Tagged: significant figures

My Way to Make Measurement Make Sense

To me measurement is one of those topics that doesn’t seem like is should be a challenge.  BUT every year and every teacher I talk to finds that measurement is a HUGE PROBLEM with students.  I’ll let you form your own theories as to why this is the case.  This post is to describe the process I use to “reteach” students how to measure correctly, with the emphasis being on using a system of 10 divisions.

Nerd Rulers

I personally treat this activity as a lab, but it can be done any number of ways.  Here is the handout I use: How Many Nerds Lab.  The lab begins with me giving each student a ruler that looks like this:
They are asked, what do you notice about this ruler?  Things such as “no markings,” “a 0 and a 10,” “10 inches,” and many other things come up.  First thing I tell them is that this is NOT 10 inches or any other unit of measurement they have ever used.  So I have given the units a name: NERDS!  I tell them that since I’m kind of nerdy I wanted a unit of my own.

In reality the credit for this name and the rulers goes to Mr. Ryan Peterson of Brillion High School.  Any unit name you see fit can work.

From there I ask: “What is this ruler good for.”  The usual response is “not much.”  I usually have to hold something up and say “really, you can’t tell me anything about the length of this object?”  Someone usually realizes they can tell the object it either shorter or longer than 10 nerds.  “Ok, so what should I do to report the length of the object?”  “Guess” seems to be a popular response, but I quickly say guessing is random, like how many Jelly Beans are in a Jar.  Someone will finally say “Estimate.”

“Ok, now go and measure 8 things around the room.  Try to get at least 2 or 3 things that are longer than 10 nerds.”

I give them about 5 minutes to make their 8 measurements.  Then I hand out this ruler:
After a very brief discussion revolving around the fact that this ruler has the whole numbers marked, and that in science WE DO NOT USE FRACTIONS, I send students to remeasure the same 8 objects.

Finally comes the gold ruler:

They are off to remeasure the same 8 objects they have already measured.

Before I explain my discussion, here is how these rulers were created.  Again credit for the original Excel file goes to Ryan Peterson.

Nerds Rulers.  This Excel file contains 3 tabs, each containing 6 rulers on each tab.  Print from each tab.  I then copied enough sheets to make a class set of ruler on colored paper.  Finally I sent the pages through the laminater and cut out each ruler.  The set I made 5 years ago is still going strong.  The only real issue I’ve had is kids fold the blue ruler in half.

Back to the lab.

Once all measurements are made the students see a pattern of increased precision with each ruler (they don’t call it precision, but that’s what it is).  We quickly discuss things such as which ruler is best for measuring different objects, what makes a measurement “correct,” and how to handle measurements that we right on a marking.  The next thing is to have small groups discuss and whiteboard a rule or rules for using a ruler.

I give groups about 10 minutes to discuss some things and come up with what they think we should consider as “rules.”  Most groups come up with things like:

  • Start at zero
  • Measure the same object
  • Measure twice

I tell them that those are good and true and all, but how do make sure we are using a ruler correctly?  I sometimes have to give them the hint about what they did with the 3 different color rulers.  Most of the time that helps them realize that they ESTIMATED something in the answer.  Ultimately, after all groups have shared their rules I want the class to agree that we should:

  • ESTIMATE ON PLACE MORE THAN WHAT THE RULER TELLS YOU FOR SURE

This usually makes sense to the class, and they agree this the #1 rule.  PERFECT!

I extend the discussion with what that estimated number means.  I show them how the number they estimated is really reporting a range of numbers.  So for example a measurement of 3 is likes saying somewhere between 2 and 4.  A measurement of 3.6 is the range of 3.5 to 3.7.  Reporting 3.65 is something like 3.64 to 3.66 (this range might be a bit larger depending on the ruler).  Something I don’t stress at this point.

Metric Rulers

The next step is to introduce our more typical units of measurement: METERS!  Students know it is coming but they grumble and complain, and raise a fit over why we can’t just do it the “easy” way (inches, feet, etc.)  I then proceed to ask them something like; “so what increment comes after 7/32?”  Someone says 8/32.  “Ok, but I’m not going to find a wrench marked that.”   4/16, 2/8. oh 1/4.  “Congrats that just took you 1 minute to figure out.”  “What wrench comes after 7 mm?”  8mm.  “1 second, nice job, now which way is easier?”  I will then of course have the discussion about english vs. metric.

I usually like to get on my soapbox a little bit and complain about how “the US is the greatest country in the world, and by gosh we aren’t going to change our ways for anybody.”  I play that angle up a little bit, and we get to the point that metric is easier to work with, just not as common for us in America, but they hopefully see the point.

I now show the class a dowel that is marked as 1 meter long, but that is it.  Kind of like the blue ruler from above.  I then discuss and demonstrate how you can split a meter into 10 equal sections called decimeters, show them a ruler with only dm marks.  Each decimeter into 10 centimeters, show a cm ruler, and finally centimeters into 10 millimeters, show a typical meter stick with mm markings.

My lab hand out is here:  How Many Meters

Basically, I have a set of different metric rulers, and whatever the smallest markings are on the ruler is the “type” of ruler it is.  So a standard metric ruler is a mm ruler even though the numbers are cm.  This is just another complication in the process we have to deal with.  I have each student measure the same stuff with the same type of ruler so that we can compare results.

This is all for practice, and the whiteboard discussion is pretty minimal.  I usually focus on the different prefixes used in the SI system.  Why it is called SI and why it is important we have a standard unit of measurement for scientists.

Uncertainty of Measurement Activity

The questions I usually get about all this emphasis on measuring is “Why is this important?”  To me this activity nails it and relates why estimating measurements is so important as well as demonstrating exactly why significant figures work the way they do.

Uncertainty in Measurement

I distribute a index card and blue ruler to each student.   (something smaller than 10 nerds is a must, otherwise the uncertainty becomes too great.)  Each student measure the length and width of the card.  They then calculate the area, and I instruct them to write down exactly what the calculator says.

They remeasure with the pink ruler and again calculate and record area exactly as is displayed.  Finally they measure a third time with the gold ruler.

After all measurements and calculations are made I have each student list their calculated areas on the board.  One column for each ruler.  I forgot to take a picture of the entire set of class data, but it was something like this:

Notice how sweet this data is!  The discussion revolves around where the uncertainty in each calculation shows up.  For the Blue ruler it is in the tens place.  The Pink ruler the ones place, and the Gold ruler is in the tenths place.  Coincidence?!  I don’t think so.

Because of our “rule for using a ruler” the significant figure in each measurement comes out to be where our uncertainty shows up in our calculations!  Amazing!

Bottom line is the blue ruler gives us calculations that are uncertain plus or minus 10, thus we can only report an answer that is rounded in the tens place, which just so happens to coincide with 1 sig fig!

The pink ruler is uncertain plus or minus 1, thus rounded to the 1s place; 2 sig figs!

The gold ruler is uncertain plus or minus 0.1, thus rounded to the tenths place; 3 sig figs!

I think it is really cool how nicely the data comes out for students to “see” significant figures really work out.  In all actuality I not huge on make significant figures a huge issue as the year goes on, but I tell students, you better never ask me: “Where do I round?”  That is one of the biggest pet peeves I have and when the do I’m going to tell them to look at sig figs.

WOW!  That is a lot of information in one post.  Please use it for your benefit, and as always, if you have any questions or comments please leave them below.

HAPPY MEASURING!