NEW JERSEY'S MATHEMATICS STANDARDS
Nine Vignettes
This section contains nine vignettes which suggest how New Jersey's
Mathematics Standards can be effectively implemented in
classroom settings.
The table below indicates the content standards and grade levels
which each vignette particularly addresses.
The vignettes highlight, using marginal notes, how the learning
environment standards and the first five content standards serve as a
context for mathematics learning. These reinforce the emphasis that
the why's and how's of mathematics learning must be
integrated with the content.
Although these nine vignettes reflect all eighteen standards, they
certainly do not fully address all of the cumulative progress
indicators that are attached to the standards. They are intended to
be illustrations of the way that individual educators have suggested
that these standards be implemented. Teachers are encouraged to
review and discuss them, to experiment with practices that they
exemplify, and to develop their own activities consistent with the
standards.
Elevens Alive!
While Mr. Johnson is meeting with some of the children in his
firstgrade class, others are involved in a number of different
activities. At the Math Center, pairs of students have cups with
eleven chips that are yellow on one side and red on the other. As each
pair pours out the chips, they write a number sentence showing how
many yellows and how many reds they got, as well as the total. When
they have written ten number sentences each, they move on to another
activity.

The students:
work on basic facts in the context of a problem and in relation
to other areas of mathematics.
work in pairs with manipulatives.

Later in the day, as Mr. Johnson begins the math lesson, he asks
the students to recall their discussion from the previous day,
"What were we talking about yesterday in math?"

practice their number facts by writing down each
result.

"We were doing numbers that add up to eleven, like 5+6
and 2+9," answers Clark.
"Or 3+8 and 4+7," adds Sarah.
"Is there more than one way to get a sum of eleven?"

share mathematical ideas.

Mr. Johnson lists all of the children's responses on the board. He
goes on to ask them, "What were you doing at the Math Center
earlier today?"
Jackie responds, "We were tossing counters and writing number
sentences."
"We were tossing eleven counters!" says Toni.
"What can you tell me about your results?" asks
Mr. Johnson. "Did you get the same number sentences as your
partner?"
"No  we got different ones!"

connect their understanding of one mathematical idea to
another.

"Our answers were always the same  eleven!"
"I got some number sentences more than once!"
"I got 5+6 three times!"
"I didn't get 0+11 or 11+0 at all!"

report and reflect on the differences of their results.

"Why do you think you got different answers?" asks
Mr. Johnson. He listens as the students talk about fairness, luck, and
chance, pointing out that all of the counters are alike. The students
agree finally that the different number sentences are a result of
chance.
The students continue their discussion of which number sentences
appear more often than others. One of the children suggests that maybe
they should make a graph to help them see which number sentences occur
most often. Mr. Johnson thinks that this is a good idea. He goes
through their list of number sentences, asking students to raise one
finger if they got that number sentence once, two fingers if they got
it twice, and so on. For each finger raised, he puts a tally mark on
the board. When they are done, he asks whether there were any other
number sentences that anyone got. Then the children look at the
general shape of the data, noticing that most of the number sentences
were in the middle. Mr. Johnsonpoints out that not all of the number
sentences are equally likely to occur. He says that tomorrow they will
have a chance to play a game with the counters in which they will
need to select which number sentences will be winners. Tomorrow's
activity will continue providing opportunities for practicing basic
facts while building on the beginning ideas of probability.

informally explore the concepts of probability.
use different methods to display data.
make inferences about their data.

Product and Process
Mr. Marshall had assigned the following problem from the New Jersey
Early Warning Test as a homework assignment for his fourth grade
class:

Use each of the digits 3, 4, 5, 6, 7, and 8 once and only
once to form threedigit numbers that will give the largest
possible sum when they are added. Show your work.
Is more than one answer possible? Explain your answer.

The students:
are asked to respond to openended questions and present and
defend their solutions.

The students were to solve the problem and match their response
with that of Tilly Tester to see if they agree or disagree with
Tilly's response and explain why.

are asked to analyze problems for reasonableness of results and
to diagnose errors.

As the math class begins, Mr. Marshall allows the students to work
in the cooperative learning groups which they have been working with
this month to compare the results of their homework
assignment. Mr. Marshall visits each group noting who has completed
the assignment as well as the direction of the discussion for each
group. Homework assignments are important and students are given
credit for homework. Strategies such as displaying answers on the
overhead projector and working in cooperative learning groups are used
to ensure that homework review is no more than 5 to 10 minutes.

work cooperatively to assess their own and each other's
work.

Mr. Marshall then asks the students to show the level of their
agreement with Tilly's response on a 05 scale, with 0 signifying
disagreement and 5 signifying total agreement. Most students raise 4
or 5 fingers, and the discussion then focuses on how Tilly's answer
could be improved. One group notes that Tilly should have added each
pair of numbers and shown the sum for each, while another group
explains that Tilly could have also changed the hundreds place to get
754+863 and 763+854.

are willing to take a position without the fear of being
incorrect.

