USING THE ISA MATHEMATICS
RUBRIC EVERYDAY
Jonathan Katz, Ed.D, ISA
Joe Walter, ISA
Regine Philippeaux-Pierre, Ed.D, NCREST
We would like to acknowledge the following ISA coaches, teachers, and NCREST researchers for their help in preparing this
document:
Marian Mogulescu, ISA Coach
Dr. Denise Levine, ISA Coach
Marylin Cano, Teacher, Queens High School of Teaching
Matthew Sullivan, Teacher, Bushwick School for Social Justice
Maria Peralta, Teacher, Pablo Neruda Academy
Patrice Nichols, NCREST
Dr. Charles Tocci, NCREST
1
Dear Teachers and Principals,
We at the Institute for Student Achievement believe that students must develop the ability to think and reason mathematically
in order to become fully participating members in a developing global society. It is for this precise reason we are avid proponents of
inquiry-based teaching and learning in mathematics, as well as in other disciplines. Inquiry teaching and learning requires students to
think of mathematics as more than a set of detached facts, formulas and discrete skills. It allows students to have a deeper
understanding of the concepts of mathematics while learning to use its tenets to solve routine and novel problems both in the
classroom and in society.
We recognize that ISA is not alone in its on-going efforts to develop academically rigorous mathematics instruction that
prepares all students to be college ready. You, the teachers and school administrators, are at the heart of this work. ISA is appreciative
of your continuing commitment to improve the performance outcomes for our students by investing in your personal learning and
teaching.
To support your professional learning, ISA mathematics coaches and NCREST researchers, in collaboration with several ISA
math teachers and school-based coaches, have designed this tool, Using the ISA Mathematics Rubric Everyday. The document is
intended as a practical resource to help you implement a curriculum and instructional program which ensures that all students are
prepared for and experience success with inquiry-based mathematics. Throughout the document, the dimensions of the ISA
mathematics rubric are explained, and several inquiry problems and tasks are included.
Again, thanks for your support and commitment. I would also like to extend a special thanks to the group of teachers, coaches,
principals, and researchers for their part in producing this document.
Sincerely,
Gerry House, Ed.D.
President and CEO
Institute for Student Achievement
2
Using the ISA Mathematics Rubric Everyday ............................................................................................................................................ 5
Rubrics ........................................................................................................................................................................................................ 6
Problem Solving.......................................................................................................................................................................................... 8
Teaching Idea #1: Choose the appropriate problem/task ................................................................................................................. 10
Teaching Idea #2: Use problems that can be solved with multiple strategies ................................................................................. 11
Teaching Idea #2a: Selecting and applying an appropriate strategy to find a solution ..................................................................... 13
Teaching Idea #3: Value process in addition to the answer ............................................................................................................. 15
Teaching Idea #4: Answer student questions in ways that foster understanding ............................................................................. 16
Teaching Idea #5: Use Error as a tool of inquiry ............................................................................................................................. 17
Teaching Idea #6: Have students create their own problems based on their experience with solving different problems ............. 18
Reasoning and Proof ................................................................................................................................................................................. 19
Teaching Idea #1: Conjecturing ....................................................................................................................................................... 22
Teaching Idea #2: Encouraging the use of evidence and proof in daily problem solving ............................................................... 23
Teaching Idea #3: Increasing students’ metacognition .................................................................................................................... 24
Communications ....................................................................................................................................................................................... 26
Teaching Idea 1a:
Writing in mathematics gives students the opportunity to reflect on mathematical concepts and clarify their
ideas through the use of journals ....................................................................................................................... 28
Teaching Idea 1b: Writing in mathematics gives students the opportunity to reflect on mathematical concepts and clarify their
ideas through mathematical research and writing within problems and projects .............................................. 29
Teaching Idea 2:
Oral communication ........................................................................................................................................... 31
Connections............................................................................................................................................................................................... 32
Teaching Idea #1: There are common structures (e.g. patterns) that bind together multiple ideas of mathematics ........................ 35
Teaching Idea #2: The history of mathematics helps students make sense of and appreciate mathematics ................................... 36
Teaching Idea #3: Using contextual problems that are meaningful to students .............................................................................. 37
Representations ......................................................................................................................................................................................... 38
Teaching Idea #1a: Help students see how the concrete is connected to the abstract, such as hoe the concrete leads to
generalizations ................................................................................................................................................... 40
Teaching Idea #1b: Use concrete structures or examples of concrete structures to have students examine mathematical ideas ...... 42
3
Teaching Idea #1c:
Teaching Idea #2:
Teaching Idea #3:
Use additional mathematical modeling and design projects to move students from the concrete to the abstract
and develop mathematical understanding .......................................................................................................... 43
Moving from the arithmetic to the algebraic ...................................................................................................... 44
Using mathematical representations to investigate, analyze, interpret, explain, and justify.............................. 45
Appendix A ............................................................................................................................................................................................... 47
Appendix B ............................................................................................................................................................................................... 49
Appendix C ............................................................................................................................................................................................... 51
Appendix D ............................................................................................................................................................................................... 52
Appendix E ............................................................................................................................................................................................... 53
4
USING THE ISA MATHEMATICS RUBRIC EVERYDAY
Mathematics is a discipline filled with the beauty of interconnected ideas and patterns that continue to amaze and awe mathematical
communities. Its uses are numerous and its influence, substantial. Mathematics is a science and an art.
