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OCR for page 231
Bail
- ~
F1F2li~
-- ~
it'
Cases of Mathematics Inst:ruct:ion to Enhance Teaching
This case was developed by the collahora-
tive team of Margaret Smith, Edward
Silver, Mary Kay Stein, Marjorie
Henningsen, and Melissa Boston under the
auspices of the COMET Project. COMET;
funded by the National Science Foundation,
is a project aimed at developing case
materials for teacher professional develop-
ment in mathematics. The project is co-
directed by Edward Silver, Margaret Smith,
and Mary Kay Stein and is housed at the
Learning Research and Development
Center at the University of Pittsburgh. For
additional information about COMET
contact Margaret Smith by phone at (4129
648-7361 or by e-mai! at .
OPENING ACTIVITY
Solve
For the pattern shown below, compute
the perimeter for the first four trains,
determine the perimeter for the tenth
train without constructing it, anti then
write a (lescription that coul(1 be use(1 to
compute the perimeter of any train in the
pattern. (Use the edge length of any
pattern block as your unit of measure.)
The first train in this pattern consists of
one regular hexagon. For each subse-
quent train, one a(l(litional hexagon is
a(l(le(l. The first four trains in the pattern
are shown below.
OCR for page 232
Catherine Evans Talks About Her
Class
~ have been teaching the new curricu-
lum for about six weeks now and ~ have
found that my sixth graders are not
always prepared for the challenges
presented. The tasks in the curriculum
generally can't be solved by just using an
algorithm, the solution path is not immedi-
ately evident and usually involves explor-
ing and reasoning through alternatives,
and most tasks involve providing a written
explanation. If my students can't solve a
problem imme(liately, they say, "I (lon't
know," and give up. They have had limited
experience in elementary school actually
engaging actively with mathematics and
expressing their thinking and have found
this to be very difficult.
Seeing students give up has caused me
great concern. ~ can't buy the idea that
kids don't feel bad starting off with what
they perceive to be failure. When they
have work they can't do or don't have the
confidence to do, then ~ have to intervene.
~ decided to help kids do more verbaliza-
tion in class, get to the kids who didn't
volunteer and guarantee them success by
asking them to do things they couldn't fail
to do right. ~ can't ignore the fact that
success breeds success. Too many are
starting out with what I'm sure they
perceive to be failure.
In order to ensure student success, ~
have started to make some modifications
in the curriculum, at times putting in an
extra step or taking out something that
seems too hard; rewriting problem
instructions so that they are clearer and at
an easier reading level; and creating easier
problems for homework. In addition,
during classroom instruction ~ try to break
a task into small subtasks so that students
can tackle one part of the task at a time.
We have been talking about patterns for
a few weeks. The new unit that we started
APPE N DIX
last week uses trains of pattern blocks
arranged in a geometric sequence. The
unit is supposed to help students visualize
and describe geometric patterns, make
conjectures about the patterns, determine
the perimeters of trains they build, and
ultimately, to develop a generalization for
the perimeter of any train in a pattern.
This unit really lays the groundwork for
developing the algebraic ideas of generali-
zation, variable, and function that students
will explore in gra(les 6 through 8. Expe-
riences like these lay the foundation for
more formal work in algebra in eighth
gra(le.
We spent a lot of time in the beginning
of this unit just making observations
about the trains the number of pattern
blocks in a train, the geometric shapes
that comprise a train, and the properties
of a train (e.g., each train has four sides,
opposite sides of the train are parallel).
Students got pretty good at making
observations about specific trains once we
ha(1 (lone a few, but ~ ha(1 to keep remin(l-
ing them that the observations nee(le(1 to
be mathematical. For some patterns ~ got
some really weird responses like "it looks
like a squished pop can" or "it looks like a
belt buckle." But once ~ reminded students
that the point in making observations was
to be able to predict what larger trains
were going to look like, they were able to
move beyond these fanciful responses.
The Class
Yesterday for the first time we starte(1
(letermining the perimeters of the trains
using the side of the square as the unit of
measure. Homework last night ha(1 been
to fin(1 the perimeters of the first three
trains in the pattern shown below. ~ also
aske(1 students to fin(1 the perimeter of the
lOth, 20th, anti lOOth trains in this pattern.
My plan for class was to begin by (liscuss-
ing the homework and then having
students explore another pattern.
OCR for page 233
train 1 train 2
train 3
As students entered the classroom and
got their papers out, ~ made a quick trip to
the back of the room to check on the
video camera. My colleagues and ~ have
decided to videotape some of our classes
this year so that we could use the tapes to
reflect on how things are going with the
new curriculum and to talk about various
issues that arise in using the materials.
This was my first day of taping, and ~ was
a little nervous about being on film.
Students asked about the camera as they
entered the classroom but seemed
unfazed by the idea of being taped. ~ just
hoped ~ could forget that it was there.
Discussing the Square Pattern Trains
In order to get things started, ~ asked
students to make observations about the
pattern. Shandra said that she had noticed
that all of the trains were rectangles. lake
said that he noticed that the perimeter of
the first train was 4. ~ asked him to come
up and show us. When he got to the
overhead he took a square tile (black) and
laid an edge of the square next to each
side of the train as he counted the sides.
