inconstant, so our speedometer might read thirty miles per hour when we set out at nine o’clock, and eighty a minute later after we’ve gotten on the highway.”
“Speed limit’s only seventy heading out of town,” piped up a male student in the back row. “So you’ll be going zero after you get pulled over by the state trooper hiding behind the Buc-ee’s billboard.”
He smirked proudly as several of his classmates laughed, and Mia put him down as the cocky class clown. Class clowns could be useful for keeping the other students engaged, but they could also pose a distraction if you let them derail the conversation.
“What’s your name?” she asked him.
“Cody, ma’am.”
She winced inwardly at being called ma’am as she gave him an approving nod. “That’s a great point you’ve brought up. When the police set up a speed trap, they don’t care about your average speed over time, do they? They use a radar detector to clock your speed at the moment you happen to be passing them. In calculus, we call this instantaneous speed, which is speed at one specific instant in time.”
The nods were a little less vigorous now that she’d introduced the first new mathematical term, but most of the class seemed to still be with her. Good, because they were about to jump right in—hopefully before most of them noticed it was happening.
“What if your car didn’t have a speedometer?” she asked. “But it did have an odometer to tell you how many miles you’d traveled. Using just your odometer and a clock, could you figure out your speed?” Madison’s hand shot up again, but this time Mia pointed at a young Latino man who’d only tentatively raised his hand. “What’s your name?”
“Antonio.” He looked as nervous as Mia was, and she felt an immediate sense of kinship with him.
“Tell us how you’d do it, Antonio.”
His gaze darted around the room before he answered. “I’d subtract my starting mileage from my ending mileage and divide that by the time it took to get there.”
“Excellent.” Mia gave him her most encouraging smile. “Antonio’s just described the formula for calculating rate of speed, which you may remember learning in one of your previous math classes. It’s expressed mathematically as r, for rate of speed, equals d—distance—over t for time.” She wrote the formula on the chalkboard behind her, circled it, and then wrote it again as an equation, filling in the variables. “So if, for example, between nine o’clock and nine thirty—a period of one-half hour—you’d traveled thirty miles, your average speed would be sixty miles per hour.”
She turned around to make sure everyone was still with her. This was middle school math, but it was good to review it before introducing the next concept, which would be their first real calculus lesson. Mia was pleased to see most of the class was still with her, aside from one or two who seemed to be reading their phones. Overall, not a bad engagement rate.
Now it was time to do some calculus.
“What if I wanted to know the car’s speed at exactly 9:05?” she asked the class. “Is there a way to estimate that using only our odometer and a clock?”
The question was met by silence, but Mia was gratified to see a lot of furrowed brows and other evidence of mental wheels turning as they contemplated the problem. Several students shifted in their seats, sitting up straighter or leaning forward in concentration.
Now that she had them hooked, she showed them how to do their first approximation, writing out a formula for estimating the miles per minute for a five-minute interval. Once they seemed to have grasped that, she walked them through a second approximation, reducing the interval even further, and then a third, using a fraction of a minute.
By the end of the first fifty-minute class, she’d introduced them to limits, derivatives, and functions, which would form the basis of much of their work over the upcoming semester. Even better, most seemed to have followed along, judging by their body language. There were some groans when she assigned homework to reinforce the lesson they’d just learned, but that was to be expected. Overall, it felt like a success.
After the students had all filed out, Mia felt both drained and exhilarated. She practically ran back to her office, where she closed the door before collapsing into her chair in an exhausted heap.
She’d done it. She’d survived her first class.
And it had actually gone…well?
Maybe this job wouldn’t be so