Our Flavor of Conceptual Physics

Meet the New Student

Many of us high school Physics teachers are used to teaching Physics as the third or fourth course in a sequence of science classes. Our courses are often named College Prep Physics, Honors Physics, or even AP Physics. The students who populate our classes are generally fluent in mathematics, quite capable of algebraic manipulation of Physics formulae, and have an above-normal comfort with numbers. Physics may be the course in the science sequence for which only 25-50% of the student body ever enrolls. We teach a privileged group of students. 

To understand this project - the Conceptual Physics Course Pack - you will need to realize that there are some schools (more like many schools) that are broadening their curriculum offerings and are faced with the task of offering Physics to a different type of student than what many of us are accustomed to teaching. In such schools there will be students taking a Physics class who ...

• ... have a vague understanding of slope.
• ... would have great difficulty solving for t when given the equation v=d/t.
• ... can look at two simple numbers like 12 and 6 and not recognize that one number is twice the other number.
• ... do not have a sense of what doubling, tripling, quadrupling, and halving is.
• .... do not understand that dividing or multiplying by 10 or 100 or 1000 involves moving a decimal point a given number of places.
• ... will use a calculator to multiply 2 by 3 (and will exhibit signs of panic when you take their calculator away in the middle of process)
• ... have great discomfort, a lack of confidence, and a mild to intense phobia with mathematics

How do we know this? We know this because we have taught Physics to such students ... in a school where 90-95% of the student body enrolled in a Physics course. We've watched students pull out calculators to multiply 2 by 3. We've observed blank stares after giving our best explanations for how to solve for t in v=d/t. 

Nearly all of us would find discouragement in the possibility that there are students with such math shortcomings. Some of us would view the task of teaching Physics to such students as a lost cause. But we view it as a welcomed challenge and an awesome opportunity. This is a chance to be an educator. A chance to meet students where they are at. A chance to do a bit more hand holding to help them through their mathematical struggles. A chance to increase our patience level and compassion for sincere students who have difficulty learning. And a chance to introduce students to the world of Physics in spite of their lack of math prowress. 

 

There's Always a Spectrum

OK. We'll fess up. Not all students in this "new Physics class" that your school is offering will exhibit the level of math deficiency described above. There will likely be a wide spectrum of math abilities. There will probably be students who are comfortable with basic algebraic manipulations. There will be students who don't complain about math (... but you should see their writing ... and their backpack). Don't be fooled into thinking that all students in this new course offering are weak at math. There will be a wide spectrum.

Even with the spectrum, if your school makes an effort to reach a larger percentage of the student body with a Physics course, it would be reasonable to expect that the mathematical ability of the students in the course are considerably lower than what you are accustomed to with your College Prep and Honors Physics courses. And if that's the case, spending four weeks (or more) attempting to get every student comfortable using the kinematic equations may not be the best curricular decision. Helping students use v_f² = v_o² + 2•a•d to solve for d may not be the hill to die on. There are better ways to spend students' time. And there are more appropriate mathematical skills to emphasize ... assuming that you will be emphasizing some mathematical skills (we would suggest that you do). What we hope to make clear on this page is the curricular approaches that we have taken with respect to teaching Physics to students with relatively low mathematical abilities.

 

Dissenting Opinions

There is one thing we have noticed when the phrase "Conceptual Physics Course" is mentioned. To our surprise, there is an unusually large percentage of teachers who get a little perturbed and edgy. "What? You mean a high school is teaching a Physics course and not including math in it?" A number of objections are put forward in protest of such an endeavor. Let's take some time to address the elephant in the room.

The Physics Must Include Math Argument
Let's begin with the "If there's no math being taught, then it is not Physics" argument. For us, the argument is a bit of a red herring as we are not advocating removing the math from Physics. We are suggesting that we need to think differently about  how to use math in Physics if our learners are not proficient at algebraic manipulations and have very little to no exposure to trigonometric functions. For us, this does not exclude a student from learning a vast amount of Physics. It won't be learned in the same way or to the same mathematical level; but it will still be a lot of Physics.

