TS: My name is Tanya Sanchez and we're at Garcia Elementary School, which is a K-8 school in Phoenix, Arizona.
Good morning. Today we're going to continue talking about a couple of things that we've been learning last week.
I've been using mathematics in context for the last two and a half years.
Today
what we will be doing is, we are going to be measuring again, we are going
to be looking at the properties of a ladder. Can anybody tell me where
we might use a ladder or what a ladder might be necessary for? Ris?
Ris: To like get somewhere real high?
TS: Okay, to get somewhere real high, that's right. Who might need to climb up real high or for what purpose might we need to get up really high? Paul?
Paul: A janitor?
TS: A janitor might need to get up really high. Who else? Victoria?
Victoria: A fireman?
TS: That's right. Okay, so today we're going to investigate what a ladder does. Okay, we're going to investigate three things about a ladder. Okay? And guess what? I have a ladder outside. There's a ladder outside, come on, let's go outside.
MEG MEYER: Mathematics in context was developed in response to an identified need for curriculum materials that would embody the NCTM standards. I think mathematics in context is a unique curriculum, because students are starting with a context. We start with a context and extract the mathematics and then later on apply it new situations.
TS: Okay, this is Mr. Sossa, how many of you know Mr. Sossa? You all know him? Okay, what he's going to do is he's going to help us to raise the ladder. Okay? Yes? Now, there are a couple of things involved with the ladder. If I climbed up here, do you think this would be a safe way to climb up?
Class: No.
TS: What do you think we should do? Jennifer?
Jennifer: I think you should push it more out this way where you can hold yourself, so it can hold more weight, because if you just sat under it right there it's going to push up this way.
TS: So it might fall if we try and climb on it on in this position?
Jennifer: Yes.
TS: Okay. So let's try and pull it out just a bit. Right there?
Students:: No, more.
TS: Watch out. Is it stable here?
Students:: No.
TS: Why isn't it stable?
Student:
It's not stable because if you go, if you step on it it might fall down,
if you put like more (?) in front it will be better.
TS: Okay, so we need to push it in so that -
Student: The angle -
TS: So that what?
Student: The angle (?)
TS: Okay, so it will be at a better angle, right? Okay, can we push it in?
TS: Does this look good?
Students:: Yes.
TS: George.
George: Because the angles are like the angle on this side was smaller and the angle on the other side it was a little bit bigger and maybe when they step on it maybe you'll fall down.
TS: When the ladder was really close to the wall the angle was -
Students:: (?).
TS: Was larger. And when we pulled it out the angle was -
Students:: Smaller.
TS: Smaller. And in either case will it fall?
Students:: Yes.
TS: Okay, one will tip over and the other one will kind of get smashed? Okay.
The lesson we're working on today is one of the lessons from a unit called Looking at an Angle. Using the context of a ladder and steepness, the students are investigating the ratio between the height and the distance and how the angle changes when the height or the distance changed. This lesson is a preliminary lesson into investigating the tangent ratio. The students do this by investigating a triangle formed by a ladder propped up against the wall. The height along the wall, the distance along the ground and then the angle formed by the ladder.
We're going to go inside and with your partner, okay, or if you are in a group of 3, I want you to investigate what happens when that ladder moves out, tilts out, and when it tilts up. Thank you, Mr. Sossa? Let's go back in.
When
we come inside we begin to design the height, distance and angle of the
ladder.
With your partner, I want you to take a ruler, and find a place in the room, what type of a place are you going to be looking for in order to investigate this triangle, Paul?
Paul: The floor and the wall?
TS: Okay, something that would represent the floor, and the wall. Okay? With your partner you are going to look at these three things, the angle of the ladder to the ground, okay, this is going to be our main concern. The angle of the ladder to the ground. Even though this one does change as well. Okay? The distance between the ladder and the wall and the height of the ladder from the ground. Okay? I want you to look at the changes that happened and tell me, what types of relationships or what things happened between those as you move the ladder. On your white sheet of paper, I want you to write down what it is you and your partner are talking about and you can make some sketches of the triangle, as well, to give examples.
Student: As the ladder goes up the height of the angle or of the right angles gets longer, but the base of the right angle gets shorter, right? So that means the distance decreases after we expand it longer.
Student: That makes the height of the right angle -
Student: Longer.
Student: And the height gets shorter.
Student: Yeah.
