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An unexpected angle to teaching congruent triangles


Created on 10 May 2019 by ncetm_administrator
Updated on 15 May 2019 by ncetm_administrator

TEACHER VOICE

An unexpected angle to teaching congruent triangles

About the author

Ben Waine teaches at The Buckingham School in Buckingham, where he is Assistant Curriculum Leader. In 2018/19 he is leading a professional development Work Group for the Enigma Maths Hub, ‘Challenging Topics at GCSE’.

 

There I was, explaining how no one wants to look like an ‘ASS’ by claiming ASS (Angle, Side, Side) / SSA triangles are congruent (bar the one, Right-angle/Hypotenuse/Side (RHS) exception we’d learned). We’d all had a giggle about the anatomical homonym; we’d identified an ass as being a donkey not a horse. This was about Angle/Side/Side triangles, however…

…when a small, quiet Year 8 (who I’ve recently been attempting to encourage) put his hand up.

‘This is positive…’ (I’m thinking) ‘it was worth putting correct words into his mouth then awarding him with a house point for using them a few days ago.’

But, was it positive? Thus far, I’d been intentionally emphasising the symmetry, demonstrating that the same ASS information could generate two very different triangles (notwithstanding the RHS exception*) and so, generally, one could not rely on ASS data to generate a unique triangle.

A classic ASS triangle (Figure 1) had been lovingly drawn up, freehand, on the board (using colour whiteboard markers no less!) Where was this boy coming from?

figure 1

‘But there’s another exception, sir.’

Hold on… another exception? not that gets a mention in any of the GCSE textbooks or guides I’ve read or watched in the past ten-plus years.

‘Really, Eric?’

What was he going to say?

‘Yes, sir. What if the two sides are the same length and the angle is 60 degrees.’.

‘Poff!’ Fog bomb goes off in my brain. Or rather my brain starts modelling and checking the unfamiliar situation as I simultaneously attempt to follow Eric’s instructions to draw a diagram up on the board while managing the class.

‘Two successive lines the same length?’

‘Yes, sir.’

‘Moving round clockwise: Angle, Side, Side?’ (The moving round clockwise part was simply our way to give a common starting point in imagining things.)

‘Yes, sir.’

It was now up on the board (Figure 2).

figure 2

And it looked good.

When the angle was sixty degrees and the two sides were equal in length, the only valid mirror-image of the second side disappeared (was subsumed into the first side). So there was only one valid triangle: an equilateral one!

Bingo! Congruence! But was it guaranteed? We checked. We thought.

‘Well done, Eric! My head went a little foggy for a while there; but it looks like you are correct. I’ve not seen that example in ten or more years’ teaching: not in a textbook; not from a pupil; not on an online system; not on a tutoring video. So, I’m still cautious; but the geometry suggests we’ve all been missing something!’

‘I think everyone will agree that has to be worth two house points!’

Everyone did. They loved it.

And so, we left it there and they returned to the investigation: hunting congruent triangles. Our conclusion?

There appeared to be other circumstances (aside from RHS) where ASS might provide conditions for congruence. Were there others, in addition to the one Eric found? This was a question for future problem-solving sessions*.

*Footnote:

In fact, there are further circumstances where ASS provides conditions for congruence. It can be interesting to explore what these are, with students.

Specifying an angle and two sides in the order ASS is not a sufficient condition for congruency, as there will not be a unique example (see Figure 1 above). However, congruency is assured by adding one of the following conditions to ASS:

  • A=90° (as highlighted earlier in the lesson, the same as RHS)
  • S1=S2 (as Eric spotted) – there is actually no need for A=60°, but A<90° is implied
  • S1<S2 (where S1 is the side adjacent to the given angle).
  • A>90° (but this implies S1<S2)

Therefore, generally: ASS provides sufficient conditions for congruency only when
S1≤S2 or A=90°.

enlarged figure 3
(click to enlarge)

The specific conditions in which ASS does generate congruency are, as Ben points out, usually ignored by Key Stage 3 and 4 maths textbooks. However, this could potentially be explored through reasoning and problem-solving aspects of GCSE questions.

What other conceptual depths are overlooked when we focus on set content? How can these be explored through reasoning and problem-solving? Let us know what you think in the Comments box below.

Or maybe there was a similar moment in your lesson, where a student opened a door that you had not previously been through? We’d love to hear your stories.

 

 
 
 
 

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Comments

 


27 May 2019 20:12
DavidChandler - agree. Geogebra Apps to be praised, if used carefully in preparation for year 7 and year 8 level assessments at whichever differentiated level is appropriate.

However, do look out for the case where S2 is greater than S1: so the triangle is not isosceles; but is unique. Unique seems an important word.
By BW_2012
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24 May 2019 19:54
As soon as you know two sides are equal, regardless of the angle, the triangle is isosceles, so you know two angles, which means you really know all three angles and SAS applies. A GeoGebra (or other dynamic geometry) applet is worth a thousand words in exploring all the variations of this problem.
19 May 2019 16:44
Erudite (mathematically) though you may be, I think you are rather missing the point (and being rather condescending to Ben; no matter your praise of Eric's genius).

