Teaching objectives
Did you know that a mathematical mirror can create flowers, stars, and mandalas? This lab takes primary-school students from their very first reflection of a point all the way to drawing their own mandala. It is a gentle lab with beautiful animations and visual effects. We recommend pairing it with our Tank Battle game.
What you'll learn
Students build the concept of line symmetry step by step: first with a single point and a single axis, then with several axes at once, and finally by identifying and counting axes in real geometric shapes.
- They practise reflecting points across vertical, horizontal, and diagonal (45°) axes on a grid, counting steps to the axis.
- They explore what happens when two axes of symmetry are combined: perpendicular axes produce 4 images; axes at 45° produce 8.
- Using an angle slider, students discover that the reflection pattern is finite only when the angle between the axes is a divisor of 360°.
- They identify whether shapes such as a circle, a scalene triangle, an isosceles triangle, or a parallelogram have line symmetry, and count the exact number of axes (5-pointed star → 5, regular hexagon → 6, etc.).
- The lab ends with a creative room: students draw freely and the "digital doily" tool replicates each stroke with however many axes they choose, producing their own mandala.
Key mathematical ideas
- The reflection of a point across an axis is equidistant from the axis: if the point is 3 steps away, its image is 3 steps on the other side.
- Two successive reflections across intersecting axes are equivalent to a rotation: the angle of rotation is twice the angle between the axes.
- A two-mirror pattern is finite when the angle between them divides 360° exactly (divisor of 360).
- The number of axes of symmetry of a regular polygon with n sides is exactly n.
Room-by-room contents
Room 1 · Symmetry creates beauty
Visual opening: an animated kaleidoscope draws colourful patterns using only symmetry. The student observes how a simple gesture multiplies into a beautiful figure.
Student tasks
- Watch the kaleidoscope and enjoy the effect.
- Advance to start exploring symmetry.
Room 2 · Symmetry is a mirror
The student clicks on the canvas and sees how the point appears reflected on the other side of a vertical axis, as if there were a mirror in the centre.
Student tasks
- Click in different places on the canvas.
- Observe where the reflected point appears each time.
Room 3 · Counting the steps
An animation shows how a point moves away from the vertical axis counting steps, and the symmetric point appears the same number of steps on the other side. The student is asked to locate that symmetric point.
Student tasks
- Watch the animation of steps from the axis.
- Click on the correct symmetric point on the grid.
Room 4 · Find the symmetric point
Free practice with a vertical axis: three points are marked and the student must click on the three corresponding symmetric points.
Student tasks
- Locate the symmetric of each of the 3 points.
- Click on the three correct points (vertical axis).
Room 5 · Horizontal axis
The same step-counting mechanic, but now the axis is horizontal: the point is reflected up-down instead of left-right.
Student tasks
- Watch the horizontal axis animation.
- Click on the correct symmetric point.
Room 6 · Practice: horizontal axis
Three points to reflect across a horizontal axis. The student consolidates the difference between vertical and horizontal axes by finding the symmetric points.
Student tasks
- Click on the three symmetric points (horizontal axis).
Room 7 · Diagonal axis (45°)
The axis is now tilted 45°. The animation shows how reflecting a point swaps its X and Y coordinates.
Student tasks
- Watch the diagonal axis animation.
- Click on the correct symmetric point.
Room 8 · Practice: diagonal axis
Practice with three points and a 45° axis. The student confirms that reflecting across the diagonal swaps the coordinates.
Student tasks
- Click on the three symmetric points (45° axis).
Room 9 · What happens with two axes?
Introductory text screen: so far we have used a single axis; the next stage explores what happens when reflecting across two axes at once.
Student tasks
- Read the introduction and advance.
Room 10 · Two perpendicular axes
Two axes crossing at 90° divide the plane into four quadrants. The student drags a blue point and sees 4 symmetric copies appear simultaneously.
Student tasks
- Drag the blue point to different positions.
- Observe that exactly 4 images are always generated.
Room 11 · Two axes at 45°
The same two axes but with a 45° angle between them. Now the blue point generates 8 images that form an 8-pointed star.
Student tasks
- Drag the blue point and count the 8 symmetric images.
- Compare with the previous room (4 images vs. 8).
Room 12 · Change the angle
A slider allows changing the angle between the two axes. When the angle is a divisor of 360° the pattern closes; otherwise the images accumulate indefinitely.
Student tasks
- Move the slider and observe when the pattern is finite.
- Answer: when do a finite number of reflections appear? (when the angle is a divisor of 360°).
Room 13 · Two mirrors = one rotation
Conclusion screen: reflecting across two intersecting axes is equivalent to a rotation. The angle of rotation is twice the angle between the axes. Explanatory room with animated text.
Student tasks
- Read the conclusion and the numerical example.
Room 14 · Which shapes have axial symmetry?
Six shapes are shown (circle, scalene triangle, rectangle, asymmetric trapezoid, isosceles triangle, parallelogram). The student must mark which ones have at least one axis of symmetry.
Student tasks
- Examine each shape.
- Select all those that have at least one axis of symmetry (circle, rectangle, isosceles triangle).
Room 15 · How many axes does the 5-pointed star have?
The student sees a 5-pointed star and must state how many axes of symmetry it has. They can press "Show axes" to check the answer.
Student tasks
- Think about how many axes the 5-pointed star has.
- Enter the answer (5) and confirm by viewing the drawn axes.
Room 16 · How many axes does the equilateral triangle have?
The equilateral triangle has 3 axes of symmetry, one through each vertex. The student confirms this with the animation.
Student tasks
- Enter the number of axes of the equilateral triangle (3).
- View the 3 drawn axes to confirm.
Room 17 · How many axes does the isosceles triangle have?
The isosceles triangle has only 1 axis of symmetry (the perpendicular bisector of the unequal side). Contrasts with the equilateral triangle from the previous room.
Student tasks
- Enter the number of axes of the isosceles triangle (1).
- Observe that the axis passes through the vertex of the special angle.
Room 18 · How many axes does the rectangle have?
The rectangle has 2 axes of symmetry (one horizontal and one vertical), but NOT the diagonals. A concept that often confuses students.
Student tasks
- Enter the number of axes of the rectangle (2).
- Confirm that the diagonals are not axes of symmetry.
Room 19 · How many axes does the regular hexagon have?
The regular hexagon has 6 axes: 3 connecting opposite vertices and 3 connecting midpoints of opposite sides. Closes the series of regular shapes.
Student tasks
- Enter the number of axes of the regular hexagon (6).
- View the 6 drawn axes and distinguish the two types.
Room 20 · Move the axis yourself
The student drags the ends of an axis freely and sees how a point is reflected in real time according to the axis position.
Student tasks
- Drag the ends of the axis to rotate it.
- Observe how the reflected point changes with each axis position.
Room 21 · 10-pointed star
Challenge: using a rotating purple axis (angle slider), the student must find the exact angle that generates a 10-pointed star. The answer is 18°.
Student tasks
- Rotate the purple axis until the pattern forms a 10-pointed star.
- Confirm that the correct angle is 18° (360° ÷ 20 = 18°).
Room 22 · Create your mandala
Creative ending: the student draws freely on the canvas and the program replicates the stroke with multiple radial symmetry, generating a unique mandala. Free expression room.
Student tasks
- Draw on the canvas and watch how the mandala forms.
- Explore different strokes and colours to personalise the design.
Rooms to project
The most striking ones to show and discuss in class.