MM: One of the important ideas of geometry in mathematics in context, is that things happen from the perspective of the child. The child is the center of his or her universe and this perspective means that the child explores questions like, what do I see.
Student: Right here it makes the right angle.
MM: And why do I see it that way? What don't I see? And why is that occurring? And what happens as I move, as I turn? And this perspective and the geometry that evolves from this means that the child gets a vocabulary that allows them to begin to describe their world, and how they interact with it. And I think this means that the child begins to see mathematics as a way to communicate, as a way of making sense of their world.
Student: So when the ladder's closer to the wall the length gets taller and the width on the bottom gets smaller and the angle got bigger. And when it's lower the length from the bottom got bigger and from the top got smaller and the angle got smaller.
TS: Now, how does that have anything to do with the angle, what happens with that?
Student: The angle gets smaller when it's further from the wall and it gets bigger when it's closer to the wall.
TS: Do you all agree?
Student: Yes.
TS: I noticed when you guys were working in your groups that there was a little confusion as to what changed or what was happening with the triangle, with the ladder, and the sides of the walls. Right? I want to talk about some of the things that were changing. In a triangle, there are six parts. There's the height, the distance, the ladder, which were three sides, and then there were three angles, the one from the wall, and the one to the ground, which is the one that we were really concerned about and I'm going to put a symbol here for this angle, that's a Greek letter Alpha, okay? If I wanted to shorten the height I would just put H and the distance D. There was also the 90 degree angle. On this triangle that we were investigating, what were the things that changed. Erica?
Erica: The length of the wall and the floor?
TS: Okay, what else changed? George?
George: The height.
TS: So this height changed? What didn't change? What stayed the same all the time. Victoria?
Victoria: The ladder, the ladder length?
TS: That's right. What else never changed? Tabitha?
Tabitha: The 90 degree distance?
TS: The 90 degree angle? This angle never changes. What about these two angles, George?
George: They changed, the one in the bottom, it changed, it gets smaller and the one on the other side gets a little bit bigger.
TS: Depending on what?
George: On the, depending on when you pull it back?
TS: Okay, I'm going to give you 30 seconds with the people at your table, discuss what it was that you discovered when you moved the ladder. What was happening with the ladder? You've got 30 seconds.
Student: Here's the floor and here's the ladder and then this little thing right here, it stays the same?
Student:
Yes, so like each time you're moving it something is going to get bigger
and something's going to get smaller each time, but this little box always
stays the same.
TS: I want everybody to understand what it is that your group understood, okay? Who would like to go first, which table would like to go first? Okay, Isaac, (?)
Isaac: If you move over here the distance gets longer and this keeps getting shorter and shorter.
TS: Okay, so the height keeps getting shorter.
Isaac: Yes.
TS: When you moved it, what happened to the angle?
Isaac: This one gets smaller and this one gets wider.
TS: Okay, let me see if I have this right? If I change the distance of the ladder from the wall and I make it shorter, my angle becomes larger. And my height is also larger. And then if I move it away from the wall the distance gets longer, my angle gets smaller and the height gets smaller. Did I get that right?
Class: Yes.
TS: Okay, when I have the ladder and the distance I can sometimes measure the angle of the height using my compass card. We can use this to measure the angle from the floor to the ladder, okay. So let's say that I have this ladder here, okay and it's some kind of distance from the wall, I can measure that. I can also measure the height. Using my compass card, I need to place the center point, does everybody see the center point on the compass card?
Class: Yes.
TS: And the (north arrow?) It needs to be placed right on the angle. Right on the line, so that it lines up with the zero. Can somebody come up here and help me read what this says. Rais(?)
Rais: 60.
TS: 60. Okay, so this angle here is a 60 degree angle. I want you to open up your books to page 32.
Then the students measure an angle in a triangle that's in their book. This allows me to see if they are grasping the concepts that I'm demonstrating at the board.
Okay, what was the measurement of that angle? Erica?
Erica: 50?
TS:
How many people got 50 degrees? Okay, we can measure
the height, the distance, or the angle in order to get our triangle. If
I gave you a height of 4 and a distance of 3, what would my angle be?
Okay, I need a trustee assistant, Lisette?
Lisette: It's 60 degrees.
TS: It's 60 degrees?
Student: What?