Teachers have a wonderful opportunity to help students appreciate this beauty and wonder. Polya (1944) once said:
Thus, a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine
operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges
the curiosity of his students by setting their problems proportionate to their knowledge, and helps them to solve their
problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.1
In an effort to support teacher’s work in helping student learn and understand mathematics, we have created this document “Using the
Rubric Everyday.” The document originally began as a composition of ISA teachers’ ideas and suggestions for inquiry math teaching
and learning. Through the composition of their work, it became obvious that focusing on problem solving, mathematical reasoning
and communication, making deep mathematical connections, and using multiple representations to express an idea were all important
aspects of an inquiry mathematics classroom for teachers. As such, this document is meant to support teacher’s on-going efforts to
incorporate inquiry lessons and units in their everyday teaching.
In this document, we elaborate on the definition and important components of each of the dimensions: Problem Solving, Reasoning
and Proof, Communications, Connections, and Representations. For each dimension, we provide overall teacher and student goals to
guide a reader’s understanding and we define several teaching ideas that are essential in instruction. Within the document there are
also questions, prompts, and suggestions that may be helpful to teachers as they think about, plan, and write mathematical lessons and
units. Lastly, we provide example problems within the dimensions that teachers may use in their classrooms. It is our hope that this
document will be an introductory guide to inquiry in the mathematics classroom.
1
Polya, G. (1971). How to solve it. New York: Doubleday Anchor, p. v.
5
ISA Mathematics Rubric
Scorer’s Initials ____________
Student __________________________________
1
NOVICE
PROBLEM SOLVING
No strategy is chosen or
a strategy is chosen that
will not lead to a solution.
Little or no evidence of
engagement in the task is
present.
2
APPRENTICE
A partially correct strategy is
chosen or a correct strategy for
only solving part of the task is
chosen.
Evidence of drawing on some
relevant previous knowledge is
present, showing some
engagement in the task.
3
PRACTITIONER
4
EXPERT
A correct strategy is chosen
based on the mathematical
situation in the task.
An efficient strategy is chosen
and progress toward a solution is
evaluated.
Evidence of applying prior
knowledge to the problem-solving
situation.
Adjustments in strategy, if
necessary, are made along the
way, and/or alternative strategies
are considered.
Begins to build new mathematical
knowledge by extending prior
knowledge.
All parts of the solution may be
correct.
Evidence of analyzing the situation
in mathematical terms. Constructs
new mathematical knowledge by
extending prior knowledge is
present.
REASONING AND PROOF
All parts of the solution must be
correct.
Arguments are made with
unsystematic testing and
without reasoning or
justification.
Deductive arguments are used to
justify decisions.
Correct reasoning or justification for
reasoning is present when testing
and accepting or rejecting a
hypothesis or conjecture.
Deductive arguments are used to
justify decisions and may result in
more formal proofs: systematic
trial and error; paragraph proof;
two-column proof; algebraic proof;
or geometric proof.
Deductive arguments are used to
justify decisions and may result in
more formal proofs: systematic trial
and error; paragraph proof; twocolumn proof; algebraic proof; or
geometric proof.
Evidence is used to justify and
support decisions made and
conclusions reached. This leads
to:
Evidence is used to justify and
support decisions made and
conclusions reached. This leads to:
1. Testing and accepting or
rejecting of hypotheses or
conjectures.
2. Explanation of phenomenon.
6
1. Testing and accepting or
rejecting of a hypotheses or
conjectures.
2. Explanation of phenomenon.
3. Generalizing and extending the
solution to other cases.
4. In some cases, multiple ways of
reaching the same conclusion.
SCORE
COMMUNICATION
Responses are not clear,
thorough, or detailed.
Little or no communication
of ideas is evident.
Some communication of an
approach is evident through
verbal/written accounts and
explanations, use of diagrams or
objects, writing, and using
mathematical symbols.
Responses are clear, sequenced,
thorough, and detailed manner.
Communication of an approach is
evident through a methodical,
organized, coherent, sequenced,
and labeled response.
Formal mathematical language is
mainly used throughout the
solution to share and clarify ideas.
Student uses mostly familiar,
everyday language with some
formal mathematical language
interspersed.
