KH: My name is Kelly Hankins. I teach at Hayes Middle School in Albuquerque, New Mexico and I teach 7th grade math.
Hayes is a wonderful school. Our children come from very diverse backgrounds. We have 22 different dialects spoken here. This is my fifth year of teaching at Hayes.
KH: Okay, good morning my mathematicians. I want to talk about the lobos. Who in here likes the Lobos? This is my first year using Math Thematics and last year at the math department we really spent a lot of time going through the different books that were offered to our district and Math Thematics was the one that really stood out. We love the idea of how all the math concepts would be centered around a certain theme.
Student: The New York Nicks.
Student: I like the Lobos because they are New Mexico’s team, how can you not like them.
KH: Good point. Okay, I know you guys know about basketball, but let’s make sure everyone’s on the same track. Here’s our basketball court. What are the different points that you are able to score in basketball? Adrian?
Adrian: The free throw.
KH: A free throw. Tell me what a free throw is and where you would stand and how many points it is. So put it up there.
Adrian: You’d stand right here and you’d get it if you have the ball and someone touches you.
KH: When do they get to just take one shot?
Adrian: When they make a foul, which means they make contact with you when you have the ball.
KH: Okay, thank you Adrian. So I’m going to put up your 1 point. Okay, what else, what else is possible?
Student: 3 point shots? The 3 point line.
KH: Can you come write that on the board?
Student: Yes. Right here. Anywhere right there.
KH: Do we have to be right on this line right here?
Student: It can’t be on the inside of that line? It has to be outside of the 3 point line.
KH: So 3 points. Any other possibilities? Alicia?
Alicia: You can score within the paint, which would be 2 points, if you make it.
KH: Okay, come show me. In the paint. You play basketball.
Alicia: Yes. Anything, right here, would be considered in the paint. And that would be 2 points if you make it.
KH: Okay, thank you. Let me get that 2 points. Someone says you can make a lap, what’s a lap? Fracinda?
Fracinda: Where you run and you make the basket.
KH: And that was worth how much?
Fracinda: 2 points.
KH: 2 point? Okay. So that would be in the paint, like you said?
Alicia: Yes.
KH: Okay, so everything outside the circle is 3 points, everything inside the circle is 2 points, and right here 1 point, for a free throw.
Right? But now what we’re going to do is we’re going to play a little basketball game of our own. So let me hand you a piece of paper and scrunch them up, make your basketball.
JW:
Mathematics is developed in the thematic modules, rather than chapters.
In the Math Thematics you have modules such as search and rescue for recreation
which is the module that the lesson that you’re seeing today was taken
from. We look at the various ways that people spend their recreation time
and the mathematics that’s related to that. So hopefully the children
see that the mathematics is really related to their lives, that it’s useful.
We initially chose the mathematics that needed to be developed at that
grade level, so the mathematics came first and then we chose our themes.
KH: So the game we’re going to play is trash ball and I’m going to give you each get a frequency table, remember when we did frequency tables?
Students: Yes.
JW: The lesson that you see today is from the statistics strand, the data analysis strand in Math Thematics. It builds on the material that was first introduced in the 6th grade. Students worked with bar graphs and also took a look at frequency tables, frequency distributions.
KH: I’m going to set the trash cans up around the room. Now you’ll notice these orange sheets around the room. This is one point, you have to stand behind it and then you shoot towards the baskets, and then we have our 2 points. So the 3 point is the furthest one. Our 2 points is a little bit closer and 1 point is very close. No laps, no dunks, just one shot from the line. Whistle!
The
object is to shoot for 1 minute and ½, as fast as you can, everybody in
your group and you will shoot from a point into the basket. And depending
on where you are standing, every time you make a basket you get 1 point,
2 points or 3 points. Then, when I blow the whistle, you have to stop
and you rotate to the next station where you would repeat the process.
We have one 1 point basket, a two 2 point basket and two 3 point baskets.
Every time you make a basket you must record it on your frequency table.
