AP Physics C: Students designed their own experiments to determine the type of dependence between speed and air resistance. It was the traditional coffee filter lab, but groups could collect data however they wanted. Some groups used motion detectors, others used meter sticks and stopwatches, and others used video analysis. Then we put the new regression feature in Desmos to the test. Our results so far:
NGSS Science and Engineering Practices:
#3. Planning and carrying out investigations
#4. Analyzing and interpreting data
#5. Using mathematics and computational thinking
Today is a quiz day, so I’ll share 2 other things we did this week.
AP Physics C: Students designed their own experiments to determine Young’s Modulus for 3 different types of marshmallows: store-brand jumbo marshmallow, store-brand regular marshmallows, and Jet Puff marshmallows.
Students took data, used Desmos to create a position-time graph for the sound to travel down the line of kids, and fit a trend line to the data to determine the speed of sound.
NGSS Science and Engineering Practice #3: Planning and Carrying Out Investigations
NGSS Science and Engineering Practice #4: Analyzing and Interpreting Data
College-Prep Physics: Modeling Instruction’s standard lab practicum for the constant velocity unit is colliding buggies. Lab groups take data to determine the speed of their buggy, then the buggies are quarantined and groups are paired up. Each group pair is then given an initial separation distance for their buggies and are asked to predict the point were the buggies will collide. Once they calculate the answer, they are given their buggies back to test their prediction.
It’s fun, but there are some frustrations. Groups that have poor experimental design or data collection techniques won’t calculate the correct buggy speed, which means they won’t accurately predict the collision point. Also, since only the separation distance is given, there isn’t much focus on the position of the buggy and students are less likely to use a graphical method to find the collision point. They try all sorts of equations instead. In the end, one person in the group typically does the calculations while her partners just copy her work.
This year, I decided to shy away from the calculation aspects of the buggy collision lab and instead use the activity to get students more familiar with some of the digital tools we’ll be using throughout the year.
Logger Pro: Students used a motion detector and Logger Pro to find the speed of their buggies. They learned how to select portions of the graph and how apply a linear fit. This also reinforced the concept that the slope of a position graph represents velocity. They printed a copy of the graph and taped it into their lab notebooks. Then I quarantined the buggies.
Position, not distance: Pairs of groups were then assigned a starting position relative to an origin (marked on the floor) and a direction of motion. Careful advance planning let us have a variety of collision scenarios — some head on, some where a fast buggy catches up to a slow buggy moving in the same direction.
Desmos: Groups were then required to model the collision scenario in Desmos in order to determine the collision point. For me, the physics is in formulating the correct models to type into Desmos, not actually solving the set of simultaneous equations or graphing them by hand. Surprisingly, there were some interesting mistakes in this stage: Some groups didn’t use the proper sign for the slope to indicate a buggy heading north/south. Some groups just used the sign from their Logger Pro graph (positive or negative, depending on whether they made their buggy move towards or away from the motion detector). And still some groups used the y-intercept from their Logger Pro graph as their starting point instead of the starting point they were assigned. Once the mistakes were realized, it was a quick fix in Desmos — much less frustrating than reworking a set of simultaneous equations. Then they tested their predictions and included their Desmos graph in their notebooks.
It went well this way, and took about 40 minutes from start to finish. It was something that even students with weaker math/algebra skills could find accessible. Plus, there was more reasoning and discussion about what the slopes and intercepts mean and how to model the situation rather than a focus on solving equations.
NGSS Science and Engineering Practice 2: Developing and Using Models
AP Physics C: Students are determining the relationship between the mass of spring oscillator and the resulting angular frequency. Linearization of data is an expected skill on the AP exam, and this lab is good for that because the data must be linearized twice.
The black data points are the original data (mass, angular frequency). Since it looks like an inverse relationship, we attempt to linearize it by making a new table and graphing (1/mass, angular frequency) — which are the red data points. This clearly isn’t linear, but it looks like a side opening parabola (ie, square root function). So we linearize again by making a third table and graphing (sqrt(1/mass), angular frequency — which are the blue data points. This looks straight! So we add y = mx to the window and use the slider to find the slope that fits the blue data points (0.49 in this case). Now we use our linearized equation to write an equation to fit the original data (black curve, y = (0.49)sqrt(1/x)) which works!
