Come on you guys!!

My name is Amy Doherty and I teach 8th grade mathematics.

Okay, we're going to continue our talk about means, median, modes and ranges. We worked a lot with that yesterday and now the problem that we have is we have 3 basketball players, player A, B and C. They played 7 games. Now you have this on your sheet in front of you.

Student: Where?

SJ: In our unit behind the numbers. It's our first unit in the MathScape program in 8th grade. They are coming from the 6th and 7th grade program where they have done a lot of work with data and statistics, so they have worked with means, medians modes and ranges. It's not a new topic for them.

AD: Could we take a quick second, get into your partners and we're going to fill out the rest of this chart.

SJ: The real focus for this particular unit is to take that data to calculate all the different measures and really analyze the data and look at it with a very critical eye. The focus is really on using the statistics to make a comparison and to use the data to support a particular argument.

Student: We put 16 there, so for player C what's the highest score?

Student: So what did you get for the --

Student: I got for the mean 17.

Student: For the mean 17.

Student: And so for player C's highest score, what is it?

Student: The highest score is 25.

Student: Okay, you do the mean and median and I'll do the mode and range.

SJ: MathScape is a curriculum that was written to capture mathematics and the human experience, that was kind of our guiding principle. To look for ways in which the everyday person used mathematics in their lives and to build those into mathematical experiences that would have some real relevance to middle school students. That we try to develop investigations and activities for the students that would provide a really accessible entry point for a student at every ability level, so that every kid could find a way to get started.

AD: Okay, now that you've all calculated the statistics for each of the basketball players, let's put it up here on the board. So for player B, can someone list the scores for us?

Student: The high score was 21. The median was 18. The mean was 17. The mode was 18 and the range was 8.

AD: Good, for player C.

Student: The high score was 25, the median was 16, the mean was 18, the mode was 14, and the range was 14.

AD: Now that we've calculated all of the statistical measures, we're going to use this information to compare the three basketball players and find out who is the best player. So what I'd like you to do is on the next sheet we're going to rank the players according to the different statistical measures. Which player had the best score, the highest score? Grecia(?).

Grecia: Player C and he had 25.

AD: Okay, and who was second best?

Grecia: Player B, he had 21.

AD: Okay, third best?

Grecia: Player A, he had 20.

AD: Okay, now what I would like you to do with your partners is to rank the rest of the measures. So rank by median score, by mean score, by the mode, and by the range. Take a minute to do that with your partners.

Student: I have player C first.

Student: And the second best would be player B with 17.

AD: Okay, let's fill in the rest of this chart. For the median, who was the best in the median, Amy?

Amy: B 18?

EH: We really enjoyed how problem solving drove this curriculum and how the children would be involved in problem solving and in inquiry-based experience. We felt that the mathematics would be more meaningful to them and that they would understand what was behind the algorithms instead of just rote learning and learning formulas and plugging into formulas, they would understand why we have formulas, where the formulas came from and how they were developed. So we really liked getting underneath and into mathematical understanding and problem solving.

AD: Okay, now for the mode, Gregor?

Gregor: B had 18?

AD: Yup.

Gregor: C had 14. And A had 12.

AD: Okay the last one we need to rank is the range. Jackie?

Jackie: The best was C 14, the second best was A with 9 and the third best was B 8.

AD: Why do we have so many hands up? Derrick?

Derrick: Because the best score is supposed to be B with 8 and A with 9 and C with 14.

AD: How come?

Derrick: And on the range you want the lowest, because that means that they are the most consistent and they score around the same number of points a game.

AD: Okay, I'd like you to take a look at this chart, and look at the way we ranked everybody. Depending on which measure we look at, the ranking is a little bit different. So I want you to think about that when we start thinking about who is the best player.

Coach: All right, guys, I need your help. I know you've been studying statistics and I know you have a better idea of the data than I do. As the coach of the girls basketball team I need to pick the MVP. I have one trophy, one winner, I need your help, go over the statistics, give me your best shot and tell me who you would recommend for the MVP award for this year?

AD: Coach, do you think we might be able to give it to two different players if they came out to be about the same?

Coach: If you can statistically prove that you think that two players are equally deserving of that award, then we'll consider it. But if certainly the statistics show that one player is far more deserving, then we're going to go with one.

