Contributor
Based on the reputation of the author and his work, I'd say it's more likely that his explanation is weak, and his intention was to describe the procedure which actually works (which, if you're familiar with compass and straightedge construction, is a simple as:
To construct a line at 45o from a point O on a line L:
1. Construct a perpendicular to L at O, by the usual method.
2. Bisect the resulting right angle, by the usual method.
Contributor
I actually liked geometry. Much better than trig or calculus but the geometry class was in 1965. I use the method Jim explained in post #14
[OP]
Contributor
This is something that bugs me about the book. Many things are left unexplained, it sometimes isn't clear what the author(s) are presenting.Originally Posted by Steve Demuth
I am one who likes to know how or why a process works.
I've always disliked it when shown, especially at work, how to do something and then find out it all came out wrong because a few of the steps or needed procedures where not covered in the training.
jtk
"A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty."
- Sir Winston Churchill (1874-1965)
Contributor
Going back to the OP.... "A simple way to draw a 45 deg angle..."
45 deg.jpg
"What you see and what you hear depends a great deal on where you are standing.
It also depends on what sort of person you are.”
[OP]
Contributor
Yes, but sometimes it is quicker to use a compass and straight edge, especially in a shop where one might have to search for a drafting triangle or one might not have a triangle.
jtk
"A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty."
- Sir Winston Churchill (1874-1965)
Contributor
Ummm.... it was a joke....see the emoji?
"What you see and what you hear depends a great deal on where you are standing.
It also depends on what sort of person you are.”
[OP]
Contributor
Oh, sorry, I used to be a drafter many years ago. Have a drawing board and some triangles that fit into dog holes on my bench to support it Even one of those swing arm drafting set ups:
Drafting Arm.jpg
jtk
"A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty."
- Sir Winston Churchill (1874-1965)
Contributor
Many people, in fact, the majority of people that I know, think that if a person is very, very good at something, a master at it (a sport, math, writing, playing an instrument cooking, woodworking, jewelry making etc) the person is able to teach someone how to do it, play it , write it, make it, etc. That is almost never the case.
Most people cannot teach something "from square one", no matter how well they know the subject or how skilled they are at an activity.
If the student already knows something of the subject, or has some skill at an activity (sport, playing an instrument) then the teacher will probably have some success.
But most are unable to teach someone who is starting out with a "blank slate".
They invariably assume the student knows things that he in fact doesn't know, and so skip steps, or give overly general instructions or explanations.
The student gets frustrated because he knows he is smart enough to learn the material. The teacher gets frustrated because the student isn't learning.
The teacher thinks the student is stupid and gives up.
The fault lies with the teacher, not with the student. The teacher does not know how to meet the student where he is.
Last edited by Patty Hann; 12-14-2023 at 3:41 AM.
"What you see and what you hear depends a great deal on where you are standing.
It also depends on what sort of person you are.”
[OP]
Contributor
One of the best instructors (teachers) in my life was at a training school of my employer at the time. He taught students to work on, repair & trouble shoot blue printing machines and some other equipment.
He would almost never pick up a tool to demonstrate. He would have the students perform each task.
The one time I did see him doing the actual work was when there were two teams of students, two to a team. The project was to dismantle a machine and then put it back together. My partner and I were taking our time and analyzing how all the parts interacted. The other team was treating it as a race. When we came back from lunch my partner and I noticed the other team, in their haste, had installed some parts incorrectly. They messed up their machine. That was when Tony, the instructor, picked up some tools and tried to repair their mess. On my next visit to the center, instead of two of this machine in the classroom, there was only one.
Sometimes when one makes a race of a learning experience, you loose the race and don't learn as much.
jtk
"A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty."
- Sir Winston Churchill (1874-1965)
Contributor
There are many videos showing how to get a 45* using only a straight edge and a compass.
None of them use different size circles.
"Life is what happens to you while you're busy making other plans." - John Lennon
Contributor
Most that I've looked at does exactly what Jim McCue and I are suggesting is the necessary approach: 1) construct a perpendicular; 2) bisect the resulting right angle. There is a neat way to do that using a single compass setting that constructs the guide point for the perpendicular making use of the hexagonal division properties of a circle that Jim Koepke remarked upon as being relevant, but it's really just a variant of drawing arcs at a longer distance than your initial radius choice when constructing the perpendicular. There is another method, that does involve using two separate arc lengths, in which you construct two legs of a square directly, and then draw the diagonal. But again, that means first constructing a perpendicular.
The theory of what you can and can't construct with a straightedge and compass is fascinating, and played an important role in the development of a full understanding of real numbers. It would tax my mathematics enormously to recall that - those classes were close to 50 years ago now for me - but it's fun stuff.
Contributor
In college I started as a math major, honors program, top 1%. I was on a work-study program, helping professors in the math department.
My sophomore year, as I was sitting in a cubicle sorting punch. cards I questioned if this is what I wanted to do for a living.
I was boarding at my sister’s and brother-in-law’s house to save money, helping out by babysitting their two boys. I really enjoyed that.
Next quarter I switched major to elementary education.
I sat down with my advisor and mapped out my curriculum.
I balked when I saw “eledmath 101 Addition and Subtraction of Cardinal Numbers “
“I’ve already covered integrals, differentials, polar geometry, Boolean logic, discrete math, topology…. You want me to do adding and subtracting?!”
”Knowing math and teaching math are two different things.”
He was right. I was two years behind the curve wrt to child psychology, child cognitive development, error analysis and stuff.
My math background helped a lot in error analysis, but it took a while to put everything all together to effectively teach stuff to kids.
Comments made here are my own and, according to my children, do not reflect the opinions of any other person... anywhere, anytime.
Contributor
Yep. I'm always fascinated by parents who try to help their grade school kid(s) understand fractions or long division.
9 out 10 parents have no clue as to how to explain these things to a kid who is struggling.
ANd because I had a "math block" in algebra and trig, and was helped past it (while in the Navy) by a Japanese-Am. engineer who tutored evening math classes I gained insight into "blocks" and how to deal with them.
I have successfully tutored "hopeless" kids and adults in math (up to Calc I) in high school physics and College level Latin.
"What you see and what you hear depends a great deal on where you are standing.
It also depends on what sort of person you are.”
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