V-PATTERNS
Cheryl Anderson: I’VE BEEN USING MATHEMATICS IN CONTEXT FOR FOUR YEARS. I CAN SEE THAT WITH A LOT OF THE KIDS IN MIDDLE SCHOOL, THAT THEY ARE ALREADY AFRAID OF MATH BY USING THIS PROGRAM IT HELPS THEM TO SEE THAT MATH IS EVERYWHERE AROUND THEM.
Michael Comella: I LIKE TO PLAY THE ROLE AS KIND OF NOT SO MUCH TELLING THE STUDENTS EVERYTHING, I’D RATHER HAVE THE STUDENTS FIGURE THINGS OUT FOR THEMSELVES.
JOE
DAURAY: I MEET WITH THE OTHER TEACHERS TEACHING THIS FOR A VARIETY
OF REASONS. … THE BIGGEST REASON, I HAVE TO SAY, IS THE FACT THAT I GET
SOME FEEDBACK.
CHERYL ANDERSON: I THINK THAT IF THEY CAN DRAW THE PATTERNS VERY EASILY AND TO SHOW, LIKE, THE PATTERNS CONSECUTIVELY SO THEY HAVE SOME KIND OF ORGANIZATION OF WHAT’S GOING ON IN THEIR HEADS, THAT’S GREAT.
JOE DAURAY: WELL, WE ACTUALLY DID SOMETHING LIKE THAT JUST RECENTLY, AND IT WAS A, WE HAD THIS V-PATTERN TYPE THING, AND THEY COULD SEE THE CHANGE FROM ONE PATTERN TO THE NEXT. THEY KNEW WHAT THE CHANGE WAS AND THEY WERE ABLE TO VERBALIZE IT.
JD: The other day, I come out of my house and … I heard this.
(Geese honking). It was a sure sign of fall. Does anybody know what that might be? … Jesus?
Jesus: It would probably be the birds that fly in a pattern, with the V-line. I forgot the name.
Student: Geese.
Jesus: Yeah, geese.
JD:
I’ve got these geese right here. Can someone come up to the board
and show me how you might see them flying in the air?
JD: THE GOAL OF THIS LESSON IS TO INTRODUCE STUDENTS TO PATTERNS AND …SYMBOLS … BASIC ALGEBRAIC THINKING. THE FACT THAT THE KIDS SAID RIGHT AWAY … THEY FLY IN A V, THAT REALLY … SURPRISED ME.
JD: Say I had just three geese. Can someone come up and put them on the board, someone else, Ashley, please?
JD: It would look like that, that’s great
What happens if .. other birds fly in? What would it look like then?
JD: So what happened to the number right there when we went from this one to this one?
John: It grew.
JD: It grew? What did it grow by?
John: 2
JD: What happens if another couple birds fly in? What would it look like then?
John: It will grow more.
JD: OK, these are called V patterns. The way the birds fly is in a V pattern.
This happens to be the first V. This is number 1, OK, the first V. Another V pattern, this is the second V, we’ll call it V2. And then we get this next one, this is the third V pattern. And it looks like this. How many birds are in the first V pattern?
JD: Jenny?
Jenny: There are 3 birds in the first V pattern.
Meg
Meyer, U. of Wisc: I THINK THE BIGGEST OVERARCHING GOAL OF MATHEMATICS
IN CONTEXT IS THAT CHILDREN UNDERSTAND WHAT THEY ARE DOING WHEN THEY ARE
DOING MATHEMATICS AND UNDERSTAND AT WHATEVER LEVEL THEY ARE AT…
ONE REALLY IMPORTANT IDEA THAT I THINK IS MOST EVIDENT IN THE ALGEBRA STRAND IS WHAT WE CALL PROGRESSIVE FORMALIZATION …
IN MATHEMATICS IN CONTEXT THERE IS THIS PROGRESSION OVER THE COURSE OF THE UNITS FROM THE INFORMAL TO THE PREFORMAL TO THE FORMAL
AT THE INFORMAL STAGE, CHILDREN ARE WORKING WITH PICTURES, THEY ARE WORKING WITH MANIPULATIVES, THEY ARE MOVING THINGS AND THEY ARE PLAYING WITH PATTERNS. AT THE PRE-FORMAL LEVEL, STUDENTS ARE BEGINNING TO SUMMARIZE THE PATTERN OR WHAT THEY SEE, TO BEGIN TO EXPRESS IT IN WORDS AND IN FORMULAS.