1 ~3
This was the procedure we had estab-
lishe(1 yesterday, anti ~ was please(1 to see
him use it. ~ thanked him anti he returne
to his seat.
Since lake had started talking about
perimeters, ~ (leci(le(1 that we might as
well continue in this (Erection. ~ aske(1
Zeke what he foun(1 for the perimeter of
the second train. Zeke said he thought it
was four. ~ asked him if he would go to
the overhead and show us how he got 4.
He explaine(l, "the train has four si(les
just counted them I, 2, 3, and 4." (See the
iagram below.)
2
1=
4
3
~ saw what Zeke was (loin". He was
counting the number of sides, not the
number of units in the perimeter. The
number of si(les anti number of units were
the same in the first figure but not in the
secon(1 figure. ~ aske(1 Zeke to stay at the
overhea(1 anti ~ aske(1 the class if someone
could review what perimeter is. David
said that it was the sides all the way
around. ~ asked if anyone had another
way to say it. David's definition really
supported what Zeke had done, and ~ was
looking for a definition that would cause
students to question Zeke's solution.
Finally Nick said that the perimeter would
be six. Nick explained, "I used Jake's way
and measured all the way around the
outside of the train with the square tile.
It's not 4 because the top and bottom each
have two units." Although this was not
the definition ~ was looking for, ~ figured
that this explanation would help students
see why the perimeter was 6 and not 4.
At this point ~ decided to ask Desmond
to come up and measure the perimeter of
APPE N AX
OCR for page 234
the third train for us using the procedure
that Nick had just described. I have been
trying all year to get him involved. Irately
I have been asking him questions that I
was sure he could answer. They were not
meant to challenge him in any way, just
help him feel successful. These experi-
ences have had an immediate positive
effect on Desmond he would actively
participate in class following these epi-
sodes. So Desmond came up to the
overhead, and I gave him the black square
and asked him to measure the third train.
I really thought that this would be a
simple task, but Desmond did not seem to
know what to do. Since this experience
was supposed to be about experiencing
success, I took his hand and helped him
move the square along the outside of the
train, counting as we proceeded.
·~.
2 3 4
·1 ~ ~ ~5.
~ 7 6
·~.
I thanked Desmond for his help. I was
sure that this would clear up the confu-
sion. I told Zeke that a lot of people make
the same mistake that he did the first time
they do perimeter. Just to be sure that
Zeke understood the way to find perim-
eter, I asked him if he could build the
fourth train in the pattern. He quickly laid
four squares side to side. I then asked
him if he could find the perimeter by
measuring. He proceeded to count the
si(les while moving the si(le of the square
APPE N DIX
along the perimeter of the train 1, 2, 3, 4,
5, 6, 7, 8, 9, 10. He looked up when he
finished and announced, "It will be ten!" I
thanked him for hanging in there with us,
and he returned to his seat.
Before moving on to the next part of
the assignment, I asked if anybody had
noticed anything else about perimeter
when they (li(1 just the first three. Angela
had her hand up, and I asked her what
she had noticed. She explained, "on the
third train there are three on the top and
three on the bottom, which makes six,
and one on each end." I asked her if she
would go to the overhead and show us
what she meant. She restated, "See there
are three up here (pointing to the top of
the train) and three down here (pointing
to the bottom of the train) and then one
on each end."
3
1~L: 1
3
I was surprised by this observation so
early on, but knowing that it would be
helpful in (letermining the perimeters of
larger trains, I asked Angela if she could
use her system to find the perimeter of
the fourth train. She quickly said "10." I
asked her to explain. She proceeded,
"four on the bottom and four on the top
and one on each end."
Class can be pretty fast paced some-
times, with individual students, the whole
class, and me going back and forth in a
rapid exchange. A good example of this
happened at this point as I tried to put
Angela's observation to the test and see if
I could get the whole class involved in
OCR for page 235
the third train for us using the procedure
that Nick had just described. I have been
trying all year to get him involved. Irately
I have been asking him questions that I
was sure he could answer. They were not
meant to challenge him in any way, just
help him feel successful. These experi-
ences have had an immediate positive
effect on Desmond he would actively
participate in class following these epi-
sodes. So Desmond came up to the
overhead, and I gave him the black square
and asked him to measure the third train.
I really thought that this would be a
simple task, but Desmond did not seem to
know what to do. Since this experience
was supposed to be about experiencing
success, I took his hand and helped him
move the square along the outside of the
train, counting as we proceeded.
·~.
2 3 4
·1 ~ ~ ~5.
~ 7 6
·~.
I thanked Desmond for his help. I was
sure that this would clear up the confu-
sion. I told Zeke that a lot of people make
the same mistake that he did the first time
they do perimeter. Just to be sure that
Zeke understood the way to find perim-
eter, I asked him if he could build the
fourth train in the pattern. He quickly laid
four squares side to side. I then asked
him if he could find the perimeter by
measuring. He proceeded to count the
si(les while moving the si(le of the square
APPE N DIX
along the perimeter of the train 1, 2, 3, 4,
5, 6, 7, 8, 9, 10. He looked up when he
finished and announced, "It will be ten!" I
thanked him for hanging in there with us,
and he returned to his seat.
