To be more concrete, Figure 4.3 (pages 42 – 49) shows how to use the Competent” Problem-solving Framework to solve a problem suitable for introductory students at the beginning of an algebra-based physics class. The solution is on the left (even) side of the page. The right (odd) side of the page contains commentary about how the details would support particular goals in an introductory course. [See Chapter 14, pages 175 for a problem-solving
framework for the calculus-based course.]
As you read this example, you may want to take some notes to help you personalize a problem-solving framework for your own students and situation, as described in Chapter 14. What parts of this Competent Problem-solving Framework matched the goals for your introductory course? What parts did not match your goals? What parts are not needed by your students? What is
missing that your students need? What wording needs changing to better match your students? If so, how would you change it?
Chapter 4: Building Physics into Problem Solving
45
Figure 4.3. Example of an “ideal” student problem solution, with commentary
The Problem: You are driving at 50 mph on a freeway when you wonder what your stopping distance would be if the car in front of you jammed on its brakes. When you get home you decide to do the calculation. You measure your reaction time to be 0.8 seconds from the time you see the car’s brake lights until you apply your own brakes. Your owner’s manual says that your car slows down at a rate of 6 m/s2 when the brakes are applied.
Step 1: F
OCUS ON THEP
ROBLEMDraw a picture, identifying the useful quantities.
Question: What distance did the car travel from when the brake light is seen to when it stopped?
Approach: Use the definitions of velocity and acceleration.
The velocity is constant until the brakes are applied. In this time interval, the average velocity is equal to the instantaneous velocity.
The acceleration is constant after the brakes are applied.
In this time interval, the average velocity is not equal to the instantaneous velocity. However, the average
acceleration is equal to the instantaneous acceleration.
vo = 50 mph a = 6m/s2 vf = 0
stopped applies
brakes sees
light 0.8 sec
stopping distance
46
Part 1:Teaching Physics Through Problem SolvingFigure 4.3 (continued). Example of an “ideal” student problem solution, with commentary Commentary
The Problem. This problem emphasizes the basic definitions of velocity and acceleration in one dimension. It also emphasizes the difference between instantaneous kinematics quantities and their average values, a concept is very difficult for students. The problem gives the students practice in contrasting constant velocity motion with that at a constant acceleration. Students also need practice using the units of physical quantities. We give students problems with mixed sets of units so they will pay attention to them. Every specialty uses its own set of units for either historical reasons or convenience. From our questionnaire, we found that the faculties from departments that require their students to take our physics course want the physics course to show their students how to convert and manipulate units.
Picture. Students’ difficulty with visualization can be immediately seen by their difficulty in drawing a useful picture. At the beginning of the course, many students need to draw very realistic looking objects such as cars while others are satisfied with more expert-like drawings using a simple symbol, such as a rectangle, to represent a car. Almost all beginning students have difficulty drawing multiple images on a single picture to represent an object at multiple positions of interest. They have difficulty making a decision, as to where those interesting positions might be. Many can verbalize these features before they can indicate them on a drawing. Students also have difficulty associating their pictorial representation with quantities that represent the object’s motion. We emphasize to our students that once they have drawn the picture, then they should not have to read the problem again. This makes for better time efficiency in solving the problem.
Question. Writing the question in their own words helps prevent students from solving for some quantity that is not desired. Here the student must decide on a reasonable definition of “stopping distance.”
Approach. Writing an approach helps students concentrate on the important physics in the problem.
Chapter 4: Building Physics into Problem Solving
47
Figure 4.3 (continued). Example of an “ideal” student problem solution, with commentary
Step 2: D
ESCRIBE THEP
HYSICSMake a diagram of the situation, defining the quantities that physics uses to describe motion (velocity and acceleration at each interesting position and time on a coordinate system).
Target quantity: x2
Possibly useful equations:
v x x
t , for constant velocity v x vx for constant acceleration v x vi vf
2 a x vx
t , for constant acceleration a x ax vo = 50 mph
a = 6m/s2 a = 6m/s2
v1 = 50 mph v2 = 0
xo = 0 to = 0
x1 = ? t1 = 0.8 s
x2 = ? t2 = ?
+ x
48
Part 1:Teaching Physics Through Problem SolvingFigure 4.3 (continued). Example of an “ideal” student problem solution, with commentary Commentary
Make a diagram of the situation. Students have a great deal of difficulty associating an object with a specific acceleration and velocity at a specific position and time. This diagram helps them because it requires the drawing of an idealized object, now a point, at each interesting position. The diagram also begins the process of getting students to define an appropriate coordinate system for a situation. It gives the students practice in the process of going from real objects to idealized objects. For experts there is no significant difference between the picture and the diagram, but for beginning students there is. The picture gives practice in visualizing the behavior of real objects. The diagram, on the other hand, gives students practice with the visual relationships of physical quantities. As time goes on in the course, more and more students become more expert-like and combine these two types of images. At the beginning of the course, most of our students are not ready to combine these two types of visualization. Those who try to combine the picture and the diagram tend to become more and more confused as situations become more complex.