JD: I want you to copy this chart, and I want you to extend the chart for a V pattern of 4, 5 and 6. And I want you to tell me what it looks like. Discuss in your groups, what do you think you see happening?
Lauren: See these four, it’s always going to be odd because of the one in the middle, and it’s always going to be adding up by two.
Student: How do you know that?
Lauren: Because if it’s even, with four, if you put four geese right, here, the first one is always going to be odd, because the more, it’s always adding up by two. Because the pattern up on the board, three, and if you add 2, it gets to 5, and if you add 2 by 5 it gets to 7. The more you add 2, the more, it’s just like a timesing two pattern.
Student: Is it like a timesing two pattern?
Lauren:
It’s like an adding two pattern. Adding two, not timesing.
Student: What about number 2?
Oscar: You multiply it by 2, and it equals 4, plus the one in the middle, and it’s five. And you keep on going for all of them.
JD: Did you even need to put your geese out?
Kyle: No, I didn’t need to.
JD: Why didn’t you need to put your geese out?
Kyle: Cause, um, the more patterns, you know, you keep adding two at the wing, and the one in the middle so it’s always going to be an odd number.
JD: Does anybody see any pattern that’s happening here? What seems to be happening? Oh, everybody seems to be seeing something. Ayesha?
Ayesha: Two is adding to every one.
JD: Two is adding to every one? What do you mean by that?
Ayesha: One plus two is three.
Ayesha: 2 X 2 is 4, plus one.
JD: And where are you getting that plus one from?
Ayesha:
The middle.
JD: What do you mean, can you show us? What do you mean by the middle? (waits)
Josh, can you help her out?
Josh: The um, two on the one side, two on the other side, that’s four, plus the one on the bottom.
JD: … does anybody see another pattern?
Sulanette: I see that they’re all odd numbers.
JD: ... Why do you think they’re all odd numbers?
Sulanette: Because it starts off with the V1, right, it shows 3. Well, 3 is an odd number. If it was an even number, I don’t know where it should go in the V pattern, so they’re all odd numbers.
JD: I think Sulanette was saying something pretty interesting. She was saying that she didn’t know if it could be an even number. Can there be an even number. Can I say, if there were 84 birds, what V pattern would that be? Could there be 84 birds? …
Ashley: I think that it can never be an even number because as in the V pattern of number 1, that is actually a complete V, and an even number couldn’t make that.
JD: Why not?
Ashley: Because the V is made up of 3 birds.
JD: OK the V is made up of three birds.
Ashley: And if you keep going and adding two, it’s made up of all odd number of birds, the V patterns.
JD: So here’s the first one right here (draws line around it). And then you say that two more birds come in like that, and so it’s going to be odd why?
Ashley: It’s going to be odd because if you add two more birds to three, that’ll make five, and five’s an odd number.
JD: Can someone tell me the number of geese that would be in the V number 10, the 10th pattern along?
Oscar: I got 21.
JD: And how did you find that information?
Oscar: I multiplied 10X2
JD: Why did you do that?
Oscar: Because I was thinking that every time the two extra birds come, they come and then they have a group ...
Then I multiplied it by 2 and it comes out to 20, and then I added a 1, the one in the middle, then it came out to 21.
Jenny: What they are saying is that there are two geese in a pair and there are 10 pairs, so he would try and multiply it by 2 X 10 and then add the leader would be 21.
JD:
.. Does anybody see another relationship on how they could have found
out 10 without doing … if they didn’t know that it’s times 2 plus 1.
Alande?
Alande: I think that from the beginning, like if you put one in the top of the 3 and you keep adding two, and you keep going on and then you get to the 10th V-pattern and you get to 21.
JD: So we’re going to go back to what Oscar said originally and Oscar said you take the V-number, Oscar, and what do you do with it?
Oscar: Multiply it by 2 and add 1.
JD: … Does everybody agree, anybody want to try it out? Let’s try it out for … a V-number of 15. How many geese would that be? Use your geese right there.
