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LESSON 8: Introduction & Collection of Class Measurements
DAY 16 (3/10)
This begins section III of this unit. The purpose of this section is to have students take a closure look into how statistics is used to predict the likelihoods of events or outcomes. Students thus, far have explored how probability is used as a tool for prediction. They have also conducted surveys and explored how data collection can effect conclusions. The next, step is to move further into statistics, specifically exploring correlations and their interpretations. My goal for the first day was to discuss with students their thoughts and ideas of certain relationships that they may have heard of previously, but really never explored.
I handed out the Relationship/Correlation Packet (Appendix M). I asked them to read the first page. This immediately struck up a conversation:
Teacher: Do you think the detective is right to claim this relationship between height and foot? Jessica: not all tall people have big feet-so no its not true. Teacher: So b/c not everyone may fall to this description, you are claiming theres no relationship what so ever? Jessica: No there may be one but it's not always true. Teacher: I can agree with that. I know you guys know people who are tall with small feet and vice versa. So how do you think we could find out what kind of relationship exists between height and foot? Ben: We'd probably have to take allot of measurements and do something like find the average. Teacher: An average, so what compare the average height with the average foot size. I don't know really how much that will give us-it looks like it will just tell us the average height and foot of allot of people. Well, we are going to try to sort of tackle this topic. We are going to figure out if there's a way of determining how strong a relationship maybe between two variables like height and foot. Teacher: Do you guys think that this relationship is strong? Class: "maybe" "not very strong" "no". Teacher: well, lets see what you guys have to say about these relationships: |
The following poster paper was placed in the front of the classroom:
| Strong Moderate Week No Relationship |
| foot/height |
| foot/forearm |
| smoking/lung cancer |
| height/running speed |
| studying/GPA |
| jump height/reach height |
| age of car/worth of car |
| income/IQ |
| hair loss/age |
| items on sale/#of buyers |
Teacher: Now, thus far we have been taking a closure look into data collection and utilizing it to provide us with information. However, we haven't had an opportunity to really see how statisticians interpret data. Moreover, how do they see statistics as a tool for prediction? What we will begin to do for the next couple of weeks is talk about variables that are sometimes said to hold a relationship amongst one another. Take a look at the poster up here. I want you guys to tell me to what degree you think these relationships hold. Let's first take a look at the amount of studying someone does and their GPA. Jackie: I don't think that's fair to say people who study are the only ones who do well. I mean I'm an A student and I never study. Brian: yeh, I know people who don't open a book and get the grades. The class in general seems to be agreeing Teacher: okay, but guys, think for a minute when we talk about relationships you need to go above and beyond using yourself and your friends as the norm or the population. We learned this when we did our surveys. I want you to think of "in general". Relationships that exists only do so in a general sense-they never represent every single person. Christian: Well, than I think what we are saying is that it has a moderate or week relationship. Jackie: I say its a week one. Teacher: Jackie what do you think college students need to do to get the grades. Jackie: oh, they study day and night-but that stuff is tougher than the 8th grade. Teacher: yes, but don't you think that college students are in this picture as well? Brian: Okay lets say its moderate. Teacher: okay, I'll agree with that. |
The 8th graders were asked to complete the rest of the chart as part of a class discussion. More often than not, they kept referring to their own personal experiences and the exception to the rule. At certain points I asked students how it was that they had arrived at their decisions for how strong or week something was:
Teacher: How are you guys determining the strength of these categories? Ame: We think about if its always the case to if its not always. Like its all different. Teacher: I agree, notice that even if you say something has a week relationship, you are still saying that there is a relationship but it is not very strong-there's allot of variance. Like let's take a look at money and your IQ. Larry: yeh, we said that it's a week relationship. Teacher: okay, why did you say it was week. Brian: Because there are some people that do have high IQ's and make allot of money. But we can't say that everyone who makes allot of money has a high IQ. Teacher: Exactly so even though we know the case can't hold for all-we are still saying that there is a slight relationship. Thus, this is why relationship are thought of for "general cases" not all cases. Therefore you guys don't need to be thinking about your exceptions all the time. Teacher: I noticed here that you said the height and speed is a moderate relationship. How come not strong? Kristen: well, because there may be other things like the weight of a person. Tall people can also be fat and thus run slow. Ben: yeh and what about how athletic someone is. Brian: also if someone is light they will also run faster than the average person. Teacher: See you guys already are thinking in terms of questions that you need to ask when considering strengths of relationship. Really what causes people to run faster? You guys made some good points. |
The class ended up with a chart like this:
| Strong Moderate Week No Relationship |
| foot/height x |
| foot/forearm x |
| smoking/lung cancer x |
| height/running speed x |
| studying/GPA x |
| jump height/reach height x |
| age of car/worth of car x |
| income/IQ x |
| hair loss/age x |
| items on sale/# of buyers x |
After we had finished the above graphs there was very little time left. Since this was a lesson to warm up the students for what coming up next, I decided not to give any homework.