AD: Wow, there we have it. We have a challenge. We need to write a letter to the coach and let her know who we think is the best player and who should get the most valuable player award. We could give it to one player, we could give it to two players, we could give it to all three players, or we could argue that no one deserves the award because they are all equally as good. Or it doesn't make sense to give out this type of award. So you need to work with your partner. We're going to work on some paragraph templates and we're going to write out a letter to the coach. So the topic sentence up here, in our paragraph template, will be the statement of who you think should get the award. Okay? Let's, to make it a little bit easier, let's call player A, Ann, we'll call player B, Betty, player C, Carol, okay, so we're all talking about the same three basketball players. Player A is Ann, B is Betty, C is Carol. So in the first topic sentence you have up here, it's the only one, so of course, it's the first. In the topic sentence, in your paragraph template, you are going to write who you think should get the award. Then you need to have three supporting details and your supporting details must include statistics. You cannot say, I think player B should get the award or I think Betty should get the award, because she tried hard every single game. Is there any statistic, any numbers that we calculated that shows how hard she tried? No. Okay, so these need to be based on the mathematical data that we have. I also want you to use the correct use of mathematical terms, words such as mean, median, mode, range and so forth. Okay? Once we you have all three of those statistical data and your clincher at the bottom is going to be an opinion that will persuade the coach to go with your argument. Any questions on what we need to do? Okay, let me pass out some paragraph templates.

Student: Dear Coach, in my opinion I think Betty should win the award. Betty was always ranked one of the highest in the statistics, her scores were always consistent and she showed this through her range, which was 8. Whenever she came in second, she was in fact, behind(?). Her high score was 21 and Carol's was 25. And her mean was 17 and Carol's was 18. I remember the time when she gave pointers to the team and stayed after and helped them work on their 3-pointer shots and now the team is doing much better. In conclusion, I think Betty should win the award. So do you see any mistakes?

Student: I think I should change, whenever she came into.

Student: Was ranked.

Student: Ranked.

SJ: The curriculum incorporates a lot of use of student writing. One of the main goals of the curriculum is to bring out student thinking about mathematics, both to inform the teacher about what the students are understanding and to help students begin to articulate the ways that they are understanding the mathematics and writing is one way that we can get out some of the student thinking.

Student: I think that Player B is the best player out of the three and should get the award because of her stats. The stats clearly show that she's the best, because her median is the highest and that shows consistency in her points. Secondly, her mean is only 1 point behind C and that proves nothing there for C. Plus her mode is 18. C's middle score is only 14, concluding that B is the best, not only by stats, but by how hard she tries and her consistency to make baskets. Do you see any mistakes?

Student: This is wrong because the median doesn't show, it's the range that shows if it's low and you have to (it's?) consistently if the range is low, but you wrote median.

AD: The peer editing is a really nice process that I have incorporated into the program because so often students get stuck in their writing and I find that sometimes if they work with a partner and read each others work and share each others ideas, it might give them enough spark to say, oh yeah, I really need to include that in my particular piece, or, oh yeah, I forgot to write that and that's a really great idea.

Student: Why don't you write those names instead of B and C -

SJ: When you are working as an individual and you don't have that time to peer edit or consult with a peer, then you really miss out quite a bit.

Student: Dear Coach, I believe Betty deserves the award more than anyone else. Statistics alone prove her to be the best. She is second on the team in mean, average, after Carol, but that's only by 1 point. That can't prove anything. However, if you look at the mode, a number that appears the most, then she clearly has the lead. Carol has scored 14 points the most times, but Betty has scored 18 points 3 times. Might I point out that 18 is also the highest average, which Carol has. That means Betty has matched the highest average in three different games. If you look at the range the difference between the highest and lowest numbers, then Betty has the smallest, which means she is the most consistent player on the team. Carol's range is 14. That's 6 more points than Betty. That is not a consistent player. In conclusion she is not only the best, but she is a team player and a hard worker and I think she deserves the award. Sincerely. Clint Mafizi.

AD: Great. Anyone else want to share? Kelly, come on up.

Kelly: Dear Coach, in my opinion, I think that Betty should be the player to win the award. Betty is a skilled player which shows in her ranks. She is a consistent shooter. I know that because her range, the difference between the largest and the smallest numbers and a set of data is 8. She also has the highest mode, the number that occurs most often in a set. Betty's mode is 18, which is 4 points higher than her teammate Carol.

