Overview

This lesson set uses the LEGO Education SPIKE Prime set to teach measurement and geometry for a 14‑year‑old. Each activity connects physical LEGO building + SPIKE sensors/actuators to mathematical ideas in ACARA Year 9 (measurement & geometry) and supports problem‑solving skills from AoPS Prealgebra, AoPS Introduction to Algebra, and Beast Academy.

Learning objectives

• Measure length, area, perimeter and volume using LEGO modules; convert between LEGO units and metric units.
• Apply the Pythagorean theorem to real LEGO builds and verify experimentally.
• Use coordinate geometry and motion commands to navigate a grid and represent movements with vectors.
• Explore angles and basic trigonometry by measuring rotations and using sensors (gyro/distance).
• Relate scaling to area and volume and form algebraic expressions describing measurements.

Materials

• LEGO Education SPIKE Prime set (hub, motors, distance sensor, color sensor, gyro/motion if available).
• LEGO plates, beams and bricks for building shapes.
• Grid mat or large sheet marked in 1–2 cm squares (or use LEGO baseplates as a grid).
• Ruler or tape measure (metric), marker, calculator, paper for notes.
• Computer or tablet with SPIKE app to program the hub.

Important note about units

Decide a unit: either "1 LEGO stud" or a metric equivalent, e.g. 1 stud ≈ 8 mm (or measure it exactly for your bricks). Use consistent units in calculations and conversions.

Activity 1 — Area and Perimeter using LEGO studs (30–45 minutes)

Goal: compute area and perimeter from LEGO builds and connect to algebraic expressions.

1. Build several rectangular plates or frames made of bricks whose interior is a grid of studs. Record dimensions in studs: length L studs, width W studs.
2. Calculate: perimeter = 2(L + W) studs, area = L × W studs². Convert studs² to cm² using your stud → mm conversion.
3. Algebra link: if you scale every side by factor k, show area scales by k² and perimeter by k. Example: original L=8, W=5, scale k=1.5 — compute new area and perimeter.
4. Challenge (Beast/AoPS style): Given area A = 120 studs² and width W = 6 studs, solve for length L (A = L×W). Show integer/non‑integer answers and discuss remainder studs.

Activity 2 — Wheel circumference, distance and algebra (45 minutes)

Goal: predict robot travel distance from wheel rotations and test it experimentally.

1. Build a small wheeled vehicle using SPIKE motors for drive wheels. Measure wheel diameter D (in cm). Compute circumference C = π×D.
2. Math model: distance traveled = rotations × circumference. If the motor controller reports degrees turned, rotations = degrees/360.
3. Example problem: If circumference = 12.6 cm, how many rotations are needed to travel 1.5 m? rotations = 150 cm / 12.6 cm ≈ 11.9 rotations. Solve algebraically for rotations = distance / C.
4. Program the motors to run the calculated number of rotations, then measure actual distance with a tape measure. Discuss sources of error (slippage, measurement error, rounding of rotations).
5. Extension: derive an equation to solve for wheel diameter given a measured distance and known rotations (use algebra from AoPS Intro to Algebra).

Activity 3 — Pythagorean theorem with a LEGO right triangle (40 minutes)

Goal: build a right triangle frame, measure legs, compute hypotenuse, and verify physically.

1. Build a right triangle using beams or bricks so legs are easy to measure. Let legs be a and b (in cm or studs). Measure exactly.
2. Calculate hypotenuse c = sqrt(a² + b²). Predict c in cm, and then measure the diagonal with a ruler or use a distance sensor placed across the ends.
3. Example: a = 24 cm, b = 10 cm → c = sqrt(576 + 100) = sqrt(676) = 26 cm. Build and confirm.
4. Algebra challenge: If you want an integer hypotenuse with a given integer leg (use Pythagorean triples). Relate to AoPS problem solving: find integer solutions (a,b,c) where a² + b² = c².

Activity 4 — Coordinate geometry & navigation (60 minutes)

Goal: represent robot paths as ordered pairs and vectors, program the robot to go to coordinates on a grid.