Ashley: In V-pattern #3, the V means that there are three groups, or three pairs, plus that extra one. So let’s say there’s 15 pairs in the 15th pattern, plus this one here.
Alande: I think that if you like make it from the 6th V-pattern, and to the 15th, you keep adding two and going down, and when the V-pattern gets to 15, we could check it to see if it’s right.
Meg: ANOTHER REALLY CENTRAL IDEA IS THE MULTIPLE STRATEGIES THAT CHILDREN CAN BRING TO A GIVEN PROBLEM, AND THOSE MULTIPLE STRATEGIES REFLECT DIFFERENT LEVELS OF UNDERSTANDING AND DIFFERENT LEVELS OF ABSTRACTION. BUT THE REALLY IMPORTANT IDEA IS THAT THEY ARE ALL VALUED.
JD: There’s been some confusion with people thinking there’s 15 geese. I heard this table say, well, we each have 15 birds, but it was the V-number was 15. This 15 tells us what, and I think I just heard it over there, Ashley?
Ashley: The V-number tells us how many pairs there are.
JD: And what do we do after we know how many pairs there are?
Ashley: You add the one, which is the leader.
JD: …. So what did your group come up with as an answer for the V-number 15?
Ashley: We came up with the answer of 31.
JD: How many groups got 31 geese when they did this out? That’s great, that’s great. I think we’re all on the same page.
JD: If our V-number is 100, Oleg, what would our number of geese be?
Oleg: 201.
JD:
How many people know or understand that if the V-number was 100, there
would be 201 geese? John, how do you know?
John: 100 is half of 200, and with the leader, I just added that.
JD: So 100 X 2 plus the leader, and there you go. How many people got that? That’s great.
JD: Great, all right, we’ve learned a lot of math today….
BUILDING FORMULAS
Cheryl Anderson: Good morning. Today we’re going to talk about construction, beams construction. …Has anyone seen anything being built lately? Yes, Unika?
Unika: The Providence Place Mall. When they were getting built, when I drove past it there was like triangular beams being built.
CA:
OK, let’s open our books to page 10
Iannes: Construction work has begun on a large building that will be used for a part of a movie set. The framework consists of metal beams on concrete columns. Beams can have different lengths. The length of a beam is the number of rungs along the underside.
CA: Does anyone know what that’s a picture of at the top of the page? Yes. Irene?
Irene: It’s chart of the length of the beams.
CA: So let’s just look at the table for one moment here. The first picture has the length of beams of 1, so there’s one rod along the bottom. How do you think they got the number of rods? Where does that number come from? Leore?
Leore: The 3 rods that are in the one triangle.
CA: Okay, can anyone tell me, how many rods would be in the second length of beam?
Christian: Seven.
CA: …Okay, so what I would like you to do is start drawing the chart and filling it in down into the length of beam 8.
CA: … As you are filling out the table, I would like you to write down how many rods it would take to build all of those lengths of beams that you’ve already drawn. Fill in the last column on your table, and at the bottom of your paper I would like you also to write down as many patterns that you see in that table. … If you see something and someone in your group doesn’t see it, show them it. Show them that pattern that you’re talking about and you all can jot them down either at the bottom of the page or on the back of the page.
Student: In the number of rods, every time it increases here by 1 here it increases by 4 and here, every time we do like another beam it increases by 2 like for this one right here.
Unika:
There’s another pattern here, and it goes, every time there’s a triangle
there’s always one less upside down triangle, so it’s like the regular
triangles minus one, which make up the upside down triangles. Like say,
for instance, number 8, there’s 8 right side up triangles and then when
you look at the top there’s only 7.
Student: So you always have to subtract one when you go to the upside down one.
Unika: Like number one, there are none, because 1 minus 1 is zero. So we subtract the number.
Student: That’s another pattern, right?
Unika: Yes.
CA: I would like anyone to share one thing that you saw in the patterns. Anyone have something to share? Leore?
Leore: The number of rods goes up by 4 every time the length of the beam goes up by 1. .
CA: Any
other patterns that you see? Cassandra?
Cassandra: Every time the length of the beam goes up by 1, like 1, 2, you add 2 triangles.
CA: Anyone else see anything else in those triangles? Yes, Beverly?
Beverly: Every time you make a new beam, you add a rod and a triangle.
