Prompt 33: Creating and Solving Linear Equations in the Curriculum

This prompt is designed to get departments thinking about where in the curriculum students must create and solve linear equations. Can you think of any other concepts this connects to? Are you teaching creating linear equations explicitly? Do the department have a common approach to solving linear equations?

Further questions:

  • For cell 3 and 5, are you explicitly teaching cross multiplication as a trick, or is it something students develop themselves? What the the issues with cross multiplication?

See this twitter thread started by @DrFrostMaths on cross multiplication.

  • Further topics: Direct/Inverse Proportion, Simple Interest, Similar Shapes, Angles in Parallel Lines

Prompt 31: Distance-Time Graphs (Starter)

An object moves away from a point, X. Describe what it is doing in each section of the graphs below.

After a first lesson on Distance-Time Graphs, I gave this as an independent starter. After 30 seconds, I said, “There may be parts of the graph that you are struggling to describe – keep thinking and you will have the opportunity to discuss this with your partner in 2 minutes.

Would you do anything differently?

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