Friction
The absence of the banana leaf marble ramp and the associated RipStik velocity versus height exercise has diminished the demonstrations impacted by friction. Friction does not really arise as complicating the data gathered. Friction makes a first appearance during Newton's first law - an object in motion tend to remain in motion unless acted upon by an external force. And friction is an example of an external force that tends to work against perpetual motion.
This term I realized that the grit measurements are made easier by weighting the sled. The empty sled as a very low pulling force, the weight brings the pulling force up and permits the use of a midrange scale. This is useful as the low end scales are usually in use by the groups measuring weight versus frictive force.
WiFi issues meant that the Desmos data I picked up in the 11:00 class failed to save, I was unaware of this data loss until the next day, Friday. In retrospect one solution would be to not shut down the laptop until verification of file save had been accomplished. The laptop could even be left in sleep mode until the next day.
On Friday I asked the groups to share whatever equation they might have obtained. The weight groups all found an equation fairly easily as their data is well fitted by a direct y=mx relationship.
The slope will be the coefficient of friction, but I do not mention this at this time. The average coefficient usually lands somewhere around 0.3.
The groups dealing with grit and surface area were unable to determine the nature of the equation. I used long pauses after queries to provide time for students to think about the possible equations. The only suggestion was a y=mx equation, and I showed using Desmos on the fifty inch television at the back that this did not work well for the grit and surface area data.
I then shifted to approach the issue from the question of slope - what is the slope for the surface area, the open circles in the graph above. That too met with silence. "What is slope?" I asked. Bear in mind that while the students are shy to respond, at this point in the term shyness is not stopping them from hazarding a guess. The students were genuinely perplexed. A student volunteered that slope is rise over run. I then asked what was the rise? This time more than one student responded, "There is no rise" and "None" This led to zero rise over run, which is zero and thus a slope of zero.
I think math teachers, in a push to cover curriculum, trundle through zero slope rather quickly and without realizing how hard it is for the human mind to first wrap itself around the concept of zero slope. That a horizontal line has a zero slope is usually one of many factoids tossed out in section 1.1 of some text, when the concept deserves at minimum a whole period of exploration. But the curriculum will not be covered, might counter the math teacher. Yes, the curriculum will not be covered. But the students are clueless when faced with horizontal data: they have learned nothing. Or as told my students at this point, "Mathematics classes cannot teach you mathematics." My students have almost all taken MS 100 College Algebra or MS 101 Algebra and Trigonometry. All took Algebra I in high school and most took Algebra II. And they have nothing to show for this effort. The information did not stick. The cognitive hooks on which to hang the information did not exist. Math taught in isolation is throwing uncooked spaghetti at a wall and expecting the spaghetti to stick.
I circled back around to the full slope intercept form, y = mx + b. I suggested we let Desmos figure out the slope and intercept for the surface area data. The result was m=0 and b=400 grams. And the line went straight through the data! I pointed out that this means that y = 400 grams. A slope of zero means "m" is zero, and zero times anything is zero, including the word anything. That term can be omitted, thus y = 400 grams is the equation. A strange looking equation, but one that fits the data.
The next step was to try to show that the 400 grams is actually the result of the 1520 gram weight of the surface area sled multiplied by the coefficient of friction. This is a messier matter as none of the coefficients produce exactly 400 grams of pulling force. That leaves me in a hand waving position that the coefficient which does produce 400 grams, 0.2632, is within the range of coefficients found by the weight groups: 0.25, 2.28, 0.33. 0.2632 is a possible value given the span of coefficients. I then attempted to show that the same basic argument can explain the grit data. Note that one group provided a 0.45 coefficient but I lacked data for that group. I did use their reported value at this point to calculate the range of expected pulling force for the 394 gram grit sled. All but one data point was within this range.
That the pulling force is the result of the coefficient multiplied by the sled weight is an inferential argument and is a mathematical jump too far for some students, perhaps many students. My sense is that I need another way to segue into this matter that surface area and grit are not affecting the force of friction for the glass louvers.
By now the board is filled with calculations and equations. The students would be forgiven for thinking they are in an algebra class and must be despairing of the class ever being what they think is a science class. All I could do to wrap up the period was to tell the students to step back from the mathematical details. What we learned is that weight and only weight affects the force of friction for the glass louvers. Neither sandpaper grit nor surface area affected the force - the force was essentially constant or bounced around in a narrow horizontal range, that is, was effectively constant. And that is the big picture, non-mathematically. The math simply provides support.
