Reflection: Problem Solving
If done correctly, problem solving should engage people in the task of finding a solution and the method to finding the solution should be unknown to the problem solvers. Problem solving in mathematics causes people, no matter what their age, to draw upon prior knowledge to find a solution. In the process of solving the problem, often times people are deepening their mathematical knowledge as well as gaining a new awareness of mathematical concepts. Perhaps the most wonderful thing about problem solving is that it is often a puzzle/challenge in which many people become immediately engrossed in finding the solution.
As a mathematics teacher, problem solving offers a way to engage students from all grades in various mathematical topics (e.g. probability, geometry, calculus). Besides meeting the standards for mathematics, problem solving is essential to teach students because it helps build new mathematical knowledge, makes previous mathematical knowledge more concrete, and allows a variety of problem solving strategies to be applied and adapted. The following problem I would use to enhance my students' understanding of 2-D & 3-D shapes as well as basic geometry concepts and formulas.
Favorite Problem
A cylinder 120 cm high has a circumference of 16 cm. A string makes exactly 4 complete turns round the cylinder while its two ends touch the cylinder's top and bottom. How long is the string in cm?
Hints (in the order that they should be given):
1. What shape would you get if you unroll the cylinder? Think of a piece of paper that has been rolled up into a cylinder? What happens when you unroll the piece of paper? It is in the form of a rectangle. Cylinders can be unrolled to form a rectangle. Think about what the dimensions are for the rectangle created by unrolling this cylinder.
2. Draw your unrolled cylinder. Look at your picture and see if you notice any shapes that will help you determine the length of the string or part of the string.
Additional hint: Focus on the first diagonal created by the string on your drawing of the unrolled cylinder. What shape is created by the diagonal of the
string and the top portion of the unrolled cylinder?
3. If you look at your drawing of the unrolled cylinder, you should see a right triangle formed by the string and the top portion of the unrolled cylinder.
Think about what the dimensions should be for this right triangle.
Additional hint: One side of the triangle will have the same measurement as the cylinder’s circumference. Think about what the other side of the right triangle will be.
4. Since the string circles around the cylinder 4 times, the height of the right triangle will have a measurement value equal to ¼ of the cylinder’s height. Once you have the measurements for the base and the height of the right triangle, what formula can you use to find the value of the third side?
5. Using Pythagorean Theorem you will discover the value of one diagonal. Since the string circles the cylinder four times, the total length of the string will be 4 * (the value of one diagonal).
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1. What shape would you get if you unroll the cylinder? Think of a piece of paper that has been rolled up into a cylinder? What happens when you unroll the piece of paper? It is in the form of a rectangle. Cylinders can be unrolled to form a rectangle. Think about what the dimensions are for the rectangle created by unrolling this cylinder.
2. Draw your unrolled cylinder. Look at your picture and see if you notice any shapes that will help you determine the length of the string or part of the string.
Additional hint: Focus on the first diagonal created by the string on your drawing of the unrolled cylinder. What shape is created by the diagonal of the
string and the top portion of the unrolled cylinder?
3. If you look at your drawing of the unrolled cylinder, you should see a right triangle formed by the string and the top portion of the unrolled cylinder.
Think about what the dimensions should be for this right triangle.
Additional hint: One side of the triangle will have the same measurement as the cylinder’s circumference. Think about what the other side of the right triangle will be.
4. Since the string circles around the cylinder 4 times, the height of the right triangle will have a measurement value equal to ¼ of the cylinder’s height. Once you have the measurements for the base and the height of the right triangle, what formula can you use to find the value of the third side?
5. Using Pythagorean Theorem you will discover the value of one diagonal. Since the string circles the cylinder four times, the total length of the string will be 4 * (the value of one diagonal).
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