The teacher has a student share a conjecture with the class: If the rate is seconds per step, Alex wins; if the rate is steps per second, Joe wins. Students talk in groups to try to figure out how she got this conjecture and whether it makes sense. During class discussion a student talks about 20 seconds per step and 20 steps per second. The teacher has two different students demonstrate each rate. A final student concludes that the goals are different for each rate.
00:00 Steps per second.
00:03 And if I use a seconds per step, Alex wins.
00:10 Seconds per step. Alex wins.
00:18 So that people will have numbers to work with were you working with these numbers right here?
00:24 O.K. Two minutes. Talk about her conjecture. Talk about what she believes is true.
00:32 Write down this on your paper. What is the steps per second. What is the seconds per step. Where did she get this idea from?
00:45 I know you want to share out right now. Talk about it.
00:49 Twenty seconds per step then you would be really slow but if you had more steps per second, then you would go faster.
00:57 Because if you had twenty steps in one second, then you would be like way faster.
01:05 I don't really disagree but I thought that Joe wins and that Sarah wins because I multiplied what Sarah got by two.
01:21 So twenty seconds times two is forty seconds and thirty-two steps times two is sixty-four seconds.
01:29 And then with Joe I did ten times for to get to forty seconds. So you had to do sixteen steps times four to get to sixty-four steps so they equaled the same.
01:45 O.K. Alex what did you do?
01:48 Well, the steps per second and the seconds per steps isn't any different
01:56 because for steps per second you are quantifying how many steps you can take per second and for the seconds per step
02:06 you are trying to see how many seconds for one step.
02:11 So if you had a higher seconds per step would you be faster or slower?
02:17 You'd be the same.
02:18 No you'd be slower.
02:19 You'd be the same because you're just switching around.
02:22 O.K. if you had twenty seconds per step then you'd…it would take twenty seconds to take one step.
02:30 If you had twenty steps per second it would be one second and you would walk twenty steps.
02:44 Don't you think.. do you still think it's the same?
02:48 Yeah if you turn it around. It would take you twenty steps for one second.
02:55 So Carlos what did you get? I already shared. What did you get? I'm just curious.
03:01 Well, how Michael was saying if you get the twenty seconds per step…
03:08 All Right. Let's share this out. Obviously people had ideas right away. Right away. So anybody like to share? Anyone? No one at all?
03:21 All right, let's start with Michael. Go ahead.
03:24 I think you wouldn't want to have higher seconds per step because if you had twenty seconds per step then it will take you twenty seconds to go one step.
03:37 But if you had…
03:38 Hold on. Hold on.
03:40 Mikey, stand up and show us what twenty seconds per step would look like.
03:56 Would you agree that is close to twenty seconds per step? Aiden, stand up in the back and show us what…
04:03 Twenty steps per second.
04:04 Thank you.
04:11 Twenty steps per second.
04:18 Pretty good. Pretty good. All right.
04:28 Hold on for a second. Go ahead Michael.
04:32 And it would be better to have, if you are in a race, twenty steps per second because you would, obviously, go a lot faster
04:40 and…a lot faster than twenty seconds per step.
04:45 O.K., O.K. What do you think was the confusion here? Aiden?
05:01 I think the confusion was with the idea of if you switch it around it's one hundred percent different.
05:15 I think that was the confusion.
05:18 I'm not sure I understand. How many of you would like Aiden to explain it in a different way? O.K. Try again.
05:25 Well, with the idea if you switch it around the goals become different. I think that was what was confusing.
05:40 I agree with you. I think that's…yeah. Did you want to add something to it.
05:43 No I had something for steps per second.
05:46 O.K. Go ahead.
05:48 Well, I think that Joe didn't just win. That Sarah did too because, what I did was…
05:56 For Joe I did ten…no…for Sarah I did twenty times four…I mean, twenty times two to get forty seconds.
06:10 So Sarah was thirty-two steps in twenty seconds. Uh-huh, and then what did you do?
06:18 And I times twenty seconds times two to get forty seconds. So I had to do the same with thirty-two steps.
06:27 Thirty-two steps times two and I got sixty-four. So it was sixty-four steps in forty seconds.
06:36 So for Joe I did the same. I did ten seconds times four to get forty seconds. So I had to do the same for sixteen steps. Sixteen times four is sixty-four.
06:59 All right. Go ahead.
07:04 So then I did sixteen steps times four and that got me sixty-four. They each had sixty-four steps in forty seconds.
07:22 O.K. So Sarah and Joe are equal. Is this what you are saying? That's a nice comparison. O.K.
COACH LINDA FISHER: The research by Bell and Swan, Shell Centre at Nottingham, suggests that students benefit by looking at misconceptions. This is a good demonstration of how everyone benefits from exploring the thinking behind the different rates and how the goals of fastest is different depending on the rate chosen. The teacher takes the student thinking and uses it to pose a problem for the whole class to re-engage with the idea of the meaning of rates and how to compare them. Are the larger numbers faster or are the smaller numbers faster? Why?