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?
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?
Solution
The cylinder can be unrolled to form a rectangle with the dimensions 120 cm by 16 cm.
The string creates four diagonal lines. The measure of each diagonal line can be found by considering the right triangles that they form.
Note: The base of each right triangle has a value equal to the measurement of the cylinder’s circumference. The height of each right triangle will have a value equal to ¼ of the cylinder’s height.
From the right triangle, we can get the measure of one diagonal line by using Pythagorean Theorem.
(Diagonal's length)^2 = (30^2)cm + (16^2)cm = 1156 squared cm
Diagonal's length = 34 cm
Since there are four diagonals created by the string, the full length of the string is:
4 * 34 cm = 136 cm
The string’s length is 136 cm.
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The string creates four diagonal lines. The measure of each diagonal line can be found by considering the right triangles that they form.
Note: The base of each right triangle has a value equal to the measurement of the cylinder’s circumference. The height of each right triangle will have a value equal to ¼ of the cylinder’s height.
From the right triangle, we can get the measure of one diagonal line by using Pythagorean Theorem.
(Diagonal's length)^2 = (30^2)cm + (16^2)cm = 1156 squared cm
Diagonal's length = 34 cm
Since there are four diagonals created by the string, the full length of the string is:
4 * 34 cm = 136 cm
The string’s length is 136 cm.
Click here for: Problem with Hints for Solving
Click here for: Favorite Problem Main Page