1. Scaling, dimension, & estimation

2. A note on usage:
The clicker slides in this booklet are meant to be used as stimuli to encourage class discussion. They are intended for use in a class that attempts to help students develop a coherent and sophisticated understanding of scientific thinking. They are NOT intended as items to test whether students are “right or wrong” or “know” the correct answer by one-step recall if enough cues are given. This has a number of instructional implications that are reviewed in general on the next four slides. The individual slides also contain annotations discussing their intended use.

3. Usage: 1
Feedback
One of the most important values of a clicker-response system is to provide instructors with some understanding of what students are thinking.
Good clicker questions can be highly revealing (and surprising). But the critical fact is not that the students make mistakes but to use those mistakes to probe their thinking and find out why.
This raises the importance of a rich subsequent discussion well above “letting the students know what the right answer is.”

4. Usage 2: Student-student interactions
The critical value for student learning occurs in what happens after a clicker question has obtained a mixed response from the students. The standard next cue is, “Find someone who disagreed with the answer you chose and see if you can convince them.”
After a minute or two of discussion, a second click may show students having moved dramatically towards the correct answer. A brief call for who changed their answer and why can lead to a useful exchange. When they have not moved significantly, more discussion is called for.

5. Usage: 3 Incompletely specified questions
Some items have questions that are simple if idealized assumptions are made, subtler if they are not. Part of the discussion of these items are intended to include issues of modeling, idealizations, and hidden assumptions.
Questions where answers are not provided.
In these items, the intent is to have students come up with potential answers and have the instructor collect them and write them on the board.
Occasionally, especially at the beginning of a class, it may take some time before students are willing to contribute answers. It can help if you have some prepared answers ready, walk around the class, and put up the answers as if they came from the students. This can help students get more comfortable with contributing.

6. Usage: 4 Cluster questions Problem solving items
Some questions are meant to be used as part of a group of questions. In this case, resolving the answers to individual questions is better left until the entire group is completed. The value of the questions are often in the comparison of the different items and in having students think about what changes lead to what differences and why.
Problem solving items
In these items (indicated by a pencil cluster logo), the intent is to have students work together to solve some small problem. After a few minutes, ask the groups to share their answers, vote on the different answers obtained, and have a discussion.

7. Consider two mathematical models of real world things
9/5/13
Distance
Time
We map positions and times into numbers. What kinds of numbers are we mapping to?
Integers A. All numbers (+,0,-)
Rational numbers B. Non negative only
Real numbers C. Positive only
Physics 131

8. Which equation represents the quantity on the left?
The area of a circle.
The volume of a sphere.
The circumference of a circle.
The surface area of a sphere.
Do this as 4 separate clicker questions. After you are done analyze the dimensions so that students can see what works.

9. Which equation could represent the surface area of a cylinder?

10. The diffusion constant D, describes how molecules jiggling around in a fluid drift. It has dimensions [D] = L2/T
We have good reason to believe (we’ll see it in a reading later) that D depends on the average distance a molecules travels, λ, and it’s average speed v.
If [λ] = L and [v] = L/T guess an equation that expresses D in terms of λ and v.
9/2/16
Physics 131

11. You know that 1 cubic centimeter of water has a mass of 1 gram
You know that 1 cubic centimeter of water has a mass of 1 gram. What’s the mass of one cubic meter of water?
10 g
102 g
104 g
106 g
1 kg
10 kg
100 kg
1000 kg
None of these

12. Measurement is basically about counting – but counting what?.
Perim
Area
Vol 1 2 3
Consider the perimeter, area, and volume of a cube that is made up of 1, 2, and 3 small cubes. Show them how to count the perimeter, area, and volume of a 1x1x1 cube, taking as unit one edge, one side, and one volume of this small cube. P = 12 (the cube has 12 edges); A = 6 (the cube has 6 sides); V = 1. Note in each case you are counting something different. As you go to 2 and 3, let the students fill this table out for themselves, answering the question that you only want total outer edges and faces. Collect answers from around the class and click, especially for 3. You are likely to get incorrect countings. Don’t suppress this! Let them discuss it and convince each other. Then note how the different types of dimensions scale up differently. The point is: WHEN YOU ARE MAKING MEASUREMENTS YOU ARE JUST COUNTING – BUT IT MATTERS WHAT YOU ARE COUNTING. SPECIFYING THAT “WHAT” IS DIMENSIONAL ANALYSIS.

