurately shows the arm and the motion; key parts are labeled.Diagram is partial or unlabeled; the start-to-end motion is not clear from it.
6. Show the MathIK solved at BOTH endpoints with this student's real numbers, every step shown in order — r → cos(θ₂′) → θ₂′ → θ₁ — with the Law of Cosines identified where it was used and why it applied there, and the angle unit stated — including any degree/radian conversion the work actually required.The full sequence is shown at both endpoints and the calculations are mostly accurate; the connection to the design is evident.Some math shown; a step is missing, only one endpoint is solved, or the work is copied without values of their own.
7. Did It Work?Reports predicted AND measured position at each endpoint against the stated criterion, says whether it passed, and explains one specific change made in response to a measured result or to feedback — including whether that change improved the number.Predicted and measured results are reported and compared to the criterion; at least one specific revision is described.A result is described without both numbers, or a change is reported without the evidence that prompted it.
8. What Did You Learn?Names something specific learned from the project, connects trigonometry to what the arm actually did, identifies one other place position → joint angles would apply, and names one improvement with a reason drawn from their own results.States what was learned and connects trigonometry to the project.Reflection is general; the connection to the math or to the results is not made.

Generated from the live curriculum · npm run gen:printables