I’ve written before about the app “Brilliant“, which is well worth getting, and I also follow their Facebook page which provides me with a regular stream questions. Occasionally I have to think about how to tackle them, and they’re excellent. More often, a question comes up that I look at and think would be awesome to use in a lesson.

Earlier this week this question popped up:

What a lovely question that combines algebra and angle reasoning! I can’t wait to teach this next time, and I am planning on using this as a starter with my y11 class after the break.

The initial question looks simple, it appears you sum the angles and set it equal to 360 degrees, this is what I expect my class to do. If you do this you get:

*7x + 2y + 6z – 20 = 360*

*7x + 2y + 6z = 380 (1)*

I anticipate some will try to give up at this point, but hopefully the resilience I’ve been trying to build will kick in and they’ll see they need more equations. If any need a hint I will tell them to consider vertically opposite angles. They should then get:

*2x – 20 = 2y + 2z (2)*

*And*

*3x = 2x + 4z (3)*

I’m hoping they will now see that 3 equations and 3 unknowns is enough to solve. There are obviously a number of ways to go from here. I would rearrange equation 3 to get:

*x = 4z (4)*

Subbing into 2 we get:

*8z – 20 = 2y + 2z*

*6z = 2y + 20 (5)*

Subbing into 1

*28z + 2y + 6z = 380*

*34z = 380 – 2y (6)*

Add equation (5) to (6)

*40z = 400*

*z = 10 (7)*

Then equation 4 gives:

*x = 40*

And equation 2 gives:

*60 = 2y + 20*

*40 = 2y*

*y = 20.*

From here you can find the solution x + y + z = 40 + 20 + 10 = 70.

A lovely puzzle that combines a few areas and needs some resilience and perseverance to complete. I enjoyed working through it and I’m looking forward to testing it out on some students.

*Cross-posted to Cavmaths here.*

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I think you’ve made it too complicated. There are also linear pairs, so I can more easily find x by setting 3x +2x – 20 = 180.

But it might work to show your methods to students to analyze and find other possible methods.

I have to check out this app soon!

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Aye, someone else pointed out the linear pairs, I did say there is more than one way…. my brain has a habit of defaulting to the most complicated method for some reason!

The app is well worth checking out.

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I sometimes love it when students come up with complicated methods. It’s a great way to discuss why some methods are faster, but all are valid.

Even better when a student gets part of an answer correct with a completely faulty method. Happened this week and created a really good discussion on what is “right”.

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Yes, those sorts of things can provide the best opportunities for great in depth discussions.

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“Finished already? Go write your own problem”

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Yep, a fantastic way to get them thinking, especially if you specify “with integer solutions” etc.

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That’s a bit cruel !

If they are ok with statements involving several variables then maybe you could show them Heron’s formula for the area of a triangle, and it has delightful symmetry (structure) if rewritten with (a+b+c)/2 substituted for s.

Sides a,b,c

Area = sqrt(s(s-a)(s-b)(s-c))

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Indeed. And it’s always a joy to meet another fan of Herons Formula, it’s a favourite of mine and lots of folk have never heard of it! I’ve written on it before:

https://cavmaths.wordpress.com/tag/herons-formula/

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The linear pairs allow solving it in your head.

3x+2x-20=180; 5x=200; x=40

3x+2y+2z=180; 2x+2y+2z=180-40; x+y+z=70

Opposite angles to check: 2x-20=60; 2y+2z= 3x+2y+2z-3(40)=60

3x=120; 2x+4z= (need to solve for z, not worth the bother)

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Indeed, a lovely concise solution.

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