CA: Do you want to show us a little example on the board … because I didn’t see it until you showed me, so I’m going to let Beverly show.
Beverly: To make length number 7, you need to add a rod and another triangle to make length number 7.
CA: How could we figure out by looking at our table now, what would be length #12? … I don’t want the answer, I want to know, by looking at our table, how we can figure it out. Yes Unika?
Unika: You could figure it out by taking, like, if you have number 6, you can keep … adding 4, and when you keep adding 4 you end up finding the total of rods in #12.
CA: … What if you didn’t have that table at all, what if I wanted you to do length 50 right now? What if I took all your papers and I say, do length 50? Would you want to do it?
Christian: No.
CA: So we need to come up with another way of figuring this out, without just knowing the one above it gets the next one. We talked about already, next current formula. What is the next current formula we’ve been using so far? What’s the next current formula?
Diana: You go up by 4.
CA: So to get to the next one is the current one plus 4. But we don’t want to know every single one – we don’t want to have to go through 49 to get down to 50.
CA:
What I’m going to have you do in pairs is on the construction paper
build length 6. And what I want you to try to do when you build length
six is come up with a way to figure out the length without having to know
the length right before it, without having to know the number of rods
in length five.
CA: What surprised me about … this math program is that … traditionally you always start with the formula and have kids substitute in … But … they actually came up with the formula and knew where they got it from and what it meant, and that was impressive to me.
Beverly: Wouldn’t it be easier just to multiply the beam by 4? Like if you need a new high number instead of writing it out wouldn’t you just multiply it by 4?
Student: If we multiply by 4, look at #1, there’s only 3. So we have to subtract 1.
Student: Yeah, but that row goes down one.
Beverly: No, you see, it goes outward like that.
Student: Yeah, this one don’t got a rod.
Beverly: Yeah, so one of them. If we multiply (4 – 1)…
Student: 2 X 4 = 8, minus 1 is 7.
Beverly: And then if you multiply 3 X 4, that’s 12, minus 1 is 11. So would that work? Like multiply it by 4 – 1? Is that your pattern, times 4 – 1? You use 6 X 4, so that’s 24, minus 1 is 23, and that’s the total, that’s the number of rods altogether. You could use another one, because 5 X 4 is 20, minus 1 equals 19. It works, any which way you look at it.
CA:
You want to be able to go from the length being six to get an answer
of 23. How could you do that? Look at your picture and figure out if you
just knew the length was six, picture in your head where you would put
down all the toothpicks and how you would put them down.
Mary: The length on the bottom, so if they are six on the bottom then there will be five on the top.
CA: So we got the L along the bottom and it will always be whatever it is along the bottom. That will always be the length right? So we can just use L for that. If it was 100, how many would be along the bottom?
Mary: 100.
CA: And
if the length was 20, how many along the?
Mary: 20.
CA: and if the length was 30, how many along?
Mary: 30
CA: So it always stands for that. Now, OK, what did you say about the top?
Mary: The top, that’s always minus 1.
CA: So how can we write that?
Mary: L minus.
Leor: Length minus 1.
CA: OK, let’s write that down, then.
CA: OK, so now we’ve got this and we’ve got this. What do we have left to talk about?
Leor: The top.
CA: Well, we talked about the top, we talked about the bottom.
Leor: The middle.
CA: How could you talk about that, the middle? Remember, talk about it in terms of the length. Look at the length and how do they relate to these two things.
Mary: You always add 2.
CA: So say I knew the length was 5, how many of those toothpicks would I need for 5?
Mary: 10.
CA: If I knew the length was 6, how many of these toothpicks do I need?
Mary: 12.
CA: If I knew the length was 10, how many of these toothpicks do I need?
Christian: 20.
CA: Why 20?
Christian: Because you have to double it. You double the length of the beam, times 2, and it gives you the middle.
CA: How am I going to write that in my formula?
Leor: L X 2.
CA: Now we have three pieces, because we talked about the bottom, the top, and the middle … how would we put all of these three things together?
CA: Could I have 2 groups share their formulas and how they came up with their formulas. They have very different looking things, so I would like Christian’s group, Leor and Mary, to come up and explain where this formula came from.