Before moving on to the next part of
the assignment, I asked if anybody had
noticed anything else about perimeter
when they (li(1 just the first three. Angela
had her hand up, and I asked her what
she had noticed. She explained, "on the
third train there are three on the top and
three on the bottom, which makes six,
and one on each end." I asked her if she
would go to the overhead and show us
what she meant. She restated, "See there
are three up here (pointing to the top of
the train) and three down here (pointing
to the bottom of the train) and then one
on each end."
3
1~L: 1
3
I was surprised by this observation so
early on, but knowing that it would be
helpful in (letermining the perimeters of
larger trains, I asked Angela if she could
use her system to find the perimeter of
the fourth train. She quickly said "10." I
asked her to explain. She proceeded,
"four on the bottom and four on the top
and one on each end."
Class can be pretty fast paced some-
times, with individual students, the whole
class, and me going back and forth in a
rapid exchange. A good example of this
happened at this point as I tried to put
Angela's observation to the test and see if
I could get the whole class involved in
OCR for page 236
using her observation to predict future
trains. Once Angela's pattern became
obvious to her ~ wanted to make sure that
everyone in the class saw it too. So ~
proceeded with the following question and
answer exchange:
Angela:
Me:
Angela:
Me:
Angela:
Me:
Angela:
Me:
Angela:
Me:
Angela:
Me:
Tamika:
Me:
CLASS:
Me:
CLASS:
Me:
CLASS:
Me:
CLASS:
Using your system, do you
think you could do any number
say? What would you do for
10? How many on the top and
the bottom?
10.
How many on the ends?
2.
How many all together?
22.
Let's do another one. Listen to
what she's saying and see if you
can do it also. Angela, in train
12, how many will there be on
the top and bottom?
12.
And then how many will there
be on the ends?
2.
How many will there be all
together?
26.
Tamika, what's she doing?
She's taking the train number
on the top and bottom and
adding two.
OK, let's everybody try a few.
can pick any number. Train 50.
How many will there be on the
top and bottom? Everybody!
50 [with enthusiasm]
How many on the ends?
2.
How much all together?
102.
Train 100, how many on the top
and bottom?
100. [louder and with even
more enthusiasm]
Me:
CLASS:
Me:
CLASS:
Me:
CLASS:
Me:
CLASS:
Me:
CLASS:
How many on the ends?
2.
How much all together?
202.
Train 1000, how many on the
top and bottom?
1000. [loudest of all]
How many on the ends?
2.
How much all together?
2002.
At this point ~ asked if they could
describe anything ~ gave them. Another
resounding '~ES" answered my question.
One of the things that ~ have found is that
responding in unison really engages
students and helps their confidence.
When they respond in unison they feel
that they are part of the group. Everyone
can participate and feel good about
themselves.
Angela's observation had really led us
to finding the perimeters for any train, so
decided to continue on this pathway.
asked if anyone had figured out the
perimeters using a different way. ~ looked
around the room no hands were in the
air. ~ wanted them to have at least one
other way to think about the pattern so ~
shared with them a method suggested by
one of the students in another class. ~
explained that she had noticed that the
squares on the ends always have three
sides they each lose one on the inside-
and that the ones in the middle always
have two sides. ~ used train three (shown
in the diagram below) as an example and
pointed out the three sides on each end
and the two sides on the middle square.
F:
l
1 1 1
1 ~ ~ ~
APPE N DIX
OCR for page 237
wanted to see if students understood this
so ~ asked how many squares would be in
the middle of train 50 with this system.
Nick said that there would be 48. ~ then
added that there would be 48 two's,
referring to the number of sides that
would be counted in the perimeter, and
three on each end. ~ asked what 48 two's
would be. Carmen said it would be 96. ~
then asked what the perimeter would be.
Shawntay said that it would 102. She went
on to say that that was the same as what
we got from train 50 when we did it
Angela's way! ~ told the class that was
right, there isn't just one way to look at it.
Consiclering a New Paffern
We had spent nearly 20 minutes on the
square pattern, and it was time to move on
to another pattern. ~ quickly got out my
pattern blocks and built the train shown
below on the overhead. ~ told students
that ~ wanted them to work with their
partners and build the first three trains in
the pattern, find the perimeters for these
three trains, and then to find the perimeters
for the lOth, 20th, and lOOth trains. ~ put
the pattern of square trains we had just
finished back up on the overhead under-
neath the hexagon pattern and suggested
that they might want to see if they could
find anything that was the same for the
hexagon pattern and the square pattern
that would help them. Since the generali-
zations for the perimeters of these two
trains had some similarities, ~ thought this
would help them find the perimeters for
the larger trains in the hexagon pattern.
