Help! My daughter is struggling with proofs in Geometry
Posted 08 October 2008 - 02:38 PM
Posted 08 October 2008 - 03:48 PM
You start with what you know already. You call this what you are 'given'.
So you might say, 'Given--an infinate set of non-negative numbers, each ending in the digits 0, 2, 4, 6, or 8.'
Then the assignment is what you have to prove: 'Prove--that these numbers are all multiples of 2'.
So you start with the given, and add other information that you know, step by step, until there is no other possible conclusion than the 'prove' part. Then you say, Therefore.............'[the prove part]' QED.
OR, you can do a negative proof. That is where you start by assuming that the 'prove' is NOT true, and step by step you show that that leads to absurd conclusions. Then you say, Therefore, 'the prove part' is true, by RAA.
(RAA is the initials for reduce to absurdity in Latin.) (QED is the initials for a Latin phrase as well, but I don't remember what it is.)
Posted 08 October 2008 - 03:56 PM
Quit and Eat Dinner.
(QED is the initials for a Latin phrase as well, but I don't remember what it is.)
Okay okay it's Quod Erat Demonstrandum (that which was demonstrated? something like that)
I think of proofs as using all the rules you already know to build a new one. So if you're proving that a regular (equilateral) triangle's interior angles add up to 180, you could start with:
1. all polygons' exterior angles add up to 360 (because you have to go full-circle to draw all the way around them)
2. An equilateral triangle has three identical vertices.
3. If the 360 degree sum of the exterior angles is divided into three, it's 120 per angle.
4. The interior angle at each vertex is supplementary to the exterior angle, so it has to measure 180-120.
5. 180-120 = 60
6. There are three of these 60 degree angles, and 3x60=180.
7. Thus, the sum of the angles of an equilateral triangle MUST be 180, which is what we were trying to prove (QED).
That's not the most useful proof, but it shows the steps.
Posted 08 October 2008 - 04:06 PM
Posted 08 October 2008 - 06:41 PM
Hope this helps!
Denise in NE
Posted 08 October 2008 - 08:10 PM
I have a degree in math. I love math. Always have. The only math class I hated was Geometry, because of the proofs. I also have a degree in computer science. I have worked as a programmer for 20+ years. Programming involves logic, but you would think that a programmer would do well with proofs. I dread reaching Geometry with ds.
IMO, it's okay if she muddles through.
Posted 09 October 2008 - 08:05 AM
Posted 09 October 2008 - 08:27 AM
What if she wrote it out more casually (but still step by step) and then went back and matched up the theorems?
The only type of proof my husband and I did in our schooling, was more of a casual either paragraph or step by step, where you just "talked" through it. TT shows the formal two column approach, where rather than state why something is true, you have to state the proof as being the theorem. Poor girl, she just gets so confused because she has to hold in her mind not only the steps, but also the theorem.
Posted 09 October 2008 - 09:11 AM
How does TT teach proofs? Is it your typical 2-column proof? My ds would never have had the courage to tackle these without working on filling in the blank spaces on completed proofs (we're using the older edition of Jacobs). Could you do that? Or have her view the solutions CD for the first ones to get the hang of it?
This is what my ds's Glencoe book does - once we worked through some of those, he seemed to get it better. She is going to have to memorize the terms as well.
Posted 09 October 2008 - 09:27 AM
in the beginning, they just leave some blanks which you fill in. She can manage to muddle her way through them, but when it gives the given facts and asks her to prove, she doesn't know where to begin, after she lists the given facts. The only type of proof my husband and I did in our schooling, was more of a casual either paragraph or step by step, where you just "talked" through it. TT shows the formal two column approach, where rather than state why something is true, you have to state the proof as being the theorem. Poor girl, she just gets so confused because she has to hold in her mind not only the steps, but also the theorem. She has made index cards with the theory, but then gets confused as to the actual meaning of some of them in a practical application. Thanks everyone for your suggestions. Sue in St. Pete...I'll pass your encouraging comments along to dd. The only problem is that TT incorporates proofs in every chapter, so she will never be done with it.
My dd is having the same problems. I am working through each proof with her helping her to see how to come up with the correct answers. If your dd watches each and every proof being solved, she will probably eventually be able to work through them. Plus, the proof in the lesson is usually very similar to the one worked in the practice problems. She can refer back to it when she is working on the lesson problem. Even if your dd does not get the hang of doing the proofs by the end of the book, she will have gained a lot of knowledge just by watching the process for every proof. It is really an exercise in logical thinking. Plus, by having to write out the terms and proofs, they are ingrained in the brain over time. I won't be overly concerned if my dd isn't able to do the proofs on her own by the end of the book. She will know all the terms and theorems by then!
Just a little funny, my dd insists the reason behind most of these mathematical things is "because she knows it in her soul". She can look at it and just tell. To be quite honest, that is how she has done math for her entire life. She just knows the answers many times. She is not a linear thinker. She learns and thinks in great jumps; proofs just go against her grain. They are, however, an excellent opportunity for her to learn to slow her brain down a little bit and try to think in a different manner.
Posted 09 October 2008 - 09:46 AM
PLEASE NOTE: There is NOT one correct answer for a proof. Some people take a more indirect route to get there, but that doesn't make it wrong. What makes it correct is if she starts with the given information, ends with what she is trying to prove, and each step along the way logically builds upon the steps before it. Also, note that there is NOT a universally accepted numbering or other delineation of Theorems and Corollaries, nor are they all worded exactly the same, nor are they presented in the same order. This can be problematic when using other sources to help with a tough concept.
You could point out to her that she has already been doing proofs -- she just didn't know it. When she did Algebra, and solved an equation for x, she was proving that x equals whatever. She was, essentially, doing the left-hand side of the proofs as she applied the principles of Algebra to manipulate the equation so that she could isolate x on one side. In formal proofs, she would have been listing those principles in the right column --- Distributive Property of Multiplication over Addition, Additive Inverse, etc.
It might help to not think of the problem as a proof at first. Turn it into a question. For example, rather than "Prove that the exterior angle of a triangle is equal to the sum of the remote interior angles" think "What is the measure of the exterior angle of a triangle?" (I'll assume here that she knows what an exterior angle is, that the sum of the angles in a triangle is 180, and that a linear pair sums to 180.) Then her thought process might be something like:
I know that the sum of the angles (a + b + c) in the triangle is 180 because that's the Triangle Sum Theorem.
I know that angle b and its exterior angle (d) are a linear pair because that's the definition of an exterior angle.
I know that (b + d) equals 180 because that's the definition of a linear pair.
If (a + b + c) = 180 and (b +d) = 180, they must be equal to each other (a + b + c = b +d) because of substitution.
Now I see that (a +c) = d because that's what's left when I subtract b from both sides. And that's what I was trying to prove -- the sum of the remote interior angles is equal to the exterior angle.
Sometimes it helps to work from both ends towards the middle. Look at what you're trying to prove, and think about what you need to know to make that statement.
Posted 09 October 2008 - 12:17 PM
So in a two column proof, you do the operation (in the right hand column) and then you say what theorem gave you the 'right' to assert that the operation is allowed/correct/doable (in the left column). So in an elegant proof, you don't just throw in all of the theorems at the start. Rather, you introduce the theorem when you do the step that requires it. And you introduce that theorem each time you do that kind of step.
Posted 09 October 2008 - 07:44 PM
Otherwise do as previous posters said and let her watch the CD's on how to solve the problems. Perhaps she could watch the Odd problems and try to do the Even problems. I used TT Algebra 2, and so many times I had my ds do either the odds or evens. However, I always had him watch every sample problem.