Student: Dear Coach, I think players Betty and Carol deserve the YWA Basketball VP. Betty ranked the highest in median, mode and range. Carol ranked the highest score in the average. Ann doesn't get it because she came in last in median, mode, mean, but except the range that she came in second.

Student: Dear Coach, I believe that Player B and C should have the MVP award. I believe this, because out of the 7 games played she got the high score with 25 points, but Player B got 21. There is only a 2 basket difference. Another reason is the median, out of the 7 games, for B is 18 and C is just under, but the basket was 16 points. Again, does the basket really matter? Also, again, out of the 7 games the means plus C is 18 and B is right there with the 17. 1 point difference, wow!

AD: Okay, those are great letters. What I'd like to do now is collect all your work and I'm going to grade them according to the criteria that we went over and I think you really got a good sense of why we're using the statistics to really look at real-world data and make some sense out of the statistics that we gather.

AD: So we've been talking about means, median and modes for a couple of days now and we're going to work on another activity. But before we start I want to do a little bit of a review and we're going to talk about the number of hours you watch the Olympics. Okay, are you ready? Get into your groups. I'm going to give you a recording sheet, and what I want you to do is put the name and the number of ours you watch the Olympics and then for your group calculate the mean, mode, median and range for your group. What's the mode of our whole class? Derrick?

Derrick: 4?

AD: And now what is our range? Ho?

Ho: 9.

AD: How did you get 9?

Ho: 9 is the biggest number, 0 is the smallest number, 9 minus 0 is 9.

AD: And I'd like to know, how does that compare to the mode and the range that you found in your group?

Student: The range in our group was 5, and the mode was 3.

AD: And why do you think there's a big difference between your particular group and the whole class?

Student: We're only working with a few numbers. So there's less like, less of a variety of numbers because of other people.

AD: Did anyone else have a different mode and a different range from the class set? I think Jennifer's group had a really tiny range, didn't you?

Jennifer: Our range is 1.

AD: Okay, so what was the data set you had in your particular group? You had four numbers, what were they?

Jennifer: 1 hour, 1 hour, 2 hours, and 2 hours.

AD: Let's find out the mean for our whole class set. So if you could, at your tables, or in your groups, calculate the mean for all of the 21 students that we surveyed. Okay, what did we all get? Michael, what did you get?

Michael: 3.6 repeating.

AD: Okay, does everyone agree with that?

Class: Yes.

AD: The one thing we need to find now is the median. How do we find the median? Alex, why don't you come up and show me how you got the median? You can just rip them off. So you take one from each end, another one from each end.

Alex: The median is 3 hours.

AD: Great job. If we just looked at this group's data, it would not reflect the description of our whole class and if we look at the data that we collected for our whole class it looks very different than one particular group. The larger the sample, the better description that we can have of our whole class. When Alex was finding the median and taking off all the post-its, I took 5 pieces of data, okay and what happened was I looked at a group of 5 students, and I found the mean and the mean was 4. So my question to you now is what could a possible data set be that would come up with a mean of 4? I don't want an answer right away, what I want you to think about is some strategies. Throw out some ideas of how we could go about finding an answer that would fit this particular description here, the mean is 4, there were 5 students, how could we come up with the numbers that would make that set true?

Student: Maybe we should just try a random set of numbers. Maybe 1 plus 2, plus 3, plus 4, plus 3. And it comes out to 13.

Student: And we divide it by 5.

Student: The answer's 2.6.

Student: So we need a higher number.

Student: Yeah, we need higher numbers.

Student: All right, so we need to try one higher, so let's try 5 plus 5 plus 3 plus 3 plus 1.

AD: I think most of the students actually start with this guess a check and adjusting their answers and then checking again and it's a really nice strategy for them to fall back on. It will enable them to come to a solution.

AD: Okay, how are you guys doing?

Student: Good.

AD: What kind of strategies did you come up with?

Student: Well we found out that we needed to add five numbers to get 20.

AD: How did you know that?

Student: There were 5 students and the mean was 4.

AD: Right.

Student: So we had to figure out a way to get 20 with the 5 students.

AD: Where did you come up with the number 20?