(?) modules, recreation –
Student: Last one, I’m going to make a count!
KH: The original lesson in the book was about running, however right now it’s basketball season and we love basketball here in Albuquerque and so we chose trash ball for a recreation for our students to gather data.
JW: Math Thematics allows teachers to really adapt the program and lessons to their students both the things that the students are interested in as well as the needs of their students.
KH: Okay, back to your seat please, no more shots! Everybody has their tallies filled out. So let’s complete the rest of the chart. We have frequency. What does frequency stand for?
Students: The number of tallies.
KH: So do that right now. Add up the number of tallies, and then once we have our frequency filled out we need to fill in points, so let’s say that I had a 2 on the 2 point under frequency. What is going to be my total points?
Students: 2!
Students: 4!
KH: I hear 4 and 2. Okay, Michelle, how do you get 4?
Michelle: Because if every tally is worth 2 points, so if you have 2 tallies that would be 4 points?
KH: Right, perfect. So when you have 3 points, what am I going to do there?
Student: You multiply its tally against how many you have and how much points it is.
KH: Perfect, so go ahead and do that and fill in your total scores.
Student: I got 23.
Student: How?
Student: I got 14, 6 and 3.
Student: 14, 15, 16, 17, 18, 19, 20, 21, 22, 23.
Student: Plus 14, 6 plus 6 which is 12 and 14 and 14 plus 12 is (26?)
KH: So what we’re going to do now is I need to know your data, so I’m going to call on you and I’m going write it up on the board. So let’s start with Adrian. What is yours?
Adrian: I had 30.
KH: Go ahead, Joel?
KH: Math Thematics has really had an influence on my teaching. I feel that I can be more creative with the ideas, because I will go through the book and get some ideas, but then more ideas will just generate from there.
KH: Robert’s table, quantos (?).
I find that I’m a more interesting teacher, because my students bring in their own experiences and we build on top of that.
Is that everyone?
Students: Yes.
KH: Now we need to look at our data. What do you notice about the data? Leonardo?
Leonardo: (in Spanish) 52 points.
KH: So 52 would be the highest, and what is mass chiquitos, the lowest?
Michelle: Leonardo: 10.
KH: So 10 is our low and 52 is our high, so what would be a good score in here?
Students: 30.
KH: 30 would be good. So then 52 are in the 40s, what would that be? Alicia?
Alicia: It would be average?
KH: Are we having a hard time seeing this data right now?
Students: Yes.
KH:
Okay, so let’s go ahead and organize it. So I am going to, put a line
here. I am going to put in a zero, did anyone score 10?
Students: Yes.
KH: I’m going to put a 1 up. Looking at our data, is there anyone
in the 20s?
Students: Yes.
KH: So I’ll put a 2 up. What about the 30s?
Students: Yes.
KH: Should I put a 3 up?
Students: Yes.
KH: 40s?
Students: Yes.
KH: And anyone in the 50s?
Students: Yes.
KH: Should I put a 6 up here?
Students: No!
KH: Why not?
Students: Because there’s isn’t(?) more than (60?).
KH: So it would be okay to stop right here?
Students: Yes.
KH: So if I don’t put a 6, everybody’s score will still be represented on this.
Students: Yes.
KH: Okay, so I won’t put a 6 up. So now what I want is each of your individual scores to be represented up here. So let me show you what I did. I scored 38 when I played it. So I’m going to write 38, and what I want you to do is in your brown box, you have some stickets, some post-its and I want you to take the marker and to write your score, write it large, so we’ll be able to see it. Look at my –
JW: Throughout all of the material, the focus and the statistics, data analysis strand is on being able to use statistics to really answer a question. Students started with a question about whether a particular score in a basketball game would be a good score or an average score. And how do you answer that question? How can the statistical display help you to answer that question, and that led into the fact that you had to organize the data in some way.
KH: Okay, now we’re going to take the information again and I’m going to come over to this chart. I’m going to do a bar graph over here, what should I start off with?
