PrepShorts · Teaching notes · Class 7 Mathematics · Chapter 6, Constructions and TilingsPrepShorts

Chapter 6 · Constructions and Tilings

Constructing 60° from an equilateral triangle, and the arches built on it

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9 min.

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What to assume they know

What they should be able to do

  • Explain why an arch has to be reduced to support lines before it can be drawn
  • Identify the equalities a trefoil arch's supports require, and build them
  • Build the supports of a pointed arch from two equal segments and their midpoints
  • Construct a 60° angle with a compass and an unmarked ruler
  • Explain why the two arcs of that construction force an equilateral triangle
  • Derive 30° and 15° from 60° by bisection
  • State which angles in this chapter are constructed, and trace each back to 90° or 60°
  • Say where exactness ends and design choice begins in an arch drawing

Where it usually goes wrong

  • "An arch is a semicircle." Neither of the chapter's two arches is. The trefoil is three lobes and the pointed arch has a corner at the top; both are built from arcs whose centres are not the middle of the span.
  • "There is one right radius for the arch." The chapter says outright that the radii can be adjusted so the arch looks better. The geometry fixes the supports; the curves are a design decision. Keep the two apart.
  • "60° must be measured with a protractor." It is the one angle that arrives free: three equal sides, hence three equal angles, hence 180 ÷ 3.
  • **"The 60° construction needs a specific radius."** Any opening works, as long as the same one is used from A and from B. That reuse is what makes the three sides equal.
  • "So a compass can build any angle." It cannot. Everything this chapter constructs comes from 90° or 60° by halving and combining, and a student who believes otherwise will look for a 65.5° construction that is not there.
  • "The two base angles of the trefoil have to be measured." They are copied — which is exactly the construction of Copying an angle, and why triangle congruence proves it works, and the reason that topic comes first.

Questions to check understanding

  • Construct a 60° angle with a compass and an unmarked ruler and justify the value
  • Construct 30° and 15°, showing the bisection stages
  • Build the support lines of a trefoil arch to stated conditions, then complete the arch (the shape of the task on p.149)
  • Build a pointed arch from two equal segments (Figure it Out question 1, Part II, p.151)
  • Design an arch of your own and state the equalities its supports need (Figure it Out question 2, Part II, p.151)
  • Reproduce any of the five compass designs (Figure it Out question 1, Part II, pp.154–155)
  • Explain what makes the p.155 optical illusion work (Figure it Out question 2, Part II, p.155)
  • The 60°-angle bullet of the SUMMARY (Part II, p.163) is the statement the chapter expects back

Examples worth working on the board

  • The two photographs (Part II, §6.1, p.149). Checked against the printed page. On the left, a row of three white carved arches captioned Diwan-i-Aam, Red Fort; on the right, a red stone underpass arch captioned Central Park, New York City. Both carry their Wikimedia source URLs in the printed caption.
  • The trefoil supports (Part II, §6.1, p.149). Checked against the printed page. A horizontal base with A at the left end and D at the right; B and C marked above it, B over the left part and C over the right. Conditions printed: AB = CD, and the angle at A equals the angle at D. The chapter's own build order puts the two matching base angles down first, and only then marks B and C so that AB = CD.
  • The finished trefoil (Part II, §6.1, p.149). Checked against the printed page: a large central arc over BC with two smaller arcs springing from A and from D, so the head reads as three lobes. The chapter says the radii may be adjusted for looks.
  • Fig. 6.11 (Part II, §6.1, p.150). Checked against the printed page. Two panels. The upper one is the finished pointed arch with its two supports drawn through it in a second colour. The lower one is the bare skeleton: two straight segments of equal length leaning together to a point, each carrying a marked dot part way along it. Those dots are the midpoints the chapter asks about.
  • The 60° steps (Part II, §6.1, p.153). Checked against the printed page. Step 1: the segment AX with an arc struck from A at any opening, cutting AX at B. Step 2: the same opening struck from B, meeting the first arc at C; then AC is ruled. The chapter states the result ∠CAX = 60° and immediately asks the reader to say why, pointing at the triangle.
  • The 60°/120° pair (Part II, §6.1, p.152). Checked against the printed page: a single line with a ray rising from a point on it, the acute side tagged 60° and the obtuse side tagged 120°, captioned so that building one is said to deliver the other. This is the bridge into the hexagon topic.
  • The p.154 and p.155 design set (Part II, §6.1, "Figure it Out", question 1). Checked against the printed page. (a) an inflexed arc, drawn as an upright frame closed at the top by two curves that meet in a point; (b) a six-petalled flower around a central circle, with the chapter noting it can be done with a compass alone; (c) a regular hexagon inscribed in a circle; (d) six equal circles in a ring; (e) a hexagon filled with a lattice of small triangles. All five are 60°-family constructions and make good closing visuals.
  • The optical illusion (Part II, §6.1, "Figure it Out", question 2, p.155). Checked against the printed page: three notched discs placed at the corners of a triangle that is not drawn, with three open V shapes between them. The chapter asks what is interesting about it and how it happens, and prints no answer.

Figures to have open

  • The trefoil support skeleton, able to be shown moving so the two equal angles can be copied into place and B and C stepped off. This is the topic's key image.
  • One skeleton carrying three different arch heads, to make the radius-is-a-choice point. Built fresh; the chapter prints only one finished arch.
  • The pointed-arch skeleton of Fig. 6.11 with its midpoints, able to be shown moving so the two arcs can be swung.
  • The two-step 60° construction, able to be shown moving, with the equilateral triangle appearing when AB and BC are joined.
  • Photographs of a trefoil arch and a pointed arch. The book uses three credited Wikimedia photographs — two on p.149 and one on p.150; source fresh images or draw them.
  • The five design exercises of pp.154–155, redrawn.

Where this sits in the book

The book

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