Spreadsheet Math: Hundreds of Angular and Trigonometric Calculations in Seconds!

Some students grow to hate math because it’s hard (why they find it hard is a subject for another time), others hate it because of its repetitive redundant nature -e.g. answering 20 questions of the same problem, but with different numbers. Don’t get me wrong, extensive practice is certainly a legitimate method for learning new skills or formulas, but not everyone needs 20 questions; some need less while others need more. Ultimately what is important, is being able to use an appropriate formula to obtain a desired result for the situation at hand.

Furthermore, in today’s Google-Siri-Alexa-DigitalAssistant world, there is increasingly less need for memorizing formulas and more need for being able to discern which ones to use given a particular situation and how to manipulate them accordingly. Even the College Board has adopted this paradigm as they provide students with formula sheets for their AP Physics and Calculus Exams rather than burdening them with having to memorize dozens of formulas and their corresponding applications.

For these reasons, I have adapted a unit on angular measures that uses multiple formulas to reflect these skills. Rather than giving a typical practice worksheet for each formula, I task my students to complete 50 calculations for each of eleven formulas in about 45 minutes -this works out to be a bit over 500 calculations, or about 12 calculations per minute, or one problem every five seconds! Using the traditional approach of a pencil, paper, and calculator, this is inanely unreasonable; even for adults. However, we don’t use pencil, paper, and calculator in my class (except to verify a few random results when we’re done for quality assurance purposes), we use spreadsheets.

The Task:

What the students are given to fill in. The angles actually go up to 720° (4π) at increments of 15° (π/12) plus one unspecified angle determined at a later time, for a total of 50 input values. NOTE: due to rounding errors in Google Sheet’s calculations, I have blacked out a few cells that cause the following issue: trigonometric functions that should yield undefined values return very large numbers instead since the calculator is effectively dividing by really small numbers rather than actually dividing by zero.

At this point in the year, my students have already learned how to do basic mathematical operations, drag or copy-and-paste formulas into multiple cells, and reference other cells as input variables. As a result, once they successfully enter the formula in the first cell, it should take no more than two seconds to complete all subsequent calculations for that quantity -this is true whether I give them 50 problems, or 5000 problems!

Nonetheless, this manner of thinking is still new, so I walk them through the steps for the above and test them using an application example a few days later. Typically I try to include accurate real numbers, but do occasionally throw in random numbers to prevent students from getting lucky off a quick Google Search¹. In this particular instance, I have my students determine information about various celestial bodies in our solar system based on some preliminary information I provide.

The Assessment:

An example of what an assessment looks like for this project. Orbital time, Radius, and Time elapsed are provided, but have been omitted here for test security purposes.

For the trigonometric functions, I typically pick four angles of π/12 increments between 0 and 4π, and one other angle fitting no particular criteria that the students then have to find the trigonometric value for that angle. After allowing about 20 minutes to fill in these values, I then use an online software (I prefer to use MyHaikuLearning LMS, but Google Forms is also a good option) for collecting a random subset of answers like so:

Submission and Grading:

Actual planet names and angles have been replaced to for test security purposes — sure, I could easily change them, but then I would have less time to write this blog and share these cool tricks with you.

Most students are able to recognize this as a direct application, or replication even, of the previous generic assignment. Of those students however, I have learned that sadly, many of them have forgotten how to use fractions and proportions to obtain the relevant radian value, while a few others have yet to make the connection altogether resulting in fairly low scores the first time around. Historically, a teacher in this situation might just “curve” the assignment, call it good, and move on, as grading a second attempt would be way too time-consuming. The key word there though was “history”; we live in modern times and computers make modifying and re-grading easier than ordering your frappuccino at Starbucks on your phone.

A Second (third, and even fourth) Chance for those who need it!

A quick “duplicate” tab, update sheet to have names of rodents instead of planets, enter new numbers (reasonable or random), and you have a new test!

Some Background Information

Converting degrees to radians

Given an angle in degrees, to convert to radians is a simple formula: θrad= θdeg*π/180. It’s not a particularly difficult formula to remember² or use, but it is an especially powerful one.

Calculating arc length and sector area

With all angles converted to radians, calculating things like arc length and sector area also follow rather simple formulas s = rθ and A = rθ²/2. This means that students can now add a column in which to create a formula that references their recently obtained radian value to calculate arc length and sector area respectively given any angle and a specified radius.

To emphasize the importance of using variables rather than numbers, I also prompt students to think about how they could make a better formula to accommodate my devious habits of changing what is usually a fixed value (in this case the radius or time elapsed). There are a couple ways of doing this but I will leave that as the topic of a later entry.

Calculating angular and linear speed

Angular and linear speed also follow simple formulas ω = θ/t and v = rθ/t = rω = s/t. Because the last formula has three different expressions, there is more than one way students can solve for it and each of them equally correct³.

¹This is the number one concern I receive from teachers about letting students do tests online; especially since I allow students to use open book, open notes, AND open internet. The only restriction is they can not use any service that allows them to communicate directly with another person -e.g. messenger, screen share, etc.

²Although it’s not particularly difficult to remember by itself, when thrown into the list of other formulas students are required to know, it is easy to mistake for its inverse.

³Computationally speaking, one version will actually be more efficient than the others in terms of the computer’s processing power, but I do not find the distinction worthwhile in this particular instance as the effect is negligible here.