TS: 60 degrees. Okay, thank you? I can use the height and the distance to give me a ratio, okay? So if I get a ratio height to distance it will be expressed as 4 to 3. What is another way of writing this ratio? Victoria?
Victoria: 1 and 1/3.
TS: That's right, 1 and 1/3. If I expressed this as a decimal, what would that be? George?
George: 1.33.
TS: That's right. So I can change my ratio to a decimal. But what we're going to be doing today, we'll only need to go to the 1/10 place. Okay, now on page 32 there are several angles that are given, there are also heights and distances. I want you to look at those five things, there are three things that are given. There's either the height and the distance or the angle. If you're given the height and the distance what are you going to have to find? Netsa(?).
Netsa: The angle?
TS: That's right. What happens if we're given the angle? Rolando?
Rolando: The height and the distance?
TS: Okay, if I give you the angle, you measured the height and the distance from the wall. Okay, one person from your group, please go over to the table and pick up a piece of graph paper. I'm going to give each of you a letter to do one of those problems. You are going to draw it as a group on your paper, and you're going to find the height to distance ratio for that triangle and write it as a decimal to the nearest 1/10. So I'm going to start up here at the front, Rachel, your group will do A, George B, Jennifer C, you guys will do D, okay, Juan, D? E? Sochi, your group will do an angle, of 75 degrees. Okay? Can you draw that triangle and find the other three things that I'm asking for.
Group work is valuable to the students. They are able to interact wit their peers and bring to each other a collective amount of knowledge. Each student brings something special to the group. As the students verbalize their concepts, they are able to understand it more.
Student: So it's 2, divided by 1, equals 2. So the answers are 2.
Students:: 60 degrees, or 70, no 65.
Student: Yeah, it's right around there.
Student: It's 64 (?)
Student: 65.
Student: (?).
Student: (?).
Student: Spanish.
TS: One of the strengths of the program is that students are able to interact in their native language so they can speak Spanish in the small group setting and are able to carry away the same content that an English speaker is able to.
TS: Okay, which one did you do?
Student: 35 degrees.
TS: Okay, now did you get your height and your (?)
Student: 75 degree angle and then we measured it out, how we wanted to try it.
TS: Okay. What does this represent right here?
Student: The wall.
TS: And how did we label that?
Student: H.
TS: Oh, you did label it H? Okay, and this is our L? Okay. So what was your ratio?
Student: 842(?) 8 divided by (?).
TS: Okay, good job.
Student: (?)
Student: Round it?
TS: The kids are really willing to continue to work through the process of learning math. They don't give up in the middle of a problem, they know that while working in groups they have a support system, the process is an important element of what we do. It's not just getting a right answer.
Group
work is also a valuable way of assessing the students. As I walk around
the classroom I can listen in on the conversations that are happening
at each group and I'm able to pick up on key words key concepts that the
students should be learning. Once the groups have completed the problem,
each group shares their answers and how they arrived at their solution.
Since I assigned a different problem to each group it allows the class
to get a sense of what the different problems were about.
TS: D, group D, let's see what you have.
Student: We saw that one was the height and two was the distance then we divided 1 by 2 and we got .5
TS: Okay, so we got .5 for your ratio. What was your angle?
Student: 25 degrees.
TS: Okay, there are a couple of things when you measure. I'm not sure if you've seen that, but sometimes when you measure on the angle, okay, the angle that you've begun, the triangle, because of the thickness in the line, I mean, I could measure like on the inside or on the outside. Are some of you having a little bit of difficulty deciding what that measurement was, like how? It gets kind of confusing. So these are sort of estimates of what that would be. Next group, E.
Student: The depth and height and distance, we measured it it was 60 degrees, then we put up the (?) that goes to here and we've got a 7.5 or 7 ½ and we got a distance of 5. So then we divided (?).5 divided by a 5 and it gave us 1.5.
TS: Okay, so your height was 7.5 and the distance was?
Student: 5.
TS: And your ratio?
Student: 1.5.
TS: 1.5.
MM:
The students were using the compass card to measure the angles today.
Now the compass card can be a very accurate tool when dealing with a small
triangle with a finely drawn line. As soon as the students started to
draw big triangles and to draw the lines with a very thick marking pen,
the compass card became limited in terms of how accurately it could measure,
it and the students could measure the angles. Likewise, with the large
grid paper the students were rounding those answers off in many cases
to the nearest whole unit. That will give quite a variation sometimes,
especially when using a ratio of the height to the distance. In this lesson
we are interested in the understanding of the relationship between the
height to distance ratio and the angle. Limitations by the measurement
tools should not compromise this understanding. In the next few days as
students continue with this unit there will be other opportunities for
them to increase the accuracy of their measurements.