DAY 17 (3/11)
Today I wanted to have them perform their measurements. However, I wanted them to provide some insight on how would one go about finding the relationship that we had talked about yesterday.
Teacher: So picking up from yesterday-you guys said that there is a moderate relationship between foot and height, and a strong between height and forearm-how would we test to see if they are true? Ben: We need to get the measurements of our forearm, height and foot. Teacher: Sound like a plan. Why don't you guys pair up and get the requested measurement that are in your Relationship/Correlation packet. Hand them into me before the end of class and I will compile them for you on a master chart. |
The remaining time was spent getting the students' measurements.
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LESSON 9: The Scatterplot
DAY 18 (3/12)
To begin the class, I passed out the compile list of class measurements (Appendix O). Up to this points, the students have only been able to define relationships using words such as "week", "moderate", or "strong". The idea today, is to try to get the students familiar with scatter plots which can viewed as a way of drawing the adjectives they have been using thus far.
Teacher: You guys are looking at the class data. Tell me, does this chart tell you what kind of relationship exist between say reach height and jump height? Class in general try to follow the chart and see if they can find a relationship amongst the numbers. Soon they realize that it is simply to difficult to do. Teacher: Well, lets think about this, how can we, get our data to make more sense to us we need to organize in a way that will allow us to see whether a relationship exists or not. Any ideas? Class begins to mention things like "bar graph" , "histogram", and line graph. Teacher: Let's take that bar graph idea first. Do you guys think a bar graph will tell us the kind of relationship two variables have? I started to make a bar graph. Showing the students that it really didn't tell us how strong the relationship was. Teacher: Pretend this is the bar graph for reach height and jump height-what is this bar graphs telling you. Morgan: only how high people jump in the range of a certain reach height. Teacher: Right. Does it tell you that this is a strong or week relationship? The class in general sees that it doesn't make sense. Teacher: Someone suggested to make a line graph? Well lets try it and see where it takes us. Now if we need to make a graph first what do we need to do? Class: Plot the points. Teacher: okay take a look at these graphs I have up here on the poster paper (Appendix N). Suppose these were the graphs of some of the relationships we are trying to compare. Just by looking at these graphs, which one would you say has a strong relationship. Class: Graph A Teacher: Why? Erin: Because its like the points are closer together. Teacher: I agree, what are these points trying to form or close to forming. Christian: Oh we did this in science, it's trying to be a straight line. Teacher: very good so you guys are familiar with this type of relationship. Where have you seen it before? Ame: It's like speed and acceleration graph-direct relationship. Teacher: Exactly. Why do you guys call graphs that have this sort of straight line direct? Ben: Because its like as speed increases the acceleration increases in proportion-we did this like a month ago. Teacher: Well, tell me what you think is so "idea" about a straight line-why does it make something a "strong" relationship? Brian: It's like exact or perfect-you can read it easily. Teacher: Very good, now tell me something, if these graphs are to represent our data points-how come I just don't go ahead and connect the dots. I mean you have been connecting the dots all your life. How come we don't connect the dots on these graphs up here (App. N) The class really doesn't know why connecting the dots is not an option. Teacher: well, Dave why don't you come up here and connect the dots to this graph. Dave began to start connecting the points, within in 1 minute he realized that there were too many dots and that he couldn't tell which way to go. Teacher: Dave, why did you stop? Dave: Theres too many points-I can't connect them-it will be just a bunch of zig zags. Teacher: So what does this tell you about connecting the dots with this stuff? Kristen: It won't get us anywhere. Teacher: Exactly, the zig zags are only going to tell where it increases and where it drops. Its not going to tell you the strength of the relationship. So let's go back to these 8 graphs, tell me how you would order the strongest graph to the weakest graph. Matt: It would be A is the strongest and than D, C, B. Teacher: How did you know that? Matt: The farther the points are the weaker the relationship. Teacher: Does everyone agree with that. How far do the points have to be to still consider a relationship. Brian: Well, the points can't go everywhere b/c than your data is everywhere and there is no relationship. Teacher: Good. Would you guys agree that the closer the points are to a line the stronger the relationship. Class agrees. Teacher: Now tell me what is different between graphs a-d and e-h? Morgan: The direction of the points are different. Teacher: In what way? Morgan: One goes up the other goes down. |
