I'm Up and Down, and Round and Round: common misconceptions

If your child says… “A circle has only two symmetry lines.”

Ask together

What prevents a matching fold between your existing creases?

What’s true

Every direction through the centre gives a symmetry line, so there are infinitely many.

If your child says… “Only one circle can pass through two points.”

Ask together

Do both circles pass through your original points?

What’s true

Every position along their perpendicular bisector provides a possible centre.

If your child says… “The circumcentre is always inside the triangle.”

Ask together

In the right triangle, is O inside the triangle or on its edge?

What’s true

A right triangle’s circumcentre sits on its longest side, at that side’s midpoint.

If your child says… “Turning a chord changes its centre angle.”

Ask together

Which triangle lengths change when the chord turns around the centre?

What’s true

Within one circle, turning a chord changes neither its length nor its centre angle.

If your child says… “Any line from the centre halves a chord.”

Ask together

Does a sloping line through the centre meet the chord where your crease does?

What’s true

For a chord away from the centre, only the perpendicular from the centre halves it.

If your child says… “The 4 cm answer is the whole chord.”

Ask together

What must you do to the calculated half-chord to find the whole length?

What’s true

The perpendicular splits the chord, so the right triangle contains only half its length.

If your child says… “The centre angle is half the edge angle.”

Ask together

If the centre angle is twice 40°, can it be smaller than 40°?

What’s true

Any point on the remaining circle gives half the angle that arc makes at the centre.

If your child says… “Opposite angles in a cyclic quadrilateral are equal.”

Ask together

Do A and C face equal arcs, or arcs that together complete the circle?

What’s true

Opposite angles total 180° and are equal only when both are 90°. Conversely, opposite angles totalling 180° guarantee a circle through all four corners.