Surface Area of Cylinders and Spheres

Many of my past and current Geometry students have had trouble memorizing the formulas for volume and surface area. Traditionally, teachers have asked students to memorize all of the formuals below.

Name Lateral Area Surface Area Volume
Prism L.A. = Ph S=L.A. + 2B
= Ph + 2B		 V = Bh
Cylinder L.A. = Ph
= Ch
= 2πrh		 S.A. = L.A. + 2B
= 2πr + 2πr²		 V = Bh = πr²h
Regular/Square Pyramid L.A. = ½Pl S.A. = L.A. + B
= ½Pl + B		 V = ⅓Bh
Cone L.A. = ½Pl
= ½Cl = ½(2πr)l
= πrl		 S.A. = L.A. + B
= πrl + πr²		 V = ⅓Bh
= ⅓πr²h
Sphere S.A. = 2πrh
= 2πr*2r
= 4πr²
 V = 4/3πr³

This is a lot for anyone to remember. Our jobs as math teachers is to help students make connects between all of these formulas. The connections between them allow students to memorize very little and to generate these formulas based on their knowledge of how they are related. This also provides students with a much deeper understanding of the math.

Here are just a couple visual representations of some of these connections that I commonly share with my own students.

And in that vain, I created a Desmos graph that shows a visuallization of the connection between the latteral area of a cylinder and the surface area of a sphere. Click on the graph below to see the math behind it on Desmos.com.

Triangle Inequality Theorem

Common Core State Standard: Know the triangle inequality theorem to determine the possible side lengths of a triangle.

Description: In this activity, students will discover and explore the triangle inequality theorem. It is similar to a popular activity where students are given different lengths of spaghetti and are asked to find relationships between the sides when trying to form triangles. It is a great way to still do this discovery lesson without needing the large amount of prep time needed to measure a bunch of side lengths.

Teacher Guide: Click Here

Desmos Activity: Click Here

Below is a sample of how students may try and create a triangle out of specific side lengths.

Volume of a Sphere

Common Core State Standard: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Description: In this activity, students will discover and explore the use of the formula for the volume of a sphere. Students will already need to know and how to use the formula for the volume of a cylinder.

Teacher Guide: Click Here

Desmos Activity: Click Here

DohCubes 3 Act Lesson

This 3 Act Lesson was created by Jenn Vadnais. You can visit her website at jennvadnais.com. In this lesson, students will exploring finding the number of cubes inside a PlayDoh container.

My students really enjoyed this activity and asked a lot of questions during Act 2. My favorite was "How many cubes does the ball 'poop' out?" It was a great question, and I rolled with it. The lesson Jenn Vadnais created was very engaging and helped my students easily see that the space inside of a shape can be described by the number of cubes that fit inside of it.

Click Here to View my lesson on Desmos.com

As the Crow Flies

Students practice finding distance 'as the crow flies' on a coordinate grid. Students will use the Pythagorean Theorem and the Distance formula to find distance.

Click Here to view my lesson

I used this Desmos activity to help show my students that the Pythagorean Theorem and the distance formula perform the same task. We also discussed why and how the Pythagorean Theorem could be turned into the distance formula. We had some great math talks about this one.

Measuring Angles

Students practice using a protractor to measure angles.

Click Here to view my lesson

I had used real protractors with my students the day before I made this Desmos Activity. I found that students really struggled with understanding which scale on the protractor to use. The questions at the end of the activity allows students to analyze another student's mistake and analyze their own mistakes in the process.

CSCS Open House 2016

CSCS Open House 2016 Website

Agenda
Desmos.com

Presentation Download

Linear Regression

Computer Supported Collaborative Learning (CSCL) is an effort by California State University, Northridge to support science teachers in the San Fernando Valley. The concepts being developed by CSUN can be applied to all subject matter at all grade levels.

Click here to visit the CSCS website

“Demand for Computer Systems Analysts with big data expertise increased 89.9% in the last twelve months , and 85.40% for Computer and Information Research Scientists.”
— Forbes.com 2014

Data collection and analysis is a growing industry. Being able to pull ideas out of big data is what drives many of the newest and most successful technology companies.

Go through the steps below to collect, view, and analyze data about height and shoe size. Linear regression is one way that data can be analyzed to find patterns/relationships between two sets of data.

  • Data Collection
  • View Data
    • Google Sheet
    • Make sure you copy this data to use in the next step
  • Calculator
  • Analysis
    • What can this line tell us about the relationship between the data?
    • In regression, the R2 coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R2 of 1 indicates that the regression line perfectly fits the data.
    • Use your linear equation to predict what height someone would be if they had a size 15 shoe.

Linear regression lines can be used to help predict future results. The linear equation models what is happening in the real world. The better your R2 value, the better your equation will model the real world situation.

Linear Function Transformations

Computer Supported Collaborative Learning (CSCL) is an effort by California State University, Northridge to support science teachers in the San Fernando Valley. The concepts being developed by CSUN can be applied to all subject matter at all grade levels.

Click here to visit the CSCS website

What is a Linear Function?

Linear functions are those whose graph is a straight line. A linear function can be written in the following form. 

y = f(x) = mx + b

This is called slope-intercept form. Where m represents the slope of the line and b represent the y-intercept.

What happens when m or b changes?

  1. What happens as the value of m changes?
  2. What happens as the value of b changes?

Distance Between Two Points

Computer Supported Collaborative Learning (CSCL) is an effort by California State University, Northridge to support science teachers in the San Fernando Valley. The concepts being developed by CSUN can be applied to all subject matter at all grade levels.

Click here to visit the CSCS website

How do you get from one point to another?

  • Open the map and the Google Form. Answer the questions on the form.
  •  The shortest distance between two points is a straight line. To find that distance, it may be easier to find the distance by "taking the streets" first. These distances for a right triangle. The shortest distance is the hypotenuse of the triangle. Open the example.
    • Example
    • This example shows that you can easily count the distance of the legs of the triangle, but it is impossible to count the distance of the hypotenuse. 
  • To find the distance between two points, we use the Pythagorean Theorem. Open the Desmos calculator to view the points and the distances between them. 
    • Desmos
    • Type the Pythagorean Theorem in for the hypotenuse. It solves for the length automatically!
    • Now create your own right triangle and find the length of the hypotenuse by solving the Pythagorean Theorem for c. 
    • This is the distance between the two points
    • Take a screenshot of your triangle and answers and add it to our class Google Slides

    Favorite Desmos Activities & Graphs

    “Desmos wants to help every student learn math and love learning math. But “every student” is a lot of students so we create digital math tools and let the Internet take them to anyone who wants them.”
    — Desmos.com

    Direct Variation

    Students practice creating direct variation equations of the form y=mx while playing a game.

    Click here to view my lesson

    This was my first attempt at creating a MarbleSlide activity using Desmos. In this activity students practice creating direct variation equations while trying to launch marbles into stars. I found that this activity really motivates my students to keep trying and to persist. Turning a difficult task into a game allows more of my students to access the material and provide them with instant feedback.

    Pythagorean Theorem

    This was my first attempt at using a Desmos activity in my classroom. It is also the first activity I created myself.

    Click Here to view my lesson

    In this activity, students use the Pythagorean Theorem. They will discover that the theorem only works for right triangles. Then use their knowledge to solve some real world problems.

    This lesson was created because I noticed that my students were trying to use the Pythagorean Theorem for all sorts of triangles. Acute, Right, and OBTUSE! It occurred to me that I had never told my students that it only worked for right triangles. Then I asked myself, "Why should I tell them when they can discover it for themselves?"