This is a challenging investigation most likely not suitable for a whole class, but rather for a small group of confident pupils.

They should have completed The Tilted Squares investigation, before this extended investigation.

• SOHCAHTOA
• Simultaneous equations (systems of three)
• Solving quadratic equations
• Comfort talking about vertices, edges, faces and in being given two new formulas to consider in quick succession.

Inspect the attached teacher sheet to assess whether investigation is appropriate for your group.

Section 3 is perhaps the most challenging, and you may want to remind pupils of method to solve a system of 3 simultaneous equations.

Some 3D shape manipulatives may be useful when introducing Pick's and Euler's Theorems.

IMPORTANT: for this investigation pupils need to distinguish between $m , n$ being the number of points on a given side length and the actual true side length of the shape where 1 dot vertically or 1 dot horizontally is considered as 1 unit.