Christian:
We got the L from the length of the bottom, then we did the L minus
1 because if you do the length number minus 1 it gives you the top number.
Then L times 2, the length of the bottom times, you multiply it by 2 and
it gives you the middle.
CA: How do we know that gives us the middle, how does the length compare to the middle?
Christian: Because there’s twice as many in the middle than there is on the bottom.
CA: Can you read your formula, Beverly?
Beverly: 4L – 1 = the total.
CA: Very short, 4L - 1. Where does that come from? That’s what we want to know.
Beverly: Okay, for pattern number 6, if you multiply that by 4, that would equal 24. But since the last one doesn’t have an extra rod you have to minus 1 and that equals 23. And all together there’s 23 rods.
CA: Where do you get the 4 from and what do you mean about minusing one rod, how come it's 4? (28:54)
Beverly: So the 4 comes from this, this, this, and then this. And it goes all the way to the last one.
The last one doesn’t have this, so you have to minus one. And that’s how we got 4L – 1.
CA: Very good. Thank you.
CA:
Okay, today we worked on coming up with some formulas for our beam
and we came up with two formulas right here from two different groups
and there are two formulas that give us the same answer so we call these
formulas equivalent formulas. Tomorrow I’m going to have some more formulas
for you to look at and for you to tell me if they are or not are they
equivalent to these formulas?
MC: WHEN THEY COME TO ME, ONE OF THE THINGS, ON TOP OF BUILDING THE EQUATIONS AND THE FORMULAS, IS THAT THEY ARE GOING TO HAVE TO LOOK AT A GRAPH AND INTERPRET THE GRAPH, AND ALSO WRITE AN EQUATION FROM THE GRAPH, SO A DIFFERENT REPRESENTATION.
LINEAR PATTERNS
MC: Now as you can see … when Jason went to the barber the other day, that was his expression when he came home. So he was in shock, wasn’t he. … And I think Louie was the next guy in the barber seat.
MC: Jason, how long do you think your hair is?
Jason: About 3 millimeters.
MC:
Now I want you guys to take a look at Taisha’s hair. Stand up for
one moment. How long did it take, Taisha, to grow your hair?
Taisha: I never cut it.
MC: You never cut it? Since when?
Taisha: Since I was a baby.
MC: … How long do you think Taisha’s hair is? … Chelsea?
Chelsea: About two and a half, three feet.
Charlie: I think it’s about 4 to 5 feet.
MC: 4 to 5 feet? That’s maybe how tall you are, Charlie. All right, does someone want to come up and measure it?
Charlie: We got 31 inches.
MC: 31 inches? Would you measure it in centimeters now?
Jessica: 76.5
MC: OK, can you estimate, if Jason decided to grow his hair one day, how long would it take for it to grow as long as Taisha’s? Louie?
Louie: Probably about how old she is, because she never cut it … so it probably would take him 13 years or 14 years.
Heather:
Below is a table that shows the length of Paul’s hair in centimeters as
he measured it each month.
MC: What I’d like you guys to do now is break up into groups … and I want you to work on questions 14 to 16 …
So we’re going to see in this case how long it would take Paul’s hair to grow.
Christa: We have to figure out the pattern of how the numbers go, and so –
Student: See how much each month. So let’s like subtract for the first month for the length, the zero, how much it grew 3.5, subtract about 2, and you get 1.5, so I guess that’s how much he keeps adding each month.
Barry: Then it says if it’s 10 centimeters long at some point, how long will it be one month later?
Student: Add 1.5, so it’s 11.5.
Student: Because you just added another month to it, do you get it?
Barry: Yeah.
Student: How long will Paul’s hair be after a year, if it keeps growing at the same rate and he does not get a hair cut?
Student: If we keep the chart going to 7 months, 8 months, up to like 12 months, then 7 will be 12.5, 8, 14 and then so on and so then 12 is 20 centimeters.
MC: So like we pointed out earlier, the pattern is every month it’s growing by –
Kids: 1.5 centimeters.
MC: Very good … Question 18 says if you know the length of Paul’s hair in the current month, you can use it to find his hair length in the next month. Write a formula using next and current.
What would that formula be? And why? So when you are writing the formula, be sure you are able to explain it.
