Mathematical reasoning in physics in middle

school and high school

Suzanne Brahmia, Jim Flakker, Jeff Goett, Hector Lopez, Eva Thanheiser, Alan Van Heuvelen and Eugenia Etkina

July 21, 2008AAPT Summer MeetingEdmonton, Alberta

•differentiated so can be adapted to diverse classroom and learning levels.

•based on ISLE (active learning; mirrors science processes, emphasis on developing scientific abilities)

•focused on mathematical literacy.

•physics curriculum for middle school and high school in its pilot phase.

Scientific abilities + mathematical reasoning

What is Physics Union Mathematics?

•middle - high school continuum

Improved learning of both physics and mathematics

Mathematically literate students should be fluent in-• integers (positive and negative numbers)

– invented tool used in science for differentiating between quantifiable opposites: up and down, right and left, electron charge and proton charge, etc.

• zero and its several meanings, like – empty set (no acceleration, zero Kelvin, etc)– reference point (temperature, potential energy, etc.)– equal number of positive and negative (balanced forces,

electrical neutrality, etc.)

• variables that represent physical quantities, and

equations as a shorthand way to represent cause-effect relationships between them:

I=V/R a=F/m Density=m/v

Mathematically literate students should be fluent in-

• techniques of rewriting and evaluating equations that help develop proportional reasoning.

• multiple representations that are self-consistent.

– verbal, symbolic, graphical, diagrammatic, etc.

• measurement of the quantities associated with SI units – length, time, liquid volume, mass etc.

Two pieces of copper have the same mass, 240.3 g. The side of the cube is 3 cm long. The rectangular solid is 1 cm high, 3 cm wide and 9 cm long.

1. Find the amount of space each piece occupies, then draw a Micro-View of both pieces.

2. What is the mass of 1 cm3 of this copper?

3. What is the mass of 100 cm3 of this copper?

4. If you cooled down the copper by putting it in the freezer, do you think either the mass or the volume would change, or do you think it would stay the same? Explain.

MS Example: Multiple Reps, Proportional Reasoning, Math Reasoning in Physics

You rub a plastic tube with rabbit fur and thereby transfer 6 negatively charged particles from the fur to the tube.

1. Draw sketches that show the net charge on the tube and fur before and after rubbing them together.

2. Write a math statement that describes the charge-transferring process for the fur.

0 - (-6) = +63. Write a math statement that describes the charge-transferring process for

the tube. 0 + (-6)= -6

4. Explain why the fur and the tube are attracted to each other after having been rubbed together even though they don’t interact at all before they are rubbed together.

MS Example: Integers, Zero as Balance, Multiple Reps, Math Reasoning in Physics

Before+ + + + + +

_ _ __ _ _

After

Charles Coulomb used a torsion balance to measure the force that one charged ball exerts on another charged ball to find how the force between two electrically charged objects depends on their separation. The table provides data that resemble what Coulomb might have collected. Find patterns in the data and devise a mathematical relationship based on these observations. Use graph paper to help.

1. How does the electrical force depend on the separation (r) of the objects? (pick one)• Possible model 1: F12 1/r • Possible model 2: F12 r• Possible model 3: F12 1/r2

2. Would you describe it as proportional? Inversely proportional? Or something else?

3. If you doubled the separation, what would happen to the force?

4. If you halved the separation, what would happen to the force?

Charges (q1, q2)

Distance (r)

Force

1, 1(unit)

1(unit)

1(unit)

1,1 2 1/4

1,1 3 1/9

1,1 4 1/16

MS Example: Multiple Reps, Variables and Equations, Proportional Reasoning

0

-

Ki + Ug,I + Us,I + W = Kf +Ug,f+Us,f+∆Uint

+

Before After

Integer statementBar chart for a process.

Describe a process.

30 + (-20) =10

MS/HS Example: Integers, Multiple Reps, Math Reasoning in Physics

“Learning mathematics should mean finding one’s way

through problems of new sorts and taking responsibility

for the results.”

