Lesson type: Core — the opening lesson of the constructions unit, where drawing becomes exact.
Stepping Stone: understand a circle as the set of points a fixed distance from a centre, use compasses to copy a length, and construct an equilateral triangle from a given side.
You will need: the idea of measuring a length, what a triangle is, and the optional ✏️ Scratchpad. No calculator today — the compass does the measuring.
NEO Cornerstones: Connection · Movement · Reflection · Creativity · Rest · Nutrition
45-minute shape: Connect 10 · Move 14 · Reflect 9 · Create 8 · Rest/Nourish 4
Choose only the support that helps now. Every route returns you to this lesson.
Most drawing is a little approximate. Construction is exact — and the tool that makes it exact is a pair of compasses. The secret is what a circle really is. A circle is not just "a round shape": it is every point the same distance from a centre. That fixed distance is the radius. A compass holds that distance while you swing it, so every mark it makes is guaranteed the same distance from where the point sits. Once you see that, every construction in this unit grows from one idea: equal distance.
What to do: the centre O is fixed. Drag the point P around it, trying to keep it the same distance away (the readout shows the distance). Watch what shape the "same-distance" points make.
What you are trying to notice: when you keep the distance fixed, the points you visit trace out a circle. The circle is the set of all points that far from O.
Carry this forward: a circle is every point a fixed distance (the radius) from its centre. The compass is a machine for "the same distance".
What to do: you have a side AB. Follow the steps to construct an equilateral triangle (all sides equal) with a compass — leaving the arcs showing, exactly as in an exam. Each arc has radius AB.
What you are trying to notice: because both arcs are swung with radius AB, the point where they cross is the same distance from A and from B — and that distance is AB. So all three sides are equal, with no measuring at all.
Carry this forward: to construct an equilateral triangle on AB — swing an arc of radius AB from A, another from B, and join their crossing to A and B. Drag A or B: the whole construction stays exact.
You have seen why it works. Now build the equilateral triangle for real in the Geometry Studio — with the compass and straightedge, without leaving the site. Your task travels with you, and a Return to lesson button brings you straight back.
📐 Open in the Geometry Studio →What to do: the triangle looks equilateral — but why is it exact, with no ruler? Click each side to check how long it must be.
What you are trying to notice: the arc from A had radius AB, so AC = AB. The arc from B had radius AB, so BC = AB. And AB is AB. Three sides, all equal to the one radius — that is the proof.
Carry this forward: the construction is exact because both arcs share the same radius, AB — so every side equals AB.
What to do: the same idea builds any triangle from three given side lengths (this is called SSS). Choose a set of three side lengths, then press Construct: a base, an arc of one length from each end, and the crossing is the third corner.
What you are trying to notice: the two arcs still do the work — each keeps a corner the right distance from an end of the base. Watch the warning: if two sides together are shorter than the third, the arcs never meet and no triangle exists (the triangle inequality).
Make it your own: on the Scratchpad, construct a triangle with your own three side lengths — and try three that cannot make a triangle.
Choose three side lengths and construct the triangle for real with the compass — then try three lengths that cannot make one, and watch the circles miss.
📐 Open in the Geometry Studio →You have made something exactly right with two humble tools and one idea. That shift — from "near enough" to "provably exact" — is worth a quiet moment. Take a breath before moving on.
Notice one thing: did it feel different to construct a triangle rather than measure and draw one? What changed? There is no right answer — just what your own attention noticed.
You can jot it on the Scratchpad, say it aloud, or simply hold it in mind. You can pause here and return whenever you are ready.
Mode: intellectual. A construction is a mathematical idea, not the nourishment itself — so we name honestly what the learning feeds.
What it feeds: today feeds a taste for certainty from constraint — the deep satisfaction that two simple tools, used honestly, can produce a result you can prove, not merely eyeball. And it feeds the elegance of one idea doing a lot of work: "equal distance" gives you the circle, the equilateral triangle, and (soon) every other construction in this unit. Learning that a whole subject can grow from a single clear idea is one of the most nourishing experiences mathematics offers.
Carry one idea forward: when something must be exact, look for the constraint that forces it — here, the shared radius.
Your task: construct an equilateral triangle with ruler and compass, and explain why it is exact.
Submission choices: an annotated Scratchpad image, a photographed hand construction with the arcs visible, a construction built and saved in the Geometry Studio, a labelled poster, or a spoken/video walkthrough.
Construct your equilateral triangle in the Studio, save your construction and a sentence on why it is exact, then return here. Real compass, real straightedge, no measuring.
📐 Open in the Geometry Studio →Filename pattern: NEO_MATH_CONSTR_COMPASS_FirstName_Date
| Criterion | Approaching Proficiency | Developing Proficiency | Proficiency | Refining Proficiency |
|---|---|---|---|---|
| Understanding | Knows a circle has a centre and radius. | Sees the two arcs share the radius AB. | Explains why the crossing is equidistant from A and B. | Explains the equilateral result as a consequence of one shared radius. |
| Application | Draws a circle or arc with compasses. | Swings both arcs of radius AB. | Completes the equilateral construction with arcs shown. | Constructs a triangle from three given sides (SSS). |
| Communication | Labels the centre and radius. | Labels A, B and the crossing C. | Leaves a clear construction with arcs. | Explains the method another learner could follow. |
| Reflection | Names one thing noticed. | Notices the arcs are the same size. | Connects equal radius to equal sides. | Recognises the triangle-inequality limit for SSS. |
Use support in this order: 🤔 Socratic prompt → 💡 hint → ✅ check → 👀 worked solution. The worked solution unlocks separately for each question, once you have used the prompt, used the hint and checked two answers.
Use the Scratchpad to predict, sketch, test and revise.
You can use this board, paper or another way of recording your thinking. The Scratchpad is optional.
Angles are in degrees. Examples: 9 × sin(37) = 5.42; sin⁻¹(0.5) = 30