Triangles and 10 year old

[Kezia]

My 10 year old has learned that the sum of the 3 angles have 180°
 

He also learned that a quadrilateral can be divided into 2 triangles and how to figure area of a triangle. 
 

We learned that the sum of the 4 angles in a quadrilateral add up to 360°. 
 

Then he went off track. He insists that if you can break a quadrilateral with 4 angles consisting of 360° into two triangles  and put them together into a big triangle then that triangle must have 360°.  

 

I keep telling him that each individual triangle has angles totaling 180°, the larger triangle that has been divided into 2 triangles has THREE angles totaling 180° When you take the dividing line away. If that big triangle can be divided to make two smaller triangles and those two smaller triangles can be put together to make a FOUR sided quadrilateral then that new 4 sided figure will have 4 angles totaling 360° By splitting the triangles into a 4 sided figure, one gains another angle not previously counted. He can tell me quadrilaterals have 360° and triangles 180° but then will proceed to argue over this point.

What am I teaching wrong here. Is there some magic set of words I need to say?

[wendyroo]

The very first thing I would do is give him paper, pencil and protractor. It is one thing to "learn" that the angles of a triangle add up to 180°, and another to draw a bunch of triangles, measure their angles, and convince yourself that really, truly it can't be any other way.

Then do the same with quadrilaterals. You obviously can't rigorously prove by drawing and measuring that all quadrilaterals have angles adding up to 360°, but you can form a strong intuition about it when you have drawn all sorts of wacky ones and they all add up to 360°.

Next, start physically cutting some of those quadrilaterals into triangles. He will find that in most cases you can't cut a random quadrilateral into two triangles and then rearrange those two triangles to form a bigger triangle. That only works if you can take an edge of each small triangle and arrange them to form one continuous edge of the new triangle. And that can only be done if they share an edge length and their angles are supplementary. And at that point, those two angles "merge" into a straight side on the new, big triangle and are no longer counted in the sum of the vertex angles...because they are no longer vertices, but now jointly form one of the edges.

But all of those last few sentences were just words that he can disagree with. I think the physical manipulatives are going to be a much stronger teacher in this case. He can try until the cows come home, and he still won't be able to physically make a triangle with angles measuring 360°. He needs to really experiment with physical triangles and quadrilaterals until he convinces himself of that.

[Kezia]

15 minutes ago, wendyroo said:

The very first thing I would do is give him paper, pencil and protractor. It is one thing to "learn" that the angles of a triangle add up to 180°, and another to draw a bunch of triangles, measure their angles, and convince yourself that really, truly it can't be any other way.

Then do the same with quadrilaterals. You obviously can't rigorously prove by drawing and measuring that all quadrilaterals have angles adding up to 360°, but you can form a strong intuition about it when you have drawn all sorts of wacky ones and they all add up to 360°.

Next, start physically cutting some of those quadrilaterals into triangles. He will find that in most cases you can't cut a random quadrilateral into two triangles and then rearrange those two triangles to form a bigger triangle. That only works if you can take an edge of each small triangle and arrange them to form one continuous edge of the new triangle. And that can only be done if they share an edge length and their angles are supplementary. And at that point, those two angles "merge" into a straight side on the new, big triangle and are no longer counted in the sum of the vertex angles...because they are no longer vertices, but now jointly form one of the edges.

But all of those last few sentences were just words that he can disagree with. I think the physical manipulatives are going to be a much stronger teacher in this case. He can try until the cows come home, and he still won't be able to physically make a triangle with angles measuring 360°. He needs to really experiment with physical triangles and quadrilaterals until he convinces himself of that.

We did actually go through the initial learning with a protractor and triangles, that is how the curriculum led him to understand 180° in a triangle as well as by ripping the corners off triangles to make 180° (from 0°-180° on his protractor) making a straight line. After some practice the curriculum moved on  and it had him work through the quadrilaterals and the angles of it. 
 

I will try all the manipulatives again. We worked together a few times, making him prove it may show him what he needs to see. 
 

Thanks!

[Drama Llama]

When you stick 2 right triangles together to make a big triangle, the two 90 degree angles disappear.  They combine to make a straight line so they are no longer measured as angles and don't contribute to the total.  

I would make a quadrilateral, and cut it into two congruent right triangles.  

Then color each pair of congruent angles a different color.  So, the right angles are blue on both triangles, and one pair of other angles is yellow and another red.   Measure the 3 angles of each triangle, and add them up 

Then tape the angles together into a bigger triangle.   Now, measure the 3 angles on the bigger triangle.  Note that you aren't measuring the two blue right triangles any more, because they have formed a line.  That's where the other 180 degrees goes.

Then cut the triangle apart, and tape it back together as a rectangle again.  Notice that you're measuring all of the colors again.  

That's the best I can do.  