After about 5 minutes students seemed
to be getting restless. Since most seemed
to have made progress on the task, ~
decided to call the class together and see
what they observed about the pattern.
Although this is not exactly what ~ asked
them to do make observations ~ felt
that it provided a more open opportunity
for all students to have something to say.
asked Tracy what she had noticed. She
said that every time you add one. "Add
one what?," ~ asked. "A hexagon," she
responded. ~ then asked about the
perimeter. Darrel said that he discovered
that it was six. 'what was six?," ~ asked.
Darrel clarified that six was the amount
around the hexagon around the edges
on the first train. ~ asked Darrel about the
second train. He explaine(l, "the hexagon
has six around it anti then you take away
one for each side in the middle so it is 5 +
5 or 10. Then on the thir(1 one you still
have 5 + 5 for the end ones and you add
four more sides for the new hexagon you
a(l(le(l."
~ wanted to see if Darrel realize(1 that
his observation would lead to a generaliza-
tion. ~ aske(1 him if what he ha(1 (liscov-
ere(1 would tell him anything about
building another train. Darrel sai(l, 'yeah.
On train 4 there would be four hexagons.
The end ones woul(1 each have five anti
the two mi(l(lle ones woul(1 each have
four." "If you were to buil(1 train 10,"
asked, "coul(1 you tell me how many
would have four sides and how many
would have five si(les?" Darrel appeared
to think about it for a few seconds anti
train 1
APPE N DIX
train 2
train 3
OCR for page 238
then responded that eight hexagons
would have four sides and two hexagons
would have five sides. ~ wanted to make
sure that students understood what
Darrel was saying so ~ asked him where
the two with the five sides would be. He
looked at me as though ~ were crazy and
said, "Mrs. Evans, they would have to be
on the ends!"
Again, ~ wanted to see if students could
use Darrel's method on any train. ~ asked
Tommy if he could describe the 20th train.
Tommy explained, "For train 20 you'd
count the sides and count the ends. You
subtract 2 from 20 and that would be IS
and then you multiply IS by 4, because all
the hexagons in the middle have four and
then you would add 10 from the ends."
was impressed with his explanation, and
he seemed to be pretty proud of himself
too. ~ wanted to make sure that everyone
had all the steps that tommy had so
nicely explaine(l.
~ then asked Jeremy if he could do the
30th train. He said that he didn't know.
felt that he could probably do this if ~
provided a little structure for him. ~ asked
him how many hexagons would have five
sides. He said in a questioning tone,
"two?" ~ nodded and said that this was
correct. ~ then asked how many hexa-
gons would be in the middle. He wrote
something down on paper that ~ could not
see and indicated that there would be 28.
~ then asked him how many sides each of
the 28 hexagons would have on the
perimeter. He responded more confi-
dently this time with four.
~ then asked the class how we could
write 2 five's and 28 four's. No hands shot
up immediately anti ~ glanced at the clock.
Where ha(1 the time gone the bell was
going to ring any minute. ~ told the
students that for homework ~ wanted
them to come up with a way to calculate
the perimeter of the 30th train anti any
other train we could come up with. ~
thought that this would push us toward
more formal ways of recording calcula-
tions an(l, ultimately, generalizations.
Reflecting on Class Later That Day
The lesson was all I could have aske(1
from the kids! They found the perimeters
of the trains and were even making
progress on finding generalizations. ~
have ha(1 this kind of a lesson about five
times this year, and it is very exciting. ~
want to see the tape as soon as possible to
find other things ~ could have done. The
kids were very proud of themselves
think anti so was I!
Reflecting on Class Several Weeks Later
A few weeks after this class I ha(1 the
opportunity to share a 10-minute segment
of a videotaped lesson with my colleagues
at one of our staff (levelopment sessions.
~ decided to show a segment from the
pattern block lesson since ~ thought it had
gone so well. Although they (li(ln't say so
directly, ~ think they felt that ~ was too
leading. Maybe they were right. Itis easy
to be too leading anti feel OK about it
because the kids seem happy. After all,
many kids are happy with drill and practice.
~ decided to go back and watch the
entire tape again anti see if ~ could look at
it objectively. The lesson containe(1 too
much whole group teacher questioning
anti students explaining anti not enough
time for students to stretch and discover
independently/collaboratively. ~ won-
dered, in particular, what most students
really un(lerstoo(1 about Angela's method.
Sure many of them answere(1 my questions,
but were they just mindlessly applying a
procedure that they ha(1 rehearsed? Did it
mean anything to them? Although choral
response might make kids feel good, it
really masks what individual students are
really thinking anti what it is they un(ler-
APPE N AX
OCR for page 239
stand. Just because they could come up
with answers to my questions doesn't
mean that they really understand or that
they have any idea how to apply it. ~ am
now left wondering what they really
learned from this experience.
