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Isosceles triangles and deeper understanding

When marking paper 3 of the Edexcel foundation Sample Assessment Materials recently I came across this question that I found interesting:

image

It’s a question my year tens struggled with, and I think it is a clear marker to show the difference between the current specification foundation teir and the new spec.

The current spec tends to test knowledge of isosceles triangles by giving a diagram showing one, giving an angle and asking students to calculate a missing angle. This question requires a bit of thinking.

To me, all three answers are obvious, but clearly not to my year 10s who do understand isosceles triangles. The majority of my class put 70, 70 and 40. Which shows they have understood what an isosceles is, even if they haven’t fully understood the question. They have clearly mentally constructed an isoceles triangle with 70 as one of the base angles and written all three angles out.

What they seem to have missed was that 70 could also be the single angle, which would, of course, lead to 55 being the other possible answer for B. One student did write 55 55 70, so showed a similar thought process to most but assumed a different position for the 70.

I already liked this question, and then I read part b:

image

Now students are asked to explain why there can only be one other angle when A = 120. Thus they need to understand that this must be the biggest angle as you can’t have 2 angles both equal to 120 in a triangle (as 240 > 180), thus the others must be equal as it’s an isoceles triangle.

The whole question requires a higher level of thinking and understanding than the questions we currently see at foundation level.

In order to prepare our students for these new examinations, we need to be thinking about how we can increase their ability to think about problems like this. I think building in more thinking time to lessons, and more time for students to discuss their approaches and ideas when presented with questions like this. The new specification is going to require a deeper, relational, understanding rather than just a procedural surface understanding and we need to be building that from a young age. This is something I’ve already been trying to do, but it is now of paramount importance.

There is a challenge too for the exam boards, they need to be able to keep on presenting questions that require the relational understanding and require candidates to think. If they just repeat this question but with different numbers than it becomes instead a question testing recall ability – testing who remembers how they were told to solve it, and thus we return to the status quo of came playing and teaching for instrumental understanding, rather than teaching mathematics.

What do you think of these questions? Have you thought about the effects on your teaching that the new specification may have? Have you any tried and tested methods, or new ideas, as to how we can build this deeper understanding? I’d love to hear in the comments or social media if you do.

Further Reading:

Teaching to understand – for there thoughts in relational vs instrumental understanding

More thoughts on the Sample assessment materials available here and here.

Cross-posted to Betterqs here.

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  1. Sheila Greenaway
    April 6, 2016 at 3:32 pm

    Having trouble finding the third answer here. What am I missing?

    • April 6, 2016 at 3:54 pm

      The three possible answers are 70, 40 and 55. Which did you miss?

  2. April 10, 2016 at 11:48 pm

    Wow. Maybe the same one I missed? I saw it as Angle A could be “in a corner” at 70, so B matches, giving 70. Alternately, Angle A could be unpaired “at the top”, so B could be 55. Did NOT see the third possibility (40), which moved B “to the top”. Even after reading this post, which said “55 is the other possible answer for B” I was thinking, yes, yes, I know that… 70, 55, and what the heck is the third? Just goes to show how we can build certain assumptions into how we look at things. (By the way, I also like the part b extension.)

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