[caffeineandbooks]

You have a kid who's curious, who's actively thinking about how to extend and apply his learning - I don't think you're doing *anything* wrong!  When he gets this, he will get it deeply and forever because he has engaged and struggled with it.  Go you!!

With the previous posters, I'd ask him to show you he's right.  If he can't easily come up with a quadrilateral that can be cut in two pieces and rearranged into a triangle, show him that you can do it with a square (tangrams might be helpful for this) and let him measure the angles and see that the two right angles, as @BaseballandHockey said, "disappear" and become sides.

You might also show him that a large triangle can be cut into four smaller ones, to make a tetrahedron net.  We don't measure/include the internal angles of the net when talking about the sides of the larger triangle, though - only the three at the corners.  That's what TRIangle means; three.

[Kezia]

Great! Thanks all! I have a couple of new things to try with him tomorrow. Physically seeing it with manipultives in a slightly different way than I have tried and I can read it to him verbatim from other people’s words. They sound like exactly like what I have been saying but sometimes word choice makes a huge difference. 😀

[Not_a_Number]

I'm curious how this went! Updates? 

Personally, if my kiddo was saying this, I'd invite them to demonstrate how to do it. They could try to convince me 😉 . I'd only get annoyed if they'd be unwilling to try or impossible to talk to about their attempt. 

[Caroline]

One thing I like to do is cut out a bunch of triangles. Then cut the vertices off the triangle and physically add the angles to get her to show you get a line, or 180 degrees. Don’t use a protractor. Literally cut the angles off the triangles and add them together. 

[Monica_in_Switzerland]

15 hours ago, Caroline said:

One thing I like to do is cut out a bunch of triangles. Then cut the vertices off the triangle and physically add the angles to get her to show you get a line, or 180 degrees. Don’t use a protractor. Literally cut the angles off the triangles and add them together. 

 

I agree with this, and doing the same for quadrilaterals.  I would suggest tearing the angles off though, or else highlight the angles before cutting up the shape, otherwise it's easy to get lost on which ones were the actual interior angles of the shape and which were formed when you cut/tore it apart!  (personal experience talking...)

 

 

[Not_a_Number]

35 minutes ago, Monica_in_Switzerland said:

I agree with this, and doing the same for quadrilaterals.  I would suggest tearing the angles off though, or else highlight the angles before cutting up the shape, otherwise it's easy to get lost on which ones were the actual interior angles of the shape and which were formed when you cut/tore it apart!  (personal experience talking...)

Am I the only one who has NEVER had any trouble believing the beautiful parallel line proof that the angles of a triangle add up to 180 degrees?? I mean, it's so striking! It's more much striking than actually measuring the angles... you can literally SEE those angles add up to a line... 

Edited February 7, 2021 by Not_a_Number

[Monica_in_Switzerland]

26 minutes ago, Not_a_Number said:

Am I the only one who has NEVER had any trouble believing the beautiful parallel line proof that the angles of a triangle add up to 180 degrees?? I mean, it's so striking! It's more much striking than actually measuring the angles... you can literally SEE those angles add up to a line... 

 

I don't think anyone here is claiming this?  The parallel line proof is great, assuming kids are quite comfortable with parallel lines and how they work.  Tearing the corners off a triangle take 10 seconds and works regardless of a kid's prior knowledge of geometry.  It seems like one segues right into the other.  

I would never recommend measuring with protractor though.  That's a recipe for "Wait, this triangle adds up to 183°, and this one to 179°!"

[Not_a_Number]

2 hours ago, Monica_in_Switzerland said:

I don't think anyone here is claiming this?  The parallel line proof is great, assuming kids are quite comfortable with parallel lines and how they work.  Tearing the corners off a triangle take 10 seconds and works regardless of a kid's prior knowledge of geometry.  It seems like one segues right into the other.  

I would never recommend measuring with protractor though.  That's a recipe for "Wait, this triangle adds up to 183°, and this one to 179°!"

I guess I would worry about the "this isn't exactly a straight line!" thing even with tearing corners off. But then I wouldn't teach the sum of the angles of a triangle before a kid was comfortable with parallel lines 😉 . In my experience, parallel lines actually come relatively easily to kids, so waiting for that makes sense to me. 

[lewelma]

When I get questions like this, I say "Great question!  Let's investigate."  Then we play around with how something might work. We go completely off the script of any curriculum. We get rulers, blocks, protractors, measuring cups, or just old fashioned pencil and paper. By using  the word 'investigate' at least once a week over many years, my students and my kids have come to understand that mathematics can be 1) fun, 2) more than memorizing what the book says, 3) an opportunity to explore a question and find an answer through many methods.  For my students who are in school, this has been a shocking revelation.

[Not_a_Number]

8 minutes ago, lewelma said:

For my students who are in school, this has been a shocking revelation.