1. Place a 10×10 grid mat (1 square = 10 cm or 1 stud) on the floor. Mark origin (0,0).
2. Program the robot to start at (0,0). To get to (x,y) where movements are axis‑aligned, command sequence: go forward x squares, turn 90°, go forward y squares. Record the vector form → <x,y>.
3. Introduce diagonal move as vector addition: to move from (0,0) to (x,y) you can move along the diagonal — measure that diagonal and compare time/distance vs axis moves. Use Pythagoras for the diagonal length.
4. Transformations: perform translations (shift all points by a vector), rotations (90°, 180°) and reflections. Program robot to follow a transformed path and observe coordinates change. Tie to algebra by writing equations for transformations: (x,y) → (x+3,y−2) etc.
5. Challenge: give the robot a sequence of moves (a short program) that places it at (2,−1). Work backward to find initial moves (introduces solving equations and inverses).

Activity 5 — Angles and basic trigonometry using the gyro and distance sensor (45 minutes)

Goal: measure rotations, compute angles and relate height – base using tangent.

1. Use the hub's gyro (motion sensor) to measure rotation angles. Program: rotate the robot 90° and read the angle reported; compare to expected 90°.
2. Build a right triangle ramp or use a beam at base b and height h. Compute angle θ = arctan(h/b). Example: b = 40 cm, h = 30 cm → θ = arctan(30/40) ≈ 36.87°.
3. Program the robot to climb the ramp and use the gyro to measure the robot's tilt; compare to computed θ. Discuss friction and mounting errors.
4. Algebra link: rearrange tan θ = h/b to solve for h given θ and b — practice algebraic manipulation from AoPS Intro to Algebra.

Activity 6 — Volume and surface area (30–40 minutes)

Goal: count unit cubes, compute volume and surface area, and explore scaling.

1. Build a rectangular prism using bricks: measure length L, width W, height H in unit bricks. Volume V = L×W×H unit cubes. Surface area S = 2(LW + LH + WH) (units²).
2. Scaling challenge: if you double each linear dimension, compute the new volume (8× original) and new surface area (4× original). Show algebraic proof using variables.
3. Extension (AoPS): give an integer volume and one dimension, solve for other dimensions so all are integers — discuss factorization (links to number theory & problem solving).

Assessment tasks (checks for understanding)

• Given a LEGO rectangle 12×7 studs, compute area in cm² if 1 stud = 8 mm. (Practice unit conversion + area.)
• Robot wheel diameter = 6 cm. How many rotations to go 2.5 m? (Apply circumference formula and algebra.)
• Build a right triangle with legs 9 cm and 12 cm. Find the hypotenuse and check by measuring. (Pythagoras.)
• Robot starts at (1,2). Commands: forward 3, turn left, forward 4. Where is it? Represent as coordinates and vector. (Coordinate geometry.)

Differentiation & tips

• For students who need reinforcement: focus on counting studs, simple area/perimeter, short programs with clear step counts.
• For advanced students: introduce algebraic proofs of scaling laws, Pythagorean triple searches, or optimization (minimize material for a given volume), and relate to contest‑style problems from AoPS/Beast Academy.
• When programming motors, expect some error. Encourage students to calculate predicted values and compare to measured ones, then estimate percent error and discuss causes.

Sample pseudocode examples

These are general steps you can adapt to the SPIKE Prime app (block or Python).

// Move robot a given distance using wheel circumference
wheel_diameter = 6.0  // cm (measure your wheel)
circumference = PI * wheel_diameter
distance_needed = 150.0 // cm
rotations = distance_needed / circumference
degrees_to_turn = rotations * 360
motor_pair.run_for_degrees(degrees_to_turn, speed=50)

// Turn 90 degrees using gyro feedback (pseudocode)
gyro.reset()
while gyro.angle() < 90:
    left_motor.run(speed=30)
    right_motor.run(speed=-30)
left_motor.stop()
right_motor.stop()

Extensions & projects

• Design a treasure hunt: place objects at coordinate locations and program the robot to collect them. Use algebra to compute optimal path lengths.
• Investigate measurement error systematically: vary wheel load or surface and plot error vs surface type. Connect to data analysis.
• Challenge (AoPS style): create LEGO puzzles that require minimal perimeter for a given area or integer solutions for given area/volume constraints.

Wrap up (what students should be able to do)

• Measure and compute perimeter, area, surface area and volume from LEGO builds and convert units accurately.
• Use the Pythagorean theorem and trigonometric relationships to predict distances and angles, then verify with sensors.
• Represent robot motion with coordinates and vectors, translate between program instructions and algebraic expressions.
• Apply algebra to solve for unknown measurements (distance, rotations, dimensions) and explain scaling effects.

If you want, I can create a printable worksheet with step‑by‑step build instructions, measurement tables, and 6 assessment questions (including answers) tailored to this lesson and the SPIKE Prime exact parts you have.