Last Thursday, I spent about 10 minutes with one of my students on what I thought started out as a simple question.
She had written, "200 cents" as an answer for one of her questions. I asked her, "What is it another way we could say that?"
She stared at me uncomprehendingly. I tried to clarify. I said, "That answer is not wrong, but usually, when the number is that high, we use a different unit for money. Can you say 200 cents a different way, using that other unit?"
She thought for a moment and then responded, "200 dollars?"
Since cents and dollars are not the exact same thing, her response began the long, complicated discussion of how many cents are in one dollar. Getting that answer out of her was like pulling teeth. Teeth that apparently would have wildly fluctuating values for the tooth fairy.
I asked her to imagine that she had one dollar in her pocket, and I asked how many cents that would equal to. She corrected me, and informed me that she actually had seven dollars in her pocket. My mistake.
I tried a different tact. If she were to give me one of her dollars, how many cents what I give her for it to be a fair trade? One cent was her answer. I questioned that logic.
"So you could give me one dollar bill, and I could give you one penny, and that would be fair?"
Her head said no, but her eyes said she didn't know.
Her next guess was that one dollar was equal to twenty five cents. So I took a quarter out of my pocket, placed it on her desk, and asked, "This equals one dollar?"
"Four cents?" was her reply. Although she was grossly wrong, I thought I understood where that answer had come from, and that it meant she was at least stumbling towards the right path. Sure enough, when pressed further, she confirmed that she had gotten that last answer by adding the quarter four times.
So I asked her to write down twenty five cents four times on her paper and add them up. She did that and came up with -- gasp -- 100 cents.
"Yes, one dollar equals 100 cents," I confirmed. "So how many dollars with 200 cents be worth?"
At that moment, I honestly felt like I was stuck in the middle of a MasterCard commercial.
"Everyday Math Journal -- $1.50
Demonstration Quarter -- $.25
Incomprehensible Mathematics Conceptual Error -- $7.00
Bleeding Head Wound, Caused by Pounding My Head on the Surface of the Desk -- Priceless"