Yeah, I believe it. And isn’t that sad???

[EKS]

On 2/4/2021 at 7:46 AM, Kezia said:

Then he went off track. He insists that if you can break a quadrilateral with 4 angles consisting of 360° into two triangles  and put them together into a big triangle then that triangle must have 360°.  

Have him actually do it. 

What happens when you break the quadrilateral into two triangles?  Try it with a square and you'll see that the right angles are cut into 45 degree angles.  Now put those triangles together to form a big triangle.  What happens to the angles?  Two of them add together to become a 90 degree angle, but the other two stay at 45 degrees. 

[lewelma]

2 hours ago, Not_a_Number said:

Yeah, I believe it. And isn’t that sad???

I've just started with a new calculus student who is very far behind with lots and lots of gaps.  We were working on factoring and quadratic equations. And I asked him why he switch the sign at the end. He had no idea. "No one ever told me why, they just said to do it." This appears to be true with every question I ask him. 😞  Lucky for me, he is an incredibly fast learning and has just been very very badly taught over many years. 

[Not_a_Number]

27 minutes ago, lewelma said:

I've just started with a new calculus student who is very far behind with lots and lots of gaps.  We were working on factoring and quadratic equations. And I asked him why he switch the sign at the end. He had no idea. "No one ever told me why, they just said to do it." This appears to be true with every question I ask him. 😞  Lucky for me, he is an incredibly fast learning and has just been very very badly taught over many years. 

Oooh, poor kid! I'm glad he has a good tutor now and has a chance to catch up 🙂 . 

[Caroline]

9 hours ago, Monica_in_Switzerland said:

 

I agree with this, and doing the same for quadrilaterals.  I would suggest tearing the angles off though, or else highlight the angles before cutting up the shape, otherwise it's easy to get lost on which ones were the actual interior angles of the shape and which were formed when you cut/tore it apart!  (personal experience talking...)

 

 

Yes, you’re right about tearing or highlighting.  Been there, done that! 

[Kezia]

On 2/6/2021 at 5:19 PM, Caroline said:

One thing I like to do is cut out a bunch of triangles. Then cut the vertices off the triangle and physically add the angles to get her to show you get a line, or 180 degrees. Don’t use a protractor. Literally cut the angles off the triangles and add them together. 

Beast Academy suggested that as well and we did experiment with tearing off the angles. He could see they added to 180°. 
 

I printed off two identical triangles (so he could cut one and still have an original), told him to measure them, color them different colors, tear angles off again, gave him a blank paper to draw different triangles on (he drew them freehanded and had a hard time with one being 183°, I told him his line was crooked, he should have used straight edge), gave him a page of various quadrilaterals but we did this at the end of his math time, he tried for a bit and was just done. We moved on to other concepts after we came to this stall. It appears to be only the triangles that can be cut and moved to make a parallelogram that bothers him. I will have him try again one day this week when he is fresh. Maybe have him teach me all about triangles...
 

I may have to just give it time and revisit this concept again in a few months. I have LoF liver and mineshaft (don’t remember seeing geometry concepts in these two, but maybe). It looks like BA won’t come back to it.  

[Not_a_Number]

1 minute ago, Kezia said:

I will have him try again one day this week when he is fresh. Maybe have him teach me all about triangles...

I think that's a good idea. Try to get him to demonstrate this new idea he discovered 😄 . And when he sees what's wrong, have him find where the "extra" angles went, so he doesn't feel silly. 

Does BA show the proof that the sum of the angles is 180 degrees? 

[Kezia]

1 minute ago, Not_a_Number said:

I think that's a good idea. Try to get him to demonstrate this new idea he discovered 😄 . And when he sees what's wrong, have him find where the "extra" angles went, so he doesn't feel silly. 

Does BA show the proof that the sum of the angles is 180 degrees? 

They had him add several. Called it to their attention “Have you noticed this?” And did the same thing with 4 sided figures. What have you noticed here? I really think his confusion comes from the disappearing 90° angles and that overall that big triangle has 3 angles. 
 

This kid was bugging me in kindergarten (public school) every day for months on end about the numbers. “When do the numbers end???!!! When can I finally stop counting! How high do I have to count before I get to the end?”

”The numbers don’t just end.” We eventually looked up googolplex. He was fascinated. 

 

 

[Not_a_Number]

12 minutes ago, Kezia said:

They had him add several. Called it to their attention “Have you noticed this?” And did the same thing with 4 sided figures. What have you noticed here? I really think his confusion comes from the disappearing 90° angles and that overall that big triangle has 3 angles. 
 

You could try showing him the parallel lines proof! 😄 Does he know anything about parallel lines? If you introduced that, it could convince him. But if he doesn't know how that works, that's too much at once.