Student: Because 20 is the only number that you can divide by a 5 to get 4 and we need to get 4 because of the mean.

AD: Okay. So what did you come up with for a solution set?

Student: We pretty much just picked a few numbers, see what they added up to and then see what numbers we needed to add up to 20.

AD: Okay, so what did you start off with?

Student: We started off with 2, 4, 8, 4, 2.

AD: Okay, how did you pick those numbers?

Student: Bessie and I just decided to try five different numbers that came off the top of our heads and added up to see if it came up to 20.

AD: Okay.

Student: And then when it did, which it did, we divided it by 5 and it was 4.

AD: Okay, it looks like you have 3 different solutions here.

Student: Yes, well actually Mike and Gregor thought of those other two ones.

AD: Okay, are there any others?

Student: It's so easy to think of any-- there's so many possibilities to pick from.

AD: Even though all four of these will work, because they all have 5 numbers, they will all come out to a mean of 4. There are two sets in here that can't possibly work due to the data that we collected in our class. Can you find which ones don't work and tell me why? Jennifer?

Jennifer: The one with 3, 3, 3, 10 and 1. And 6, 4, 3, 5 and 2.

AD: Why can't we use these?

Jennifer: Because on the chart we have nobody chose 6 or 10 for how many hours was watched of the Olympics.

AD: Even though this data set 3, 3, 10, (1?) will actually come out to a mean of 4, on our chart, on our line plot, no one chose, watched 10 hours worth of Olympics. That was not part of the numbers that I could have picked from. So this one is not a solution set for our particular class data. Now, I've collected some data about some different sporting events and it's all mystery data and I'm going to give you some information, some statistical measures, and you have to come up with the data sets that will match that particular group of information. So now you might get the mean, the mode, or the median or the range, all different types of measures. So what you'll need to do is come up with a data set that matches the measures in your mystery game. Okay? So what I'd like you to do is break up into pairs and I will give you your own mystery game.

Student: Sara competed in 7 gymnastic games, so 7 games total.

Student: Games -

Student: Are 7. 7.

Student: At each meet her performance is rated from 1 to 10, 10 is the best. Her mean was 8.8.

Student: She didn't get any 10's which is a perfect score. None of her scores were integers. So they can't be whole numbers.

Student: Yeah, so everything has to be like a decimal and a fraction (phrase?).

Student: So if her mean was 8.8 -

Student: And then there were seven games total, so let's multiply 7 times 8.8 with 1.6.

Student: And then what we have to do is find 7 numbers that can be averaged into 8 points.

Student: And they have to be less than 10, because she didn't get any 10s. We have to do 7.

Student: We have to just choose the number (?).

Student: (?) that are lower than 5.5.

SJ: Mastery of a basic is still a very important component of the MathScape curriculum, but we try to develop the curriculum in such a way that kids would be able to practice some basic skills in the context of some other mathematical investigation.

Student: Betty competed in 7 gymnastic meets, at each meet her performances rated from 1 to 10. 10 is the best. Her mean was 8.8. She didn't get any 10's, perfect score. None of the scores were intergens. What might each of the scores be? (Show it at?) work.

Student: So we have the mean of 8.8. And so we have 7 meets. How are we going to do it?

Student: I don't know.

Student: Okay, we'll have to balance it out then, so it's 8 right there. So it's 1, 2, 3, 1, 2, 3 -

Student: 8.9.

Student: 8.9. And then we'll go over to the other side?

Student: 8.7.

Student: 9.1.

Student: Okay, why would it be 9.1.

Student: It can't be 9, because there are no integers, so we'll only have 9.1. We skipped one over there. On this side it will have to have -

Student: 8.5.

Student: Yes, and then 9.2. 8.4. So check out the calculator if it works.

Student: 61.6.

Student: You want to check that that's right?

Student: 8.8 times 7 (meets?) it goes 61.6. (?)

AD: Okay, you guys all set? What did you come up with?

Student: 8.4, 8.5, 8.7 and (?) 7.

AD: Okay and what strategy did you use?

Student: We needed to balance out, since it was 8.8, in which 8 is a median also, we thought we needed to find out the other different numbers.

AD: Okay, what do you mean, balance it out?

Student: When I added these two numbers, 8.7 and 8.9, add them up, divide by 2, it came to 8.8, as far the same, these two numbers, 8.5 and 9.1, add them up and it came to 8.8 and so on.