Students: Zero.
KH: And then?
Students: 10.
KH: Okay, and it’s going to be the 10s and what is it?
Students: (?).
KH: And remember any time you’re setting up your data you want to do it in equal intervals. I’m going to take my sheet and here’s my score, 38. I’m going to fold it in half, and then I’m going to take my scissors and cut it. Now, remember my score is 38. Let’s say I take my 8. Where do I need to place it along here so that my original score would be represented?
Students: By the 3.
KH: Okay. Why do I need to put it by the 3?
Students: Because your original number was 38.
KH: So I’m going to put the 8 right there. Now, what’s my score?
Students: 38.
KH: So what do you think this side stands for?
Students: 10.
KH: Okay. So then I’m going to take my three and what was it again?
Students: 30.
KH:
And I’m going to put it over here, on this graph. And when you come
up I want you to place your score so it will be represented over here
and also so that on top of here so that you will make a bar graph. Okay?
Any questions?
When we collect the data it was random, so we had a pretty chaotic mess up there, so it was important for the kids to be able to organize it themselves and see how we could interpret data by putting it into a stem and leaf plot.
KH: What did the person score?
Students: (?).
KH: Okay, what about this person?
Students: 22.
KH: And you guys got it, right? So what does this side represent?
Students: 10s?
KH: And this side?
Students: 1.
KH: In math this is called a stem and leaf plot. Now, why do you think we would call it a stem and leaf plot?
Students: The data is like kind of coming out of the 10s, so it kind of looks like a plant or something.
KH: Good. So what do we call this side?
Students: Stem.
KH: And what will we call this side?
Students: Leaf.
KH: So stem and leaf plot, what does that remind you of?
Students: A tree.
KH: What does this represent?
Student: (?).
KH: And what would this represent?
Student: 1.
KH: Okay, good. So now let’s go over some information that we have here. Let’s look at this and what did these 4s represent?
Students: 40.
KH: And how many 40s, how many people scored in the 40s?
Students: 3.
KH: And how many people scored in the 30s?
Students: 1 (?) 5.
KH: 1, 2, 3, 4, 5. Okay, and what do you notice, how many people scored in the 20s?
Students: There’s a mistake.
KH: There’s a mistake in here. Okay, what is the mistake?
Students: 6.
KH: Monica, why don’t you come up here and fix it.
So how did you know that the 6 was wrong?
Monica: Because there was only one 2 over here and the 6 was over there.
KH: Good job, so let’s go ahead and go back to our data here and every time we’ve done a plot or a graph or a line graph or a bar graph, frequency table, what do I always tell you everything needs? Kate?
Kate: A title and labels.
KH: So what do you want to call this?
Students: trashketball.
KH: Okay, trashketball. Okay? Now, if someone didn’t know what this was about, would they be able to tell by your title trashketball?
Students: No.
KH: Okay, so we need to add something. Because I don’t know if this is the number of players, the numbers of teams, we need to add something and be real specific.
Students: Scores?
KH:
Now we know what this is about. Trashketball scores. What about someone
who has never seen this before, how would they know that this number right
here represents 52? Michael?
Michael: You could put like on the side of it like 10s, 20s, 30s, 40s and 50s?
KH: So I need to indicate that this is 10s, and this is 1s. So when I do a bar graph or when you do a bar graph and let’s say that you added color to it and a certain color represents something different, what would you put on that graph to help the people reading it?
Michael: Put a key?
KH: Okay, so let’s make a key. Now, I’m going to take a score from here and we have, let’s just take the 52 and the 5 and the 2 and let’s write what it stands for. Would that be clear. Good, let’s move over to this, because we don’t know what this is about. So what do you want to call this?
Michael: A trashketball graph.
KH: Sure. Do I need to put the scores also?
Michael: I guess.
KH: Scores, is that good?
Students: You forgot the (?).
KH: Okay, thank you guys.
So let’s go ahead and go back to our data on our stem and leaf. Yes, Michelle?