At this point, Mr. Marshall discusses Tilly's understanding of
place value and uses the opportunity to summarize the students'
responses and lead into the objective of the day which focuses on
place value and multiplication.

use their knowledge of numeration to help solve
problems.

"Let's work on multiplication today, and to get started, let's
do some mental math with multiplication. On the back of your homework,
number 1 to 10. Write the answers only for my mental math
flashcards."
Individually, the students write answers for 8000x3, 6000x7+50,
300x7, etc. After the ten problems, Mr. Marshall has the students
exchange papers, and they correct and discuss the answers. The papers
are collected, and Mr. Marshall poses the following problem for his
students:
Use four of these five digits and construct the multiplication
problem that gives the greatest
product: 1, 3, 5, 7, 9

use mental math regularly throughout the curriculum.
demonstrate their understanding of mathematical concepts in a
variety of ways, each of which provides valuable assessment
information to the teacher.

Before allowing the students to start work on the problem, he asks
them to estimate what the largest product obtained in this manner
might be. Students offer estimates ranging from 3000 to 10,000 and
provide explanations for their guesses. When allowed to, the class
works in their cooperative learning groups. Calculators are
available, and some students start guessing and checking with their
calculators.

are encouraged to estimate solutions before actually determining
answers.
use calculators to aid in the problemsolving process.

One group begins to discuss which digits to use, wondering whether
there would be a reason not to use the four largest digits. Another
group is discussing whether a 2by2 or a 1by3 arrangement would be
the best for getting a large product, an aspect of the problem that
some groups have completely missed. Most of the groups get around to
trying out sample problems of a variety of sorts to get some
parameters worked out. Toward the end of the class session, the
groups share the specific answers they have come up with. The three
examples that are suggested are:
753 93 953
x 9 x 75 x 7
  
6777 6975 6671

using mathematical reasoning to formulate strategies and
solutions.

It is clear to everyone that the 2by2 digit problem is the one
with the greatest product, but Mr. Marshall is looking for some
generalizations that can be made. He points out that none of the
groups used the digit "1" in their examples. Can the
lowest digit always be ruled out? He asks the groups that arrived
at the 2digit problem to explain how they decided where to put the
individual digits. Does it matter where they were placed?
Where does the largest one go? The smallest? Do you think it would
always work that way regardless of what the individual digits
were? How can you check? The students reflexively pick up their
calculators and begin to formulate other versions of the problem that
use other digits and to check which arrangements of the digits give
the largest product. One student asks his partners what they think
would happen if two or three digits were the same.

approach numerical operations from a holistic point of view
rather than only through paperandpencil manipulation.

For homework, Mr. Marshall asks the students to use the same five
digits, but to find the smallest possible product. They are then to
write a paragraph describing their solution and the reasoning they
used to show it is, indeed, the smallest product. Specifically, they
are to consider the question: Can you just turn your
thinking about the way you got the largest product upside down and use
it to get the smallest product?

write paragraphs describing and justifying their
positions.

Sharing a Snack
Today is November 12 and Maria, a student in Miss Palmer's
second grade class is very excited. Today is Maria's birthday,
and as is the custom in her class, she is bringing in a birthday snack
to share with her classmates. Maria and her father spent much of the
previous evening making a batch of chocolate chip cookies and she
proudly walks into class carrying a cannister full to the brim. Miss
Palmer realizes that she can use mathematics to help the class divide
the cookies.
Before the afternoon snack time, Miss Palmer poses the problem to
the whole class.

The students:
use mathematics to devise a solution for realworld
problems.

Miss Palmer states, "Today is Maria's birthday and she
has brought in some delicious chocolate chip cookies for all of us to
enjoy at snack time. Maria told me she baked a whole bunch of
cookies. I would like us to think about how we could determine the
number of cookies each student in the class should get. Discuss it
with your partner."

use cooperative work to generate potential solutions.

The students begin to discuss all of their ideas. After a few
minutes, Miss Palmer calls on a few of the students. As they share
their ideas, the teacher records them on the language experience
chart.
Sarah states, "Well, me and Mario think that the first thing
we have to do is count the cookies to find out how many there
are."
Jerome adds, "Yeah, and we also need to know how many children
are in the class today."
"That's easy. I did the lunch count this morning and there
are 22 children in school today," Maria volunteers.
Luis chimes in. "Once we know how many cookies and how many
children, then we can figure out a way to solve the problem."
The children all agreed that since they know there are 22 children
in class today, the next step was to determine the number of cookies.
Miss Palmer highlights that idea on the language experience chart and
gives each pair of children a bag of counters which represents the
number of cookies.

regularly share their ideas publicly.

Miss Palmer says, "Each pair of children received a bag of
counters. I want you to pretend that these are Maria's cookies. I've
counted the cookies. The number of counters in each bag is equal to
the number of cookies Maria brought for a snack. With your partner,
use your counters to first decide how many cookies Maria brought to
school and then determine how many cookies each student will get if
the cookies are to be shared equally among everyone in the class.
When you are finished, each pair will need to write a story which
explains how both of you solved the problem."

use manipulatives to model realworld situations.