CONNECTIONS
No attempt to connect
mathematical ideas.
REPRESENTATION
Responses are somewhat clear,
sequenced, thorough, and detailed
manner.
No attempt is made to
construct accurate
mathematical
representations.
Student does at least one of the
following:
Student does at least one of the
following:

Student observes a connection
between mathematical ideas.


Student observes a connection
between the task and past
experience, past problems, or
other subjects.
Student observes and
explains a connection
between mathematical ideas.

Student observes and
explains a connection
between the task and past
experience, past problems, or
other subjects.
An attempt is made to construct
accurate mathematical
representations to record thought
processes.
Appropriate and accurate
mathematical representations are
constructed to show growth in
thought processes.
7
Responses are very clear, well
sequenced, thorough, and detailed
manner.
Communication at the practitioner
level is achieved, and
communication of arguments is
supported by mathematical
properties.
Formal mathematical language and
symbolic notation are used to
consolidate mathematical thinking
and to communicate ideas.
Student does all of the following:

Student observes and explains
coherence between
mathematical ideas.
 Student observes and explains
a connection between the task
and past experience, past
problems, or other subjects.
Generalized and accurate
mathematical representations are
constructed to analyze
relationships, extend thinking,
and/or interpret thought processes.
DIMENSION I
PROBLEM SOLVING
8
I. PROBLEM SOLVING
Defining a Problem
A problem is any situation, task, or question that students find interesting and/or challenging. Problems can be procedural,
conceptual, and/or they may center on a real world situation.2
Goal(s) of Problem Solving
Teachers:
Since problem solving is the essence of mathematics, teachers may use problems as an entry point to help students learn new
ideas, construct new knowledge, and connect concepts with procedures or an application to a learned idea.
Students:
Students are encouraged to learn, construct, connect, and apply knowledge and/or mathematical concepts to solve problematic
situations.
We will discuss six teaching ideas in problem solving. We will also provide examples of problems, teaching prompts, questions, and
suggestions that may help teachers develop and implement these ideas.
Teaching Ideas in Problem Solving
1. Choose the appropriate problem/task
2. Use problems that can be solved with multiple strategies
a. Selecting and applying an appropriate strategy to find a solution
3. Value process in addition to the answer
4. Answer student questions in ways that foster understanding
5. Use Error as a tool of inquiry
6. Have students create their own problems based on their experience with solving different problems
2
Please see Appendix A for examples of procedural, conceptual, and real-world problems.
9
Problem Solving
TEACHING IDEA #1: Choosing the appropriate problem
Teaching Idea
Choose the appropriate problem
Teachers please take some time while
preparing your lesson or unit to think about
which problems are appropriate. Please think
about what makes a task appropriate. For
example:
1. Entry points for all students
2. The task is mathematically rich: new
concepts and/or procedures are
embedded within the problem/tasks.
3. Students will find the task interesting
4. The task asks students to construct new
knowledge.
Questions/Prompts
For the teacher:
We encourage teachers to think about the
following set of questions when choosing
problems for students to attempt:
a. What is your goal for the lesson? (The
lesson can be for more than one day.)
b. What do you want students to leave
with after the lesson?
c. Will the chosen problem help you
attain your goal(s) for the lesson?
d. Will the problem/task interest the
student? Why?
10
Suggestions
Try the task or problem on your own
before using it in the classroom
Anticipate where students will struggle
and what questions you would ask to help
them further engage in the task.
Problem Solving
TEACHING IDEA #2: Use Problems with multiple strategies
Teaching Idea
Use Problems that can be solved with
multiple strategies
Questions/Prompts
Suggestions
For the Students:
How else can you do this?
Did anyone do it differently?
Why did you solve the problem in the
manner you chose?
Do you see a connection between the
different approaches your classmates have
proposed?
Anticipate the various strategies students
might use to solve a problem or task.
In order to produce multiple strategies
teachers are encouraged to let students do
the discovering.
Give students the freedom to solve a
problem anyway they want. Avoid
imposing your methods on them, if
possible. Impositions may shut students
down and give the impression there is
only one way to solve a problem. Often if
students do not understand that one way,
they are lost.
EXAMPLE PROBLEM:
Display Dilemma Problem
I visited friends in New York City during the summer. They took me to this HUGE Wal-Mart store. There was a display of cookie boxes that I
could not believe! The display was in a pyramid shape with at least 100 boxes as the base. I had to stand back and wonder how many boxes were
in the whole display. I imagine when they started building the display it might have looked like the pictures below.
11
How many boxes of cookies are in a display with a base of 5 boxes? 10 boxes? Can you come up with a rule for finding the number of boxes in
a display that is 100 boxes in the base like the one I saw?