Michelle: I have a question on the stem part, on the board. How would like they know if it was like 18 and 15 and 16 and 13 and 10 and 6? How would they know it wasn’t 4,302, but without the lines, because it’s 1 big number.
KH: Okay. Do you think it would be better if we chose, let’s say the 4 and the 3 in our key. Do you want me to write 4, 3 instead of 4, zero?
Student: That way it could show that you have to have a line in between the numbers. So it’s like 40 and 42, so that way it shows the lines. So you don’t think it’s all one big number.
KH: Okay, so Michelle, you wanted 40, right? 40. And then we’ll have to change our points over here. That’s 40. That’s a great point. Do we all see what she was saying about that? Okay, so we’re going to make it 40 so nobody thinks that this is 4,302. And so if we write 40 equals 40, then they’ll know that the 4 and the 2 would be 42 also.
So let’s look at this data over here and we went from this mess(?) mass(?) and we put it over here. What do you think, is this better?
Students: Yes.
KH: It’s more organized. Okay, well could we make this organized anymore? Kevin, what do you think?
Kevin:
To put them in order, from least to greatest.
KH: Kevin can you come up and put it in order for us?
KH: Now is this right, does everyone agree?
Students: Yes!
KH: Okay, so this is a better organization of our data.
Students: Yes!
KH: And now we can make some observations because it’s easy to read?
Students: Yes!
KH: Okay. Let’s talk about the shape of the data now. Let’s compare this graph shape compared to this graph shape. Is there anything you notice? Is there anything you want to say? Robert?
Robert: It’s like you just flipped like instead of being strait like this, you turn it to have it like that.
KH: Right and what that is, this makes a bar graph. But what do we need in order for it to be a real bar graph? We always need to do something. Rocinda?
Rocinda: You need to put like zero and then 5 and then 10.
KH: Good! So we’ll have to label this access. Now that our data is organized in this plot we can take some time to analyze it. It’s not easy to analyze it when it’s like this. So let’s go ahead and erase this. And we’re just going to use our plots to talk about our data now.
From looking at our stem and leaf plot, can we tell what the high and the low, the highest and the lowest score is?
Student: 10 and 52.
KH: So we have our high as 52 and we have our low as 10. So what’s the amount of points, the difference between the point? Adrian?
Adrian: 42.
KH: 42, tell me how you got that Adrian?
Adrian: I just subtracted 52 to 10 and got 42.
KH: So we have our high and low score, and the difference between the 2 is the range. So our range is 42. Now, let’s look again over here and see if we can notice anything about the data, their score that occurs more often. Martine?
Martine: 26?
KH: Okay and anything else, is that it?
Martine: 23.
KH: Wow, 23 happens twice and –
Martine: 16.
KH: So we have 26, 23, and 16. And in math we call this the mode. When a number appears more often than the others they call it a mode. Now, I’d like for you to open up your math books, and look up the word median.
I use the glossary often and we use it in different ways. Sometimes I’ll have the students draw pictures of the words and write the definition below them, so when they are studying it they have a visual example of what the vocabulary word is, also the Math Thematics provides a Spanish/English glossary and so I have found that very helpful. I think it’s important that a lot of the Spanish speaking students who are strong in their first language will read the definitions in their first language.
Kelsey, can I have you read that?
Kelsey: The middle item in a data set ordered from least to greatest. If there’s no single middle item the number halfway between the two data items, closest to the middle.
KH: Okay, can you put that in your own words?
I think that’s very important that the students formulate the concept in their own words, so that you know that they are really getting the idea of the definition.
Kelsey: If there’s no single or medium item, like I think it’s the number that is the closest is the median.
KH: What is it to you. What does that word remind you of.
Kelsey: Middle, I think.
KH:
So looking up here on the board, is there a middle?
Student: Yes.
KH: What would it be. Go ahead and take some time and look and see what you think it would be. And everybody else, I want you to look and find what would be the median.
Kelsey: Would it be 26?