Teacher: Does anyone know how to interpret what Morgan is saying using other words-try to remember your science class-when she told you about speed and acceleration what were both those variable doing at the same time. Class quickly understand what I'm referring to and they respond "increasing". Teacher: Great, now which graphs show both variable are increasing? Class: the first set a-d. Teacher: Then what are the other graphs doing? Class: "decreasing". Teacher: Wait a minute both variables are decreasing?? Let's pretend I put age on the x-axis and hair on the y-axis. What do you guys think happens to hair when you get older? Class: You have less. Teacher: So as age increases, what happens to hair? Class: It's decreasing. Teacher: So tell, me what graphs e-h are doing? Ben: As one increases the other decreases-I see it now. |
Since it seemed that most of the students understood the difference between the two sets of graphs, I proceeded to give the graphs formal words-negative and positive relationship. The remaining of class time, allowed student to begin graphing their data and the starter graphs that were provided in the packet (Appendix M). For homework they were to complete the last page of the packet. This was to get them familiar with drawing scatter plots and detecting which graphs show strong relationships and those that show weaker relationships.
The class seemed to go pretty smoothly. I think what they had learned in their science class prepared them well for this class. I noticed that there were some students who were not so ready to give up the line graph! Clearly this material was new to the students. Even though they have seen the direct positive relationship in the science class, they are only aware of it in that particular context. Moreover, they have never seen scatter plots before. Yet, they were able to pick up this introduction rather quickly. Again, allot of important information surfaced today. However, nothing has been written down. Thus, I decided tomorrow would be a good day to go to the notebooks to try to formalize these concepts.
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LESSON 10: Analyze Data and Reflect on Correlations
DAY 19 (3/13)
Today was dedicated to writing notes from what we had learned yesterday. Again, I had developed the notes and asked the students to interact with the questions that I pose. The following notes were compiled:
There are two kinds of correlations: 1. Positive Correlation: When both variables will increase. The cloud of points on the scatter plot will slant from lower left to upper right corner. Examples of the graphs are drawn here (similar to those in App. N). Example are discussed: travel time and distance, studying and GPA. 2. Negative Correlation: When one variable decreases while the other increases. The cloud of points slant from upper left to lower right. Example of the graphs are drawn here (Appendix N). Examples are discussed: product on sale and number of buyers, age and worth of the car, sports training per week and pulse rate. |
(3/14) SNOW DAY-NO SCHOOL
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LESSON 11: The Best Fit Line & The Equation of a Line
DAY 20 (3/17)
I started class by passing out the Reach Height/Jump Height Packet (Appendix P). The students took out their graphs (Appendix O). I explained that the new packet would act as their guide to what we will be doing the next couple of days. Before introducing the concept of the best fit line the following took place:
Teacher: If someone told you that scatter plots like the one you are looking at can be used to predict the jump height of your favorite basketball player, what would you say? Class: "no way", "may be-but how they are all taller than us". Teacher: Well, guess what guys I challenge you and will prove to you that indeed these graphs can help us predict, data that is not even there. Teacher: Okay, you guys are looking at your graphs. Tell me where you think someone with a reach height of 87 would jump? The students began looking at their graphs only to notice that no one had a height of 87 inches. Thus, they tried to estimate where the point would fall. Class: "100, 103, 98, 102". Teacher: How did you guys determine that? Class: "We just tried to go along with the rest of the points", "the data is going up, so we need to follow it". Teacher: Exactly, Remember when we had talked about the importance of