Meg
Meyer: EVEN WHEN A CHILD IS WORKING … AT A PREFORMAL LEVEL, THERE
WILL BE TIMES AND ACTIVITIES WHEN THEY ARE WORKING AGAIN INFORMALLY. SO
IT’S NOT JUST A STRAIGHT, LINEAR PATH, BUT RATHER A CYCLING, AND EVENTUALLY,
WE HOPE THAT ALL KIDS MOVE INTO A MORE FORMAL UNDERSTANDING OF ALGEBRA.
Student: The next one, after length 20, is 21.5, so that’s the next one, the current.
MC:
Now can you write that as a formula, though, without the numbers,
what would it look like? … Think about how we’ve written formulas before,
we first describe them, right? So if you’re going to describe –
Barry: In words?
MC: In words. And the words you’re going to use are next and –
Barry: Current.
MC: Current. So –
Matt: Next plus current –
Jayla: Current plus 1.5 equals next.
MC: OK?
Matt: OK.
MC: Now what we’re going to do using the information on the table, okay, we’re going to show the information in a different way on a graph. So each group will get graph paper, the markers, a ruler, and in your groups I want each group to come up with a graph based on Paul’s hair growth.
MC: THE TERM LINEAR ACTUALLY COMES OUT BEFORE WE DO THIS PARTICULAR ACTIVITY … THERE WAS ONE ON TREE GROWTH AND SO THE STUDENTS KNEW … THAT IF THE CHANGE, IN THIS CASE PER MONTH, IS THE SAME, THEN THE POINTS SHOULD FALL INTO A STRAIGHT LINE.
Student: For 6 it’s going to be 11.
Student: Then it would be 12.5.
Student: Add 0.5, so it’s 12.5.
Student: 12 is going to be –
Student: Right there.
Student: 15.5
Student: The circles are going in a straight line as you can see, so it’s probably the right pattern.
Student: Now we don’t need to figure it out.
Student: Now we just put the last circle right there.
Student: 21.
Student: Now you draw a straight line.
MC: If we look at the graph for a moment, looking at the similarities and differences, what’s one thing that they all have in common, Cathy?
Cathy: All the lines on each graph goes diagonal so you can tell that … they are all right, yeah, I guess that they are all right because we all got the same results.
MC: Okay, are all the lines pointing the exact same direction?
Kids: No.
MC: No, okay. So we will look first at Barry’s group, okay? Now Cathy you said same direction. All right, you described it as diagonal. But again, looking at how, looking at the shape of the line -
Chelsea: I was just going to say that Crystal’s line is more vertical and Barry’s is more horizontal.
MC: Why is that?
Chelsea: I guess because of the way they spaced it. Barry spaced his two each on the horizontal and the vertical and Crystal spaced it a half on the vertical and 2 on the horizontal.
MC: Excellent observation Chelsea, does everyone see that? OK, so if we go back to what Cathy was talking about, the direction of the line. And Chelsea described Barry’s as moving more toward the horizontal and Crystal’s as moving more toward the vertical.
MC: I just want to call your attention to one more graph, that Chelsea, Louie and Charlie had worked on originally. What do you notice that’s different with their graph? What do you notice that’s different with their graph? OK, Jusenia?
Jusenia: On all the other graphs, the time is on the x axis, and on theirs the time is on the y axis.
MC: Louie, why did you make a change?
Louie: We thought that it would be easier, but then we figured out it’s harder because our group, we read the time first before the growth, and that’s why we had to do it all over again.
MC: Why did you read the time first? Someone else in the group talk about that, because you all had a part of it. Jason?
Jason: Because we saw the time above length on the paper, so we thought time would go onto the vertical axis and the length would go on the horizontal axis.
Chelsea: I want to compare the two, OK?
MC: You want to compare what?
Chelsea: The slope and everything.
MC: OK.
Chelsea:
This one, it’s 2 and 3 and if you divide 2 into 3 you get 1 and a
remainder of .5, so it would be 1.5. And that’s for the growth of the
hair every month and then you find the slope, besides that point down
there, if you find it right here it goes over 3 and up 2 as you divide
3 into 2 you get a decimal and that doesn’t show the hair growth in every
month, the right one does.
