A lovely angle puzzle

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:

image

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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10 thoughts on “A lovely angle puzzle

  1. 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!

    Liked by 1 person

    • 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.

      Like

      • 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”.

        Liked by 1 person

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