Prompt 30: Interesting Fraction Methods

Note: Cell 1 provides interesting methods to show that two fractions are equivalent. The ideas have been known for quite some time, but I came up with them independently quite recently before finding out that they already existed. The more general idea being that any point lying on a straight line provides the entire set of equivalent fractions.

I learnt Cell 2 as the first proof by contradiction presented at University.

I believe that I came across the naive sum in Cell 3 on Richard Perring’s (@LearningMaths) twitter feed.

I’m not sure if, how and when I might insert any of these methods into the curriculum. I’m quite sure that I would develop exploratory tasks, possibly to deepen and stretch the knowledge of higher attaining learners. The prompt acts as a reminder that there are always creative and interesting avenues to solve even the most familiar of problems.

Additional links: for cell 3:

1. Here is a geogebra applet I made to visualise the naive sum.
2. A lovely little proof of the naive sum by Matt Parker (@standupmaths).

Prompt 29: Angle Sum of a Triangle Proofs

Questions

  1. When do you choose to prove this? Do you begin intuitively and then only prove when students have knowledge of angles in parallel lines? Could you use the proof in cell 1 immediately?
  2. How do you choose to prove that the angles in a triangle sum to 180 degrees? One of the techniques above, all of them?
  3. Could you utilise the idea of goal free problems so that students identify all angles they can in the 4 cases above and then you reveal that they have just proven that the angles in a triangle add to 180 degrees in 4 ways?

To have students complete a proof, I developed one that provides multiple avenues in the final step, so as to increase the chances of a proof from more students:

Prompt 28: Expanding Double Brackets

Questions:
  1. Do you have an agreed departmental structure?
  2. Which method is slowest? Which method is quickest?
  3. Should we teach more than one method? i.e. On first learning, teach cell 3, but then transfer to cell 2?
  4. Which method extends most naturally to three or more brackets?
  5. Should cell 2 and cell 4 come with a discussion of the distributive property, or an intermediate step showing this? i.e. (x + 3)(x – 5) = x(x – 5) + 3(x – 5)?
  6. Additional thinking:

Prompt 27: Common Misconceptions in Algebra

Questions:

  1. Are all of these misconceptions directly addressed through explicit curriculum objectives, or is it luck as to whether they are addressed whilst doing algebra? i.e. Is there a specific lesson or sequence of lessons about the difference between (a+b)/a, and a/(a+b)?
  2. For each of the prompts, how exactly would you set up a pedagogical sequence to address and teach students the correct manipulations?
  3. Are there any other common misconceptions you can think of?

Additional common misconceptions:

Additional sources:

  1. @MathCurmudgeon shared: Do this and a Bunny Dies
  2. @ProdctvStruggle shared: Twitter Comment
  3. @greg_roberts86 shared: Classic Mistakes

Prompt 24: Length Scale Factors

Questions:

  1. Which methods leads most seamlessly to area scale factors and volume scale factors?
  2. Is there an argument for method 2 being helpful to practice algebraic manipulation?
  3. Imagine you had a scenario in which there was a missing length on both similar shapes. Which method might be least confusing in this case?
  4. Which method connects most effectively to scientific concepts learnt in school regarding scaling things up or down – i.e. Antman is made 100 times smaller in length. How will this affect the strength of his muscles?

Prompt 21: Solving Linear Equations

This is a task I gave to a Year 8 mixed attainment class which I think would be a nice pedagogy prompt given the question set below:

Questions:

  1. How do you show balance operations between steps? Is this consistent between department members? Should all department members use a consistent approach?
  2. Do you explicitly reference all variants of linear equations in curriculum documentation? i.e. I can solve a linear equation with unknowns on both sides when one or both unknown terms are negative, or, I can create a linear equation given the solution.
  3. Do you teach only one method for an equation like this? How do you approach different methods? Is this student led or do you explicitly instruct on a range of methods?