Good Proof Problems for a High School Student?
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High school student here...
So recently, I have accomplished what I am considering the most exciting breakthrough of my mathematical journey so far...
I WROTE A PROOF!
I proved that the sum of any two consecutive odd numbers is always a multiple of 4
I posted my proof here and received feedback on how to improve it and where I made logic errors. I've also managed to prove that the sum of two consecutive odd numbers is always even as well (a lot easier once you get the hang of it). At this point, I don't want to stop and feel a need to prove something else - hence why I'm posting this at nearly four in the morning. Does anyone have any good ideas of something a high school student would understand and be able to prove? I tried to look for some problems online, but I keep coming across the same things over and over again.
proof-writing advice learning
asked Sep 10 at 7:51
CaptainAmerica16
350112
 |Â
show 2 more comments
up vote
1
down vote
favorite
High school student here...
So recently, I have accomplished what I am considering the most exciting breakthrough of my mathematical journey so far...
I WROTE A PROOF!
I proved that the sum of any two consecutive odd numbers is always a multiple of 4
I posted my proof here and received feedback on how to improve it and where I made logic errors. I've also managed to prove that the sum of two consecutive odd numbers is always even as well (a lot easier once you get the hang of it). At this point, I don't want to stop and feel a need to prove something else - hence why I'm posting this at nearly four in the morning. Does anyone have any good ideas of something a high school student would understand and be able to prove? I tried to look for some problems online, but I keep coming across the same things over and over again.
proof-writing advice learning
asked Sep 10 at 7:51
CaptainAmerica16
350112
1
A classical example of a first proof that one does in first year university, that I think is perhaps challenging enough for a high school student, would be to prove that $sqrt2$ is irrational, i.e. cannot be written as a/b for two integers a,b.
– Christopher.L
Sep 10 at 8:00
@Christopher.L Starting a google doc for it right now!
– CaptainAmerica16
Sep 10 at 8:01
1
Geometry is a marvelous field for practicing writing proofs. There are so many things you can show about circles and triangles.
– Arthur
Sep 10 at 8:06
@Arthur That sounds fun, I want to do something more abstract than the two-column proofs I had to do in school though.
– CaptainAmerica16
Sep 10 at 8:11
1
@CaptainAmerica16 Two-column is a good technique though. It is a good way to keep track of what you know and what you need for the times when the path forward isn't clear and you're just trying things. It is not so mych a type of proof (like induction, or contradiction), but more of a mental help on the go. As such, it really shouldn't be visible in the final proof, but rather in your own notes. But, just like studying techniques, school some times goes overboard on the how without really helping students understand why.
– Arthur
Sep 10 at 8:17
 |Â
show 2 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
High school student here...
So recently, I have accomplished what I am considering the most exciting breakthrough of my mathematical journey so far...
I WROTE A PROOF!
I proved that the sum of any two consecutive odd numbers is always a multiple of 4
I posted my proof here and received feedback on how to improve it and where I made logic errors. I've also managed to prove that the sum of two consecutive odd numbers is always even as well (a lot easier once you get the hang of it). At this point, I don't want to stop and feel a need to prove something else - hence why I'm posting this at nearly four in the morning. Does anyone have any good ideas of something a high school student would understand and be able to prove? I tried to look for some problems online, but I keep coming across the same things over and over again.
proof-writing advice learning
asked Sep 10 at 7:51
CaptainAmerica16
350112
High school student here...
So recently, I have accomplished what I am considering the most exciting breakthrough of my mathematical journey so far...
I WROTE A PROOF!
I proved that the sum of any two consecutive odd numbers is always a multiple of 4
I posted my proof here and received feedback on how to improve it and where I made logic errors. I've also managed to prove that the sum of two consecutive odd numbers is always even as well (a lot easier once you get the hang of it). At this point, I don't want to stop and feel a need to prove something else - hence why I'm posting this at nearly four in the morning. Does anyone have any good ideas of something a high school student would understand and be able to prove? I tried to look for some problems online, but I keep coming across the same things over and over again.
proof-writing advice learning
proof-writing advice learning
asked Sep 10 at 7:51
CaptainAmerica16
350112
asked Sep 10 at 7:51
CaptainAmerica16
350112
asked Sep 10 at 7:51
CaptainAmerica16
350112
asked Sep 10 at 7:51
CaptainAmerica16
350112
asked Sep 10 at 7:51
CaptainAmerica16
350112
350112
1
A classical example of a first proof that one does in first year university, that I think is perhaps challenging enough for a high school student, would be to prove that $sqrt2$ is irrational, i.e. cannot be written as a/b for two integers a,b.
– Christopher.L
Sep 10 at 8:00
@Christopher.L Starting a google doc for it right now!
– CaptainAmerica16
Sep 10 at 8:01
1
Geometry is a marvelous field for practicing writing proofs. There are so many things you can show about circles and triangles.
– Arthur
Sep 10 at 8:06
@Arthur That sounds fun, I want to do something more abstract than the two-column proofs I had to do in school though.