MC: ONE OF THE STUDENTS NOTICED, SHE ACTUALLY USED THE TERM SLOPE, WHICH WE TALKED ABOUT BEFORE IN ANOTHER UNIT, BUT WE DIDN’T REALLY GET INTO ITS MEANING OR HOW TO FIND SLOPE, BUT SHE UNDERSTOOD THAT MOVING TO THE RIGHT TWO AND UP THREE SHOWED A CONSTANT CHANGE.
Meg Meyer: THE CONNECTIONS BETWEEN THE STRANDS OF THE CURRICULUM ARE AS RICH, IF NOT RICHER, THAN THE CONNECTIONS ACROSS GRADE LEVELS WITHIN A SINGLE STRAND, AND WE HEAR TEACHERS TELLING US, KIDS ARE GETTING THESE THINGS.
MC: On the next page, we have someone who, you can see in the picture is a girl named Sonya and she’s having her hair done. It says Sonya’s hair grew both 14.4 centimeters in one year. It is possible to write the following formulas, and I have them up here on the board: Next = Current + 14.4, and Next = Current + 1.2.
Can
we explain what each formula means? Jayla?
Jayla: The first one, this is next equals current plus 14.4 is the current plus, they’ve got the 14.4 for the whole year.
MC: So 14.4 represents her hair growth in one year.
Jayla: And the next one is the 1.2 represents each month.
MC: Good, how did you get that?
Jaila: Because 12 divided by 14.4 equals 1.2.
Student: The formula for Sonya’s hair growth does not include information about how long her hair was at the beginning of the year. You knew that the beginning length of Paul’s hair was two centimeters. That’s why it is possible to make another formula for Paul’s hair growth.
MC: Matt, can you just read the formula for us, please?
Matt: L equals 2 plus 1.5 T.
MC: Let’s talk about what each letter and number in the formula, what it represents. OK, so the L represents ?
Evelyn: The L represents the length and the T represents the time.
MC: But by how long? Paul’s hair grew every, what?
Evelyn: Month
MC: That’s good, the time is in months. How about the length, the length is in what?
Evelyn: In centimeters.
MC: And how about the two and the 1.5
Samantha: The two represents his hair after he got it cut, in centimeters.
MC: How about the 1.5?
Samantha: How much it grew every month.
MC: Now what I’d like you to do, looking at the next question is that Sacha’s hair – now we have another one – Sacha’s hair is 20 cm long and grows at a constant rate of 1.4 cm a month. Write a formula with L and T to describe the growth in Sacha’s hair.
Charlie: So we got L equals 20 plus 1.4 and then we got that because length was 27 meters and it grew 1.4 centimeters every month. So do you get it?
Louie: Yes, we got the same thing.
MC: What has a formula that they would like to share? Evelyn?
Evelyn: L = 20 + 1.4T
MC: Did any other group get a different formula? So you all got the same? Well done. Can anyone describe their formula?
Jusenia: L equals length of hair, 20 equals how much hair she has to begin with. The 1.4 is how much her hair grows each month, and the T equals how much time in months.
MC: And once again, the length is just what?
Jusenia: Centimeters.
MC: Very good, the length is in centimeters. Excellent.
MC:
OK, the next time you guys go to get a haircut, I just want you to
keep in mind this gentleman from Thailand, Hoo Sateow, I think that’s
how you pronounce it, his hair is 16 feet, 11 inches long. So to get an
idea of that I need … Jason and Taisha … Take this end please and just
roll it out. And you guys think Taisha’s hair is long!
MC: For homework tonight, I want you to work on page 36, questions 25, 26, and 27, and it talks about a different type of growth – fingernail growth. And perhaps for extra credit you might also want to find out who holds the world record for the longest fingernails. OK, good luck.
MC:
Because they’ve all gotten haircuts … it’s some connection to the
math for them, so it’s not just a formula, it’s not just a graph, it’s
some interest in what they are working on.
JD: So even if it’s not something that’s happening to them, like the haircut for example, the building is something they’ve seen, the geese, they’ve all seen fly in a V-pattern
CA: With a little bit of discussion, you can get everyone up to the same level as far as what we’re discussing at that time.
JD: And then they understand that this math makes sense.
MC: We get less and less of that “why do we need to learn this”.
CA: There is always something to answer the “why” – the picture or the story always tells them some reason why they are learning this kind of math.