Posts Tagged ‘Significant Digits’
This is not a new topic for me, it’s been a burr in my saddle for some time now. All of the introductory physics textbooks address significant figures in much the same way. The problem is – nobody in the “real world” uses sig figs. At the same time, introductory physics isn’t the time to introduce complex error analysis models.
I’m having this discussion with Andy Rundquist of Hamline University. I asked Andy how they handled this at the college level. He told me they don’t teach significant figures and pointed me to a very lengthy article discussing why significant figures are all wrong. The article suggests the use of Monte Carlo analysis its place. That may make sense on a lab, but not on classwork and homework problems. The uncertainty article did have a suggestion; use six significant figures for calculations and round the final answer to three sig figs. The article does a good job explaining the reasoning, and I’m fine with it. The three extra “guard digits” preserve the accuracy, and the rounding makes the answer more reasonable.
- I will project an archery target on the board.
- Students will move back about 20 feet and shoot a round of Nerf darts at the target. They will be far enough back that most of them will shoot a 6, or 7 and not a 9 or 10, at least at first. Each student will take a turn.
- We will plot the overall results. We should get something resembling a normal distribution curve, but I won’t tell them that.
- I will ask the kids to average the data and come up with a value of x.x +/- y.y and start a discussion on whether or not that represents the data.
- We will then put a ring or other object on an electronic scale and write the mass with the error in the same way.
- After some discussion, I will bring up slides of normal, rectangular, triangular, and maybe exponential distribution curves. I want them to discuss the fit of the models to the data.
- My goal is that they understand that error is probability.
- About a week later we will drop rulers and calculate individual reaction times. This would be a good time to bring back the distribution graphs and perhaps even input our data into a statistical analysis program to find the best fit.
I think this will work and go over well. I’d love some feedback. It’s a first pass, what did I miss?
Teachers – how are you handling significant figures? I’m a bit at odds with my textbook and I’m wondering what the rest of the world is doing. I’ve discussed my issues with our chemistry teacher, he tends to agree with me, but it’s just the two of us. Let me explain.
I teach from Holt Physics. The book treats sig figs mostly okay. When they provide numbers for problems, they are always precise. Usually the numbers are in scientific notation, so you know where you stand with your given information. The book does state that 1500 could be 2 or 3 or 4 significant figures because we don’t know about the two trailing zeroes. I tell the students to err towards caution in those cases and treat that number as though there are 4 sig figs. The book correctly states that the answers are rounded to the least significant number of figures. You all know what I mean.
Here is where we part ways. I teach my students to carry an extra place while doing calculations. For example, if I’m dividing 35 by 62, my working answer is 0.565. If this is my answer to the problem, I would round this to 0.57. If I’m using this number in another calculation, I would use all three digits. The textbook rounds this here along the way even when it is used later. I’ve even seen problems where they have rounded more than once in the same problem. (There is no way I can remember the actual problem right now.) The results are often an error of about 10% difference between my answer key and my calculations.
While we are at this, I have a question I’m stuck on. Suppose you read a meter stick and you get a reading of 8.65 cm. That is three significant figures. Now you move a little ways up the ruler and read 22.40 cm. The accuracy of the ruler hasn’t changed, but I’m now working with 4 significant figure versus 3 before. I wouldn’t round the second number, it is as accurate as the device, but the first number isn’t 8.600. How do you account for this when you are dealing with the significant figures of a problem?
If you are new to teaching Physics, you are probably going to expect the students to know how to manipulate numbers and variables. I’ve found that to be a big mistake. Yes, some of my students are very good with math, but I’ve learned not to assume this. As a result, my first two weeks of class are spent working on the skills they are going to use the entire year. Those skills are:
- Scientific Notation
- Significant Digits
- Unit Conversions/Dimensional Analysis
- Solving for variables
- Use of their TI-83 calculators
This is a big post. I have included all of the introductory worksheets, homeworks, and quizzes from the past school year. A lot of example problems I just make up on the fly, but I have attached quite a few word documents. Feel free to use them and adapt them as needed.
Last year I reviewed estimation first. That was fine until we started dealing with very small numbers, then the students were lost. This year, scientific notation comes first. I have a three part handout that I go over in class. I make up additional problems as we go, then I hand out the worksheet. They start it in class and are expected to complete it for homework.
The point of estimation is for them to be able to get order-of-magnitude answers quickly. At first, they are completely amazed that I can get within 5% or 10% of the answer faster than they can do it on their calculator. Some of them get good at this, most don’t bother even though they see that I will estimate quickly and accurately throughout the entire year. My goal is for them to realize when their answer on the calculator can be wrong and their brain can be right. It does happen and it feels good when one of them comes over to the dark side with me.
The first document is the handout they get with the guidelines. I think it is rather clear, but I think I want to change the example problem near the end with the sin( ) function. They don’t know sin( ) yet and it distracts them from the operation. I expect them to use these guidelines all year long.
Unit Conversion/Dimensional Analysis
Wow are they bad at this. Most of them almost randomly multiply or divide by the conversion to get an answer but never seem to know if it’s right or what the units should be. This is a real problem. The textbooks like to throw different units into the problems. It’s actually easy to miss, especially when the kids are struggling to learn a new concept. I’ve also found that very few of them are comfortable with the metric system. I feel like I beat this section to death, but still many cannot convert correctly.
Solving for Variables
I don’t have a worksheet made up for this section. What I have done in the past is gone to an Algebra 1 textbook and copied the problems from there. I gave up on this section last year, they just didn’t have the math skills to do this for every problem, and I had to choose my battles. If they plugged numbers into equations, most of them were able to solve for the unknown.
Here is the problem; the students are unaware of the key labeled EE that does the scientific notation. They also constantly miss the exponent that is shown as 1.2E33. Instead, they enter the number as 1.2 X 10 y^x 33. This works fine if all they are doing is multiplication. However, if they are dividing by the exponent, the 10 y^x 33 becomes part of the numerator. This totally screws up their answers and they have no idea why. There are two solutions: 1) They can use the EE function and only enter exponents in that form, or 2) they can use their longer form and put each term in parenthesis. Obviously, I prefer the first solution and I push them to use it.
I also have them solve simple sine and cosine problems on their calculator. Not all of them use the TI-83, I don’t require it, I only require a scientific calculator. It’s up to them to know how to use it, that is why I spend time in the beginning going over the functions we will use in the course.
I give them the following quiz after all of this is completed.
If you have additional material, please share it. Trudi, I hope this helps.