– CaptainAmerica16
Sep 10 at 8:11
1
@CaptainAmerica16 Two-column is a good technique though. It is a good way to keep track of what you know and what you need for the times when the path forward isn't clear and you're just trying things. It is not so mych a type of proof (like induction, or contradiction), but more of a mental help on the go. As such, it really shouldn't be visible in the final proof, but rather in your own notes. But, just like studying techniques, school some times goes overboard on the how without really helping students understand why.
– Arthur
Sep 10 at 8:17
 |Â
show 2 more comments
1
A classical example of a first proof that one does in first year university, that I think is perhaps challenging enough for a high school student, would be to prove that $sqrt2$ is irrational, i.e. cannot be written as a/b for two integers a,b.
– Christopher.L
Sep 10 at 8:00
@Christopher.L Starting a google doc for it right now!
– CaptainAmerica16
Sep 10 at 8:01
1
Geometry is a marvelous field for practicing writing proofs. There are so many things you can show about circles and triangles.
– Arthur
Sep 10 at 8:06
@Arthur That sounds fun, I want to do something more abstract than the two-column proofs I had to do in school though.
– CaptainAmerica16
Sep 10 at 8:11
1
@CaptainAmerica16 Two-column is a good technique though. It is a good way to keep track of what you know and what you need for the times when the path forward isn't clear and you're just trying things. It is not so mych a type of proof (like induction, or contradiction), but more of a mental help on the go. As such, it really shouldn't be visible in the final proof, but rather in your own notes. But, just like studying techniques, school some times goes overboard on the how without really helping students understand why.
– Arthur
Sep 10 at 8:17
1
1
A classical example of a first proof that one does in first year university, that I think is perhaps challenging enough for a high school student, would be to prove that $sqrt2$ is irrational, i.e. cannot be written as a/b for two integers a,b.
– Christopher.L
Sep 10 at 8:00
A classical example of a first proof that one does in first year university, that I think is perhaps challenging enough for a high school student, would be to prove that $sqrt2$ is irrational, i.e. cannot be written as a/b for two integers a,b.
– Christopher.L
Sep 10 at 8:00
@Christopher.L Starting a google doc for it right now!
– CaptainAmerica16
Sep 10 at 8:01
@Christopher.L Starting a google doc for it right now!
– CaptainAmerica16
Sep 10 at 8:01
1
1
Geometry is a marvelous field for practicing writing proofs. There are so many things you can show about circles and triangles.
– Arthur
Sep 10 at 8:06
Geometry is a marvelous field for practicing writing proofs. There are so many things you can show about circles and triangles.
– Arthur
Sep 10 at 8:06
@Arthur That sounds fun, I want to do something more abstract than the two-column proofs I had to do in school though.
– CaptainAmerica16
Sep 10 at 8:11
@Arthur That sounds fun, I want to do something more abstract than the two-column proofs I had to do in school though.
– CaptainAmerica16
Sep 10 at 8:11
1
1
@CaptainAmerica16 Two-column is a good technique though. It is a good way to keep track of what you know and what you need for the times when the path forward isn't clear and you're just trying things. It is not so mych a type of proof (like induction, or contradiction), but more of a mental help on the go. As such, it really shouldn't be visible in the final proof, but rather in your own notes. But, just like studying techniques, school some times goes overboard on the how without really helping students understand why.
– Arthur
Sep 10 at 8:17
@CaptainAmerica16 Two-column is a good technique though. It is a good way to keep track of what you know and what you need for the times when the path forward isn't clear and you're just trying things. It is not so mych a type of proof (like induction, or contradiction), but more of a mental help on the go. As such, it really shouldn't be visible in the final proof, but rather in your own notes. But, just like studying techniques, school some times goes overboard on the how without really helping students understand why.
– Arthur
Sep 10 at 8:17
 |Â
show 2 more comments
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1
A classical example of a first proof that one does in first year university, that I think is perhaps challenging enough for a high school student, would be to prove that $sqrt2$ is irrational, i.e. cannot be written as a/b for two integers a,b.
– Christopher.L
Sep 10 at 8:00
@Christopher.L Starting a google doc for it right now!
– CaptainAmerica16
Sep 10 at 8:01
1
Geometry is a marvelous field for practicing writing proofs. There are so many things you can show about circles and triangles.
– Arthur
Sep 10 at 8:06
@Arthur That sounds fun, I want to do something more abstract than the two-column proofs I had to do in school though.
– CaptainAmerica16
Sep 10 at 8:11
1
@CaptainAmerica16 Two-column is a good technique though. It is a good way to keep track of what you know and what you need for the times when the path forward isn't clear and you're just trying things. It is not so mych a type of proof (like induction, or contradiction), but more of a mental help on the go. As such, it really shouldn't be visible in the final proof, but rather in your own notes. But, just like studying techniques, school some times goes overboard on the how without really helping students understand why.
– Arthur
Sep 10 at 8:17