The children worked with their regular partners. The first task
they all tackled was to count the number of counters in each bag.
Most of the pairs of children counted by twos to determine the total
number of counters was 62. However, Alex and Laura kept losing count
when trying to count all the counters and decided to group the
counters by ten. Miss Palmer was delighted to see that most pairs of
children had written the total number of counters (62) on a sheet of
paper. She had been stressing the importance of collecting data and
recording information.
As Miss Palmer continued to circulate around the classroom, she
noticed the children were solving thesharing problem in various
ways.

use their knowledge of decimal place value to simplify the
task.

One pair of students begins by drawing 22 stick figures to stand
for the students in the class and then starts to "give out"
the cookies by drawing them in their stick figures' hands. Another
pair also starts with 22 stick figures but then draws 62 little
cookies on another part of the paper and is stumped about where to go
from there. Mario and Sarah begin to sort the 62 counters into 22
piles. Another pair, trying to use calculators to solve the problem,
starts by adding 22 cookies for everyone to another 22 cookies for
everyone to a third 22 cookies for everyone and then realizes that
they have exceeded the number of cookies available.
Miss Palmer, noticing that the students will be unable to finish
the problem before they have to go to Physical Education, calls the
students back together.
"I want all of you to stop what you are doing, and with your
partner write a story to tell me how you are attempting to solve this
problem," she directs.

develop their own methods for solving the problem.
use technology as a problemsolving tool.

The students eagerly write their stories. Some use pictures to
help illustrate their solutions.
Miss Palmer requests, "I would like some of the pairs to
report to the whole class how they were attempting to solve the
problem."

draw pictures to model their solutions.

Luis states, "Well, Elizabeth and I figured out that each
student could have 2 cookies and there will be 18 cookies left. We
know this because we drew a picture of the class and put counters on
each student. When we couldn't give counters to every kid, we decided
those were leftovers and we counted them."
Lisa volunteers, "We drew stick figures too. After we gave
out 2 cookies to each child, Jerome said we couldn't give out the 18
leftovers. But I think we can break the leftover cookies in half.
Then each child would get 2 whole cookies and one half cookie. But
I'm not sure how many would be left over then."
"Sarah and I used the calculator to solve the problem. We put
in 62 and I counted while Sarah subtracted 22. We got 2 with 18 left
over," Mario added.
"Alex and I got a different answer. We used the counters and
put them into 22 piles, but we got 17 leftovers," Laura said.
Lisa suggested, "Maybe you and Alex should count them again to
make sure you have the right number in each pile."

give explanations of their strategies for solving the
problem.
informally explore the uses of fractions and notions of fair
sharing.

Laura and Alex recount their piles and discover that one counter
fell on the floor.
Vanessa states, "Me and my partner thought of another way of
sharing the leftover cookies. Everyone could write their name on a
piece of paper, then put all the papers into a bag and have Maria
close her eyes and pick out 18 names. Those kids would get the extra
cookies."
Sarah protested, "We forgot about Miss Palmer. We should give
her 2 cookies and that would leave 16 left over. Maria could give
them to the principal and the other ladies in the office."
Miss Palmer wrapped up the discussion. "We've discussed many
ideas for sharing the 62 cookies Maria brought for a snack. On the
back of the sheet of paper I gave you, I would like you and your
partner to decide on how you think we could fairly share the
cookies."

are mutually supportive and regularly offer feedback to each
other.

The children work on their final summary of the problem and hand
their papers in before getting on line for Physical Education. While
the children are in Physical Education, Miss Palmer reads the
children's solutions. She makes notes on the cards she keeps for
each child. This will help her better understand various
developmental levels of her students. She notices that Vanessa has
really made progress since September. Laura and Alex still like to
"rush" to finish their work. She makes a note on their
paper encouraging them not to be so concerned about being the first
ones finished. Overall, she feels encouraged, not only about the
solutions to the problems, but also about the ways in which her class
has learned to communicate their ideas both orally and on paper. She
decides to let the class choose one of the methods suggested to
distribute the cookies at snack time.

demonstrate their understanding of mathematical concepts in a
variety of ways, each of which provides valuable assessment
information to the teacher.

The Powers of the Knight
Mr. Santos' 6th grade class has just completed a review of place
value in the decimal number system and he is preparing to start a unit
introducing exponents. He has coordinated the timing of this unit
with the language arts teacher whose class is in the midst of a unit
on fables. One fable they have read involves a knight who saves a
kingdom from a horrendous dragon. Given the opportunity to determine
his own reward, he tells the king that he would take one penny on the
first square of a chessboard, two pennies on the second, four pennies
on the third, and so forth until each square on the chessboard has
twice as many as the previous one. Mr. Santos has the students recall
the story and then asks the students to determine how much money the
knight would make with this method of payment.