12
Problem Solving
TEACHING IDEA #2a: Selecting an appropriate strategy
Teaching Idea
Have students select and apply appropriate
strategies to find a solution. There are a set
of powerful strategies that can help
students become better problem solvers if
they are given multiple experiences using
them:
a. Simplification of a problem
b. Pattern recognition
c. Working backwards
d. Creating diagrams, tables,
charts to organize data visually
and to observe patterns.
The examination of problems can lead to
students connecting their mathematical
knowledge. Encourage students to articulate
the connections they make and have them
evaluate which of these connections will help
solve a particular problem.
Teaching Questions/Prompts
What patterns do you see?
Why does the pattern exist? Where does
the pattern come from?
How did you come up with the pattern?
What do you already know?
What can you already do?
What strategies can you use to solve this
problem?
What don’t you know how to do? What do
you know how to do? Where can you get
those knowledge and skills?
To support students in selecting and applying
appropriate strategies, have students:
- Restate a problem for
understanding.
-
3
Connect the problem to prior
knowledge, ideas in math, and
skills from prior lessons,
Please see problems in Appendix A.
13
Suggestions
Provide
students
with
multiple
experiences with various types of
problems which will encourage different
types of strategies.
Problem solving needs to be embedded in
all units. To transition students into a
problem solving culture, you may begin
the school year with a problem solving
unit that gives students multiple
opportunities to learn and grapple with
different strategies. Later units will use
problems as an entry point to think about
different concepts and procedures.3
subjects, or disciplines.
-
Make observations.
-
Decide on a strategy.
Assess/test your strategy. Reevaluate your strategy if it does
not help you answer the
question.
-
Communicate a strategy in
various ways including in
diagrams, tables, graphs, etc
-
Identify pattern(s) & develop a
conjecture to answer the
problem
14
Problem Solving
TEACHING IDEA #3: Value process and answer
Teaching Idea
Value Process in addition to the answer
While the answer is important, the
development of mathematical understanding
requires students thinking about what they are
doing and why they are doing it.
Questions/Prompts
For students:
What are you doing?
Why are you doing it? Why did you do it?
Why does this approach make sense to
you?
If we view mathematics as being about skills
and procedures, then focusing on the answer
becomes central. However, if we believe that
mathematics is about ideas and concepts then
the process of problem solving and thinking
become central to our instruction. The depth of
understanding comes from the process and not
the answer.
Suggestions
What do students learn as they look at
their own process?
How can I question students to help them
think about their own process?
What can I learn as a teacher from the
different processes students use?
EXAMPLE PROBLEM:
Samantha is a farmer and the other day she was on her way to sell eggs at the market. She got into an accident and all her eggs broke. In order
for her to collect money from the insurance company, she had to give the exact amount of eggs she had. She doesn’t remember how many eggs
there were but when she was packing them she remembered when she put them in groups of 2,3,4,5, and 6, one egg was left over. But in groups
of 7 no eggs were left over. How many eggs did she have?
15
Problem Solving
TEACHING IDEA #4: Answer student questions to foster understanding
Teaching Idea
Answer student questions in ways that
foster understanding.
Respond to student questions with probing
questions that help students reason about the
problem and grasp the concepts or ideas in the
problem.
Questions/Prompts
For students:
What is the problem about? What are you
trying to find out?
What observations can you make?
Can you simplify the problem?
Can a diagram help you understand the
problem?
Can you make any conjectures?
Can you make any conclusions?
EXAMPLE PROBLEM:
a) How many squares are on an 8 by 8 checkerboard?
b) How many rectangles are on an 8 by 8 checkerboard?
16
Suggestions
Allow students to grapple with problems.
Dealing with frustration is part of the
process of becoming a good problem
solver.
How do you encourage students when
they are struggling with a problem
without showing them what to do?
Problem Solving
TEACHING IDEA #5: Error as a tool for inquiry
Teaching Idea
Use Error as a tool of inquiry
Error is a great opportunity for discussion and
developing conceptual and procedural
understanding. Through an inquiry process,
student misconceptions can be observed and
corrected.
Questions/Prompts
For Student:
Does anyone disagree/agree with the
process and solution?
Suggestions
Let students wonder about their work; do
not rush to correct mistakes. Selfcorrection is more valuable than teacher
correction.
Why do you disagree/agree?
How might you have done it differently?
Why did you do that? Does your
strategy/solution make sense? Why?
Can you show/prove that the solution is
correct or incorrect?
EXAMPLE PROBLEM:
John solved the equation in this way3x = 5x – 4
-3x -3x
____________________
2x = -4
x = -2
Do you agree or disagree with John’s method? Why or why not?
17
When you question students, the purpose
should be to have students reflect on their
process and thinking.