To emphasise the point, perhaps I should ask you to:

1. Delve into the past (say) ten years' worth of *GCSE* papers searching for questions on constructions...

2. Note that the whole article focuses on a side-line conversation away from the main (brief) investigation [which I'm led believe came from a lovely, if perhaps now-dated, Pearson Delta 2 textbook (page 207 I suspect!)].

3. Note Ben teaches at a school for pupils who may well not all have passed the 11+ (he is quite possibly being a gentle soul, teasing carefully, rather than baffling with excessive extraneous detail)...

As I wrote earlier.

"Horses for courses".

May I add the words "Working memory" too?
By BW_2012
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17 May 2019 04:11
Some brief points. I'm under time pressure due exams, marking, etc.. So apologies if any of the following points appear blunt. Nothing intended by that.

1. ASS for acute angles leads directly to, and completely explains the ambiguous angle in the sine rule. Maths teachers need to be, and should be, fully aware of this.

2. If a member of staff is not aware of this, then perhaps the lesson time is better used in a different way.

3. Collaboration within depts needs to be in place (preferably as a matter of routine) to ensure teachers of maths (and other subjects) are aware of the detail of the content to be covered.

4. The outline of Ben's lesson, above, is evidently not a lesson which involves investigation by the students. At which point, such a lesson requires the teacher to be in command of all of the details of the content matter. (see need for collaboration, above).

5. For a heavily investigative lesson, I'd expect to have low base, high ceiling type, which would encourage a wide range of results, methods & processes to be found/considered. (scaffolded, as necessary, by a well-prepped teacher). ASS in itself really is not such a task. There are only two outcomes, as far as I can see. Angle is acute -> leading to ambiguous case. Angle is NOT acute -> congruence.

Well done Eric. As I said before.



I think Ben was hinting at the idea of next time he teaches this topic, there'll be scope for all students to discover "Eric's triangles" for themselves.

The question really is whether this single result is sufficiently "WOW" to warrant a lesson devoted to finding it, or perhaps this task is better placed alongside sine rule, ambiguous angle, and might form only part of such a lesson ??
By stefn
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16 May 2019 16:14
There are many circumstances in which teachers are not fully aware of the facts before a lesson. Recently trained teachers may not be on top of the whole syllabus, or supply teachers, or the army of non-specialists currently teaching maths. Likewise there are learners classrooms who see things differently or find a topic so compelling they take it beyond the usual syllabus.

It is hugely important for learners to see their teacher modelling being a mathematician for them; being surprised, stumped, uncertain etc. and what an empowering experience for Eric.

Nobody would suggest we need teachers who know less maths but Ben's story is a reminder that we can organise the classroom to allow what the "Realistic Maths" approach calls "guided reinvention". We can devise lessons that give learners the opportunity to engage in reasoning and proof. I would wholeheartedly recommend teachers have a go at putting together a lesson where their class search for "Eric's Triangles."
By cfangier
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16 May 2019 15:49
We are all learning all the time (hopefully). To set teachers up as 'fonts of knowledge' denies students the opportunity to take the 'knowledge authority' sometimes. Teachers need to model an openness to learning, as Ben has done so well in this example. It's a brave thing to admit in front of a class that you didn't know something, even braver to write about it - but I would suggest it's a rare teacher that hasn't been there. Even if we think we understand something, there's always more to learn. Sometimes we need to be 'alongside' students to model how to learn.
16 May 2019 15:42
We are all learning all the time (hopefully). To set teachers up as 'fonts of knowledge' denies students the opportunity to take the 'knowledge authority' sometimes. Teachers need to model an openness to learning, as Ben has done so well in this example. It's a brave thing to admit in front of a class that you didn't know something, even braver to write about it - but I would suggest it's a rare teacher that hasn't been there. Even if we think we understand something, there's always more to learn. Sometimes we need to be 'alongside' students to model how to learn.
16 May 2019 15:35
Well, yes; and perhaps those writing KS3/GCSE textbooks too?

However (aside from being chastened by the wider world of mathematics!) I hope you would also agree it is good to always be chastened by the exam syllabus, the amount of available time and the need to retain (hold back) some elements of knowledge to encourage "curiosity" in years 9, 10, 11 and beyond! Not to mention to ensure one properly balances giving sufficient "shock and awe" to a pupil for "discovering" something beyond the norm against the risk of that one person's fascination taking an investigation too far off track in the available time for the other 31 pupils in the class!

"Horses for courses."

Refs:

https://corbettmaths.com/2013/04/15/congruent-triangles/

https://www.bbc.com/bitesize/guides/z9jpv9q/revision/10

https://www.drfrostmaths.com/resource.php?rid=150

+ variety of Pearson, OUP, etc textbooks...
By BW_2012
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16 May 2019 13:27
https://en.wikipedia.org/wiki/Congruence_(geometry)

... about half-way down the page under SSA providing congruence lists both of the conditions you found after Eric's intervention.

Well done, Eric.

Hopefully maths teachers will be more aware of 'facts' BEFORE lessons, for most (all) topics.
By stefn
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