TS: In your books on page 23 there is a steepness table, which tells us the steepness of the ladder as it moves. Okay? We're going to fill in this table from the least to the greatest angle, okay? You also have a hand-out and I want you to follow along with me. Okay, let's look at the angle over there. Which was our smallest angle? Rolanda?
Rolanda: 25 degrees.
TS: So we're going increase in size. What was the ratio for 25, Juan?
Juan: .5?
TS: The steepness chart will allow the students to organize all of the information that we have processed through the measurement and height and distance as well as the angles that the groups did.
Did everybody have a chance to write these numbers down?
TS: Yes. Okay, in the graphics below, we're going to graph these points. It's going to tell us about the steepness, okay, according to the angle, which will be at the bottom, and the height to distance ratio. I want you to plot it as points. Okay? So if I do, now 15 I think that 25 is pretty close, so I'll kind of estimate where 25 is and here is .5. Okay? Go ahead and plot those points.
Student: This is one.
Student: It's right here.
Student: 1, 2, 3, --
Student: Something like that.
Student: This is half. (?) is half.
Student: So it's up right here, so that's one whole. So it's 1, 2 so you could get 4 and that's where 4 is at.
Student: Yes, watch, this is half, this is (?) a half and this is a whole.
MM: Another very important idea has to do with the connections between the strands of the curriculum. There are many times where the geometry starts to look as though we're doing an algebra unit, because although we've started with the geometric context, we end up with a table of values that turned into a graph. These connections between the strands are extremely rich and they begin to let students see that mathematics is a connected whole rather than a set of fragmented parts.
TS: Juan, can you please come up here and plot the answers for us? I want you to look carefully at what he does and compare your answers with his. Okay? Why did you put that point way up there?
Juan: Because there's no (?) at all so you double 2.4 and you get 4.0 and it gets up to like right here.
TS: So we're doubling 2 and the length of 2 and going up to 4. Up high, so it's way, way up there. Connect them on your papers also if you haven't already done that. What can we say about the angle as it increases and our ratio increases. What's going to happen if I plot more angles? George?
George: It will go up.
TS: Okay, so it's going to become steeper and steeper?
George: Yes.
TS: Okay, as the angle increases, the steepness of the ladder or the height to distance ratio becomes larger and larger. Okay? And increases. It becomes steeper and steeper and it gets closer to being vertical. Okay? Now let's see what that means, to what we were doing at the beginning of class with the ladder outside. So if I have an angle of 45 degrees, that means that my height and my distance are equal. Is it about even?
Student: (?).
Student: The bottom is more (?).
TS:
Does this look safe?
Students: Yes. No.
Student: Why not.
TS: Marko, why don't you feel this looks safe?
Marko: You see the top, the bottom should be like in between the top, the bottom, you know, because the bottom is way too far, so it could collapse down.
TS: Okay, let's look at the 65 degree angle. Okay? The height is going to be 2 times the distance. Okay? So if I take this measurement here, and then do it twice, does that look about twice the size or twice the height?
Students:: Yes.
TS: Does this look safe?
Students:: Yes.
TS: Who feels like they'd like to try 25. My ratio is 25. Rais(?). Rais, we have a height of 1 and a distance of 2.
Rais: (?).
TS: Does that look like it's correct? Does the distance look like 2 times the size of the height?
Students: Yes.
TS: Does that look like that would be safe?
Students: No.
TS: Thank you Rais. So we found out that if our ratio is somewhere between 2 and 3 this ladder would be safe, but anything steeper, we'd fall over or anything lower, like a 45 degree angle we would slip and fall. So next time you see somebody climbing up a ladder I want you to check to see if they are between the 2 to 3 ratio. And see if they are safe, otherwise let them know. Okay? Have a good day!
TS:
Using mathematics in context has been very rewarding, because I've seen
students who otherwise were challenged by math and didn't quite understand
what they were doing actually begin to love math and kids come in every
day and are ready to learn and are very helpful with each other. The students
are engaged in the learning, they come up with the questions, they give
me the answers and I'm just a facilitator in the classroom.
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.
END