The students:
are connecting a language arts experience to their mathematics
learning.

Mary said, "We need to know how many squares there are on a
chessboard before we can do this problem."

are comfortable taking risks.

Lionel stated, "Give me a minute to think. I play on the
chess team, but I need to take a moment to picture it. Let's see, I
know it's square and there are 1,2,3, ...8 squares along the one side.
There are 64 squares!"
Jerry shouted, "He gets 128 pennies. Two on each
square."

use known facts to explain their thinking.

"The fable doesn't say he gets two on each square! It says
that each square has twice as many as the one before. It has to be
more than 128!" corrected Meredith.

react substantively to others' comments.

"We need to examine this situation in some organized fashion.
I want you to get in your groups of four and determine the people who
will serve the usual roles of leader, recorder, reporter, and analyzer
of group interaction," stated Mr. Santos.

use standard cooperative learning strategies.

One group decided to develop a computer program which printed a
table listing the number of the square, the number of coins on that
square, and a subtotal to that point.

use technology to help solve the problem.

Another group borrowed the class chessboard and began placing play
coins on the squares. It soon became obvious to them that they would
not have enough play money to complete this attempt. They started to
make a table with the information they had constructed and worked to
find a pattern which they could extend to the complete board. Their
table only included columns showing the number of the square, the
number of coins on that square, and a column to list patterns. They
discovered that the number of coins could be represented by raising 2
to the power which was one less than the number of the square. Using
calculators, they found the number of coins on each square and then
the total number of coins.
Another group began making a table similar to the group above, but
they also included a column showing the partial sums and another which
attempted to find a pattern in the partial sums. Eventually, they
discovered that the partial sum at each square was one less than 2
raised to the power equal to the number of the square. They could
then quickly utilize the calculators to compute the total.

concretely model the problem before they move on to more
symbolic procedures.
use selfassessment to determine the effectiveness of their
method.
analyze mathematical situations by recognizing and using
patterns and relationships.

At the end of the period, Mr. Santos reminded the groups that they
were to prepare a report of their methods which included a description
of their processes, an explanation of why they chose them, and their
evaluation of their processes. He asked each of them to consider the
magnitude of their answer and find some way to explain to another
person just how large the answer was. Students brainstormedsome ideas
such as the distance between two known points or objects, the
magnitude of the national debt, and the number of people on earth.

choose technology to reduce the computational load.
write about their approaches and solutions to problems.
connect their knowledge of mathematics to the real
world.

Shortcircuiting Trenton
Ms. Ramirez announces to her seventh grade class that in three
weeks they will make a journey to Trenton, the capital of New Jersey.
They will be visiting eight sites  the Capitol, the New Jersey
Museum, the War Memorial, the Old Graveyard, Trent House, the Old
Barracks Museum, the Firehouse, and the Pedestrian Mall. To ensure
that they spend as much time at the sites as possible, and do as
little walking as possible, the class must find the most efficient
walking tour for the trip, starting and ending at the parking lot.

The students:
apply mathematical skills to solve a realworld problem.

The first problem that the students must address is finding the
walking distance between each pair of sites. Ms. Ramirez supplies
each team with a street map and a ruler; the maps identify all the
sites to be visited and the routes joining them. She assigns each
group the task of finding the distances between one site and all the
others. This turns out to be an interesting task, since different
groups interpret it differently. Some groups, for example, measure
the straight line distance between two sites forgetting that buildings
or ponds might render that walk impossible. How to measure the
walking distance thus becomes an important topic of discussion, as
does the question of appropriate units. These questions are
eventually settled and the teacher uses the students' measurements to
write a matrix which indicates the walking distance between any two of
the eight sites; different groups occassionally have obtained
different numbers, but after discussion, they have arrived at a common
answer.

use cooperative group work to generate problemsolving
strategies.
freely exchange ideas and participate in discussions requiring
higherorder thinking.
collect and organize data needed to solve the problem.

Ms. Ramirez selects a sample route for the walking tour and through
discussion with the class explains how the total length of the walking
tour is obtained from the matrix of information that the students
generated  you find the distances between consecutive sites on
the tour, and then add up the walking distances along the tour. She
now asks her students to work in groups to decide on a strategy that
they think will produce an efficient route (which starts and ends at
the parking lot), and to assist the group's recorder in writing a
short paragraph explaining their strategy. Some groups decide to list
all possible routes and calculate how long a walk each route entails.
(Ms. Ramirez asks the students how many possible routes do they think
they will have to list.) Other groups suggest that the best route is
obtained by always going to the nearest site.

recognize there are numerous ways to solve the problem.
work in cooperative groups to develop alternative
strategies.