13. An example from a math exam
Writing the equation in this problem on a physics exam would receive 0 credit and the comment: “This is a meaningless equation!”
The population density of trout in a stream is
where r is measured in trout per mile and x is measured in miles. x runs from 0 to 10.
(a) Write an expression for the total number of trout in the stream. Do not compute it.
How would you fix this?

14. A dollar and a penny
A student makes the following argument: "I can prove a dollar equals a penny. Since a dime (10 cents) is one-tenth of a dollar, I can write:
10¢ = 0.1 $
Square both sides of the equation. Since squares of equals are equal,
100 ¢ = 0.01 $.
Since 100 ¢ = 1 $ and 0.01 $ = 1 ¢ it follows that 1$ = 1 ¢."
What's wrong with the argument?
Stuck?
Try it with
10 cm = 0.1 m

15. The rate at which an animal in a cold environment loses heat is proportional to its surface area, but its metabolism generates heat in all of its cells, so the rate of heat generation is proportional to its volume. All other factors being equal (which they often are not!), which implications would follow from these facts?
Because it is smaller, a baby seal is less at risk for hypothermia (body temperature falling to too low a value) than is an adult when it is in an ocean significantly colder than its body temperature.
Because it is smaller, a baby seal is more at risk for hypothermia (body temperature falling to too low a value) than is an adult when it is in an ocean significantly colder than its body temperature.

16. [M]=M [g]=L/T2 [h]=L [ω]=/T [v]=L/T [R]=L [I]=ML2
As part of an exam a few years ago, a student wrote the following derivation of a final result. Without knowing the problem, but knowing the dimensions of each quantity shown along the bottom can you determine:
9/2/15
Is equation D correct?
Yes No
Can’t tell
Given that equation D is NOT correct, can you tell which is the first line that has an error? A B C D A. B. C. D.
Physics 131
[M]=M [g]=L/T2 [h]=L [ω]=/T [v]=L/T [R]=L [I]=ML2

17. A student measures distance x to be 5 meters and area A to be 25 ft2
A student measures distance x to be 5 meters and area A to be 25 ft2. Discuss with neighbors which of the following are true; then vote for all that are true.
9/2/16
[x2] = [A]
[5x] = A
x2 = [A]
x2 = A
None of the above
Physics 131

18. Example
x = 5 meters
y = 27 inches

19. Is x = y?
Yes No
Wait, how many meters are in an inch?

20. Is [x] = [y]?
Yes No

21. A student measures distance x to be 5 meters and area A to be 25 ft2
A student measures distance x to be 5 meters and area A to be 25 ft2. Discuss with neighbors which of the following are true; then vote for all that are true.
[x2] = [A]
[5x] = A
x2 = [A]
x2 = A
None of the above

22. Your personal scales inches centimeters First digit of thumb
Open handspan
Forearm (cubit)
Full height
Pass out meter sticks and have students write down some personal measuring sticks for themselves that they can use in later estimation problems.

23. Estimate the thickness of a page in a textbook.
Physics 131
9/2/11
Estimate the thickness of a page in a textbook.
100 m
10-1 m
10-2 m
10-3 m
10-4 m
10-5 m
10-6 m
10-7 m
10–8 m
Prof. E. F. Redish

24. Estimate the number of cells in your body.
Physics 131
9/2/11
Estimate the number of cells in your body.
100
102
104
105
106
108
1010
1012
1014
Prof. E. F. Redish

25. Physics 131
9/2/11
Given that the average size of a human cell is about 10 microns across, estimate the number of cells in your hand.
108
1010
1012
1014
100
102
104
105
106
Prof. E. F. Redish

26. Physics 131
9/2/11
Given that the average size of a human cell is about 10 microns across, estimate the number of skin cells in your hand.
108
1010
1012
1014
100
102
104
105
106
Prof. E. F. Redish