As far as approaches to Conceptual Physics are concerned, ours is probably heavier on its mathematical emphasis than other approaches. But if we were to consider those approaches that included considerably less mathematical emphasis, we've seen as much or more Physics being learned than those that courses that spend hours and hours using Physics formulae to solve Physics word problems. Physics is a collection of ideas about how the physical world works. Using geometry to derive the thin lens equation and then using the equation to solve 20-30 lens problems of varying difficulty may have merit for a certain group of Physics students. But using a couple of simple patterns about the direction light bends as it enters and exits the lens of a magnifying glass and using the rules in a diagram to explain how lenses produce magnified images of nearby objects will take one-fifth the time and may do more to explain how the world of lenses work. In our opinion, both approaches can be described as Physics.


The If They Don't Have the Math Skills, Maybe They Shouldn't be Taking Physics Argument
We are going to admit that the "If they don't have the math skills, maybe they shouldn't be taking Physics" argument makes us a bit perturbed. We get a bit edgy about this one. We could take the edge off and agree in the middle, if we re-worded this as ...

• If students lack the math skills that are required for Honors Physics (and are unwilling to put in the extra work required to make up for that lack of skill), then maybe they should be taking a different Physics course. Or ...
• If students lack the math skills that are required for College Prep Physics (and are unwilling to put in the extra work required to make up for that lack of skill), then maybe they should be taking a different Physics course (and maybe the school should be offering it).

We agree that an Honors Physics program that works for Honors Math students does not and should not be watered down to accommodate students who would be better served by the College Prep Physics course. And the same can be said of watering down the College Prep Physics course. As long as these courses are appropriately leveled and reasonable about their expectations, students lacking the prerequisite math ability will need to put in extra time to be successful. Keep your standards high. Lowering the bar is never a good idea if the bar is at the proper height. BUT ...

Our point on this page, the reason for this project - the Conceptual Physics Course Package - is to install another bar. We're advocating for another course and a wider audience. Our hope is to serve another student who is not traditionally served. And as we will argue in the rest of this page, it is possible to serve them in a manner that presents Physics concepts and blends a mathematical component that is appropriately leveled for this new student.

 

​There's Nothing Wrong with Difficult or Challenging

We might be showing our age with these statements: There is nothing wrong with a course that is difficult or challenging. There is nothing wrong with struggle. There is nothing wrong if a student encounters some frustration. Of course we want our students to love Physics as much as we do. But if on Tuesday night, they don't have a warm, fuzzy feeling about Physics, that's OK. Do damage control during class on Wednesday. Now when large numbers of students don't have a warm, fuzzy feeling during the whole month of September, it's time to quickly address the issue(s).

If you begin teaching Physics to a greater percentage of your student body, you are going to have more students in your class (compared to what you're used to) who have not traditionally had the best school experience. There will be a higher percentage of students (compared to what you're used to) with a lower frustration tolerance. And when this is combined with the fact that they are not as strongly skilled in math, you're going to see challenge, struggle, and frustration. And you going to be tempted to avoid presenting challenges. Don't! If you're not used to this student, then you need to know that they can do more than you think. They just need more help, a slower pace, more hand-holding, and a lot more scaffolding.

We're emphasizing this point because there will be a strong temptation to avoid mathematics because you know the population of students you are teaching struggle more with math. We do not feel that the appropriate response is to remove the math just because students struggle with math. The appropriate response is to change what type of math that you present to students.

Consider changing how you present the mathematics; for instance, provide three forms of v=d/t. And use nicer numbers in your problems. Instead of F = 23.8 N and m = 10.1 kg, solve for acceleration; use F = 24 N and m = 12 kg, solve for acceleration. Use numbers that promote head math. Sure, it is a lot easie; but students more readily see the conceptual meaning in or behind the numbers. And it helps to cultivate some math confidence and number comfort.

Consider changing why your present the math. Don't present a math problem for math's sake but use the numbers to illustrate a mathematical expression of a concept or to reinforce a concept with numerical information.

Finally, consider changing how many mathematical problems you include. You're not trying to get them to master the skill of solving a problem. You're trying to highlight the mathematical meaning of a concept. So, avoid drill-and-kill approaches. You're trying to cultivate some math comfort and instill a recognition that physics concepts have mathematical expressions.

