Choose P on the semi-circle such that EP is tangent to the semi-circle. By Pythagoras on triangle EPF, EP = 1. By the tangent-secant theorem, EP squared is equal to EG x ED. Since ED was sqrt(3) by Pythagoras, EG is 1/sqrt(3). This means the scale factor is 3 and I finished off in the same way as you. ]]>

Where I have more setpiece group activities, I will generally devise the task so that several processes can or need to happen at once so loafing is harder to achieve and tends to be peer policed. I reinforce the latter by making a point during follow up discussion of asking non-leaders in groups why they made a certain choice or how they found such a value. If they don’t understand what their group did I will criticise them for not engaging but also the group leaders for not taking them with them; this becomes more interesting when the group ends up with the wrong answers!

]]>Another question, and this gets right to the core of your beliefs as a teacher. Should we be providing social opportunities in our maths lessons to work together with all the challenges that brings? Should we expose our students to social loafers and help them learn how to deal with the challenges of working in teams. Or should we ensure that every minute of every lesson is optimised for maximum learning of mathematics? If my school provides enough opportunities for the former, that helps me to justify an approach that prioritises mathematical learning opportunities in my lesson. ]]>