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# Circular Logic

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Circumference Angles

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Cyclic Quadrilaterals Proof

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Pythagoras Proofs

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Matter of Scale

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The Converse of Pythagoras

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Overlap

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Pentakite

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Kite in a Square

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Quad in Quad

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To Swim or to Run?

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Creating convincing arguments or "proofs" to show that statements are always true is a key mathematical skill.

The problems in this feature offer students the chance to investigate geometrical properties, make conjectures and create convincing mathematical proofs.

Many of the problems in this feature include proof sorting activities which challenge students to rearrange statements in order to recreate clear, rigorous proofs. These tasks aim to introduce students to the formality and logic of mathematical proof.

You can watch a recording of the webinar in which we discussed the mathematical thinking which can be prompted by these activities.

Plus magazine has a selection of interesting articles exploring proofs in which pictures play an important role.

Age 11 to 16

Challenge Level

Can you prove the angle properties described by some of the circle theorems?

Age 11 to 16

Challenge Level

Can you prove that the opposite angles of cyclic quadrilaterals add to $180^\circ$?

Age 11 to 16

Challenge Level

Can you make sense of these three proofs of Pythagoras' Theorem?

Age 14 to 16

Challenge Level

Can you prove Pythagoras' Theorem using enlargements and scale factors?

Age 14 to 18

Challenge Level

Can you prove that triangles are right-angled when $a^2+b^2=c^2$?

Age 14 to 16

Challenge Level

A red square and a blue square overlap. Is the area of the overlap always the same?

Age 14 to 18

Challenge Level

Given a regular pentagon, can you find the distance between two non-adjacent vertices?

Age 14 to 18

Challenge Level

Can you make sense of the three methods to work out what fraction of the total area is shaded?

Age 14 to 18

Challenge Level

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Age 16 to 18

Challenge Level

The famous film star Birkhoff Maclane wants to reach her refreshing drink. Should she run around the pool or swim across?

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of these resources.*