Problem Solving
TEACHING IDEA #6: Student create their own problems
Teaching Idea
Have students create their own problems
based on their experience with solving
different problems
When students engage in the process of
creating their own problems and show ways to
solve those problems, they develop a higher
level of mathematical understanding.
Questions/Prompts
For the Teacher:
When looking at student-created problems
think about what understandings the
student had in order to have created the
problem. (look for levels of sophistication;
ideas, concepts and procedures embedded
in problems)
Suggestions
IMPORTANT: Please be clear about the
goal of the activity. If we are looking for
understanding, we need to be exact about
what that understanding is. Then we can
make the assignment clearer to students.
This task may require one-on-one work
with students to probe their understanding
and prod them to go deeper.
Example Problem 1:
Create a situation in your life using one of the following: at least, at most, minimum, or maximum. Write the situation algebraically and in
words.
Examples:
- I need to be at least 21 to drink my father’s rum
- The teacher said we had to write an essay containing between 600 and 800 words.
Example Problem 2: Mini Project- Creating a Quadratic Situation
Your task is to create a situation that you can model with a quadratic function. You will use that function to answer questions about the situation
that you develop. There are two ways you can approach this. You can use situations similar to ones we’ve studied. For these, you must alter
the story and the data, coming up with a new equation. Or you can select a situation (realistic or not) that interests you and that you want to
develop into a quadratic situation.4
4
Please see Appendix C for full project.
18
DIMENSION II
REASONING AND PROOF
19
II. REASONING AND PROOF
Defining Reasoning
Reasoning is the thinking employed for the purpose of reaching a conclusion.
Reasoning encompasses the tools, skills, and ideas from each of the other dimensions.
Defining Proof
Proof is the cogency of evidence that compels the acceptance or establishes the validity of a fact, statement, truth, or conclusion5.
Goal(s) of Reasoning and Proof
Teachers:
Providing mathematical explanations and justifications ought to be a consistent part of a student’s experience in the classroom.
Thus, teachers ought to build these elements into their daily lessons.
With daily demand to explain and justify, teachers should help build students’ capacity to approach new, unfamiliar problems.
It is essential that teachers create opportunities for students to engage in inductive and deductive reasoning.6
Students:
Students ought to acquire and demonstrate competent and proficient mathematical reasoning skills and abilities.
Students ought to provide evidence for all mathematical conclusions and justify any mathematical reasoning.
We will discuss three teaching ideas in Reasoning and Proof. We will also provide examples of problems, teaching prompts,
questions, and suggestions that may help teachers develop and implement these ideas.
5
http://www.merriam-webster.com/dictionary/proof retrieved on January 29, 2009
Deductive reasoning goes from the general to the specific. For example, if all wasps have stingers, the one flying near you has a stinger.
Inductive reasoning goes from the specific to the general. One might first conduct specific observations and come to some general conclusion about a
phenomenon.
6
20
Teaching Ideas in Reasoning and Proof
1. Conjecturing
2. Encouraging the use of evidence and proof in daily problem solving
3. Increasing students’ metacognition
21
Reasoning and Proof
TEACHING IDEA #1: Conjecturing
Teaching Idea
Conjecturing
Conjecturing, making judgments and forming
a hypothesis, is a part of every daily life. For
example, as a teacher, have you ever
conjectured about a student very quickly
early on in the term about how he or she will
behave based on immediate observations?
Students do the same thing.
In mathematics we want to see how we can
use the skill of conjecturing to develop
mathematical rules, procedures, and
concepts.
Questions/Prompts
For students:
What did you observe or notice?
Suggestions
Observations, pattern hunting, and
conjecturing are all part of the inquiry
process.
Describe any patterns you see
Use your observed patterns to make a
conjecture
What is your reason for making that
conjecture?
Based on your conjecture, what is your
prediction?
One of the most useful ways to develop
rules, procedures, and concepts is to give
students opportunities to:
- Observe
- Look for patterns
- Make conjectures
22
Students need multiple experiences. They
need to be left alone to grapple with the
observed data so they can make new
discoveries and construct knowledge.
Reasoning and Proof
TEACHING IDEA #2: Evidence and proof
Teaching Idea
Encourage the use of evidence and proof
in daily problem solving
Providing proof and justifications are an
every day part of the mathematics classroom
through answering basic questions such as
how a problem can be approached and why a
solution works (or not). Offering these kinds of
evidences makes the processes of reasoning
apparent to students and teachers so that it can
be reflected upon.
Questions/Prompts
For Students:
What conjectures did you make? What is
your reason for making that conjecture?
Why does that solution work? What is
your evidence?
What is your prediction? What is a
reasonable answer?
Where am I going? Does my prediction
still make sense?
Effective proofs and justifications for
conjecture/answer should include:
1. Clear articulation/demonstration of
patterns
2. Clear description of reasoning
3. Clear examples, possibly addressing
counter-examples
4. Work or explanation showing that
students are building off already known
mathematical principles and other
knowledge.
