Prompt 38: Simultaneous Equations (Elimination and Graphing)

Lesson 1: 15 minute task

Lesson 2: Starter


  1. In which order do you teach graphing, elimination and substitution? Does it matter which order they are taught in?
  2. How do you introduce solving simultaneous equations? Which problem do you use? Is it devoid of context?
  3. What prior learning needs to be re-capped or mastered before starting with any of these techniques? How do you ensure this is retrieved and fluid? i.e. adding and subtracting negative algebraic terms for elimination, gradients of straight lines for graphing, etc. Would you re-cap in starters? homework tasks? full lessons?
  4. See the applet below. Would you have students explore solutions graphically as an independent/collaborative task, would you lead a discussion with this applet, or would you simply not use it?

Applet: Solving Simultaneous Equations Graphically

Prompt 36: Dividing Fractions


  1. If you choose to divide fractions using the standard technique of multiplying by the reciprocal, how could you use cell 2 and cell 4 as a learning experience? i.e. Would you provide these as something that higher attaining students can verify with more examples? When might it be easier to use cell 4 rather than multiplying by the reciprocal?
  2. In regards to cell 2, would you ever consider teaching fractions by always finding a common denominator first when adding, subtracting, multiplying and dividing – then have students reflect on which operation they don’t need to do this for?

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


  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: