Jump to content

Menu

I don't understand proofs!


Recommended Posts

I just started proofs yesterday and I have no idea what I am doing! I have finally hit the brick wall, and I just can't go any further. I understand how a proof works and everything, but I have no clue how to find the reasons and statements. I am completely lost and I just don't know what to do. My mom is getting me a new geometry curriculum soon, but she doesn't know which one would be better for me. I have to learn how to do proofs, or I want to learn how to do them. But I don't know how to do them!!

 

What do I do, what do I do??:confused::confused:

Link to comment
Share on other sites

I can do nothing but commiserate, sorry:( I had to take Geometry 3 times in high school. Guess what? I NEVER understood how to do proofs. I think the teacher finally passed me out of pity:tongue_smilie: Had he not, I wouldn't have graduated despite being in all honors classes otherwise. I wish you much more success than I had!

Link to comment
Share on other sites

I'm no math expert, but when my dd got stuck on proofs we briefly used Geometry for Dummies. It gave a good summary on how to build an "old-fashioned" proof. I think she just used that one chapter to get the general idea, then returned to the curriculum we had chosen (Jacobs).

 

Good Luck!

Link to comment
Share on other sites

Maybe it would be helpful if you went, step by step, through a few example proofs and familiarized yourself with the format. Take a paper and pencil and work alongside, trying to understand the logic flowing from one step to the next. You should study other people's proofs thoroughly before attempting to prove something on your own.

 

The general idea is to look at the initial statement and isolate very precisely what conditions you have to work with. (For example, if the problem mentioned a trapezoid, then the only thing you could use as a given fact would be that there are two parallel sides - you would not be able to assume anything about the angles, or about the lengths of these sides etc..)

then you try to construct a logical argument that starts by using this one fact you know and concluding things that depend on this knowledge only, and so on.

 

My DD dislikes proofs, too, but she loves working geometry problems where you just have to figure out something.

For example: this morning, she loved a problem where she had to find the diagonal in an isosceles trapezoid, but hated proving that the two diagonals have the same lengths. It helped, however, for her proof to look at the solution to the problem and to see what steps she needed to solve a specific example.

So, based on this observation, it may also be a good idea for you to begin by working a lot of geometry problems where they do not ask you to prove a relationship, but that simply let you discover and use the relationship. this will give you a better feel for geometry , and it will then be less intimidating to prove things.

Link to comment
Share on other sites

Maybe it would be helpful if you went, step by step, through a few example proofs and familiarized yourself with the format. Take a paper and pencil and work alongside, trying to understand the logic flowing from one step to the next. You should study other people's proofs thoroughly before attempting to prove something on your own.

 

The general idea is to look at the initial statement and isolate very precisely what conditions you have to work with. (For example, if the problem mentioned a trapezoid, then the only thing you could use as a given fact would be that there are two parallel sides - you would not be able to assume anything about the angles, or about the lengths of these sides etc..)

then you try to construct a logical argument that starts by using this one fact you know and concluding things that depend on this knowledge only, and so on.

 

My DD dislikes proofs, too, but she loves working geometry problems where you just have to figure out something.

For example: this morning, she loved a problem where she had to find the diagonal in an isosceles trapezoid, but hated proving that the two diagonals have the same lengths. It helped, however, for her proof to look at the solution to the problem and to see what steps she needed to solve a specific example.

So, based on this observation, it may also be a good idea for you to begin by working a lot of geometry problems where they do not ask you to prove a relationship, but that simply let you discover and use the relationship. this will give you a better feel for geometry , and it will then be less intimidating to prove things.

 

I am like you daughter, I don't like proofs! Right now I am on basic proofs and then it starts moving on to harder proofs. Do you think I should skip the proof chapter for now and continue working other people's proofs until I get it? What I really don't understand is the order you have to put the reasons and statements in and how you find them. Do you happen to have any links I could use to study proofs?

 

Thank you!

Link to comment
Share on other sites

Hi!

Regentude gave you some very helpful advice, and hopefully this will help you as well. Start with what you know about a certain type of figure. What information are you given? Where do you want to end up? Think of it as baby steps, or maybe as a flow chart of logic. Start looking for relationships and things you can say, even if they don't turn out to be what you need to say - you never know what bit of information you can use! Another good way to go about things is to work backwards, Where do you want to end up? What do you need to know to get there? Don't worry about doing proofs in the fewest number of steps right now. You can always clean them up later if you find that you have rabbit trails that are not needed. Think of your proofs as a journey of logical discovery - be curious. Also, finding parallel lines and congruent triangles often comes in extremely handy! Good luck! Also, try googling how to do proofs - you will likely come up with some video clips that may be helpful to you.

Blessings,

April

Link to comment
Share on other sites

Proofs are great! They help you to think about one of the most fundamental questions in life, "How do we know what we know?"

