Thursday, February 3, 2022
It’s the time of year to give an update as to what Y8 Math has been doing this semester. It seems like we’ve done so much since the first week in November, 2021.
We started with the multiplication and division of polynomials, which led us into solving quadratic polynomial equations. We were using a software program which graphed quadratic equations showing the students how the graph changed as we changed the coefficients for the squared variable, variable in the 1st power, and the constant term in the quadratic equation. We solved various special cases of quadratic equations and culminated with using the Cross Method and touched briefly on Completing the Squares Method for solving more difficult quadratic equations. We showed that when we set the quadratic equation equal to zero, we were really seeing where the quadratic function crossed the x-axis on an x-y coordinate plane.
We also did different classifications of triangles and identified them by both side and angle type. We identified all triangle types, but we were focused on identifying the two types of right triangle types (right scalene and right isosceles) as these were used in Pythagorean Theorem for finding an unknown side of these triangle types. We also used the Converse of Pythagorean Theorem to determine if a triangle is acute, right, or obtuse by comparing the square of the longest side to the sum of the squares of the other two sides.
We studied special right triangles, 3-4-5, 30-60-90, and 45-45-90 triangles to learn the side and angle relationships. We discovered the angle relationship for the 3-4-5 triangle is 37-53-90. We determined that the side proportions of the 30-60-90 were, 1:√3:2; while the right isosceles (a.k.a. the 45-45-90 triangle) has side relationships, 1:1:√2. We used these side proportions for these triangles to solve a right angle puzzle worksheet.
This complemented our earlier work on triangle congruence and learning about the 5 theorems for triangle congruence (SSS, SAS, AAS, ASA, HL). This helped greatly in our Universal project of measuring unknown angles in triangles with an online protractor angle measuring app. When we identified triangle congruence, we could calculate the unknown angles across congruent triangles much faster.
Currently, Y8 is working on calculating the surface area and volume of cubes, cuboids, prisms, and cylinders. We recently wrapped up our lessons on calculating the surface area and volume for the 3-D shapes mentioned above except for the cylinders. Therefore, we will focus on the calculations of surface area and volume for cylinders in the following paragraphs.
The surface area formula tends to be more difficult for students to understand as it needs to be conceptualized as a binomial. One term for the lateral area of a cylinder, which when opened and laid flat, results in a rectangle with length the same as the circumference of the cylinder and a width equal to the height or depth of the cylinder. For the purposes of calculating heights, lengths, or depths of cylinders we used the variable ‘d’ representing depth or the distance between the 2 circular faces of a cylinder. Therefore, the lateral term of our surface area equation becomes 2πrd. That leaves the 2 circular faces of the cylinder, whether it is sitting on 1 of the bases or on its lateral curved side. That term is 2πr^2 if the cylinder is closed and 1πr^2 if the cylinder is open, meaning it has no top or lid. Therefore, the complete formula for the surface area of a closed cylinder becomes: Acylinder = 2πrd + 2πr^2, or when the HCF of both terms are factored out we get; Acylinder = 2πr(r + d).
The volume of a cylinder is the easier formula as it is just the area of 1 of the circular faces times the distance ‘d’ between the 2 circular faces of the cylinder. Our formula for the volume of a solid cylinder is Vcylinder = πd(r^2).
In the rare problem where we have a cylinder that has been hollowed out we get the formula for a hollowed cylinder as Vcylinder = πd (R^2 – r^2). Here, ‘R’ is the radius of the external cylinder and where ‘r’ is the radius of the hollowed out portion.
Students have been working vigorously to master the concepts and calculations of surface area and volume of all the 3-D shapes that have been mentioned in this article. Below is an example problem we have done in a recent class on calculating the surface area of a closed cylinder that was made open by removing the top.
In summary, the Y8 students have had a good semester and have learned a lot to help them progress onward into year 9. The calculating of surface area and volume of cylinders were just a couple of the things they’ve learned. However, it is better to visualize what you have learned more that it is to hear somebody verbalize it. Therefore, with that idea in mind, we would like to leave you with one last visualization of what we’ve learned about cylinders.
Never stop learning!
Written by Mr Joseph S. Capobianco