a straight line and how you guys said its easier to read. Well, I'm wondering do you guys see a place where we can put a line through or data to try to capture as many points as possible? Christian: You mean like how you said that a correlation is strongest when the points are close together and they form a line? Teacher: yes. However our graph probably looks more like a moderate relationship-but still can you imagine a line through your points? Class in general sees that there is an upward trend and that a line can be drawn in the middle of the points. Teacher: Now, I'm going to draw a line through the middle of my data, you guys go ahead and do the same. Now use your line to predict the jump height of someone at 87 inches. Tell me what you got. Students were quick to get the hang of using their line. Class: 102,103,101,104, 98 Teacher: Compare that two the values you got without drawing that line. Ben: It's like the same. |
Teacher: Exactly see, what you guys did is predict a value by imagining a line was there-of course you may not have known you did this, but you did. Then I asked you to literally draw a line. This line is know as the best fit line. What is its purpose? Morgan: To predict data that may no be there. Brian: It suppose to go through the middle of your data. Teacher: Exactly, it is a line that tries to summarize your data points. To gain more practice with using the line. I asked students to try to predict their jump heights using the line. Teacher: How far off are you from your actual? Nate: I'm like 8 inches off. Teacher: How many people are more than 4 inches off? About 10 students are more than 4 inches off. Teacher: Why is this so? Morgan: First off, Nate is like really tall, of course he will jump allot higher than the rest of us. Ame: Yeh, besides we all have different lines anyway. Teacher: I agree, is it okay to be off from the line? Brian: Yeh, b/c its impossible for everyone to be on the line, we all jump differently. Teacher: and what is the purpose of that line again. Kristen: to give you the average. Teacher: yeh-that's one way of seeing it. The line is trying to hit as many points as it can. You usually draw your line where the average number of point are. Nice. Matt: I'm really short so. of course I won't be on the line! Teacher: Guess what Matt-the same goes for me! |
As students began to use the line, they all saw it as an average. This was an extremely important insight on their behalf and will be explained in section V of this unit. The students were also able to see that it was okay for the line not to hit all the points-which is almost impossible anyway. They provided some good thoughts as to why certain people were off the line.
Teacher: Suppose I wanted to determine how high Michael Jordan jumps. Our graphs don't go that high-what should I do? Brian: Can't you just extend the line and read off like we've been doing. Class in general agree with his suggestion. Teacher: That is definitely one way, but look I don't have enough graph for the this huge guy. Now, I deliberately used Jordan as an example to introduce to you another way of using your scatter plot to predict. Moreover, this way will be more precise than just reading it from your line. Take a look up here. What does this equation mean to you? Class: nothing. Teacher: Why nothing? Brian: Because we don't know what the heck it means. Teacher: Oh so you don't know what those terms and letters mean. That makes sense. What do we need to do in order for you to understand what this equation is? Class: Tell us what it means. Teacher: Okay, that sounds like a deal. But wait, before we move on- What's an equation? Matt: It's like a bunch of letters. Ben: It's like you put numbers in the letters and than get an answer. Teacher: What are equations used for? Tim: To get an answer or solve for something. Teacher: True. I sort of think an equation as a way of writing a picture. Like right now you have a graph in front of you with a line through it. This equation is like the blue print of your line. What 's a blue print? Matt: It's like what architects use to build a house? Teacher: Yes, what do they show you? Matt: what your house will look like. Teacher: Exactly, so let's think of this equation as a way of writing your best fit line. What we will do beginning tomorrow is break the equation piece by piece and than show you how to use it to predict anyone's jump height. |
I was not surprised by the fact that students did not know what the purpose of an equation is. However, I think that viewing the equation of a line as a blue print allowed them to meaning to something they are familiar with. The last thing that I wanted to do was build fear in the students for what was coming up ahead. I think the students felt comfortable in knowing that we would break the equation apart and kind of work through this nice and slow.
