*Seventh-Grade Geometry

Seventh-Grade Geometry Curriculum

• Scope intentionally layered; topics revisited in smaller, spaced units rather than a single long block.
• Material normally delivered in Skills Classes; only Perspective Drawing & Intro to Algebra placed in Main Lesson block.
• Geometry units broken into 2–3 mini-blocks over the year to allow digestion and reinforcement.

Sequences, Patterns & the Fibonacci Prelude

• Year opens with a numeric unit on sequences.
• Students build the Fibonacci additive sequence: 1,1,2,3,5,8,13,21,34,\dots
• Tangential explorations encouraged (e.g.
– rabbit problem, sunflower spirals, pinecones).
• Calculator used here (one of only two calculator appearances in Grade 7) to compute successive ratios \dfrac{F{n+1}}{Fn}.
– After ≈20 terms, ratio converges to the Golden Ratio \phi (≈ 1.618 033).

Calculator Policy

• Allowed only for specialised investigative work, never for routine arithmetic.
• Grades 6–8 each include two carefully chosen calculator moments; in Grade 7 they are:

  1. Generating \phi numerically.

  2. (Later) Negative-number explorations.

Review of Sixth-Grade Angle Vocabulary

• Acute
• Right
• Straight (180°)
• Reflex / all-around (360°)
• Supplementary: two angles summing to 180^\circ
• Vertical ("across-the-street") angles created by intersecting lines; always congruent.

New Seventh-Grade Angle Theorems

• Context: a transversal cutting two parallel lines. Students discover by measurement with protractor & painter’s tape.
• Alternate Interior Angles – congruent.
• Corresponding Angles – four congruent pairs.
• Same-Side Interior (Co-Interior) – not congruent; sum to 180^\circ.

Pedagogy for Angle Work

• Use blue painter’s tape on desks, floor, and board to create tactile, full-body explorations.
• Separate these three new theorems from prior angle study to make the abstractions manageable.
• Emphasise repeated exposure (“layering”) over the year rather than one-and-done.

Golden Ratio – Geometric Approach

Underlying Idea

• Translate the numeric mystery of \phi into a visible, constructible form.
• Students intuitively place a point on a line “in an interesting place”—most instinctively choose near the golden mean.

Constructing the Regular Pentagon & Pentagram (“Fifth Division”)

• Tools: Compass with sharp lead, straight-edge, optional 45°/90° triangle (but pure-compass preferred).
• Steps (concise):

  1. Draw circle; eyeball a horizontal diameter.

  2. Construct its perpendicular bisector to obtain vertical diameter (basic intersecting-arcs method).

  3. Bisect the right-hand radius precisely; verify with compass used as divider.

  4. From midpoint, swing arc through circle’s top to intersect left-hand radius—critical intersection.

  5. With new opening, step off five equal arcs around circumference; slight human error corrected by "split the difference."

  6. Join the five points for the equilateral, equiangular pentagon.

  7. Draw pentagram by linking each vertex to its non-adjacent vertices.
    • Common pitfalls:
    – Dull compass lead → cumulative millimetre errors.
    – Misplaced midpoint ruins division; insist on re-drawing if arcs do not meet.
    • Extensions:
    – Measure side vs. diagonal; verify \dfrac{\text{diagonal}}{\text{side}} = \phi.