Ms. Ramirez now asks the students to use calculators to carry out
their strategy and determine the travel time for the routes they will
be considering. After each group presents its results, the class will
together compare the various methods that were proposed and the
accompanying results. Among the questions which Ms. Ramirez will ask
are: "Do the various methods give the same result?",
"Which methods result in a most efficient route?",
"What other strategies could we have used?" Responses from
the students might include: "always use the shortest
distance", "never use the longest distance", "put
distances in increasing order and use only those that neither make a
loop or put a third edge into a vertex."

compare the variety of strategies proposed.

Mathematics at Work
As a regular feature in his class, Mr. Arbeiter has parents of each
student make a presentation about their job and how the various
educational disciplines are needed for them to be successful. Today,
Emily has asked her Mom, the owner of a heating and air conditioning
company, to talk to her class. Mrs. Flinn and Mr. Arbeiter decide to
have the students help her solve a problem similar to one which her
company faces regularly. She briefly describes her company, the work
that she does, and tells the students that they are going to help her
determine how large an air conditioner will be needed in the
classroom. She poses the following problem: What information about
the room would be most important in determining how large an air
conditioner is needed? The students quickly agree that the amount of
air conditioning would depend on the amount of air in the room, and
that in turn, depended on how much space there was in the room.
Through suggestions and hints, Mrs. Flinn had them realize that the
amount of sunlight entering the room would have an effect as well and
they quickly agreed that the area of the windows must be found
too.

The students:
interact with parents who use mathematics and other disciplines
in their daily lives.
have the time to explore a problem situation thoroughly.

Mr. Arbeiter reminded the class that there is a mathematical term
which represents the amount of space, and asked each student to write
down that term. As was his custom, Mr. Arbeiter asked six students,
one quarter of the class, to read the words they had each written.
Four read the word "volume" and two read "area."
By a show of hands, he found that about one third of the class had
written "area" and two thirds had written
"volume." In their groups, the students were asked to
discuss the difference between area and volume and to write down the
differences between them. As the groups discussed these concepts
Mrs. Flinn and Mr. Arbeiter circulated among them, making sure that
each group had focused on the difference between area and volume;
subsequently the groups read the statements they had prepared, and the
entire class discussed and commented on the groups' statements.
Mrs. Flinn had the class discuss which of the concepts were needed on
the two phases; amount of space in the classroom and how much window
space there was.

are regularly assessed through a variety of methods.
work in a variety of settings to develop concepts and
understanding.

Now that all students agreed on the difference between area and
volume and where each applied in this case, the discussion turned to
discussion centered on how one obtains the volume of the classroom and
the area of the windows. Although familiar with the concept of
volume, the class was not able to calculate volume easily, so
Mr. Arbeiter suggested that each group build a rectanguar box out of
cubes and figure out how many cubes the box contained. Most groups
discovered that they could get the answer by multiplying the number of
cubes in the bottom layer by the number of layers (the
"height"), and agreed with Mr. Arbeiter's conclusion
that V = BxH. When Mr. Arbeiter asked them how they calculate the
number of cubes in the bottom layer, all agreed that you multiplied
length times width; and when the teacher wrote V=BxH=(LxW)xH, several
other groups recognized that that was how they found the volume of
their box.
Mr. Arbeiter asked the class "How does the volume formula help
us find the volume of the classroom?" The students agreed that
the shape of the classroom was about the shape of a rectangular box,
but were quick to point out that to any answer obtained by the formula
would have to be considered an estimate, since it would not be taking
alcoves and pillars into consideration. They agreed to change the
question to "How does this formula help us estimate the volume of
the classroom?"

use concrete materials to develop a model for volume.

"All we have to do is measure the three quantities 
length, width, and height, the three dimensions of the classroom, and
multiply the three numbers together" was the prevailing
sentiment. Marcia observed that "since we're only going to get
an estimate anyway, why should we measure those three amounts
exactly?" And Mrs. Flinn noted that her sales people often
estimated the size of the room without making any measurements.
"How can we estimate the dimensions of a room without
makingmeasurements?", she asked. Paula suggested that
"maybe the salesperson estimates the three dimensions and
multiplies those estimates together." "A great
suggestion," Mrs. Flinn responded. "Let's try that
ourselves."

recognize and apply estimating to geometric situations.
are exposed to a variety of openended questions and respond
.

"Let's first estimate the width of the room. About how
many inches wide is this room?" Brian pointed out that inches is
an appropriate unit for a piece of paper, but not for a room. After a
brief discussion, Mrs. Flinn revised her question to "About how
many feet wide is this room?"

feel comfortable identifying errors.

The students wrote down their estimates and explanations of how
they arrived at them. After hearing all of the students estimates and
reasons, the students were asked to return to their regular groups and
decide as groups what they thought the width of the room was.
"Well," said Mr. Arbeiter, "you all gave good reasons
for your estimates, but now let's see whose estimate was closest.
We'll measure the width of the classroom." Great cheers were
heard for the groups whose estimate was closest to the actual
measurement. The same process was repeated for length, and width as
well as estimating the window area of the classroom. Mrs. Flinn
pointed out that estimates were getting closer to the actual
measurements each time they did it. She then showed the class a
formula used to determine the number of BTUs needed for a room in
terms of the volume of the room and the area of the windows. The data
obtained by the class for the volume and window area was entered in
the formula, and a quick calculation gave the number of BTUs needed
for the classroom. Mrs. Flinn wrapped up her presentation by making
the connection between the size needed, the cost of the purchase, and
the regular expense of running the air conditioner. She emphasized
that the success of her business rested on the sales people and their
ability to estimate the needs well.

communicate their answers and defend their thought
processes.
examine the correctness of their results.