KH: Let’s see, what does everybody else think?
Students: 23.
KH: 23 or 26? How many scores do we have up here?
Kelsey: 21.
Student: It wouldn’t be exactly even, because 21 isn’t even, so the answer would be between 10 and 11 or 10 ½, because half of 20 is 10. So that would be the closest answer, and plus 10 plus – 10 ½ plus 10 ½ would equal 21.
KH: Let’s try it. 1, 2, 3, 4, 5, 6,7, 8, 9,10, this is where we said, where you talked about, so let’s see from there. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. How many needs to be on the other side?
Student: A half needs to be on the other side to make the number.
KH: Watch again. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. We have 10 right here. Now, let’s say that we were to say, you said a half. So this would be a half, right? So you want to cut this one in half. So let’s start counting 10 from the other place then after it. After the one you want to cut in half. 1, 2 3, 4, 5, 6, 7, 8, 9, 10. Do we need to cut it in half?
Student: To make 21 we have to. Wait –
KH: Okay, imagine this, there’s –
Student: To make it exactly even we have to cut it –
KH: There’s 10 on this side, there’s 10 on this side, how many do I have?
Student: 20.
KH: And then how many in the middle?
Students: 1.
KH: Do you see it? So we have an odd number of data and so we have 10 and 10 so the 11th one is going to be our median.
We need to take the 2 middle scores and find the average and that will give you your median. And we can also find if there’s a cluster anywhere in here, so what I want you to do is analyze your data again and see if there’s a certain number that the data seems to cluster around. For example, 52. Were there a lot of scores in the 50s?
Students: No.
KH: Were there a lot of scores in the 40s? I want you to take a minute and be able to tell me where you think the data is clustering.
Students: Do you mean like where there were the most scores at?
KH: Yes, not a specific score, that’s the mode. We want to know a specific area where the data seems to be clustering.
Students: There’s 10s, 20s and 30s.
KH: Good. And that is actually one of your vocabulary words, cluster. And we have from the 10s through the 30s. But sometimes you’ll have a ton of numbers right here in the 20s. So then you would say that’s where the cluster is, it would be in the 20s, but in this case you are absolutely right it would be from the 10s to the 30s. We want to talk about the gap, but with the data being experimental there were no gaps in our data, so we’ll revisit these ideas.
So we got all this information, we’ve got the high, the low, the range, the modes, the median and our cluster. We’ve got all that from our stem and leaf plot. Now let’s not consider this plot anymore. Let’s move over here to this graph. And let’s ask ourselves the same questions. Can we find the high score? Michael? Can you find the high score?
Michael: Yes.
KH: What is it?
Michael: 5.
KH: That’s the exact high score.
Michael: I don’t know.
KH: What if you used your stem and leaf plot. Could you tell me what the highest score is?
Michael: 52.
KH: So you can’t tell what the highest score is by using this graph. Can you tell the lowest score?
Students: 10.
KH: Okay, in this hands, but can you tell what exactly the lowest score is?
Student: No.
KH: How about the range? Can you tell me the exact range?
Student: No.
KH: Because that’s the difference between the highest and the lowest. What about cluster, could you tell me where the data seems to cluster?
Students: Yes.
KH: Okay, what would it be?
Student: 20s and 30s?
KH: Okay. And we have the 20s and the 30s, is that it? Where else?
Student: 10s.
KH: 10s. Okay, so again, our cluster stays the same, because in cluster you don’t need to know the exact numbers, you want to know where the data is clustering, where it’s kind of bunching up.
When we went to the bar graph and tried to get the same information from it, that’s when we realized how powerful the stem and leaf plot is because it can give you the shape of the data, but not only the shape, it also gives you exact points from the data –
JW:
Too often, statistics is presented as just procedures. You learn how
to make a bar graph, you learn how to make a stem and leaf plot, the real
power of statistics and data analysis that you can take a problem, or
a question that you are interested in and really chose the statistical
tool that will enable you to solve that problem.
