Thinking about rates and times

As we were traveling back from track practice tonight, my son and I had a fairly natural conversation about driving on the highway.

Son: “How fast are we going right now, Daddy?”

Me: “According to my speedometer, we are going 2 miles over the speed limit. Oops!”

Son: “So if we were traveling for a whole hour, we would only go 2 miles??” Here my son is double-checking his reasoning with me and thinks I might be a bit crazy to suggest that we are going only 2 miles an hour. He has a strong sense that the speed he can observe outside the car does not match the calculation he has done.

Me: “No, I mean we are traveling 52 miles an hour and the speed limit is 50 miles an hour, so we are going 2 miles an hour too fast right now.”

Son: “Oh. Okay.”

Me: “I have a question for you. If we were traveling at 60 miles an hour, how far would we get in ten minutes?” Now, if we were not in a car and I was not trying to strike up a conversation with my son, I probably would not have asked this question. In the context of driving a car down a highway, I think this makes sense as a conversation starter – at least to a parent who is a mathematician and a mathematics teacher.

Son: “10.” (nearly instantly)

Me: “How do you know that?”

Son: “Well, there are 60 minutes in an hour and we are traveling at 60 miles an hour, so ten minutes must be ten miles. I don’t know how to explain this, Daddy!” Actually, you did explain it just fine son, but somehow you feel your explanation is inadequate.

Me: “I think I understand. The miles and the minutes have to be the same.” My son did not articulate his thinking in this way. I don’t know if this was helpful or not, but I felt that he wanted me to revoice what he had said using different language.

Son: “Yeah.”

Me: “I have a harder question.”

Son: “Okay, but I want it to be just slightly harder not a lot harder, okay?”

Me: “Okay. If we were traveling at 50 miles an hour, how far would we travel in ten minutes?” This question comes straight from a video I watched of Magdalene Lampert teaching during a workshop she ran over the summer. I was curious, how might my son approach this problem?

Son: “Hrmm. Maybe 9 miles an hour?” I love that my son’s first instinct is to estimate.

Me: “How do you know that?” I always ask this question, so this is normal for my son, rather than being like a quiz. I’m not quizzing him to find out how he understands, I am genuinely curious.

Son: “Well, it has to be less than 10 miles an hour. 50 is less than 60 and 9 is less than 10.”

Me: “How could we double-check that?” This is the teacher in me. I’m never satisfied with the first explanation.

Son: “We need to find nine times six. Hrmm. Nine plus nine plus nine plus nine plus nine plus nine.” My son was probably counting these nines in his head but he would have had to keep track of how many nines he was counting a couple of years ago.

Me: “What is that?”

Son: “Nine plus nine. Eighteen. And nine. Uh… I don’t know.”

Me: “Can you work it out? Also, I recommend paying attention to the answer.” This comes from MP7 in the Common Core and George Polya’s heuristics for problem solving — pay attention to the form and structure of your answer to problems as it can make other problems easier.

Son: “Okay. 27. Oooooohhh. Plus 9. 36. Plus 9. 45. I see the pattern. Plus 9. 54. Plus 9. Oh I can stop.” Meanwhile, while my son added on the nines, for each addition I pointed out the associated multiplication fact.

Me: “So does nine work?”

Son: “No, it’s too big. Let’s try six.”

Me: “Okay. Go ahead.”

Son: “6 and 6 is 12. 12 plus 6 is eighteen. Eighteen plus six is 22. No, 24. 24 plus 6 is 30. 30 plus 6. No, it doesn’t work. It’s too small.” I continue repeating the associated multiplication facts for each multiple. 

Me: “What’s next?”

Son: “Let’s try 7. No, that will probably be too small. Let’s try 8. 8 plus 8 is 16. 16 plus 8 is 24.” At this point, I said that 8 times 3 is 24. My son stopped adding, “Really? Hunh.”

Me: “Yes.”

Son: “24 plus 8 is 32. 32 plus 8 is 40. 40 plus 8 is 50, no 49, no 48! Too small. It must be between 8 and 9.” Do kids that understand numbers only as counting objects recognize that there are other types of numbers in between the ones we use for counting? It seems to me that kids need plenty of experience with situations that require other types of numbers to motivate an intellectual need for those types of numbers. We don’t go look for tools, even thinking tools, arbitrarily; without some clear kind of need for the tool.

Me: “Okay.”

Son: “Hrmm. Maybe it’s 8 and a half. No, that’s too big because 6 halves is 3.”

Me: “Right, and then we would have 48 and 3 which is 51. Too big.”

Son: “It can’t be 8 and a quarter either. That would be 49 and a half.”

Me: “So it must be between 8 and a quarter and 8 and a half.” This is a key idea that I am sorry I did not let my son develop himself.

Son: “What about 8 and a third? Let’s see. YES!! 6 thirds is two, so it works! I got it! Daddy, that problem was not a ‘little bit harder’ than the other problem. It was WAY harder.”

I find it hard not to turn off my teacher self when my son and I are discussing mathematics. I find myself really curious about how he will approach problems and I often find myself straddling the line between pushing him to articulate his thinking and just being a bit annoying.

On the other hand… the teacher part of me is now curious; which mathematical standards and practices did my son and I uncover in our conversation together? I remember Magdalene Lampert telling me once that when she teaches, she tries to hold all of the standards for the year in her head while working with students because she is never really working on one specific thing at once; she is trying to connect all of the ideas of the year together.