Reflecting at the En cl of the School Year
In early rune, at the end of the year
retreat, my colleagues and ~ were asked to
make a lO-minute presentation regarding
the areas which we thought had changed
most over the year. ~ began by showing a
clip of one of my fall lessons the one in
which Desmond went to the overhead to
measure the perimeter of the pattern train
and in which ~ assumed control, even
moving his hands. ~ told my colleagues,
"I'd like to start with the first clip because
feel it pretty much sums up how ~ taught
at the beginning of the year, and I'd like to
show you that ~ really have become less
directive than this tape." ~ showed the
clip without sound. Attention was drawn
to two pairs of hands on the overhead, the
large pair (mine) which seemed to be
moving the smaller pair (Desmond's). ~
explained, 'you'll see Desmond comes up
and ~ am very helpful very directive and
move the lesson along. That was a big
thing with me to move these lessons and
if they didn't get it, I'd kinda help them do
so." ~ added that ~ asked many yes/no
questions very quickly and did not provide
time for students to think. In contrast, ~
then showed video clips from the spring,
in which ~ walked around the room,
asking groups of students questions that
would help them focus their efforts rather
than telling or showing them what to do.
For me, the differences in my actions and
interactions with students on these two
occasions provided evidence that ~ had
changed.
APPE N DIX
Transition
Catherine Evans and her colleagues
continued their efforts to improve the
mathematics teaching and learning at
Quigley Middle School. They met fre-
quently to talk about their work and
attended professional development
sessions once a month and during the
summers to support their growth and
development. And their efforts were
paying off students were showing
growth not only in basic skills but also in
their ability to think, reason, and commu-
nicate mathematically.
At the beginning of the third year of the
math project a new teacher joined the
faculty at Quigley David Young.
Catherine and her colleagues welcomed
David into their community. From their
own experiences they knew how hard it
was to teach math "in this way." But
David had something that Catherine and
her colleagues did not have initially the
opportunity to work beside teachers who
ha(1 experience with the curriculum.
The case of Davi(1 Young picks up at the
beginning of David's second year at
Quigley. He has been working with
Catherine anti others anti has had one
year's experience teaching "this new way."
Catherine is now beginning her fourth
year of the math project.
THE PATTERN TRAINS
Part 2 David Young
David Young has just started his second
year at Quigley Middle School. The job at
Qu igley was at first overwhelm ing for
David. His mathematics teacher colleagues
were implementing an instructional
program based on a constructivist view of
learning Although such approaches had
been foundational to his teacher prepara-
tion program, his teaching up to this point
OCR for page 240
had been fairly traditional. The schools he
had been in for student teaching and his
first year teaching did not support innova-
tion. But at Quigley the students were not
passive recipients of what the teacher dished
out, and drill and practice seemed to have a
fairly limited role in instruction. Students
were actually doing mathematics explor-
ing, making conjectures, arguing, and
justifying their conclusions.
The enthusiasm and energy he saw in his
colleagues was invigorating hut also scary.
His colleagues all had lots of experience, hut
he had almost none. He worried about his
ability to he a contributing member of the
community and whether or not he would he
ahie to teach in a new way. David's fears
were put to rest early in his first year. His
colleagues were very supportive and
understanding They told him "war stories"
about their initial experiences in teaching
"this way" and how they had helped each
other through the tough times. They would
see him at lunch, in the morning before
school, or just in the hall way and ask
'what are you doing today?" and "How is
itgoing?" They woul~give him some
suggestions based on what had worked well
for them, hut they never told him what to do
or harshly judged the decisions he made.
Mostly they listened and asked a lot of
questions. Over time David felt that he
could ask or tell them anything It was, he
decided, the perfect place to teach.
During his tenure at Quigley, David had
been working hard to help students develop
confidence in their ability to do mathemat-
ics which he in turn felt would influence
their interest and performance in the
subject. Far too many students, he thought,
hated math in large measure because they
had not been successful in it. He had talked
a lot with Catherine Evans about his
concerns. Catherine had been quite open
about her early experiences in teaching
math the new way (just three years ago)
and her misstarts in trying to help students
feel successful. David came to believe that
developing cony dence as a mathematics
doer resulted from facing challenges and
persevering in the face of them. The key,
Catherine had often said, was trying to find
a way to support students in solve a chal-
lenging task not creating less challenging
tasks for students to solve.
Davic! Young Talks About His Class
This is the beginning of my second
year teaching sixth grade with this new
curriculum. The first year was rough-
both for me and kids as we tried to settle
into our new roles in the classroom. Me
as the facilitator anti my students as
constructors ofknowle(lge. When things
did not go well, my colleague Catherine
was always there with a sympathetic ear
and a word of encouragement. She is
such a won(lerful teacher everything in
her classroom seems to always go so well.
(She is right next (loor, anti we have a
connecting door between our rooms.
Sometimes during my free period ~ leave
the (loor open anti listen in on what is
happening over there.) Although she has
repeate(lly sai(1 that it was a long anti
painful trip from where she started to
where she is today, it is har(1 to believe. ~
guess it is comforting though to know that
if she ma(le it, ~ can too.
Catherine anti ~ are both teaching sixth
grade this year, so we touch base nearly
everyday about what we are doing. We
are only a month into the school year, and
so far we have been working with patterns.
