- Cell 1 may be the most common proof to discuss with students. Would you do this at the start of a unit, in the middle, at the end, or not at all? Do you ask students to reproduce the proof? Do you think they remember it and does it matter if they don’t?
- How and when might you introduce the proof in cells 2, 3 and 4? Are there any proofs you absolutely wouldn’t show? Is there a circumstance in which you would show this prompt, then take it away and ask students to reproduce it?
- Cell 3 provides a more general result about similar triangles. Are there any other general results which reduce to Pythagoras?
- Do you know any other proofs of Pythagoras? What’s your favourite proof of Pythagoras and why?

# Prompt 7: Interior Angles of Polygons

1. How do you introduce finding the interior angle sum of any polygon? Would you start with a formula and pick it apart? Build a formula together? or do you do it differently? i.e. through generating the sequence and finding the nth term as in cell 4? Would you have students “discover” for themselves through an investigation or inquiry?

2. Do you relate interior angles of polygons to a “real world” problem or context?

2. Would you consider not introducing a formula and only encouraging students to split polygons into triangles and/or quadrilaterals?

3. Which method do you think is the easiest for students to grasp? Is there a circumstance in which you would only explicitly teach this one method and nothing else?

4. In Cell 4, do you give “geometric” names to different formats of the formula? Do you utilise functions derived earlier in school maths to discuss new concepts such as domain and asymptote?

# Prompt 6: Prime Factorization

- Do you use prime factorisation for any of these topics? If not, is this a result of sequencing of topics? Would you consider teaching prime factorisation earlier?
- Are there any other applications of prime factorisation? (excluding the obvious HCF and LCM)
- How do you initially teach prime factorization? Factor trees?
- An interesting avenue for prime factorization: https://www.mathspad.co.uk/interactives/primeDecomposition/primeDecompositionTool.php

5. When is multiplication of numbers easier using prime factorization? When isnâ€™t it?

6. In cell 3, why does this algorithm work for finding how many factors a number has?

# Prompt 5: Pythagoras 2

- Do you tell students that the connection between areas of squares on a right-triangle also holds for any shape – as long as the shapes on all sides are similar? If not, why not? If so, when and how do you have them engage with this idea – in a unit on similar shapes or during a Pythagoras unit? How might you utilise cell 1 to interleave/review areas of different shapes?
- Could Cell 2 and Cell 3 increase transfer with science? Could science complete the experiments in an inter-disciplinary unit on Pythagoras?
- How and when might you discuss cell 4? Before A-Level/during A-Level? during a unit on 3-D coordinates?

# Prompt 4: Intergerizing the Denominator

- Would you use the term in the prompt even if textbooks or exams do not?
- Does decimal division work in a similar way?
- In regards to fraction division, do you justify the equivalent calculation of multiplying by a reciprocal in this way? Do you think it’s important to justify? If so, when would the justification happen? Would you assess a justification in a unit test?
- How do you explain why we would bother to rationalise the denominator? Is it an opportunity to discuss the history of mathematics, or is it a technique that is isolated with no obvious application or reason?

# Prompt 3: The Cosine Rule 1

1. Which formula do you use for the cosine rule? A formula that isn’t listed above?

2. Why might you stress the importance of number 2?

3. Is there a context in which you would consider teaching number 3? Is it a problem that this doesn’t appear in textbooks or formula books?

# Prompt 2: A = 1/2absinC

- Which question would be the first in your explanation sequence and why?
- Is there a question you wouldn’t include in the initial sequence but would include later in the learning journey?

# Prompt 1: Pythagoras 1

- How do you begin the teaching of Pythagoras? Would you use any of these cells? If you were to use a hook, can you think of one more appropriate to your own style/to your local context?
- Would you consider utilising a student in the room to walk 3 paces left – then turn quarter of a turn – then 4 paces up – then count how many paces it takes to walk back to the starting point? Why would/wouldn’t you do this?
- Are there any high landmarks nearby which you could apply to Cell 3? What would be the approximate view distance from the top of Mount Everest? or from the cruising height of a plane?
- If you think Cell 3 and Cell 4 should be reserved for extension work, which other extension tasks/questions might you use?
- Cell 4: Fermat’s Last Theorem (5 minute clip)