Two weeks ago, my summer school Algebra 2 students were exploring sequences and series. A problem I thought would be a routine check on students’ ability to compute the sum of a finite arithmetic series morphed into an experimental measure of the computational speed of the TI-Nspire CX handheld calculator. This experiment can be replicated on any calculator that can compute sums of arithmetic series.
Teaching this topic in prior years, I’ve found that sometimes students have found series sums by actually adding all of the individual sequence terms. Some former students have solved problems involving addition of more than 50 terms, in sequence order, to find their sums. That’s a valid, but computationally painful approach. I wanted my students to practice less brute-force series manipulations. Despite my intentions, we ended up measuring brute-force anyway!
Readers of this ‘blog hopefully know that I’m not at all a fan of memorizing formulas. One of my class mantras is
“Memorize as little as possible. Use what you know as broadly as possible.”
Formulas can be mis-remembered and typically apply only in very particular scenarios. Learning WHY a procedure works allows you to apply or adapt it to any situation.
THE PROBLEM I POSED AND STUDENT RESPONSES
Not wanting students to add terms, I allowed use of their Nspire handheld calculators and asked a question that couldn’t feasibly be solved without technological assistance.
The first two terms of a sequence are and . Another term farther down the sequence is .
A) If the sequence is arithmetic, what is k?
B) Compute where is the arithmetic sequence defined above, and k is the number you computed in part A.
Part A was easy. They quickly recognized the terms were multiples of 3, so , or .
For Part B, I expected students to use the Gaussian approach to summing long arithmetic series that we had explored/discovered the day before. For arithmetic series, rearrange the terms in pairs: the first with last, the second with next-to-last, the third with next-to-next-to-last, etc.. Each such pair will have a constant sum, so the sum of any arithmetic series can be computed by multiplying that constant sum by the number of pairs.
Unfortunately, I think I led my students astray by phrasing part B in summation notation. They were working in pairs and (unexpectedly for me) every partnership tried to answer part B by entering into their calculators. All became frustrated when their calculators appeared to freeze. That’s when the fun began.
Multiple groups began reporting identical calculator “freezes”; it took me a few moments to realize what what happening. That’s when I reminded students what I say at the start of every course: Their graphing calculator will become their best, most loyal, hardworking, non-judgemental mathematical friend, but you should have some concept of what you are asking it to do. Whatever you ask, the calculator will diligently attempt to answer until it finds a solution or runs out of energy, no matter how long it takes. In this case, the students had asked their calculators to compute values of 8,388,608 terms and add them all up. The machines hadn’t frozen; they were diligently computing and adding 8+ million terms, just as requested. Nice calculator friends!
A few “Oh”s sounded around the room as they recognized the enormity of the task they had absentmindedly asked of their machines. When I asked if there was another way to get the answer, most remembered what I had hoped they’d use in the first place. Using a partner’s machine, they used Gauss’s approach to find in an imperceptable fraction of a second. Nice connections happened when, minutes later, the hard-working Nspires returned the same 15-digit result by the computationally painful approach. My question phrasing hadn’t eliminated the term-by-term addition I’d hoped to avoid, but I did unintentionally create reinforcement of a concept. Better yet, I got an idea for a data analysis lab.
They had some fundamental understanding that their calculators were “fast”, but couldn’t quantify what “fast” meant. The question I posed them the next day was to compute for various values of k, record the amount of time it took for the Nspire to return a solution, determine any pattern, and make predictions.
Recognizing the machine’s speed, one group said “K needs to be a large number, otherwise the calculator would be done before you even started to time.” Here’s their data.
They graphed the first 5 values on a second Nspire and used the results to estimate how long it would take their first machine to compute the even more monumental task of adding up the first 50 million terms of the series–a task they had set their “loyal mathematical friend” to computing while they calculated their estimate.
Some claimed to be initially surprised that the data was so linear. With some additional thought, they realized that every time k increased by 1, the Nspire had to do 2 additional computations: one multiplication and one addition–a perfectly linear pattern. They used a regression to find a quick linear model and checked residuals to make sure nothing strange was lurking in the background.
The lack of pattern and maximum residual magnitude of about 0.30 seconds over times as long as 390 seconds completely dispelled any remaining doubts of underlying linearity. Using the linear regression, they estimated their first Nspire would be working for 32 minutes 29 seconds.
They looked at the calculator at 32 minutes, noted that it was still running, and unfortunately were briefly distracted. When they looked back at 32 minutes, 48 seconds, the calculator had stopped. It wasn’t worth it to them to re-run the experiment. They were VERY IMPRESSED that even with the error, their estimate was off just 19 seconds (arguably up to 29 seconds off if the machine had stopped running right after their 32 minute observation).
HOW FAST IS YOUR NSPIRE?
The units of the linear regression slope (0.000039) were seconds per k. Reciprocating gave approximately 25,657 computed and summed values of k per second. As every increase in k required the calculator to multiply the next term number by 3 and add that new term value to the existing sum, each k represented 2 Nspire calculations. Doubling the last result meant their Nspire was performing about 51,314 calculations per second when calculating the sum of an arithmetic series.
My students were impressed by the speed, the lurking linear function, and their ability to predict computation times within seconds for very long arithmetic series calculations.
Not a bad diversion from unexpected student work, I thought.