Here’s a link to the Desmos file so you can play around with it:
I really like how in the table, Desmos leaves the entries as is, without converting to decimal. For example, leaving it as sqrt(1/0.1) rather than 3.16. This is a nice visual reminder of how we are transforming the data in order to linearize it.
College-Prep Physics: Curve-fitting by hand can be tedious and linerization can be confusing. But curve-fitting with technology has its drawbacks, too — Excel is too unweildy and Logger Pro’s buffet of functions quickly has students blindly finding the function with the lowest R value.
Enter Desmos: While it doesn’t produce pretty labeled graphs like Logger Pro, I LOVE the slider function for curve fitting. (I know Logger Pro can do this, too, but it’s so much simpler in Desmos.) So today, everyone graphed their paragraph data from several weeks ago in Desmos. Then we reviewed the 4 types of functions we’ll be encountering this year and their characteristics:
Then we looked at several data points to see what doubling x (paragraph width) did to y (paragraph height). Did height double (linear), quadruple (quadratic), halve (inverse), or quarter (inverse square)? Now that we knew it would be an inverse relationship, we added to Desmos a “k=1″ line and a “y = k/x” line. Then we dragged the slider for k until we get a good fit for the data.
So in the example in the first picture, we get y = 6.8/x. But what does y represent? What does x represent? What are the units for 6.8? While Logger Pro automatically figures that out, I like that Desmos forces kids to wrestle with those questions. After some analysis, we see the relationship in the first picture to be height = (6.8 cm2)/width.
Now that we walked through that example as a class, we did a follow-up activity which looks at the effect of font size on the height of the paragraph, which I modified from here: http://bestcase.files.wordpress.com/2012/08/paragraphshandouts.pdf
To mix things up, I had each group member analyze a different font. My handout is here: LAB measuring paragraphs FONT SIZE 2014 Low Tech
Once the graphs are done, students get a unique URL for their graph that they can share with me, or download an image of the graph for inserting into Word documents or printing and pasting into lab notebooks.
PS: Sorry for being MIA the previous 9 days. College-prep students were working on their midterm projects for the first 4 days, and the remaining 5 days were midterm exam days where students took buffet quizzes and handed in their projects.
AP Physics C: This is a classic problem (which also was on the 1983 AP Physics C Mechanics exam). At what angle will the block lose contact with the wheel?
We cranked through the physics and got around 48 degrees. Then I showed the class the video, but it looks like it’s still in contact with the wheel at 48 degrees (screenshot below).
So we decided to take another approach for verification: If the block really leaves the wheel at 48 degrees, how far from the center of the wheel will the block be as it passes through the horizontal? The video shows around 35 cm. (screenshot below).
So again we crunched through some physics. But instead of solving a series of simultaneous equations for projectile motion, we modeled those equations in Desmos instead.
The red graph represents the vertical position of the block over time. The green graph is the time at which the block reaches the horizontal (about 0.12 second). The blue graph represents the horizontal position of the block over time. So the intersection of the blue and green graphs will tell us the horizontal position of the block as it passes through the horizontal. And look — 35 cm! So even though it’s tough to tell that the block looses contact with the wheel at 48 degrees, the resulting projectile motion holds up.
College-Prep Physics: Today we explored graphical relationships for proportional, quadratic, inverse, and inverse square functions using Desmos. Here’s the sheet: MATH RELATIONSHIPS 2014
The goal is for students to be able to determine the type of relationship by looking at data.
Proportional: If X changes by factor of N, Y will change by factor of N.
Quadratic: If X changes by factor of N, Y will change by factor of N2.
Inverse: If X changes by factor of N, Y will change by factor of 1/N.
Inverse Square: If X changes by factor of N, Y will change by factor of 1/N2.