Mr. Arbeiter thanked Mrs. Flinn for her presentation and asked the
students how they would like to practice the skills they had discussed
today. Feeling confident, the students volunteered to estimate the
data for their other classrooms. Mr. Arbeiter agreed to display the
results, so long as the students agreed to leave off estimating while
their other classes were in session.

extend their skills through practice in similar
problems.

On the Boardwalk
"It isn't fair!", Jasmine announced to her class one
Monday morning. "I used up $10 worth of quarters playing a
boardwalk game over the weekend at the shore, and I only won once.
And all I got for winning was a lousy stuffed animal!"
Ms. Buffon often told her class that mathematics was all around
them, and had encouraged them to see the world with the eyes of a
mathematician. So she wasn't surprised that Jasmine shared this
incident with the class.
"Please explain why you thought there was mathematics
here," Ms. Buffon asked Jasmine.

The students:
recognize the role that mathematics can play in explaining and
describing the world around them.

"Well, first of all, I threw the quarters onto a platform
which was covered with squares, you know, like a tile floor, so that
reminded me of geometry. And as I was throwing my quarters away, one
after another, I was reminded of all the probability experiments that
we did last year, you know, throwing coins and dice. It wasn't
exactly the same, but it was like the same."
"Those were very good observations, Jasmine," said
Ms. Buffon, "you recognized that the situation involved both
geometry and probability, but you didn't tell the class what you had
to do to win the game."

connect previously learned mathematics to the current
situation.

"Oh, you just had to throw the quarter so that it didn't touch
any of the lines!" Ms. Buffon asked Jasmine to go to the board
to draw a picture, explaining to her that not everyone will visualize
easily the game she was talking about.

use different forms of communication to define a problem and
share their insights.

Every other Monday, Ms. Buffon began her geometry class with a
sharing session. Sometimes the "mathematics situations"
that the students shared did not lead to extended discussions, in
which case Ms. Buffon continued with the lesson she had prepared. But
she was prepared to use the entire period for the discussion, and even
carry it over into subsequent days, if the students got interested in
the topic.

are afforded the opportunity to fully explore and resolve
mathematical problems.

"Why didn't you think the game was fair?" she asked
Jasmine. Jasmine repeated what she had said earlier, that she should
have won more often and that the prizes should have been better.
Other students in the class were asked to respond to the question, and
after a lively interchange, they decided that for the game to be
really fair, you should get about $10 in prizes if you play $10 in
quarters; but, considering that they were having fun playing the game,
and considering that the people running the game should get a profit,
they would be satisfied with about $5 in prizes. Jasmine listened to
the conversation intently, and chimed in at the end "That lousy
bear wasn't worth more than a dollar or two!"
Moving the discussion in another direction, Ms. Buffon said
"Now that we understand that it is possible to explain
'fairness' mathematically, let us investigate Jasmine's game
to see if it really was unfair. What do you think were Jasmine's
chances of winning a prize?"

explore questions of fairness, geometry, and
probability.

This question evoked many responses from the class, and after some
discussion the class agreed with Rob's comment that it all depended on
the size of the squares. Jasmine did not know the actual size of the
squares, so the class agreed that they might as well try to figure out
the answer for different size squares. Dalia pointed out that this
looked like another example of a function, and Ms. Buffon commended
her for making this connection to other topics they had been
discussing.

are encouraged to make connections to other topics within
mathematics.

Returning to her previous question, Ms. Buffon suggested that the
students do some experiments at home to help determine the probability
of winning a prize. Each pair of students was asked to draw a grid on
poster board, throw a quarter onto the poster board 100 times, and
record the number of times the quarter was entirely within the lines;
to simplify the problem, quarters that landed off the grid were not
counted at all. Different students chose different size grids,
ranging from 1.5" to 3.5", at quarter inch intervals.
After school, Ms. Buffon visited the Math Lab where she spent some
time trying to find materials related to this problem. When she
looked under "probability" in the indexes of various
mathematics education journals, she was led to several articles
discussing geometric probability, which she learned is a branch of
mathematics which addresses problems like Jasmine's game. With these
resources available to her, Ms. Buffon no longer feels that she has to
have all the answers, and can entertain discussions about mathematical
topics with which she is unfamiliar. Tomorrow she will be able to
tell the class what she has learned!

model problems and conduct experiments to help them solve
problems.

