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Themed maths

Maths art for IB Maths: curve stitching, tessellations and graph pictures

A calm end-of-term lesson that makes something to put on the wall: curves from straight lines (and the equation of the curve they make), tilings that fit perfectly, and a picture drawn with equations.

Level
IB Maths SL and HL (AA and AI)
Time
40 minutes, plus a 10-minute extension
Topics
Straight lines and simultaneous equations; Angles of polygons; Circles and quadratics; implicit differentiation (extension)
Equipment
No GDC needed. Task C is best done in a grapher.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A12 minCurve stitching
Main: task B12 minTessellations
Main: task C11 minGraph pictures
Extension10 minFast finishers or homework

Starter (5 minutes)

Quick questions while the paper goes out.

  1. Find each interior angle of a regular hexagon.
  2. Find the gradient of the line through (0, 8) and (2, 0).
  3. Write down the centre and radius of x2 + y2 = 25.

Main activity (35 minutes)

Task A: Curve stitching (12 min)

On squared paper, mark 1 to 9 along both axes. Join (1, 0) to (0, 9), (2, 0) to (0, 8), and so on up to (9, 0) to (0, 1). The straight lines make a curve.

  1. Find the equation of the line joining (2, 0) and (0, 8).
  2. Find the equation of the line joining (3, 0) and (0, 7).
  3. Show that the line joining (k, 0) and (0, 10 − k) has equation (10 − k)x + ky = k(10 − k).

Task B: Tessellations (12 min)

At every corner of a tiling the angles must add to 360°.

  1. For which regular polygons does the interior angle divide exactly into 360°?
  2. A tiling has a square, a regular hexagon and a regular 12-sided polygon at every corner. Show that the angles fit.
  3. For polygons with p, q and r sides meeting at a point, the angle condition becomes 1/p + 1/q + 1/r = 1/2. Check it for p = 4, q = 6, r = 12.

Task C: Graph pictures (11 min)

Draw a face with equations, on graph paper or in a grapher.

  1. An eye is the circle with centre (2, 2) and radius 0.5. Write its equation.
  2. The smile is y = 0.2x2 − 3 for −3 ≤ x ≤ 3. Find the y-coordinate of its ends.
  3. Show that the ends of the smile are inside the face x2 + y2 = 25.

Extension (10 minutes)

For fast finishers.

  1. The curve-stitching lines all touch the curve √x + √y = √10. Show that the line for k = 5, x + y = 5, meets the curve at (2.5, 2.5) with the same gradient.

For teachers

Teacher notes and full worked answers
  • Bring squared paper, rulers and coloured pencils.
  • Make it a display: curve stitching in coloured pencil (or thread on card), a tessellation tile coloured in groups, and the graph pictures printed with their equations underneath.
  • Task C: students plot the face in our free Grapher, then add their own features (eyebrows as quadratics, hair as a sine wave) and share the picture.

Starter

  1. 120°
    • (6 − 2) × 180 ÷ 6
  2. −4
    • (0 − 8)/(2 − 0)
  3. (0, 0); 5
    • r2 = 25

Task A: Curve stitching

  1. y = −4x + 8
    • Gradient = (8 − 0)/(0 − 2) = −4
    • y-intercept 8
  2. 7x + 3y = 21
    • Gradient −7/3, intercept 7: y = −7x/3 + 7
    • Multiply by 3: 7x + 3y = 21
  3. Both points satisfy it
    • At (k, 0): (10 − k)k = k(10 − k). At (0, 10 − k): k(10 − k) = k(10 − k).
    • A linear equation through both points is the line.

Task B: Tessellations

  1. Triangles, squares and hexagons (n = 3, 4, 6)
    • Interior angle = 180 − 360/n.
    • Checking n = 3 to 12 and beyond: only 60, 90 and 120 divide 360 (the angle is always between 120 and 180 for n > 6).
  2. 90 + 120 + 150 = 360
    • 12-gon: 180 − 30 = 150
  3. 1/4 + 1/6 + 1/12 = 1/2
    • 3/12 + 2/12 + 1/12 = 6/12

Task C: Graph pictures

  1. (x − 2)2 + (y − 2)2 = 0.25
    • (x − a)2 + (y − b)2 = r2
  2. −1.2
    • 0.2 × 9 − 3 = −1.2
  3. 9 + 1.44 = 10.44 < 25
    • At (3, −1.2): x2 + y2 = 10.44, less than 25.

Extension

  1. Both pass through (2.5, 2.5) with gradient −1
    • √2.5 + √2.5 = 2√2.5 = √10, and 2.5 + 2.5 = 5.
    • On the curve, dy/dx = −√y/√x = −1 at (2.5, 2.5), the gradient of the line.

The same answers are in the answers PDF. Every answer was recomputed by computer before publishing.

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