Prompt 15: Pythagoras 4

This prompt is for those teachers who may not teach beyond Middle School (Year 9/Grade 8) or beyond GCSE (Year 11/Grade 10). I find it’s always pedagogically helpful to know how a concept might be extended beyond the curriculum I teach.

Additional Resources:

  1. Cell 1: See this Gif by Idan Tal (@MagicPi2)

2. A lovely post about the connection between Pythagoras and the Cosine Rule by Andrew Stacey (@mathforge)

3. Cell 4: A little more on time dilation: Time is Relative video

Prompt 14: Nth term of a Linear Sequence

  1. Why do we often only introduce the arithmetic sequence formula post-16?

2. If you only teach one method, which method do you teach and why? Does each method lend itself to another concept in the curriculum?

3. The standard formula for an arithmetic sequence is given in cell 4. However, you can use any term in the formula u_n = u_x + (n – x)d; would you consider teaching this more general form, even though it may not appear in formula booklets?

4. In regards to cell 2, the zeroth term is equivalent to the y-intercept, and the difference is equivalent to the gradient on a co-ordinate grid. How best could you represent or teach this integral connection?

Prompt 12: Equations of Straight Lines

  1. Which method out of cell 1 and cell 3 do you think is most helpful when teaching gradient? Are both important? How can cell 3 be used to enhance understanding of perpendicular lines?
  2. When would you introduce x-intercept form? Before studying the factored form of a quadratic? as an inquiry or investigation during a unit on y-intercept form? How is it helpful in leading into transformation of functions later in school?
  3. Should gradient be reconceptualized as a vertical stretch?
  4. Would you teach the general intercept form or is it simply interesting to know? This catalyses a more general question: How do external tests influence what we teach?

Prompt 10: Surface Area of a Cylinder

1. How do you get from cell 1 to cell 2 and 3? (Geogebra File)

2. Do you show students the geometric form of a factored formula? Is it important that students know the connections between Geometry and Algebra in this case?

3. How might you bring this into a lesson on surface area of cylinders? Card sort? Plenary? Does it change how you teach areas of circles earlier on in the curriculum? i.e. by transforming the circle into a triangle to derive the area formula?

Prompt 8: Pythagoras 3

  1. 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?
  2. 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?
  3. Cell 3 provides a more general result about similar triangles. Are there any other general results which reduce to Pythagoras?
  4. 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?