Look at the set of gears. Gear B has 12 teeth and gear C has 20 teeth. Find the number of teeth that Gear A has to have so that when it rotates once, Gear B and Gear C have rotated some whole number of times. You may use the gears we were experimenting with to help solve this problem. Be sure to explain your reasoning clearly. Connect as much mathematics and use as much math language as you can.
My sixth grade class was just beginning to study gears in science class and had already studied multiples in math class. I wanted to integrate and connect the concepts that were being presented in both classes. The class had been playing with some gears we made out of cardboard, investigating what direction interlocking gears turned, and putting different size gears together to see how they turned. As they collected data on their gear ratios many were beginning to see a relationship. This task pulls together some of the investigations the students were conducting. I was hoping that most students would recognize the use of multiples in determining gear ratios. We had not studied ratios formally, but had some discussions about ratios and proportions in other problems. Some students may be able to connect those ideas also. Standard 4 in the NCTM Standards-Mathematical Connections- states that, "In grades 5-8 the mathematics curriculum should include the investigation of mathematics connections so that students can apply mathematical thinking and modeling to solve problems that arise in other disciplines such as art, music, psychology, science and business." Some students used the cardboard gears while others felt confident to imagine the gears turning. Most students used a chart to keep track of the turns the gears made. 45 minutes This task could be used with a science unit studying gears. Having gears that students can handle would be helpful, especially if this will not be coordinated with a science unit. I had templates of gears and cut them out of cardboard and used straight pins as the axle. Cardboard gears The least common multiple of 12 and 20 is 60, so gear A has to have 60 teeth.
There is no evidence of a strategy or an answer to the problem. The student shows two multiplication problems that each equal 160, but the numbers do not relate to the problem. There is no explanation of the solution. The student tries to place the gears differently, but that does not seem to help solve the problem.
The student uses a strategy that is partially useful. Their use of multiples is evidence of mathematical reasoning, but they fail to find the least common multiple and therefore Gear A will turn more than one time. There is some use of mathematical representation and language.
This student shows a broad understanding of the problem. His/her use of the chart allowed a pattern to be seen. The clear explanation shows that the student visualized the gears moving. The connection to multiples is made and used to find the solution. The student uses effective representation to collect data and find a pattern.
This student shows a deep understanding of the problem. S/he not only made the connection to multiples, but also talks about ratios and shows how the ratio was determined. The solution leads directly to a solution and the student employs refined reasoning when they simplified their findings to make ratios. There is also a clear and effective explanation using precise and appropriate mathematical language (multiples, LCM, ratio), and the chart helps communicate the pattern that leads to the solution.
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