Our Conceptual Physics Course Package includes some mathematics and possibly more mathematics than what most people expect of a Conceptual Physics course. Before you remove it or reject it, rethink it. Add more scaffolding. Reduce the amount. If it still doesn't work, replace it with something that works better for your population of students.
​

The Mathematics of Our Conceptual Physics Course Package

So what is so mathematical about your Conceptual Physics course? In the interest of full disclosure, we have identified the mathematics that is implemented in our Conceptual Physics curriculum. We have organized it by unit. In just about every case, the math skill is not used as an end in itself but as a means of highlighting a relationship or concept.

Unit 1: Graphs and Relationships

• Students analyze data and graphs and identify the type of relationship (linear, inverse, quadratic, or constant) based upon the patterns they see in the graphs or the data.
• Students use observed relationships to predict how a doubling or a tripling or a halving of one variable would affect another variable.
• Students revisit how to calculate slope and the meaning of slope; they use for every statements.

Unit 2: Kinematics

• Students calculate distance and displacement for multi-stage motion (with nice numbers).
• Students use simple numbers and tabular data to calculate average speed and acceleration values.
• Students relate the slope and the shape of position-time graphs to the type of motion an object has.
• Students use slope calculations to determine velocity values for position-time graphs.
• Students relate the slope and the shape of velocity-time graphs to the type of motion an object has.
• Students use slope and area calculations to determine displacement and acceleration values for velocity-time graphs.
• There are no kinematic equations.  

Unit 3: Newton's Laws

• Students calculate weight from mass and mass from weight; a value of 10 N/kg is used for g.
• Students predict an acceleration value that results from a doubling, a tripling, a quadrupling, or a halving of the net force or of the mass or of both.
• Students use given values of individual forces and of mass to calculate the net force and the acceleration. Numbers are considerably nice; many students will be able to solve the problems with head math. Some problems provide the free body diagrams. Other problems require that students construct the free body diagram. Problems are scaffolded.

Unit 4: Free Fall and Projectiles

• Students plot distance-time data and velocity-time data for free-fall motion and identify patterns they see in the numbers and graphs.
• Given an initial velocity (nice number like +40 m/s or + 50 m/s) for a vertically thrown object, students identify the speed and velocity after specified amount of time (a whole number of seconds)
• Students use interactive Desmos graphs with sliders for initial height and initial launch velocities to solve problems. They use their understanding of the concepts to manipulate the sliders and to read the graphs in order to determine how high a projectile will go, how far horizontally a projectile will go, and how fast a projectile will be moving at a specified time. All solutions are done by the manipulation of the sliders and the reading of the graphs.
• Students use concepts to create scaled velocity vector diagrams displaying the velocities in the x- and the y-direction for a horizontally launched projectile at 1-second intervals.
• Kinematic equations are not used; there is no algebraic manipulation of equations.

Unit 5: Momentum and Collisions

• Students calculate and compare momentum values when given m and v values.
• Students use impulse=momentum change equation to determine values of F, t, m, and ∆V  in a table format and draw conclusions from tabular data as to the effect of a change in one variable on other variables.
• Students use impulse=momentum change equation to perform calculations and then analyze results to draw conclusions regarding the effect of increased collision time and rebounding (bouncing) upon collision force.
• Students sum momentum values for a variety of collision conditions and make judgements as to whether momentum is conserved and what it means to say momentum is conserved.
• With considerable scaffolding, students analyze explosions and collisions to calculate individual object momenta and system totals before and after the explosion or collision and determine a post-explosion or a post-collision speed.
• For hit-and-stick collisions where one object is at rest before the collision, students use mass ratios and proportional reasoning to predict post-collision speeds.

Unit 6: Work and Energy

• Students predict the effect of doubling, tripling, halving of mass, speed, or height upon a KE or a PE value.
• Students use mass, speed, and height values to calculate a PE, a KE, and a TME value and to make judgements regarding whether or not total system energy is conserved.
• If given an energy diagram for several states of a complex motion (initial, final, in-between states), students determine a missing KE or PE value.
• Students calculate the height from PE and the mass value for a specified point in an energy diagram.
• Students calculate the speed from the KE and the mass value for a specified point in an energy diagram. Equation for speed is given.