5. Generalizations and applications to
other cases
6. Multiple ways to solve a problem
(possibly)
Suggestions
All student explanations should answer
these three questions:
a. What did you do?
b. Why did you do it?
c. How do you know your solution is
correct?
It is necessary for students to go from
identifying a pattern to making
conjectures to generalizing an idea or
solution.
In order for students to show why their
solution or generalization makes sense,
they need to use both informal and formal
proofs.
Verification of solutions is a way of
encouraging students to think about a
particular concept. It pushes students to
make sense of the theorem, procedure, or
idea they have learned or discovered
23
Reasoning and Proof
TEACHING IDEA #3: Metacognition
Teaching Idea
Increasing students’ metacognition
Metacognition is the ability to reflect on your
own thinking. A metacognitve student is able to
ask himself probing questions while working
on mathematical problems or tasks. These
questions or directives include:
a.
b.
c.
d.
e.
Am I on the right track?
Does my thinking make sense?
Where am I going?
Let me try another strategy
Is this question connected to another
question I’ve seen before?
Questions/Prompts
Additional questions for students:
Where am I going? Does my prediction
still make sense?
Should I change my prediction or change
my approach or strategy?
Have I hit a roadblock?
Suggestions
Encourage students to be aware of their
own thinking. Have them write down
when they are confused, questions they
might have, how and why they feel a
strategy works.
In order to support student metacognition,
it is important for the teacher to model
good questions including ones students
can ask themselves.
Hitting a block in reasoning is a
“teachable moment.” There are multiple
sources students can access to get over the
block – other students, the teacher, texts,
etc. The key is developing students’
independent ability to recognize their
blocks and how to address them.
24
The Locker Problem
There are 1000 lockers lined up numbered 1 to 1000 and there are 1000 students. The lockers are all closed. The first student, Jasmine, walks by
and opens all the lockers. Then the second student, Al, walks by and goes to every second locker starting at #2 and closes it. Then Mary walks
by and goes to every third locker starting at #3, closing the opened lockers and opening the closed lockers. The 4th student walks by and goes to
every fourth locker starting at #4, closing the opened lockers and opening the closed lockers. This routine goes on until student 1000, Michael,
goes to locker # 1000 and either closes it or opens it. After this is finished, which lockers will be open? Why?
Teaching Suggestions:
1. Have students try the problem on their own, see how they approach the problem and where they hit blocks
2. Teacher intervention point: Determine at which points you will intervene to help facilitate discussion and reasoning
3. Elicit strategies from students
25
DIMENSION III
COMMUNICATION
26
III. COMMUNICATION
Defining Mathematics Communication
Communicating mathematically means expressing the results of mathematical thinking orally, in writing, or with symbolic
representations7 in a clear, convincing, and precise manner.
Reflecting and communicating are the processes through which understanding develops.8
Goal(s) of Mathematics Communication
Teachers:
Teacher will give students multiple opportunities to demonstrate and communicate their mathematical understanding.
Students:
It is essential that students learn to clearly discuss mathematical ideas and processes, and clarify their understanding for
themselves and for others.
We will discuss writing and oral communication ideas in this section. We will also provide examples of problems, teaching prompts,
questions, and suggestions that may help teachers develop student’s mathematical communication skills.
Teaching Ideas in Communication
1. Writing in mathematics gives students the opportunity to reflect on mathematical concepts and clarify their ideas. Two possible
ways to incorporate writing are:
a. Through the use of journals
b. Through mathematical research and writing within problems and projects
2. Oral communication
7
8
For a more thorough explanation of communicating symbolically, please see REPRESENTATIONS in Section V.
Hiebert, J. et al., (1997). Making sense. Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
27
Communications
TEACHING IDEA #1a: Writing Communication -Writing in journals
Teaching Idea
Use journals to solidify and synthesize
learnings; reflect on learnings or ideas;
segue or connect previous knowledge to
new knowledge and strategies.
Journals can be used at the beginning of a
lesson, throughout the lesson to reflect on
mathematical ideas or problems, or at the end
of the lesson. With journal prompts, students
can be slowly guided to learn to justify their
ideas, proofs, reasoning, and solutions.
Journals may also be used to help students
develop multiple perspectives or strategies.
Questions/Prompts
For Student:
Explain the [math concept] that was
learned yesterday. Why/How is it
important in solving [a particular openended problem]?
Where else have we seen this idea or
concept before?
How is it similar to what we are doing
now?
How else might you solve this problem?
Describe all the strategies we have used so
far to solve [problem]. How are these
strategies different from one another?
How are they similar?
Which strategy works best? Why?
Will this strategy always work? How do
we know? How can we tell?
