This is an introduction to the quadratic formula. The fun is that it uses a 4000 year old method and places the Maths in history. Interesting questions are how it works when solutions are irrational and negative. The latter is the most interesting as it shows that a method fro a real tangible example can extrapolate to unreal imaginary rectangles with negative sides.
Teachers can remove the Hint section and try to get the students to suggest ways forward. How do they find a link between the Babylonian problem and the quadratic equation?
I do not claim credit for this, and I have no idea where I got it from many years ago. The teacher needs to give the following prompts:
What would you expect to happen to the result if we move the square down the grid to where the numbers are larger? Most of the class usually makes the wrong guess!
Can we prove it will be the same however far we go? How can you be sure it will not increase after we hit the millions?
What if we change something? Let them choose the changes. The key skill here is being aboe to organise their research.
The generalisation sets up increasingly complex quadratic expressions as you change the size of the square, the number of columns and move from a square to a rectangle. The main aim is to talk about proof.
What it says really. Activities on exploring lines and algebra. They should get to gradient, intercept and hopefully parallel and perpendicular. The setting of problems to each other is important. the writing of a final report is all about communication of Maths. We often find that students’ communication skills evaporate when they enter the Maths classroom, as suddenly theri writing age goes down by several years. We need to constantly be asking them to write in order to link literacy across teh curriculum.
This is a more open ended way of exploring some facts about quadratics, practicing the formula, looking for limits and other interesting things.
Maths Thoughts 
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