Up to this point we have focused primarily
on numerical patterns. The new unit that
we starte(1 yesterday uses trains of pattern
blocks arrange(1 in some geometric
sequence. The unit is suppose(1 to help
students visualize anti (1escribe geometric
patterns, make conjectures about the
patterns, (letermine the perimeters of
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trains they build, and ultimately, to
develop a generalization for the perimeter
of any train in a pattern.
Last year this unit did not go well. It
was too much teacher talk and too little
time for students to think. ~ moved them
through the entire set of exercises in one
period. ~ felt great because ~ had really
"covered" the material, but a week later it
was clear that the students hadn't gained
much from the experience. When ~ talked
with Catherine about it she told me about
her first time through this unit three years
ago. She said that one thing she learned
is that kids need time to think, to struggle,
and to make sense of things for them-
selves. If you make it too easy for them
they wall never learn to figure things out
for themselves. This made sense to me,
but it was hard not to step in and tell them
what to do. ~ was determined, however,
to do a better job this time around.
The CIass
Yesterday my sixth grade class spent
some time getting familiar with the
pattern blocks identifying the shapes
and determining the perimeters of the
blocks. Today they are going to make
observations about trains of pattern
blocks and determine the perimeters for
the trains. Basically ~ am just going to
follow the curriculum here. It suggests
giving students a pattern sequence and
having them compute the perimeter for
the first three or four trains and then to
determine the perimeter of a larger train
like 10 or 20. Ultimately the curriculum
suggests asking students to imagine that
they are constructing the lOOth train and
to look for ways to find the perimeter.
will see how things go, but ~ hope to be
able to follow this suggestion and use
large numbers like 1000 so there is no
way they can build or draw the trains and
count the number of si(les.
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Geffing Startecl: The Square Tile Pattern
~ started by building the pattern of
squares shown below on the overhead
and asking students to work with their
partners to find the perimeters of the first
four trains in the sequence. Emily imme-
diately asked for pattern blocks so she
could actually build the trains. This of
course started a series of requests to use
the blocks. ~ hadn't anticipated this, but ~
had no problem with it either. ~ grabbed a
few bags of blocks and dropped them off
at the tables of students who had
requested them.
train ~ train 2
Students started building the trains and
quickly seemed to realize that the fourth
train would have four squares. They then
began to determine the perimeter and
record their findings. This initial activity
seemed to be pretty easy for students.
After about five minutes ~ asked Derek to
go to the overhead and show us how he
found the perimeter for the first three
trains. Using a technique that we had
use(1 yesterday when we began exploring
the perimeter of the blocks, Derek (new
line segments parallel to the si(le of the
square as he counted (as shown below),
in or(ler to show that he hall counte(1 a
particular segment. Once he hall com-
plete(1 the count, he recor(le(1 the perim-
eter on top of the train. ~ asked Derek
what the numbers "4," "6," anti "a" repre-
sente(l. He respon(le(1 that"these are the
(listance around the outside of the train in
units." ~ aske(1 what a unit was, anti he
explaine(1 that he ha(1 use(1 the si(le of the
square as the unit. The previous (lay we
train 3
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1
P=4 P=6 P=8
2 2 3 2 3 4
3 1
14 1
4 6 5 8 7 6
had discussed that fact that we were going
to be measuring using the side of the
square as our unit. That way we could
talk about the number of units without
worrying about the actual measurement.)
~ then asked the class what they
thought the perimeter of the fourth train
would be. Crystal said that she thought it
would be 10. ~ asked her how she found
it. She explained, "I just built the fourth
one and counted the way Derek did."
lamal said that he got 10 too, but that he
just added two more to the third train. ~
asked him to explain. He said, "When you
add on one more block to the train the
perimeter only gets bigger by two more
units cause only the new piece on the top
and bottom add to the perimeter." ~ asked
the class if they had any questions for
lamal. Kirsten said that there were four
sides in every square, so how could the
perimeter only increase by two? lamal
went to the overhead and explained, "See
if you look at the second train there are
two units on the top and bottom, and one
on each side. When to go to train three
and add one more square (as shown below)
you still only have one unit on each side
cause the sides of the new square are on
the inside not on the perimeter."
15
~ then asked students to take a few
minutes anti think about what the tenth
train woul(1 look like. ~ wanted to be sure
that all students ha(1 time to consi(ler this
larger train. ~ know that sometimes ~
move too quickly anti (lon't allow enough
wait time for students to think about
things. This ten(ls to work against the
students who have goo(1 ideas but work at
a slower pace. Since ~ have been waiting
longer, more students have been involved.
~ started by asking Michele what she
thought the perimeter would be. She said
she got 22. ~ aske(1 her if she coul(1
explain to us how she got this answer.
She in(licate(1 that she ha(1 built the tenth
train and then counted. Although this was
a perfectly good approach for the tenth
figure, it was going to be less helpful
when we started considering larger trains.
I asked if anyone did it another way.
Travis said that he got 22 too, but that he
just took ten plus ten plus 2. Although
his answer was correct, it was not immedi-
ately obvious why he added this set of
numbers. I asked him why he did this.