-Hassler Whitney

7th grade math teacher

Wolf Prize Laureate

Princeton Mathematician

•For more information–http://pum.rutgers.edu–http://www.islephysics.net–http://paer.rutgers.edu/scientificabilities

In a similar experiment to the one you did in class, student B pulls Student A in a few different ways and takes data to determine her acceleration. •When she pulls exerting half as much force, A accelerates at 0.25 m/s per second (0.25 m/s/s).•When she pulls exerting twice as much force, A accelerates at 1.0 m/s/s.•When she pulls exerting three times as much force, A accelerates at 1.5 m/s/s.

(a) Represent these verbal statements with one picture, and three force diagrams.(b) Represent these verbal statements with a graph. Please explain your choices for the horizontal- and vertical- axes. What is your independent and what is your dependent variable?(c) Represent your graph with a mathematical description.(d) How does the acceleration of A depend on the net force exerted on her by other objects?

MS Example: Integers, Zero as balance, Mult Reps, Math Reasoning in Physics

In science class did an experiment. The setup for the experiment is shown in the diagram below, in which the heating of the soil and water is done using an overhead lamp. You took temperature readings every 2 min for 10 min. Your data are in the table below:

Soil Water

time (min)

temp (° C)

time (min)

temp (° C)

0 20.0 0 20.0

2 21.9 2 20.4

4 24.1 4 20.8

6 25.9 6 21.3

8 28.1 8 21.6

10 30.0 10 22.0

a) Use the graph paper provided above to graph the data. Use a solid line and dotted line to distinguish between the two sets of data.b) Describe how the soil data are different from the water data based on your graph.c) What feature of the graph describes how the soil data are different from the water data in this experiment?d) About what temperature do you think the soil would reach at 12 minutes? Explain.e) Can you use these results to make general statements about thermal properties of water and soil? If yes, explain your reasoning. If no, what can you do to improve the experimental setup?

A B

M L Y

Zara and Mona pull on cart A in opposite directions. Zara pulls with 3 units to the left, Mona – with 2 units to the right. Laila and Yasmine pull on an identical cart, B, also in opposite directions. Laila pulls with 3 units to the left, Yasmine pulls with 5 units to the right.

Zara pulls to the left

Mona pulls to the right

Picture representation

Number representation

Result

3.5 units 4.00 units

2.00 0.7

1.5 3/2

3.25 3.75

7/5 1.5

0.8 0.80

2.50 6/2

(c) If you were using the spring scale to measure forces in the table, then how certain would you be in each value? Write the values of the forces using the experimental uncertainty, assuming that the smallest division on your spring scale is 0.1 N.

(a)Represent these events and the result of the pulling of each couple using integers.(b) Here are more experiments, when Zara and Mona pull in the opposite directions. Represent these events and the result of the pulling of the couple using number statements.

You are a member of a team hired by an architectural firm to design a wheelchair accessible entrance to a building. The Americans with Disabilities Act requires the steepness of wheelchair ramps to be less than a 1:12 ratio for rise to run.a) Explain why The Americans with Disabilities Act is concerned with the steepness of accessibility ramps.b) Draw three ramps that meet their requirement.c) Decide which of the following ramps meet the recommendation.

(i)

16m2m

(ii)

46m

4m

(iii)

36m

3m

(iv)

4m0.5m

(v) 1:10 (vi) 3:42 (vii) 2:28d) Calculate the amount of work that is required to push a 20kg object up each of the ramps (i – iv).e) If the stairs are 2.5 meters tall, how long does a ramp need to be?f) The company controller argues that regardless of the length of the ramp, a person must do the same amount of work to get a wheelchair to the top. He suggests that we can save money by using less construction material by making the ramp short and steep as possible. You need to make a ramp with a rise of 1.5 m, what length will the ramp need to be? g) The controller says that is too long. Give two reasons why it is important not to make the ramp any steeperh) One of your classmates is having trouble understanding the difference between the amount of effort it takes to lift or move something and the amount of work required to lift or move something. He says, “If it is easier to do, it must be less work”. Describe how you would help your classmate to understand this idea better. Use examples like the ramps from the previous problem.

Middle School Students

Label Axes

Time on x axis

Percent Students Answering Correctly

Percent High School Students Answering Correctly

Represent Data

Label Axes Time on x axis

High School Students

Percent Students Answering Correctly

Magnitude of slope

Sign of slope

‘x’ and ‘t’ in equation

Middle School

High School