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LESSON 12: Slope of a Line
DAY 21 (3/18)
The students had their graphs and their packet ready to go. I had the equation of a line written on the board. I began class with the following:
Teacher: So yesterday we left off with the ideas that this equation is going to give us a way of writing our own best fit lines. The first word in that equation is "slope". Well you tell me, what comes to your mind when you think of slope. Jessica: Like slant. Many students agree Justin: Is like a degree of slant. Brian: yeh, like the angle defines the steepness-like when your skiing. Teacher: okay, let's take that skiing example. Because a ski slope is a incline, do you guys no what kind of distance you are covering when you go up or down the hill. Ben: It's just the height of the hill. Teacher: Well, not exactly, its more like this. |
I began drawing the vertical and horizontal distance that is covered on a line. I explained to them that slope is a ratio of the distances and modeled how to find it using my own graph. An easy way for them to remember it was "rise over run". I had students figure out the slope of their own line. This took about 15 minutes. I had assigned student helpers to go around the room when ever necessary just as long as they had displayed an understanding of how to find the slope. Once everyone had a slope, the following took place:
Teacher: Now, guys, pay attention this is how I want you to interpret slope on your graphs. First off as I was going around the room, allot told me slope is like the rate that the line slants upwards. Well, that's a nice way of seeing it. In a way, it is like a rate of change of your line. Now tell me what is on our x-axis. Class: Reach height. Teacher: the y-axis? Class: Jump height. Teacher: Matt, what did you get for your slope? Matt: 2.4 inches. Teacher: okay, here it comes, the slope that you found means that suppose that Nate over hear is one inch taller than you. For every one inch that he is taller, he will jump 2.4 inches higher At this point the class was really engaged and interested in the comparison that was happening. Brian: So wait, Nate is like 5 inches taller than Matt. So that means he'll jump like 10 inches higher! Teacher: Yup, its amazing isn't it? So Lindsey-tell me what your slope means according to your data. Lindsay: I got a slope of 1.9 so it means that for every inch taller someone is they will jump almost 2 inches higher. Teacher: Good! okay, look at your graph is that a positive or a negative correlation? Class: Positive. Teacher: Good, is your slope a positive or negative number? Class: Positive. Teacher: okay, tell me suppose I drew this graph of physical fitness, and pulse rate. Is this a negative or positive correlation? Class: Negative. Teacher: what is increasing what is decreasing. Tim: As someone is more fit they have a lower pulse rate. Teacher: okay, now tell me do you think the slope will be positive or negative? Class: Some say positive while others say negative-they were just guessing. |
Well :Suppose I got a slope of 3. What does that mean in the date. Ame: For every 3 levels of fitness you are you will get one pulse rate lower. Teacher: Well, your close but you are saying the opposite. Remember what every variable is on the y axis is what will be changing per every increment on the x-axis. Brian; For every one level increase in you fitness, your pulse will be 3 higher. Teacher: Wait, but you said as your fitness gets better your pulse gets lower? Brian: oh yeh, I mean it will be 3 lower. Teacher: okay, so guys listen up when every you have a negative correlation, you are going to have a negative slope-in this case -3. This means your pulse rate will be 3 times lower. |
At this point I went around the room asking people to give me their interpretation of slope. It came quicker for some than others which is naturally expected. However, I knew that I would give them plenty of opportunity to understand this concept. I should mention here that when I modeled the method of finding slope I did mention that they are allowed to pick any 2 points on the line. However, I did not tell them why, nor did they ask. I think this is because it is still early in the investigation. I also think that students knew that they all had different lines anyway so it didn't matter which points were chosen. I knew that this would be something I would have to show the students. At this time it did not occur to me to let this concept become a discovery. Section V, will discuss my view on this after the fact. The homework assigned was a slope worksheet (Appendix Q).
DAY 22 (3/19)
Today, the students wanted to go over the homework, graph by graph. Since this was new to them. I decided to go ahead and grant their wish. Some of the graphs had a negative slope which needed to be reviewed as well. By the end of class almost all the students were able to figure out slope on their own. More over, some students wondered how is it that most of the class is getting the same answer on the homework when they chose different points. Rather, than realizing that the students were on to something I gave it away to quickly. I explained to them that slope never changes, as long as the line never changes. Therefore they can pick any point on the line. I also used the ski slope to help them understand the concept as well. They understood that on a hill, the steepness won't change unless the hill drops or straightens.
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LESSON 13: TheY-intercept
DAY 23 (3/20)
I decided to start off the class by getting the slope into their notebooks. My cooperating teacher wanted to make sure we put this in the notebook as well.
Slope of a line is a measure of steepness. It is a ratio between the vertical an the horizontal move along a line. It can also be seen as a rate of change. Formula: rise/run + slope goes uphill - slope goes down hill |
Next, we began to discuss the y-intercept. I first explained to students by drawing a number of graphs that the y-intercept is where the line will cross the y-axis. I first explained to students what the y-intercept was using a variety of graphs. This was relatively easy. It simply involved reading where the line crossed the y-axis. Students also realized that by looking at the few exercises we did together that (x) will always be 0. However, they had not yet figured out the slope on their own graphs:
Teacher: okay, it seems like you got the change of this. Now take a look at your graphs. Tell me what the y-intercept is. The class takes a look at quickly respond: "0". Teacher: What how do you see that? Brian: Well look my line is like going in that direction. Teacher: Guys, see those zig zags at the corner of your graph. That means that your graph did not start at 0. Its a broken graph. I did that so that I can fit all your data. A whole portion of your graph is missing. That's why you can't tell where the y-intercept is. Here I'll show you. |
At this point I superimposed the graph to show them how much of the graph was missing. This worked out great because the students had a visual to help them understand why they couldn't simply read off their graphs. Moreover, they caught on to the fact that we needed another way to try to figure out how to calculate the y-intercept.