Explain how and why this solution works
28
Suggestions
We suggest that teachers take time to
respond to students’ journal responses.
If there are no responses to journals,
journals lose their impact as a tool for
mathematical communication
Journals can be a place to explore
mathematics. We suggest that teachers
scaffold questions or provide prompts for
journal writing. The hope is that the
journal will become a place where
students automatically go to write when
they are confused, and when they have
made a connection between a
mathematical concept and something in
another class or something in the real
world.
Communications
TEACHING IDEA #1b: Writing Communication -Writing in problems and projects
Teaching Idea
A writing portion can be added within a
project or open-ended problems so that
students
- explain their thinking process
- fully describe their problem solving
process
- explain their solution and why it
works
Projects may include research where a student
must probe deeper into a mathematical idea or
prove a mathematical idea.
Teaching Questions/Prompts
Suggestions
Explore the mathematical phenomenon,
Some suggestions for students writing within
theory, or concept
projects9:
Describe the problem to be solved
- Explain the phenomenon
Provide an explanation of how you will
- Use a particular problem to
approach the problem
describe how else or where else the
Provide a thorough explanation of how
concept, theory or phenomenon
you arrived at a solution. Describe your
may be used.
problem solving process.
- Provide examples
- What was your thinking
- Connect the phenomenon, theory
process?
or concept to a concept we have
- What helped you come to a
learned. How is it similar? How is
solution?
it different?
- What obstacles did you come
- What are your lingering questions
across?
about this concept?
- Where did you get confused?
- How did you resolve any
confusion or obstacle?
Explain your solution and why it works.
Clearly label your diagrams, graphs, or
other visual representations.
Clearly define any variables.
If an equation was derived, give a detailed
account of the derivation.
Use appropriate and accurate
mathematical vocabulary.10
9
Adopted from: Crannel, A. (1994) A guide to writing in mathematics classes. http://edisk.fandm.edu/annalisa.crannell/writing_in_math/guide.html retrieved
December 23, 2008.
10 We recommend that mathematical vocabulary develop within context. Definitions given to students prior to learning about and exploring at topic or concept
may have very little meaning to students even after a lesson has been taught.
29
Check for spelling, grammar, punctuation,
and mathematical mistakes.
Example Project/Research: Prime Numbers
What is a prime number? Why do we want to study prime numbers? What makes prime numbers interesting? Where are prime numbers used in
real life? How have mathematicians used the idea of prime numbers to discover new learning or solve problems?
Example Project/Research: Imaginary Numbers
What are imaginary numbers? Explain the history of imaginary numbers. Describe the significance of imaginary numbers in industry and/or
science. Compare real numbers to imaginary numbers. How do the properties of real numbers apply to or not apply to imaginary numbers.
Why?
Example Project/Research: Sum of a Triangle
Explain why the sum of any triangle is 180 degrees. Create a proof to justify your answer.
Extension: Choose an area in life where triangles are important. Explain why they are important. What properties of a triangle make them
indispensable in the specified area? Why?
Resource: Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth: Heinemann.
30
Communications
TEACHING IDEA #2: Oral Communication
Teaching Idea
Have students explain their work to peers
in small groups, pairs, or to the class.
Communicating orally allows students to
clearly exhibit their knowledge of a topic and
concept and also helps them reflect on the
concepts learned. One possible way to
incorporate oral communication in classrooms
is through explanations of work and solutions
daily in class to peers in small groups, pairs, or
to the whole class.
11
12
Questions/Prompts
For students doing or giving a presentation:
What is the problem asking?
What strategy did you employ? Why?
How do you know your solution is
correct?
What would you do if ___ was changed
to ____?
Suggestions
All the ideas discussed in every section
need to include communication, where
students share their thinking and
understanding.
Examples of Activities that promote
communication:
a. Socratic Seminars11
b. Cooperative groups and pairs through
inquiry investigations12
c. Presentation
Please see Appendix B
Please see Appendix B
31
DIMENSION IV
CONNECTIONS
32
IV. Connections
Defining Mathematics Connections
It involves seeing mathematics as a coherent body of knowledge in which ideas and concepts are bound together through big ideas
and a common structure.
A mathematical connection is also seeing the relationship between mathematics and the world one lives in and /or between
mathematics and oneself.
Goal(s) of Mathematics Connections
Teachers:
Since making connections, in whatever form they may take, is crucial for development of mathematical understanding it is
imperative that teachers create numerous opportunities for students to view and experience the connectedness of mathematics
through engaging problems and historical anecdotes/ investigations. Teachers should, thus, create mathematics units around big
ideas and essential questions.
Students:
Students will see and understand how mathematical concepts and ideas are linked and build on one another to produce a coherent
whole in order for them to see it as a discipline that makes sense.13
Students will see the meaning of mathematics in the world through engaging in interesting contextual problems.