Target quantity. Writing down this quantity gives students a focus for their mathematics. When doing mathematical manipulations many students lose track of the quantity they want.
Possibly useful equations. This is where the student explicitly connects the conceptual approach to the problem with mathematics. It represents a gathering of mental resources or a “toolbox” for the quantitative solution to the problem. Even in the beginning chapter on kinematics in most textbooks, the number of equations students believe they will need to know can overwhelm them. Here they must decide to limit those equations to only those that are independent and might be useful. At this stage, it is OK if there are some extra equations since, the student does not yet know how to solve the problem.
To compel students to concentrate their efforts on the basic concepts and discourage the student behavior of formula memorization, we only allow our students to use equations chosen by the instructor (see Chapter 3, pages 31 - 33). The choice of equations depends on the course and the emphasis of the instructor. For example, the equations used in this solution would be given to the students in the algebra based physics course. In the calculus-based course, we replace the equation
an equation directly connected to the calculus expression defining acceleration.
Chapter 4: Building Physics into Problem Solving
49
Figure 4.3 (continued). Example of an “ideal” student problem solution, with commentary
Step 3: P
LAN THES
OLUTIONConstruct the chain of equations giving a solution. Begin with an equation containing the target quantity. Keep track of any additional unknown quantities that are introduced.
Unknowns
Find x2 x2
v 1,2 x2 x1
t2 t1
v 1,2 , x1 , t2Find v 1,2
v 1,2 v1 v2 2 v1
2
Find x1
v1 x1 xo t1 to x1
t1
Find t2
a1 v2 v1
t2 t1 v1
t2 t1
Check for sufficiency:
Yes: 4 unknowns and 4 equations
Outline the solution steps. Work from the last equation to the first equation that contains the target quantity.
Solve
for t2 and put it into
. Solve
for x1 and put it into
. Solve
for v 1,2 and put it into
. Solve
for x2.50
Part 1:Teaching Physics Through Problem SolvingFigure 4.3 (continued). Example of an “ideal” student problem solution, with commentary Commentary
Construct the chain of equations giving a solution. We wish to convince students that solving a problem relies more on understanding the concepts of physics than on mathematical techniques. Students seem to trust mathematical manipulation to give them an answer and make “magical” mistakes to get one. This procedure restrains that manipulation by requiring that
sufficient equations to solve the problem have been assembled first. It does not always yield the most elegant solution, but it is straightforward, easy to
understand, and very general.
Students always know where to begin because they always start with an equation that contains the target quantity. The unknowns in that equation give the student a way to decide on next equation to be used and so on. This procedure gives students a logical way to build a chain of equations that connects the target quantity to quantities that are known. If the process reaches a “dead-end”, the student explicitly see how to revise one of their decisions of which equation is used to determine an unknown quantity. This procedure depends on having linearly independent equations, which is another reason for the instructor to control the equations that can be used (see Chapter 3, pages 31 - 33). Although we emphasize paper and pencil solutions, constructing this chain of equations is very useful for computer or calculator algebra.
Remember that working backwards from the goal is a general thinking tool (heuristic) used by experts solving real problems. See the reference in Endnote 1 for a description of working backwards and other heuristics.
Check for sufficiency. Matching the number of equations with the number of unknowns gives the students an easy way to determine if they need more information to solve the problem. After a few weeks, we also point out how you can solve the problem if there are fewer equations than unknowns provided an unknown cancels out. We give instruction on how to detect such cases and the physics implication of these cases.
Outline the solution steps. Actually writing down an outline seems to be necessary for most of our algebra-based physics students, but not for the calculus-based students. Students who do not trust their mathematics
background can become very confused when doing algebra unless they have a written plan. All introductory students often get into infinite algebraic loops when solving equations with more than two unknowns. Here when an unknown is determined, even in terms of other unknowns, it is immediately substituted in every upstream occurrence. This procedure assures that such algebraic traps are avoided.
Chapter 4: Building Physics into Problem Solving
51
Figure 4.3 (continued). Example of an “ideal” student problem solution, with commentary
Step 4: E
XECUTE THEP
LANFollow the outline from Step 3.
Solve
for t2:
t2 t1
a1 v1 Calculate the value of the target quantity.x2 50mi
x2 is the distance traveled by the car from when brake light is seen to stopping. The question is answered.
The answer is in meters, a correct unit of distance.
A car is about 6 meters long so 10 car lengths is not an unreasonable distance to stop a car going that fast.
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Part 1:Teaching Physics Through Problem SolvingFigure 4.3 (continued). Example of an “ideal” student problem solution, with commentary Commentary
Step 4 - Execute the Plan.
This is the only part of the problem containing mathematical manipulation. The mathematical solution follows the verbal outline. Here the student begins with quantities that are known and proceeds backwards through the chain of equations to the target quantity. The student can concentrate on the
mathematics because they are assured of a solution when they follow the plan.
Step 5 - Evaluate the Answer
This step reinforces the connection of the physics used in the problem solution to the student’s reality. It is the step most characteristic of expert problem solving and is the most difficult part of problem solving for introductory students.