Rodman and Eric work on the role of weight in the force of friction
Joyann pulls a constant weight sled while Ellena records data.
This term I realized that the grit measurements are made easier by weighting the sled. The empty sled as a very low pulling force, the weight brings the pulling force up and permits the use of a midrange scale. This is useful as the low end scales are usually in use by the groups measuring weight versus frictive force.
Joyceleen and Dickenson will be working on weight versus force of friction
Mayboleen record data while Susan pulls the 1520 gram multistack sled that is used to measure surface area versus the force of friction. This term the students used an eight glass stack.
On Friday I asked the groups to share whatever equation they might have obtained. The weight groups all found an equation fairly easily as their data is well fitted by a direct y=mx relationship.
The slope will be the coefficient of friction, but I do not mention this at this time. The average coefficient usually lands somewhere around 0.3.
The groups dealing with grit and surface area were unable to determine the nature of the equation. I used long pauses after queries to provide time for students to think about the possible equations. The only suggestion was a y=mx equation, and I showed using Desmos on the fifty inch television at the back that this did not work well for the grit and surface area data.
I then shifted to approach the issue from the question of slope - what is the slope for the surface area, the open circles in the graph above. That too met with silence. "What is slope?" I asked. Bear in mind that while the students are shy to respond, at this point in the term shyness is not stopping them from hazarding a guess. The students were genuinely perplexed. A student volunteered that slope is rise over run. I then asked what was the rise? This time more than one student responded, "There is no rise" and "None" This led to zero rise over run, which is zero and thus a slope of zero.
I think math teachers, in a push to cover curriculum, trundle through zero slope rather quickly and without realizing how hard it is for the human mind to first wrap itself around the concept of zero slope. That a horizontal line has a zero slope is usually one of many factoids tossed out in section 1.1 of some text, when the concept deserves at minimum a whole period of exploration. But the curriculum will not be covered, might counter the math teacher. Yes, the curriculum will not be covered. But the students are clueless when faced with horizontal data: they have learned nothing. Or as told my students at this point, "Mathematics classes cannot teach you mathematics." My students have almost all taken MS 100 College Algebra or MS 101 Algebra and Trigonometry. All took Algebra I in high school and most took Algebra II. And they have nothing to show for this effort. The information did not stick. The cognitive hooks on which to hang the information did not exist. Math taught in isolation is throwing uncooked spaghetti at a wall and expecting the spaghetti to stick.
I circled back around to the full slope intercept form, y = mx + b. I suggested we let Desmos figure out the slope and intercept for the surface area data. The result was m=0 and b=400 grams. And the line went straight through the data! I pointed out that this means that y = 400 grams. A slope of zero means "m" is zero, and zero times anything is zero, including the word anything. That term can be omitted, thus y = 400 grams is the equation. A strange looking equation, but one that fits the data.
The next step was to try to show that the 400 grams is actually the result of the 1520 gram weight of the surface area sled multiplied by the coefficient of friction. This is a messier matter as none of the coefficients produce exactly 400 grams of pulling force. That leaves me in a hand waving position that the coefficient which does produce 400 grams, 0.2632, is within the range of coefficients found by the weight groups: 0.25, 2.28, 0.33. 0.2632 is a possible value given the span of coefficients. I then attempted to show that the same basic argument can explain the grit data. Note that one group provided a 0.45 coefficient but I lacked data for that group. I did use their reported value at this point to calculate the range of expected pulling force for the 394 gram grit sled. All but one data point was within this range.
That the pulling force is the result of the coefficient multiplied by the sled weight is an inferential argument and is a mathematical jump too far for some students, perhaps many students. My sense is that I need another way to segue into this matter that surface area and grit are not affecting the force of friction for the glass louvers.
By now the board is filled with calculations and equations. The students would be forgiven for thinking they are in an algebra class and must be despairing of the class ever being what they think is a science class. All I could do to wrap up the period was to tell the students to step back from the mathematical details. What we learned is that weight and only weight affects the force of friction for the glass louvers. Neither sandpaper grit nor surface area affected the force - the force was essentially constant or bounced around in a narrow horizontal range, that is, was effectively constant. And that is the big picture, non-mathematically. The math simply provides support.
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