The next day the students reported on their results, and Ms. Buffon
tabulated them in the following chart, and, at the same time, plotted
their results on a graph:
Size of Squares 
Number of Wins 
1.25 
5 
1.5 
10 
1.75 
18 
2 
25 
2.25 
30 
2.5 
34 
2.75 
40 
3 
45 
3.25 
48 
3.5 
54 
"Do you see any patterns here?", Ms. Buffon asked the
class. They all agreed that, as expected, the larger the size of the
squares, the more frequently Jasmine would have won the game.
"What do you think the size of the squares were on the boardwalk
game?" Ms. Buffon asked next. Everyone agreed that the squares
were most likely smaller than
1.25" since her one prize out of forty quarters corresponded
to 2.5 wins out of a 100 games, which was lower than obtained for the
smallest squares in the experiment. Turning to the graph,
Ms. Buffonasked "What would have happened if we tried the
experiment with squares smaller than 1.25?" The students
laughed, one after another, as they realized that if the size of the
squares were small enough, you would never win the game. "Well,
then, what would have happened if we tried the experiment with larger
and larger squares?" Looking at the graph, the class found this
a difficult question, but Fran broke the group's mindset by saying
"Yeah, suppose the squares were as big as this room?" Then
everyone realized that if the squares were larger and larger, you
would become almost certain to win the game. "Dalia, do you
remember your comment yesterday, that it sounded like we were working
on a function?" Ms. Buffon asked. "Would you sketch the
graph of that function for the class?" Ms. Buffon made a mental
note to discuss this problem with her precalculus students, since she
had many questions to ask them about this graph.

collect, analyze, and make inferences from data.
recognize connections between numerical patterns and
functions.

"Well, we've gotten a lot of information by using experimental
methods about Jasmine's game; let's see if we can figure out the
probability theory behind it as well." At this point, Ms. Buffon
was eager to tell her class what she had learned at the Math Lab.
However, being aware that the students will grasp the solution method
better if they have an opportunity to discover it for themselves, she
asked the class to discuss the following question in their study
groups: "How can you tell from the position of the quarter
whether or not you would win the game?" Going from group to
group, Ms. Buffon listens to the discussions. When the groups have
discovered that to win the game, the center of the quarter must be
sufficiently far from the closest border, she gives the groups their
next task  to describe where in the square the center of the
quarter must be.

formulate and test mathematical conjectures.

By reasoning in different ways, the groups all arrived at the same
picture involving a smaller square inside the original square, and at
the same conclusion  that you win if the center of the quarter
lies inside the smaller square. With this information, the students
are able to calculate the probability for any particular size of the
square, and even to write an equation for the function whose graph
they sketched earlier.
Having found the probability of winning the game, Ms. Buffon
planned to return to the question that began this whole discussion
 whether Jasmine's game was fair. But that was the topic for
another day.
Note: Ms. Buffon realized that the graph of the function was not
linear, as depicted earlier, even though the data seemed to indicate
linear growth. With her precalculus students, she would have them
translate the above situation into the equation
y=(xd)^{2}/x^{2}, where d is the diameter of the
quarter. Then she would have them graph the function, enabling them to
discover that although the graph appears to be linear, in reality it
increases at a decreasing pace, and goes asymptotically to the line y
= 1.

construct a pictorial model to represent the problem.

A Sure Thing!?
Ms. Jackson is teaching her geometry students to use and identify
inductive reasoning.
She asks each student to draw a large triangle on their paper. She
then asks the students to hold up their triangles so that they can see
the wide variety that have been created. The students observe that
all the triangles are different.
Ms. Jackson then asks the students to cut out their triangle, to
tear off the corners of their triangle and to place the corners
together so that they are adjacent. She circulates around the room to
be sure everyone is on task, and tells students to record a
description of what they see in their notebooks.

The students:
use a variety of types of mathematical reasoning to solve
problems.

Ms. Jackson then asks a representative sampling of students to tell
the class what they observed after fitting the corners together. The
students report that it looks as if the corners form a straight
line. Everyone agrees.
Ms. Jackson now asks the students to write a generalization about
the angles of ANY triangle based upon the class results of this
activity. She asks another representative sampling of students to
state their generalizations. The students conclude that the sum of
the measures of the angles of ANY triangle is 180 degrees.
She gives the students a definition of inductive reasoning. They
recognize that they have used induction to reach their generalization
about the angles of a triangle. She then asks them to think about
when they have used inductive reasoning in the past and write an
example in their notebooks.

are encouraged to form generalizations based on observations
they have made.

Volunteers are asked to share their recollections with the rest of
the class. Some are funny and some quite poignant. The teacher asks
if anyone can see a drawback to inductive reasoning within social as
well as mathematical contexts.
The class decides that one drawback is that you can't check all
examples  all triangles cannot be checked to see if the angles always
add to 180 degrees. Another is that if you check too few examples you
might reach an erroneous conclusion. They discuss how this is the
reason for much of the racial and gender stereotyping that they
encounter. Ms. Jackson asks students to identify counterexamples for
racial and gender stereotypes.

are regularly asked to write about their understandings of
mathematics and its uses in the real world.

