Rectangle Diagonals: Measure and Investigate

Create the complete core lesson for Section 8.5 using ONLY Mili and Arin. The pedagogy must be investigation-led: predict → draw/measure → compare → discover. 1. INTRODUCE DIAGONALS: show a clean rectangle PQRS. Draw PR and QS by connecting opposite vertices. Label them “diagonals”. 2. INVESTIGATION 1 — LENGTH: ask whether PR is longer, QS is longer, or both are equal. Pause for predictions. Show a ruler/divider measuring each diagonal, then reveal PR = QS. Avoid fabricated numeric measurements; the source only requires comparing lengths. 3. INVESTIGATION 2 — FIRST DIAGONAL'S ANGLES: zoom to P and R. Show each 90° corner divided by diagonal PR into angles c and d at P, and g and h at R. Ask whether c = d and whether g = h. Let learners predict and then measure with a protractor. Do not assume the smaller angles are equal in every rectangle. 4. INVESTIGATION 3 — SECOND DIAGONAL: show QS dividing the corners at Q and S into angles e, f, a and b. Invite learners to identify pairs of equal angles by measurement. Keep labels attached to the correct corners and angles. 5. TEST MORE RECTANGLES: morph the rectangle into wide, tall and almost-square proportions. Draw both diagonals in each and encourage repeated measurement rather than relying on one diagram. 6. SPECIAL CASE — SQUARE: gradually make all four sides equal. Compare the diagonals again and investigate how they divide the corner angles. Treat this as a special case to examine, not as a blanket statement that all small angles in every rectangle are equal. 7. CONSTRUCTION CONNECTION: show a right angle divided by a diagonal into 60° and 30°. Display 60° + 30° = 90°. Ask what the remaining angle must be and use this as a bridge to the construction lesson. 8. MEMORY CARD: Join opposite corners → diagonals; measure → compare; PR = QS; investigate the smaller angles; a square is a special case. 9. END CHALLENGE: “Can you construct a rectangle whose diagonal divides an angle into 60° and 30°?” Transition to a construction workspace. Mathematical visuals must be accurate. Keep P, Q, R, S consistent and diagonals connected to the correct opposite vertices. Use no extra characters, no unrelated concepts, and no unverified angle claims. Mili leads curiosity; Arin organises the investigation.