Teacher: Let's look at our nifty equation. We know slope and look we are trying to figure out the y-intercept. Now, if I told you that this equation will help you predict a jump height as well as reach height, what must (x) and (y) be in that equation? Matt: What? Teacher: Well, I told you that you can predict using this equation. But if you can predict someone's reach height or jump height-they have to be part of the equation somewhere. Kacey: So the reach height as to be a letter in the equation. Teacher: Right. What is on your x-axis? Class: Reach height. Teacher: So what is (x) in our equation? Class: Reach height! Teacher: There you go, you got it. Kacey: So (y) is the jump height. Teacher: Perfect. Now we know the slope, and we know our (x) and (y), can't we use all this to find the y-intercept? |
At this point I began to model the process. The students were given the rest of the time to find their own y-intercept. They also needed to brush up on solving for equations. The one thing, I knew they understood was why we needed to use the equation to find the y-intercept. For homework I handed out Appendix R which consists of graphs that required them to find the slope and y-intercept.
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LESSON 14: Piecing it all Together
DAY 24 (3/21)
Many student came in today, very confused on the homework. This is understandable since they were getting quite deep into algebraic and graphical manipulation. Thus, I decided to spend class time going over the homework. This was absolutely necessary because if students did not have a concrete understanding thus far, they would never be able to follow along in the remainder of the unit. I found it interesting that since the students didn't have concrete variables to use in their homework (they were just plain graphs), they had a tough time plugging in an (x) and a (y). Sometimes they even got the numbers they used for slope confused with the point to use for the y-intercept equation. At this point, the students arrived at mechanical errors. I was confident that these problems would straighten out with more practice. Once everyone's' questions were answered I decided to piece everything together:
Teacher: Well guys, it looks like we've got everything in our equation that we need. Now tell me how this equation can be used to find your jump height. Brian: Don't we just need to get our reach height and put it in for x? Teacher: Class what do you think? Kristen: yeh, I guess he's right. So we just get our reach height and put it in for x. Teacher: okay guys let's try it. Brian give me your reach height so I can predict your jump height. |
I began to model the process using Brian's reach height. The students were really engaged in finding out their own jump heights. It was nice to see them so interested and not overwhelmed with all the stuff they have learned thus far.
Teacher: Brian, I got a jump height of 100, what is your actual. Brian: Wow, Ms. Maine your only one off. I got mine exactly! Ame: Oh my g-d, so did I. Katie: me too! Teacher: Any body else want to make a comment? Nate: I was about 5 inches off-but still that's not too bad! Again I'm the guy that is way out there anyway. Teacher: So what do you guys think of this method? Do you trust it? Or would you rather just guess how high someone can jump? Jackie: Ms. Maine I wasn't hear when we did our measurements, so I predicted Kacy's jump height and I was 2 inches off! Teacher: Impressive huh! Ben: This stuff really works-I just don't know if I can remember everything to do it again! Teacher: I agree guys, this was allot of stuff. We are going to be doing this stuff using different variables for the next couple of days. Moreover, I want to give you a unit test before Easter. Plan on next Thursday as your unit test. All next week we will review. Don't worry about this particular stuff-right here-that will be tested in a project. However, I want you to be able to figure out slopes and y-intercepts for the unit test. |
The students seemed pretty impressed with what they had just explored. However, there was still much confusion in the air. The best thing for these students right now was to look at something different and start from step 1 to the end! For homework they were assigned a reflection sheet (Appendix S).
DAYS 25 & 26 (3/24 -3/25)
Students worked on various exercises in Appendix T.
These few days provided the opportunity to work out the steps without any pauses or breaks. Thus, the students were able to see how each piece fits into the big picture. I think this marks the first time that the students were able to really understanding how to go from one step to the next.
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UNIT REVIEW & FINAL ASSESSMENT
DAY 27 (3/26) Students work on review sheet for unit test
DAY 28 (3/27) Unit test (Appendix U)
3/31 - May cooperating teacher takes over his class again. He decided to discuss mean, median, mode as well as the proper format in creating a histogram.
4/2 - PROJECT ASSIGNED!!! (Appendix V)
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