 

I'll bet you do a lot of algebra problems that have multiple steps. A proof is like that, except that it has two columns. One column is the 'therefore' column. It's like the algebra problem--first you do one thing, and then you do another. The right hand column is the 'how do we know this?' column.

 

For instance, suppose you wanted to prove that an supplemental exterior angle of a certain angle in a triangle has a measure that equals the measure of the two opposing interior angles of the triangle. It's easy to see that this is true, because the sum of the angles of the triangle is 180, and the sum of the angles that comprise a straight line is also 180, so if you subtract out the measure of the 'certain angle' from each, you get the same number. In a proof, you use techniques and given material to show that this is always true. You have say where you get the information in the right hand column.

 

So you might say:

 

The sum of all angles in a triangle is 180

 

Then the right hand column would cite the reference for that.

 

Then you say, a straight line comprises 180

 

Then the right hand column would cite the reference for that as well.

 

See what I mean? The left column is the information that you're using to solve the problem, in logical order. The right column is the footnotes or laws or givens--it tells where you got the information that you're asserting on the left.

 

So all the time you probably spent learning the names of properties in algebra finally pays off--those associative, distributive, and identity properties, for instance, turn into reasons in the right column for operations that you perform on the left.

Link to comment
Share on other sites

Proofs are great! They help you to think about one of the most fundamental questions in life, "How do we know what we know?"

 

I'll bet you do a lot of algebra problems that have multiple steps. A proof is like that, except that it has two columns. One column is the 'therefore' column. It's like the algebra problem--first you do one thing, and then you do another. The right hand column is the 'how do we know this?' column.

 

For instance, suppose you wanted to prove that an supplemental exterior angle of a certain angle in a triangle has a measure that equals the measure of the two opposing interior angles of the triangle. It's easy to see that this is true, because the sum of the angles of the triangle is 180, and the sum of the angles that comprise a straight line is also 180, so if you subtract out the measure of the 'certain angle' from each, you get the same number. In a proof, you use techniques and given material to show that this is always true. You have say where you get the information in the right hand column.

 

So you might say:

 

The sum of all angles in a triangle is 180

 

Then the right hand column would cite the reference for that.

 

Then you say, a straight line comprises 180

 

Then the right hand column would cite the reference for that as well.

 

See what I mean? The left column is the information that you're using to solve the problem, in logical order. The right column is the footnotes or laws or givens--it tells where you got the information that you're asserting on the left.

 

So all the time you probably spent learning the names of properties in algebra finally pays off--those associative, distributive, and identity properties, for instance, turn into reasons in the right column for operations that you perform on the left.

 

Oh! Now I get what you are saying! Ok, thank you so much. But I just don't know how to get that information. It's easy for me to understand what you are saying because you already gave me the information, but I am lost when trying to get the information that I need for the columns. The book I have kind of jumped into proofs, and it's driving me crazy.

 

Would you mind telling me how to get this information? Could you give me another example? Thank you!

Link to comment
Share on other sites

Does your book have blue (or green or whatever) boxes with the theorums and axioms in them? In general, the things you can use are the things given in the problem itself and the theorums and axioms in the blue boxes. If you go through the chapter and make a list of the theorums and axioms in the boxes, then you will have a list you can refer to when you are looking for ways to get from one step to another step. The title of the box is usually what you put in the right hand column.

-Nan

Link to comment
Share on other sites

Oh! Now I get what you are saying! Ok, thank you so much. But I just don't know how to get that information. It's easy for me to understand what you are saying because you already gave me the information, but I am lost when trying to get the information that I need for the columns. The book I have kind of jumped into proofs, and it's driving me crazy.

 

Would you mind telling me how to get this information? Could you give me another example? Thank you!

 

You are welcome!

There are two main places to get the information.

One is the 'givens'. Those are the starting position outlined in the problem itself. So in my example, the given might be a diagram with a triangle with labelled angles A, B, and C, and a line extending one of the sides of the triangle beyond angle A to form an exterior angle D. So your first step would be to describe your 'given' starting conditions: 'Assume that angles A, B, and C are an interior angle in a triangle, and Angle D is angle A's supplemental exterior angle outside of and just beyond the triangle.' That would go on the left side, and on the right you would say, "Given" because this information was given to you at the start of the problem.

 

Another place to get information is your previous study. Somewhere you should have a list of rules and definititions that you can cite. I gave you some examples before--the associative, distributive, and identity properties. Other examples include definitions--like those for supplementary and complementary angles. So say you are continuing with that same problem above, and in the left hand column you say, 'The sum of the measures of angles A, B, and C is 180 degrees.' Then you would cite in the right hand column the rule that angles of a triangle add up to 180 degrees.

 

Again, this is more of a logic issue. Ask yourself, how do I know that I can take this step? Answer it in the right column.

Link to comment
Share on other sites

You are welcome!

There are two main places to get the information.