KH: Now you guys are going to play the game one more time. But this time you have to use the opposite hand. Okay, here we go.
Then
we shoot again and this time we use the opposite hand and the kids gather
the information again. Today’s lesson was a great way to learn about analyzing
data because even though we did something fun, the math concepts were
very strong and very clear to the students. Then it was their turn to
create their own stem and leaf plot.
Now I want you to work with one person and one of you is going to turn the paper over and you are going to use this to create a stem and leaf plot. What you want to do is make sure that you organize your data. Remember when it was in all different spaces over here and different numbers, we couldn’t really analyze it, so I want you to organized your data and make sure you have a key, make sure that you put a title and then I want you to answer these questions, tell me the high and low score, I want you to give me the range which is the difference between the high and low score. I want you to give me the mode if there’s more than one mode, then you’ll do that, the median, and the cluster and one thing I want to point out, I noticed over there that we have a 4 and there’s no 10s, so how are we going to write that, let’s say that we had a score of 4 here.
Student: (on?) zero
KH: Yes. So anything to the right of zero means that it’s less than 10. so now it’s your turn, show me what you’ve learned.
Students: --
KH: As I was walking around the room, I can make the assessment that some of the kids grasped the concept right away and some still need a little more work on it, but it will be developed more as we revisit the concepts.
Students: (?) 25.
Student: 30, 35.
Student: It was added to the range and the median and the mode.
KH: Michael, what is this number?
Michael: 19?
Student: The last number’s like 9.
Student: On all of them?
KH: What do you mean by the last number?
Student: You don’t need to put 1, just the 9, because this one represents 10.
KH: And that’s what you mean by the less number?
Student: Yes.
Student: (Spanish. (?) 25, 24 (?) 30 –
KH: The Math Thematics program has worked out great with my Spanish speaking students. There’s a lot of hands-on activities that the kids can jump into and really get involved, that doesn’t really take a certain language to express when they are working together as groups, and so this is actually an English as a second language strategy, to do group work and to do more hands-on activities.
Student: Isn’t’ the median the numbers between all of them?
KH: The middle median reminds you of the word middle.
Student: So what did you do? Did you do like 4 through –
Student: You count them all up, if the middle number’s 11 and then if there are 10 numbers before that and 10 numbers after that, then 11 is the median.
Student: So what’s the height?
Student: The height is 35 and the low is 4.
Student: Okay.
Student: And then range is 31 and mode is 24, 25, 30, and 8.
And then the median is 20 ½.
Student: How did you get 20 ½?
Student: I just divided 20 in half and then like the number after that was 11, no that’s not what I did. I just like counted these until I got to the 11th ones and it was 2, but then I don’t remember. I think it was like in between 20 and 21, so I just put 20 ½ instead of –
Student: Okay, and do you know what the next one is?
Student: The clusters are the 20s, and I guess the 10s too, you’ll probably put it –
KH: It’s very important that the students gather their own data so that the data has meaning to them and sometimes when you do that though, the data doesn’t come out pretty. For example, our mode, we had many different modes and so you just work with what you got. And even though the data may have been difficult, the kids still seem to get the concept.
Okay, you guys did a great job working as a class and working in your group, tomorrow we’re going to compare your scores from one hand and we’ll compare it to your scores from the other hand. So I just want you to think tonight about the 2 different plots and the 2 different high scores and low scores and we’ll compare those. And tonight what I would like for you to do is please open your math book to page 322, please look at number 14, the skyscrapers, and notice the 20 tallest buildings and I want you to make some conclusions about that and I want you to be able to answer A through D. Okay, have a nice day.
I love teaching because I love my kids. I love being a positive influence in their life and they are definitely a positive influence in mine. Mathematically speaking, I love to watch them grow and gain confidence about their own ability in math.
For further information about the Modeling Middle School Mathematics Project, please contact www.mmmproject.org.
Major funding for Modeling Middle School Mathematics is provided by the National Science Foundation.
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