AD: Great. Can I ask why you have the 61.6? What's that number for?

Student: We added up all the numbers and we got that number.

AD: But why (do you have the price??), you were doing something else when I was coming over.

Student: (?), before we added these two then we divided by 7, 61.6 and we had to find out if the mean times the number of meets was the same.

AD: So you checked your answer using a different strategy. What was the strategy you used to check your answer?

Student: To multiply the mean by a number of meets.

AD: And that's where you got the 61.6.

Student: Yes.

AD: Okay, you know that this answer had to be correct. Great, you did a great job.

Student: Sophie kept track of how many points she got in six ping pong games. None of the scores were higher than 21. The range is 10 points. The mean is 17 points. What might each of the scores be, show your work. Okay, so we have to find the range and the range has to be 10 points.

Student: We have to find six possibilities because she played in six ping pong games total. So you have to find the possibilities where they give us this hint and see if we can find the range of 10. And we could start out by making six lines. And then we should find some sets of data that would have a range of 10, so for instance, we would do 21 minus 11, and we can just keep going from there.

Student: We can do 10 subtract -

Student: We can do 10 subtracted by 0.

Student: 15.

Student: That would also give us zero. 15 minus 5.

Student: 20 subtract 10.

Student: The highest score, it can't be higher than 21, right?

Student: Yes, it can't be higher than 21.

Student: It has to average on 17 to be the mean, which could give you something like 10 plus --

Student: 10 plus 5 which would give you -

Student: 15.

Student: 15, and then no whole number would divide it by 17.

Student: It doesn't have any factors.

Student: 17 doesn't have any factors, but 16, (?),

Student: We can use decimals, can't we?

Student: No, they said we couldn't. We have to use all integers and we can't use any rational numbers, anything like that, so 17 --try something like 7 plus 2, 7 plus 2 would give us 9. 9 plus 4 -

Student: Would give us 13.

Student: You could do 8 plus 8 equals 16, but we can't like there's no other factors that would give us 17 as a whole.

Student: That's a good point.

AD: How are you guys doing?

Student: We're doing pretty good, but we can't figure out how this mean would fit with 17 without having a factor or anything. There are six ping pong games, but like how many numbers should we use or like are we just like we did in the exercise before this, do we just want to add up numbers like 7 plus 2 plus 4 until we get to 17?

AD: Well, the mean is 17 and what does, how do you find the mean?

Student: You add them all up and then divide them by the numbers you have.

AD: Okay, so how many numbers do you have?

Student: We have 6.

AD: Okay, we have 6. So we have to add all the numbers up, all six numbers and then divide by 6.

Student: And that would give us 17?

Student: But we'd have to keep the 9.

AD: Why do you have to keep the 9?

Student: Because it's so long.

Student: Because it's the lowest.

AD: Okay. And that has to do with the range of 10. Okay?

Student: So we could turn this one to a 15. And this one to a 14.

AD: I'm going to go to another group and if you have any more trouble, just raise your hand and I'll come back. Okay? Sounds like you're on the right track.

Student: We can add these up -

Student: Let's try these again, like we did the last time.

Student: 9 plus 13 is 15 is 17, 18, 19.

Student: I got 15.333.

Student: Since we're not getting anywhere so we can change the range, these two numbers and make sure it would be 21 through 11. So we can use higher numbers.

Student: Let's try that one then.

Student: The other one didn't work.

Student: So 11 would be here and 21 would be here.

Student: 11 would be near the beginning?

Student: Yes, since that's the lowest.

Student: That's where (?).

Student: We can use 19 before 21.

AD: How are you guys doing now?

Student: We changed the range to these two higher numbers since we didn't get anywhere to these, so they could be higher, because we kept them going up and up, but they wouldn't -

AD: And it still didn't get to a mean 17? So you changed the range. Did you keep it still 10?

Student: Yes, it's still 10.

AD: But you changed the number, that was a really great idea. Yeah. So how are you doing now? Getting closer?

Student: Yeah, we have to add these.

AD: Okay, let's see.

Student: I got 15.1.

AD: 15.1.

Student: So we could change this to a 20?

AD: Yes.

Student: This to like a 19, 18, 17, --

AD: Sure, let's see how we do that.