One is the 'givens'. Those are the starting position outlined in the problem itself. So in my example, the given might be a diagram with a triangle with labelled angles A, B, and C, and a line extending one of the sides of the triangle beyond angle A to form an exterior angle D. So your first step would be to describe your 'given' starting conditions: 'Assume that angles A, B, and C are an interior angle in a triangle, and Angle D is angle A's supplemental exterior angle outside of and just beyond the triangle.' That would go on the left side, and on the right you would say, "Given" because this information was given to you at the start of the problem.

 

Another place to get information is your previous study. Somewhere you should have a list of rules and definititions that you can cite. I gave you some examples before--the associative, distributive, and identity properties. Other examples include definitions--like those for supplementary and complementary angles. So say you are continuing with that same problem above, and in the left hand column you say, 'The sum of the measures of angles A, B, and C is 180 degrees.' Then you would cite in the right hand column the rule that angles of a triangle add up to 180 degrees.

 

Again, this is more of a logic issue. Ask yourself, how do I know that I can take this step? Answer it in the right column.

 

This is starting to make a lot more sense now thank you! I think I need to review some of the things that I have already gone over just to be sure I know what I am talking about.

 

Thank you very much for your help!!!:001_smile:

Link to comment
Share on other sites

April had a great suggestion. Just try working backwards. My dd is using the Chalkdust program and in yesterday's lesson Mr. Moseley actually talked about working backwards if you get stuck. For example, to prove 2 triangles are congruent you could use SAS, AAS, etc. See what information is already given and then determine what else you need. If you have an angle and side from 1 triangle congruent to an angle and side from the other triangle then you only need to prove either one more side or one more angle congruent. Does that make sense? I have trouble with proofs as well, but the working backwards thing really helps me a lot. HTH

Link to comment
Share on other sites

All of my Geometry texts have a section in the back of the book that lists Theorems, postulates, corollaries... by section.

 

When I taught Geometry in PS I ALWAYS photocopied these pages for the students to add to their Geometry folder... students also kept a list of definitions and notes from the lessons...

 

This week I covered sections 4.3-4.5 with my online Geometry classes. Students 'know' about the list in the back... and they 'should' have a folder. They are doing proofs involving triangles. In 4.3 it introduces ONE way to prove triangles are congruent (Definition of Congruent Triangles). In 4.4 it introduces SSS and SAS proofs. In 4.5 it introduces ASA, AAS and HL proofs.

 

When my students go to work their homework this week they will only be able to use 'rules' from 4.3 or BEFORE when working the proofs in assignment 4.3

 

When working their 4.4 homework they can use any 'rule' from 4.4 or BEFORE...

 

I have known PS Geometry teachers (and I sometimes) who allow students to use their notebooks on the tests-- they give quizzes to make sure students have 'learned' the rules--and the tests are proving that they understand the PROCESS (what I feel is the most important part).

Link to comment
Share on other sites

April had a great suggestion. Just try working backwards. My dd is using the Chalkdust program and in yesterday's lesson Mr. Moseley actually talked about working backwards if you get stuck. For example, to prove 2 triangles are congruent you could use SAS, AAS, etc. See what information is already given and then determine what else you need. If you have an angle and side from 1 triangle congruent to an angle and side from the other triangle then you only need to prove either one more side or one more angle congruent. Does that make sense? I have trouble with proofs as well, but the working backwards thing really helps me a lot. HTH

 

It does and it doesn't, I am doing SAS and SSS, and then I am going to move on with the rest. I just think I need to work backwards like you suggested, thank you!

 

All of my Geometry texts have a section in the back of the book that lists Theorems, postulates, corollaries... by section.

 

When I taught Geometry in PS I ALWAYS photocopied these pages for the students to add to their Geometry folder... students also kept a list of definitions and notes from the lessons...

 

This week I covered sections 4.3-4.5 with my online Geometry classes. Students 'know' about the list in the back... and they 'should' have a folder. They are doing proofs involving triangles. In 4.3 it introduces ONE way to prove triangles are congruent (Definition of Congruent Triangles). In 4.4 it introduces SSS and SAS proofs. In 4.5 it introduces ASA, AAS and HL proofs.

 

When my students go to work their homework this week they will only be able to use 'rules' from 4.3 or BEFORE when working the proofs in assignment 4.3

 

When working their 4.4 homework they can use any 'rule' from 4.4 or BEFORE...

 

I have known PS Geometry teachers (and I sometimes) who allow students to use their notebooks on the tests-- they give quizzes to make sure students have 'learned' the rules--and the tests are proving that they understand the PROCESS (what I feel is the most important part).

 

My book has all that you listed as well, but it's missing some information as well. And I see what you are saying, you have to use what you have already learned. Ok, I just need to review more then, I think I can do this. If I have more trouble I know who to ask!

 

Thank you everyone!

Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

 Share

×
×
  • Create New...