He explained, "See when I looked at the
first four trains I saw that the number of
units on the top and bottom were the
same as the number of the train. So in
FIG
new square
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OCR for page 243
train one there was one unit on the top
and one on the bottom. In train two there
were two units on the top and on the
bottom. In train three there were three
units on the top and the bottom. So ~
figured that this would keep going so the
tenth train would have 10 units on the top
and the bottom. Then for all the trains
you have to add on the two sides because
they never change."
~ thanked Travis for sharing his strat-
egy and asked if anyone had thought
about it another way. Joseph said that he
multiplied the number of squares in the
train by 4, then subtracted the sides that
were in the inside. ~ indicated that this
was an interesting way to think about it
and asked him if he would explain. He
began, "Well, each square has 4 sides, so
in the tenth train there would be 4 x 10 or
40 sides. But some of these are in the
inside, so you have to subtract." "How did
he know how many would be on the
inside?" ~ asked. He explained, "Well,
there are eight squares in the inside of the
train, and each of those squares had two
sides that didn't count and that gives you
16. Then there are two squares on the
outside of the train and that each of those
had one side that didn't count, so that
gave you 18. So 40 - 18 gives you 22 and
that's the answer."
As he finished his explanation a few
hands shot up around the room. ~ asked
the class if they had any questions for
Joseph. Kendra asked how he knew that
there were eight squares on the inside of
the train. Joseph said that he had looked
at the first four trains and noticed that the
number of squares on the inside was two
less than the train number the second
train had zero squares on the inside, the
third train had one on the inside, and the
fourth had two on the inside. ~ thanked
Joseph for sharing his thinking about the
problem with the class. ~ was really
APPE N DIX
pleased with the two (different generaliza-
tions that had been offered and decided to
ask one more question before moving on
to a new pattern to see if the class could
apply these noncounting approaches to a
larger train.
~ asked the class it they could tell me
the perimeter of the lOOth train. After
waiting about 2 minutes for students to
consider the question, ~ asked if anyone
had a solution. Katherine said that she
thought it would be 202. ~ asked her how
she figured it out. She said she needed to
draw and came up to the overhead. She
drew a rectangle on the overhead and
asked us to pretend that it was 100 squares.
She then continued, "Like Travis said the
number of units on the top and bottom is
the train number, and then there are the
two on the side. So for the 100 train, it
would be 100 + 100 + ~ +~."
train number
1
train number
I commented that this seemed like a
really fast way to do the problem. Rather
than ask for additional ways to think about
this pattern, ~ decided to move on. ~
passed out a sheet of four patterns (see
attached) and asked students to work with
their partners on pattern ~ on the sheet.
In particular ~ wanted them to sketch the
fourth train in the pattern, find the perim-
eter of each of the four trains, and then to
see if they could find the perimeter of the
tenth train without building the train. ~
knew the last condition would be a chal-
lenge for some, but ~ wanted them to
think harder to find another way.
1
OCR for page 244
~ Pattern 1
/\ ~ m\
Pattern 2
O O O O
Pattern 3
/ / \ \
Pattern 4
A a> ~ an>
Continuing Work: The Triangle Pattern
~ walked around visiting the pairs as they
worked on the new train. Again students
seem to quickly see the pattern add one
more triangle and count the sides to find
the perimeter. ~ observed several pairs
starting to build the tenth train and asked
them to try to fin(1 another way. ~ sug-
gested that they look at the four trains
they had built and see if they could find
any patterns that would help them predict
the tenth train. In a few cases where the
students were really stuck ~ suggested
that they try to see if they could find a
connection between the train number and
the perimeter as a few students hall (lone
in the last pattern.
Once it appeared that most pairs had
ma(le progress on this task, ~ asked James
to come up and buil(1 the fourth train and
(lescribe the pattern. James quickly
assembled the triangles, changing the
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OCR for page 245
orientation each time he added one. He
explained, "you just add one more triangle
each time and every new one is turned
the opposite way of the last one." ~ then
asked Catherine what she found for the
perimeter of each train. She said that the
first one was 3, the second one was 4, the
third one was 5, and the fourth one was 6.
~ asked her what the fifth one would be.
She quickly said "7." ~ asked her how she
did it so fast, and she responded, "After
the first one you just add one every time.
The fourth train is 6 so the fifth train
would be one more."
~ then asked if anyone could tell me
what the perimeter of the lOth train would
be. lanelle said she thought it would be
12. { asked her how she found it. She
said she made a table and looked for a
pattern. Since this was the first time
anyone had mentioned making a table,
thought it would be worth having her
explain this strategy to the class. She
came up and constructed the table shown
below. She explained, "I looked in the
table and ~ saw that the perimeter kept
going up by one, but that the perimeter
was always two more than the train
number. So that for train number TO the
perimeter would be two more or 12."