Student: Should I try to add those up again?

Student: I got 17.666.

AD: Okay, so now you went a little bit over.

Student: And we're really close.

Student: So we did get 17, but are we aiming to 17, exactly 17?

Student: 17.1. So then this has to be 13. And I got 17.

AD: Yeah! All right!!

SJ: When Susie finally got that a ha moment, it was really exciting for me, because I know now that she owned that. She understands where it came from and she was able to work it through on her own, she now owns that material. She will take another problem very similar to it and be able to take her experience that she worked with and bring it to that problem and she will remember it and be able to use that information again. And as a teacher I want my students to own the mathematics and understand where it came from. So often students come through here, they know the algorithms, they know the steps, but they don't understand where it came from or what they are doing and why they are doing it.

AD: As I was walking around you were working on your problems, I saw a lot of different strategies to solve these problems. So can we take a look at Mystery data game F. Okay. I would like some people to talk about their strategies that they used to solve these problems.

Student: I just guessed some numbers and I just did them to find the right mean.

AD: Okay, we guessed and we checked.

Student: Guess and check.

AD: And then what did you do after you guessed and you checked?

Student: I just did numbers increased and decreased them.

AD: You adjust the numbers. And then do what again?

Student: Check.

AD: Checked again.

Student: And then I adjusted them until I got the right mean.

AD: And you keep going. Okay, who else has another strategy.? Kelly?

Kelly: My strategy was to put the mean, 8.8 in the middle and then for the next score I put 8.7, which was 1/10 lower than the mean.

AD: Does it make a difference where I put it?

Kelly: Put it before the 8.8.

AD: Okay, and then what did you do?

Kelly: I put 8.5, which is 1/10 lower than 8.7, or 2/10 lower than 8.7. And the for the next score above that I put 9.1, since you can't have an integer.

AD: Okay, so that's why you skipped 9.6 here?

Kelly: Right.

AD: So what are the last two numbers?

Kelly: 8.4, and 9.2.

AD: What do you think we should call this method?

Kelly: Balancing?

AD: This strategy can work if we don't have any restrictions on our median. What about if there were only 6 games being played or there were 6 matches. How would it change?

Kelly: We'd have to add the 2 middle numbers into 6 as an even number.

AD: Okay, so you'd have two middle numbers in there. So you could put the mean in twice. Or you could start off by finding two numbers that will have a mean of whatever the mean in that data set is. There were some other strategies that people used. What's another strategy that people used? Michael?

Michael: The strategy I used was, I took the mean and multiplied it by the number of games played.

AD: Why did you do that?

Michael: So that gave us the number that we would have to get to. So if we divided it by, if we found the average or the mean of it, it would come out the to the number we needed.

AD: Some of the more sophisticated strategies come as a result of doing several of these types of problems, when they say, hmm, I don't have to guess and check all the time. The only way that I'm going to get this particular mean is by doing this particular division problem. And once they make that connection, then they move to another method.

AD: Okay, so we have three different strategies to solve the problems. First we can guess and check and adjust and check again and so forth. I think that's how most of you started off the problem. Then we have the second strategy, we call it the balancing method. We first have the mean in the middle and then we picked two numbers that have a mean of 8.8 for this particular problem, or whatever mean you are looking for.

AD: This particular method has some limitations to it. So if you are going to use this strategy, you need to make sure that it will if the context of the problem as given.

AD: The third method is, mulitplying the mean times the number of games played or events and then just keep adding numbers until you get to that number. In this case it was 61.6. okay, you guys did a great job with that. What I want you to do for homework tonight is to make up your own mystery data game. So I want you to think about what are the essential things that you need to include in your mystery data game so that someone else will be able to solve it.

AD: When I started working with the MathScape units, it was amazing to me that the students were so engaged in their mathematics that they really and truly were able to sit down and work through a problem and when I saw the students working so hard and so diligently with their problems, I said, this is something that really works. Once I became a facilitator for the students and helped them work through a problem and not simply give them a strategy or a solution, then they really worked through a problem and they worked so hard and they have such a good feeling that they were able to work through it by themselves and help each other out that that's when I knew that this program is really going to work.

For further information about the Modeling Middle School Mathematics project, please contanct www.mmmproject.org.

Major funding for MMM is provided by the National Science Foundation. END