Ms. Jackson then asks the students to do another experiment. They
use their compasses to draw 5 circles. On the first circle, the
students identify and connect 2 points with a chord. They then state
the number of nonoverlapping regions into which the circle has been
divided. On the second circle, students identify three points and
draw all chords connecting these points. Once again, they state the
number of nonoverlapping regions into which the circle has been
divided. They continue this procedure until they find the number of
nonoverlapping regions formed when 5 points on the circle are fully
connected by chords. Students record their data in a table and use
inductive reasoning to predict the number of nonoverlapping regions
produced by fully connecting n points on the circle with chords:
# of Points 
# of nonoverlapping regions 
2 
2 
3 
4 
4 
8 
5 
16 

n 
2^{(n1)} ????? 
They are asked to state their conclusion in narrative form.

generate a set of data and use patternbased thinking to
formulate solutions.

The students agree that the number of nonoverlapping regions
produced by fully connecting n points on a circle with chords is
2^{(n1)}. Students then test their conclusion by carrying
out the experiment with 6 points. Many find their conclusion is wrong
for n=6. They fully expected to find 32 regions but only got 31!
As class draws to a close, Ms. Jackson gives a homework assignment
in which students will induce as well as produce counterexamples to
conclusions. Students leave class somewhat dazed by the last
experiment. Many of them tell Ms. Jackson that something must be
wrong because they are sure the answer is 32. They tell her that
they will prove her wrong by reenacting the experiment at home. She
looks delighted and encourages their pursuit.

validate conclusions by looking for
counterexamples.

Breaking the Mold
Mr. Miller wants his ninth grade mathematics class to review the
rectangular coordinate system, reinforce how mathematics is used to
model situations, and develop the concept of exponential functions.
He decides this would be an excellent opportunity to utilize a
realworld situation. He elects to build his effort around an
experiment involving mold growth found in an old SMSG book entitled
Mathematics and Living Things.
At the beginning of the unit, Mr. Miller presents the class with a
packet of required readings, each of which deals with growth patterns
of living things. There is an article on the rabbit population of
Australia, another on world population, and another on the spread of
AIDS. He explains the goals of the unit, gives the expectations for
the readings, and describes the purpose of the experiment the class
will conduct. Mr. Miller has students distribute the lab directions
and materials, and he has them prepare the medium for the mold
growth.
LAB DIRECTIONS
Materials:

1  9inch circular aluminum pie plate
2  sheets of 10x10squarestotheinch graph paper
1  rubber band
glue
scissors, ruler
saran wrap
mixture of clear gelatin, bouillon, and water
Directions:
Cut one piece of graph paper to fit the bottom of the tin as
closely as possible. Draw a set of axes with the origin as near the
center as possible. Cement the paper to the bottom of the tin with
rubber cement. Pour the mixture into the tin so as to cover the graph
paper with a thin layer. Allow the tin to sit 5 minutes, cover with
plastic wrap, and hold in place by a rubber band. Place the tin in a
dark place where the temperature is fairly uniform.


The students:
incorporate scientific applications in their study of
mathematics.

On each day over the next two weeks, students record an estimate of
the area covered by the mold, the increase in the area from the
previous day, and the percent of increase. On Fridays, they are asked
to extrapolate the growth they expect to occur on Saturday and Sunday
and then interpolate the same information from the growth they see on
Monday. They are required to maintain a graph of the percent of
increase versus the days. The extrapolated and interpolated points
are both graphed with special marks such as "X" or
"O."

estimate area of irregular figures.
collect and analyze data.

During the period of datagathering, Mr. Miller develops
exponential growth through the concept of compound interest and
uses a graphing calculator to illustrate the graph of such growth.
Each student is asked to suggest a function which would yield
something close to their data, and has the opportunity to put their
function into the graphing calculator and revise it until they are
satisfied with the estimate. Time is provided to have the students
discuss their reactions to the readings.

use technology as a tool of learning.
spend the time needed for mathematical discovery.

At the end of the twoweek period, Mr. Miller has the students
prepare a report relating the graph of their observations to the
discussions of the readings and the work on compound interest. To
extend the ideas developed in this experiment, students are given
different data sets which came from actualmeasurements of various
types of growth. Students work in groups, each group taking one of
the sets of data. The groups are expected to make a presentation
discussing the exponential function which models the growth, what
limiting factors could be involved, and the carrying capacity of the
environment.

write about their understandings of the connections between
mathematics and physical phenomena.
extend their understanding of mathematical concepts through
cooperative work and presentation.

As a closing activity, students are asked to choose a country from
around the world, examine population growth over some period of time,
and write a paper for inclusion in their portfolio discussing the
mathematical issues and biological issues involved as well as a
general discussion of the impact of such growth on the history of that
period.

are assessed through alternative means.
explore the uses of mathematics in other disciplines.

