A metronome is a device used by musicians to time beats that change the rhythms of music. This is extremely helpful if someone wants to be trained in learning to produce the correct rhythms with a consistent beat. Jody was loaned a metronome for a day and he did not want to spend the money necessary to buy his own. However, he had learned in science class that a pendulum swings at different rates of speed depending on its length. He determined that it might be possible to recreate the beat of the metronome by adjusting the length of the pendulum. Jody has a 30 cm length of string. The number of beats are to be measured in 5 second intervals (i.e. 4 beats per 5 seconds). If the string is adjusted at 2 cm intervals, what are the possible beats that could be produced by the pendulum? Jody was really excited about the number of beats that could be produced and decided to get a longer piece of string to produce more rhythms. Based on an analysis of the data you collect from carrying out Jody's investigation, predict the slowest beat that could be produced by a 60 cm length of string.
We were working on a series of tasks, which demonstrated exponential growth and had been creating graphs, including J-curves, which represented these numbers. In looking for a real-life situation where students could generate data to create a graph of interesting numbers, a friend and I, both math/science teachers, thought of this task. Students could apply what they knew about pendulums, gather data and examine the mathematical relevance of their results. Students will have the opportunity to demonstrate the ability to make charts, tables and graphs for investigative results. Students will have the opportunity to extrapolate data from their graph and the opportunity to evaluate the reasonableness of their solution. Students were asked to create pendulums given 30 cm of thread and washers. They needed to test the number of swings possible in five seconds with two cm intervals of thread. They were to predict from their findings how many swings could be expected with 60 cm of thread. They were expected to create some mathematical system for looking at their data in order to make a prediction for 60 cm. 60 minutes The students were able to complete this task in one hour. Some of them worked beyond the hour of class time to complete the write up or to work on an extension idea. This was a perfect math/science interdisciplinary task. It would also have been a great opportunity to tie in technology with use of spreadsheets to record data and create graphs electronically. I did not do this because of lack of access and time constraints. The task also lends itself well to integration with music and dance. I learned the hard way that having students hold the string leads to very inaccurate data collection. Some better ways would be to have them put a pushpin tack in a bulletin board and hang the thread by a loop from the pin to swing it. Students could also suspend a meter stick between two desks and tape the thread to the stick. Use thread instead of heavier string, as the size of some student's knots greatly affects the results. Use stopwatches if possible, although many students have good enough watches. Students have to work in pairs in order to time accurately. Even with all these tips, students will probably not get reliable enough data to be able to accurately predict the number of swings for 60 cm, but they should be able to graph these results and see a predictable curve develop. The problem is that most see a line and not a curve, predicting about one or two swings. It is interesting to see which students accept this prediction and which ones realize that it just does not make sense to expect one swing in five seconds. It helps students to realize that it is always good to verify your results with another method if possible and in this case, it is possible.
- Metal washers - I used some that were about one inch in diameter. All students should have ones the same size. Having other washers available of different sizes will suggest extension activities.
- Thread - I used button thread, which is more heavy duty than regular thread. We tried string, but the knots were too bulky and added excess mass to the washers.
- Stopwatches or a clock that ticks off seconds.
- Metronome for demonstration - although a metronome is not a perfect analogy for what the students are actually doing, as there is a spring mechanism in an actual metronome.
- Meter sticks
- Graph paper
The Expert exemplar has a correct solution.
The Novices had a great time "playing", but did not get to the math involved. Many either did not complete a graph or did not number and/or label one correctly. They had no idea how many swings to predict with 60 cm. Some of them did get 60 cm of string and try it for themselves rather than make a prediction. They really applied little or no math to the exercise beyond measuring the intervals of thread. Their measurement skills for both lengths of string and time intervals were not completely reliable. The exemplar student did gather and display data, but never answered the question asked. This student does not say what was done to solve the problem or why it was done.
These students all attempted to graph their results. Their data was rather unreliable as they were not very careful. Their graphs were unpredictable, as the points were all over the place. Rather than re-testing, they just declared their results unpredictable and quit. Most of these students had better graphing skills, but some of their intervals were incorrectly executed. Their strings tended to be measured accurately; their time intervals were slightly less accurate. The exemplar student has the basic idea and makes a fair attempt at creating charts and a graph. The student does not have reliable enough data to see a pattern and does nothing to confirm the answer given.
These students followed the directions accurately. They were careful in gathering and recording their data. They were able to extrapolate data from their graphs that indicated only one or two swings with 60 cm of string and they accepted that as fact and moved on. They did not question the validity of their data and did not extend their thinking on this task. Their tables and graphs were for the most part accurate and their solutions were quite well written. Their measurement of time and distance tended to be reliable. The exemplar student uses good mathematical language and good representation. The student uses sound reasoning up to the end of their work, but stopped too soon.
These were the students who extrapolated their data, predicted one swing on 60 cm and realized that did not make sense. They continued by testing it out and accounting for the discrepancy. They conducted more than one trial and accounted for the differences recorded. Their charts and graphs were accurate. Many of them went on to extend the investigation with other sized washers or by using multiple washers to increase the weight. They found methods to stabilize the swinging washers as well or at least acknowledged that this variable was affecting their results.
I selected this piece as the exemplar for its clear, concise content. The student uses good mathematical language to state what was done to solve the problem and good representation to demonstrate the findings. The student shows excellent reasoning in recognizing the impossibility of the most obvious pattern. The student uses a second method to solve the problem, compares the results and goes with the solution that makes the most sense to the student.
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