Unit 7: Circular and Satellite Motion

• Students predict the effects of doubling, tripling, and halving the v and/or the R for moving in a circle upon the acceleration value.
• Students use m and a values and a given free body diagram to calculate the net force and the individual force values.
• Students use the universal gravitation equation to predict the effect of a doubling, tripling, or halving of a mass and/or a separation distance upon the gravitational attraction of two objects.
• Students use a radius ratio and Kepler's third law to determine how many times greater one planet's orbital period is than another planet's.
• Students do not ever substitute values into a universal gravitation equation or a Kepler's third law equation to solve for the value of an unknown variable. Equations are only used as guides to thinking about the quantitative effect of changes in one variable on another.

Unit 8: Static Electricity

• Students use Coulomb's law and proportional reasoning skills to predict the effect of a doubling, tripling, quadrupling, and halving of a charge and/or a separation distance upon the force of attraction or repulsion of two objects.
• Students use the electric field equation and proportional reasoning skills to predict the effect of a doubling, tripling, quadrupling, and halving of a source charge and/or a distance upon the electric field intensity value.
• Students do not ever substitute values into Coulomb's law or the electric field equation to solve for the value of an unknown variable.

Unit 9: Electric Circuits

• Students perform simple calculations to relate the battery voltage to the amount of charge and the change in potential energy for a charge moving between terminals of a battery.
• Students use the definition of current to mathematically relate the current, amount of charge passing a point, and the time.
• Students use ∆V=I•R and proportional reasoning skills to predict the effect of a doubling, tripling, quadrupling, or halving of the battery voltage and/or the resistance upon the current in a circuit.
• Students use ∆V=I•R to calculate the current value or resistance value if given the values of the other two variables.
• Students calculate the equivalent resistance of a series circuit if given the values of individual resistors. They use the result to calculate the current in the series circuit, the current at all locations within the series circuit, and the voltage drops across the resistors. Problems are highly scaffolded. 
• For parallel circuits with identical resistors, students calculate the equivalent resistance; they use battery voltages and branch resistance values to determine voltage drops and current values for each branch and the overall current in the circuit. Problems are highly scaffolded. 

Unit 10: Waves and Sound

• Students calculate period and frequency from information about the number of cycles and the time.
• Students analyze a wave pattern use the given distance between two strategic points on the pattern to determine the wavelength of the wave.
• Students use v=f•wavelength to calculate the value of one variable if given the values of the other two variables.
• Students analyze situations involving periodic waves and use time and distance information to perform simple calculations of wavelength, amplitude, frequency, period, and speed.
• Students analyze standing wave patterns to determine the number of loops in pattern and to relate the wavelength to the length of the medium; they make comparisons of the wavelengths of the various harmonics and the frequencies of the various harmonics. Analyses are performed for strings and air columns (both open and closed).  Problems are highly scaffolded. 
• Students use powers of 10 and intensity values for various sounds to identify how many times more intense one sound is than another and how many deciBels different one sound is than another. Numbers are nice and problems are highly scaffolded.
• Draw standing wave patterns for resonating strings and use the pattern and wave equations to calculate wavelength and speed values.

Unit 11: Light and Color

• Use the illuminance equation and proportional reasoning skills to predict the effect of a doubling, tripling, quadrupling, and halving of the power and/or the distance upon the illuminance on a surface.

Unit 12: Ray Optics

• Use a modified version of Snell's law (expressed as n = ...) and a sine table to calculate an index of refraction value for a material if given the path of light through a prism made of that material. Problem is highly scaffolded.
• There is no use of the mirror equation or the lens equation. Students do not calculate a critical angle. Students do not use Snell's law to solve for an angle of incidence or refraction.

 

Teachers Can Add and Subtract

Of all people, Physics teachers can add and subtract from any of the above. That is to say, if the mathematical rigor included in our curriculum is too much, a teacher can easily remove it, soften it, or skip it. If the mathematical rigor in our curriculum is too weak, teachers can easily add to it or intensify it. We know that there will be a wide spectrum of students who will use this curriculum. It is impossible to precisely meet the needs of all classrooms. We have made an effort to introduce a number of mathematical manipulatives and to provide the supports that might be needed for students to successfully combine some math skills with Physics concepts. Our expectation is that teachers will modify the source docments in order to produce a program that is perfect for their classrooms.
 

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