Train # Perimeter
3
2
3
5
6
Before ~ could even ask if anyone had
done it another way, Joseph was waving
his hand. He announced that he got 12
too, but that he did it another way. He
said that the train number was the same
as the number of triangles, just like the
squares. He went on, "Since each triangle
has three sides, ~ multiplied the number of
APPE N DIX
triangles by 3. So 3 x TO = 30. But then
you have to subtract the sides that are in
the inside. It's like the square. You take
the number of triangles on the inside. For
the tenth train that would be 8. Each of
those triangles has two sides that don't
count and that give you 16. Then there
are two triangles on the outside of the
train and that each of those had one side
that didn't count, so that makes 18.
30 - ~ = 12."
' Avow," ~ said, "there are lots of different
ways to look at these trains aren't there?"
~ was ready to move on, but Darrell was
trying to get my attention. He said,
"Aren't you gonna ask us to find the
lOOth?" That hadn't been my plan, but if
he wanted to find the lOOth ~ was happy to
oblige. ~ aske(1 Darrell if he wanted to tell
us what the perimeter of the lOOth train
would be. He said, "It'll be 102. Cause
like Janelle sai(l, it will always be two
more." ~ asked the class if they agreed
with Darrell. ~ saw lots of nodding heads
that convinced me that we were indeed
making progress.
Exploring Three New Pafferns
~ told the class that they would have 15
minutes to work with their partners on
patterns 2, 3, and 4. For each pattern they
needed to sketch the next train, find the
perimeter for all trains, anti (1etermine the
perimeter for the lOth train without
buil(ling the train. ~ wanted students to
have a longer perio(1 of time for exploring
the patterns without interruption. ~
figure(1 that in 15 minutes everyone would
at least get the first one (lone, and pattern
four would be a challenge for those who
got that far since it was less straight-
forwar(1 than the previous patterns anti
that the o(l(1 anti even trains woul(1 be
described differently.
As students worked on the patterns, ~
again walked around the room observing
OCR for page 246
what they were doing, listening in on their
conversations, occasionally asking a
question, and reminding them that they
would need to be able to justify their
methods to the rest of the class. The
most challenging aspect of the task for
most students was finding the perimeter
to the tenth train without drawing it. For
pattern 2, ~ encouraged them to try to find
a way to talk about the perimeter of a
figure in terms of the train number. "How
are those two numbers related?," ~ asked
as ~ moved from group to group.
Discussing the Hexagon Pattern
After 15 minutes all students had
completed patterns 2 and 3. Since there
were only 10 minutes left in class ~
thought ~ would have them talk about
pattern 2 before the bell rang. ~ started by
asking lungsen to describe the pattern
and give the perimeter for the first four.
She explained that each train had the
same number of hexagons as the train
number and that the perimeters were 6,
10, 14, and IS. 'what would the perimeter
of the next one be?," ~ asked. lames said
he thought it would be 24 because the
hexagon had six sides and it would be six
more. Michelle said that she thought it
would 23, because it would only be 5 more
because all sides didn't count. ~ asked if
anyone had a different guess. Derek said
that he thought it would be 22. A number
of students chimed in with "I agree!" ~
asked Derek to tell us how he got 22. He
said that every time you added a new
hexagon, you only added on four more
sides. '~he perimeters were 6, 10, 14, and
IS. You just keep adding four."
~ asked if anyone could explain it
another way. Kirsten said that she
thought she could. "Every time you add
another hex", she explained, "you just a(l(1
two si(les on the top and two on the
bottom." She pointed to the trains on the
overhead and continued, "If you look at
train two, you have four si(les on the top,
four on the bottom, and the two on the
ends. If you look at train three you added
one more hex which gives you two more
si(les on the top and the bottom. That
gives you just four more."
. . ;` ,
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OCR for page 247
~ asked if anyone had found the perim-
eter of the tenth train. Carmen said that
she thought it would be 42. ~ asked how
she got this. She said that the tenth train
would have 20 sides on the top, 20 sides
on the bottom, and one on each end. ~
asked how she knew it would be 20. She
went on to explain, '~he number on top is
double the train number. See, the second
train has four, the third train has six, so
the tenth train would have 20."
~ thanked Carmen for sharing her
solution and asked if anyone had another
way. Joseph was again waving his hand.
asked Joseph if he used his method on
this problem too. He said he did and
explained that since each hexagon had six
sides you needed to multiply the train
number by 6 to get 60. Then you needed
to subtract the inside sides which would
be 18. So it would be 60 - 18 which was
APPE N DIX
42. Kirsten asked Joseph if you always
subtracted IS for the tenth train. Joseph
said that so far that seemed to work for
the squares and the hexagons, but he
wasn't sure if it always worked. Kirsten's
question was a good one. ~ made a note to
be sure to include a pattern for which it
would not work, just to push Joseph to
consider what was generalizable about his
approach and what wasn't.
~ finally asked about the perimeter of
the lOOth train. It seemed as though
everyone thought they had it this time. ~
took a quick look at the clock. The beD was
going to ring any minute. ~ told students
for homework to write down what they
thought the perimeter of the 100th one
would be and to explain how they figured
it out. We would start there the next day
and then jump right in and try pattern 4.
Representative terms from entire chapter:
shown below
