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Teaching Chapter I
Teacher-facing notes for running Chapter I in class. Covers pedagogical sequencing, common misconceptions, specific teaching moves, and extension ideas.
| Sections | 10 (2 tutorials, 8 exercises) |
| Estimated time | 2 weeks for a mixed-ability Year 9 class (8 to 10 lessons) |
| Prerequisites | Comfort with iPads, basic algebra, knowledge of angles in degrees |
| Live chapter | dbbudd.github.io/01.html |
Students leave able to: write sequences of Swift commands, describe coordinates on a Cartesian plane, compute perimeters and areas of basic shapes, and talk meaningfully about interior vs exterior angles.
The mathematics. Coordinate geometry as a grid with an origin and two axes. Angles as rotations measured from a reference direction, with the strong emphasis that positive angles are anti-clockwise in mathematics (not the intuitive clockwise a student might assume). Basic shape properties: squares, rectangles, equilateral triangles, and the star polygon notation {n/k}. The key theorem that the exterior angles of any convex polygon sum to 360 degrees.
The programming. The idea of a sequence: a list of commands executed one after another, top to bottom. No control flow, no variables, no functions. Just instructions the pen follows. This is the simplest possible programming model and it gives students a mental anchor for everything that follows. When they see a for loop in Chapter III, they already know what "five instructions in a row" means, and the loop just becomes a way to write those five instructions in fewer characters.
The 10 sections move from explanation to production, with a crucial break halfway through.
Sections 1 and 2 are pure tutorials. The student reads, hovers on vocabulary, runs no code. This is where you establish the walking metaphor: every time the pen turns, the student should imagine themselves turning. Don't skip this. If you skip the walking metaphor and jump straight to "OK, now type this code", students will memorise the syntax without internalising the geometry. You can tell which students skipped the metaphor by the third exercise: they can't explain why a square uses 90 degree turns.
Sections 3 to 6 are the core shape exercises: square, rectangle, nested squares, triangle inside a square. These reinforce the "sum of exterior angles equals 360" rule four times in four subtly different configurations. By the end of section 6, a student who's been paying attention can predict the turn angle for any regular polygon without being told.
Section 7 is a second tutorial that introduces move() (pen up / pen down) as distinct from addLine(). This is the crucial scaffolding that lets the last three sections work.
Sections 8 to 10 are production exercises: multiple shapes on one canvas, a five-point star, and a Star of David made from two overlapping triangles. Section 9 introduces star polygon notation {5/2} as a teaser for Chapter III's deeper dive into the gcd rule.
If you have 10 minutes at the end of the chapter to reinforce a single idea, make it the sum of exterior angles equals 360 degrees rule. This is the theorem that unifies every shape they've drawn and it's the foundation for everything in Chapter III. If you can get a student to say, without prompting, "the exterior angles always add up to 360 because I have to end up facing the same way I started", they've got it. Everything else is bonus.
Students naturally assume that p.turn(degrees: 90) creates a 90 degree corner. It does, but only because 90 happens to be its own supplement. When they try a triangle and use p.turn(degrees: 60), they end up with a tangled shape because the interior angle of an equilateral triangle is 60, but the exterior angle (what the pen turns) is 120.
Fix: draw a triangle on the whiteboard, walk around its edge with a marker, and physically turn at each corner. Students watch you turning 120, not 60. The walking metaphor is the cure.
Almost every student assumes this. It's the natural reading of a clock. Correct it explicitly in the first lesson: maths angles are anti-clockwise, and the curriculum follows maths not intuition. Frame it as a badge of honour: "this is the convention every trigonometry course uses, and you'll see it again in Year 11".
Students sometimes forget the final addShape(pen: p) call and are baffled when nothing appears. The addShape call is the moment the pen's buffered instructions get rendered to the canvas. Without it, everything is in memory and nothing draws.
Fix: make the "write instructions, then ask the pen to show them" distinction explicit from lesson 1. Some teachers add a three-beat mantra: "create the pen, give it instructions, add the shape".
Swift argument labels confuse students who've seen pseudocode like turn(90). Insist on the full form p.turn(degrees: 90) from day one. The argument label is how you read the code as English, not wasted characters.
The single most important teaching move. Every time you introduce a new turn, physically walk it on the floor in front of the class. When you get to section 9 (Five-Point Star) and students complain that 144 degrees is weird, walk around the front of the room in five 144 degree swivels and land back where you started. You'll look slightly silly. It's worth it.
Reinforce "sum of exterior angles equals 360" at the start of every lesson after section 2. Make it a ritual. Call on random students. By the end of the chapter, the whole class should be answering in chorus.
Section 6 (Triangle in Square) is where this clicks. If a shape closes, the pen ends up facing the same direction it started. Therefore the total rotation must be a multiple of 360. For convex polygons, exactly 360. For star polygons (teased in section 9), 720, 1080, etc. Plant this seed here; Chapter III harvests it.
Section 10: Star of David. This one uses move() to reposition the pen mid-drawing, and the arithmetic for "where should the second triangle's starting point be" is the first time students have to think carefully about coordinates. The move distance is 86.6, not 90, because of the triangle centroid formula (we fixed an alignment bug here during curriculum development).
How to teach it: before showing the code, draw the two triangles on the whiteboard with coordinates labelled at every vertex. Walk through the positions explicitly. Then show the code and connect each move and addLine call to a coordinate on the board. The walk-through is tedious but it's the first time students meet coordinate geometry head on, and it pays for itself in Chapters II and VI.
- Draw a hexagon. The curriculum doesn't have one. It's a natural extension of the exterior angle rule: 60 degrees per turn.
- Nested pattern. Take the More Squares exercise and draw five concentric squares in a colour gradient.
- Reproduce a road sign. A stop sign (octagon) or a yield sign (inverted triangle). Ask students to compute the turn angle first, then write the code.
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Star polygon
{6/2}. Classic gotcha: it doesn't produce a six-pointed star, it traces a triangle twice. This teaches the gcd rule and sets up Chapter III nicely for the students who get there.
When introducing the Pen (section 1):
"Imagine you're walking across a big sheet of paper with a pen in your hand. The Pen object in our code is you. It has a position, which is where you're standing, and a direction, which is the way you're facing. It draws a line whenever you take a step forward."
When students get confused by turn angles (sections 3 or 4):
"The number inside
turn(degrees:)isn't the angle of the corner. It's how much you swivel on your feet when you change direction. Let me walk a square for you."(Then actually walk a square.)
When reinforcing the 360 rule:
"If you walked all the way around any shape and came back facing the same way you started, you must have turned a total of 360 degrees. That's one full circle. Doesn't matter how many corners, doesn't matter what shape: if it closes, it's 360."
When explaining positive equals anti-clockwise:
"Maths does things differently to intuition. Every trigonometry equation you'll see in Year 11 and 12 and at university measures angles anti-clockwise from the x-axis. This curriculum uses the same convention. Get used to it early and future you will thank past you."
Fill this section in after your first classroom run. Leave it here as a place to collect observations, student quotes, misconceptions you didn't predict, and small tweaks you'd make next time.
Geometry Playground · a Swift Playgrounds curriculum for high school geometry · dbbudd.github.io · built by Daniel Budd