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, 2008
AAPT Summer Meeting
Edmonton, Alberta
What is Physics Union Mathematics?
Scientific abilities +
mathematical reasoning
Improved learning of both
physics and mathematics
•physics curriculum for middle school and high school in its
pilot phase.
•based on ISLE (active learning; mirrors science
processes, emphasis on developing scientific abilities)
•differentiated so can be adapted to diverse
classroom and learning levels.
•middle - high school continuum
•focused on mathematical
literacy.
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.
MS Example: Multiple Reps, Proportional
Reasoning, Math Reasoning in Physics
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: Integers, Zero as Balance, Multiple
Reps, 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.
_ _ _
_ _ _
Before
+++
+++
After
2. Write a math statement that describes the charge-transferring process for
the fur.
0 - (-6) = +6
3. 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: Multiple Reps, Variables and Equations,
Proportional Reasoning
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)
Dist
ance
(r)
Force
1, 1
(unit)
1(un
it)
1(unit)
1,1
2
1/4
1,1
3
1/9
1,1
4
1/16
MS/HS Example: Integers, Multiple Reps, Math
Reasoning in Physics
Integer statement
Bar chart for a process.
Before
After
Ki + Ug,I + Us,I + W = Kf +Ug,f+Us,f+∆Uint
30 + (-20) =10
+
0
-
Describe a
process.
“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
MS Example: Integers, Zero as balance, Mult Reps, Math Reasoning in Physics
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?
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?
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.
Mona
Picture
Number Result
A
M
L
B
(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.
Y
Zara
pulls to
the left
pulls to
the right
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
represent
ation
represen
tation
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.
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.
(v) 1:10
(vi) 3:42
(vii) 2:28
d) Calculate the amount of work that is required to push a
20kg object up each of the ramps (i – iv).
2m
(i)
e) If the stairs are 2.5 meters tall, how long does a ramp
16m
need to be?
f) The company controller argues that regardless of the
length of the ramp, a person must do the same amount of
4m
work to get a wheelchair to the top. He suggests that we
(ii)
can save money by using less construction material by
making the ramp short and steep as possible. You need to
46m
make a ramp with a rise of 1.5 m, what length will the ramp
need to be?
(iii)
3m
36m
(iv)
0.5m
4m
g) The controller says that is too long. Give two reasons
why it is important not to make the ramp any steeper
h) 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.
Percent Students Answering
Correctly
Middle School Students
Percent High School Students
Answering Correctly
Label
Axes
Represent
Data
Time on x
axis
Label Axes
High School Students
Time on x
axis
Middle
School
Percent Students Answering Correctly
High
School
Magnitude
of slope
Sign of
slope
‘x’ and ‘t’
in equation