Students will begin to wonder, appreciate, and marvel at the connectedness of mathematics and begin to see it as a creative
discipline
13
Adjusted from NCTM Connection Process Standard
33
We will discuss three teaching ideas in mathematics connections. We will also provide examples of problems, teaching prompts, and
questions, and suggestions that may help teachers develop and implement these ideas.
Teaching Ideas in Connections
1. There are common structures (e.g. patterns) that bind together the multiple ideas of mathematics
2. The history of mathematics helps students make sense of and appreciate mathematics
3. Using contextual problems that are meaningful to students
34
Connections
TEACHING IDEA #1: There are common structures (e.g. patterns) that bind together the multiple ideas of mathematics
Teaching Idea
Common structures such as patterns,
problems, and proof are present in all
disciplines of mathematics. Big ideas such
as the concrete and the abstract in algebra
and spatial relationships in geometry unite
individual disciplines.
Many mathematics students have felt that
mathematics doesn’t make sense; it is a
confusing bunch of rules, formulas and
procedures. Mathematics has often been
viewed by students as discrete pieces of
knowledge disconnected from each other.
There is no sense of a common binding
structure that ties all these ideas of
mathematics together. Meanwhile,
mathematicians have described mathematics as
“the science of patterns.” Patterns are one of
those binding structures that help to connect
the seemingly unrelated ideas in mathematics.
Questions/Prompts
For Teachers:
Do I think helping students make
connections matter? Why? If the answer is
yes, how can I help make that an everyday
experience?
How can I help my students make sense of
mathematics?
How can I use pattern hunting as an entry
point to understanding a new rule,
procedure or formula?
Can I create an interesting task to facilitate
making connections?
35
Suggestions
Students have to have multiple
experiences where they look at patterns as
a means of making sense out of rules,
formulas, definitions, and procedures.
(See procedural problem, example 2 in
Appendix A. See also Vignette 2 in
Appendix E and Display Dilemma in
Problem Solving Page 3-4)
An essential component when studying
functions is the idea that tables of values,
graphs, and equations are mathematically
identical. We must help students make
that deep connection between those three
representations.
Another big idea in mathematics is the
notion of ratio. When students have a
deep understanding of this idea, then the
development of many mathematical
concepts such as slope, trigonometric
ratios, pi, aspects of probability, etc,
makes sense.
Connections
TEACHING IDEA #2: The history of mathematics help students make sense of and appreciate mathematics
Teaching Idea
Students rarely have a notion that the ideas
they are learning have a history.
Mathematical ideas just didn’t fall from the
sky. Mathematical ideas developed at
different points in history because a need
arose for these ideas. (For example, the
notion of irrational numbers did not exist
until Pythagoras saw that in working with
diagonals in rectangles that the length of
the diagonal could not be written as a
fraction.) By helping students understand
some of that history, they will see that
mathematical ideas, like all ideas, develop
over time.
Questions/Prompts
To teacher:
How did numbers come to be?
How did the development of man’s mind
lead to the development of the HinduArabic Number System?
How can I use the story of pi to fascinate
students about this amazing number?
Using anecdotes and stories can be useful and
interesting to students. For example the
development of ideas of probability arose in
France as the owners of gaming parlors asked
Descartes to help them understand how to
insure they would be able to make a profit.
14
Suggestions
A full unit on the history of numbers exists for those who are interested. Please see your ISA math coach.
36
Look at the history of numbers including
Egyptian, Babylonian, Chinese, and
Mayan number systems in comparison to
the Hindu-Arabic number systems. It
would be an opportunity for kids to
appreciate the power of the numbers
system we use today. 14
Connections
TEACHING IDEA # 3: Using contextual problems that are meaningful to students
Teaching Idea
Questions/Prompts
We have often been told that we have to
For Teachers:
make mathematics relevant to our students’
lives. Yet, what does that really mean? We
Will the task/activity be interesting to my
can find in many textbooks “real world”
students? Why?
problems that have no meaning for students
and are not “real” to them. What matters
Does the task have multiple entry points?
most is that we create tasks that are
contextually interesting for students even if
How will this task be meaningful to
they are not part of students’ everyday
students?
experiences (Please see Nicol & Crespo,
200515).
15
Suggestions
Students love to play games. There are
many games that will engage students to
develop concepts in math (e.g. the game
of 27, the human peg game, the factor
game, etc…)16
Students love to create. There are many
opportunities in mathematics for students
to apply mathematical ideas through
projects (e.g. Designing a suspension
bridge in Appendix D)
Nicol, C. & Crespo, S. (2005). Exploring mathematics in imaginative places: Rethinking what counts as meaningful contexts for learning mathematics. School
Science and Mathematics, 105(5), 240-251.
16
These games and others are available upon request. Please see your ISA math coach.
37
DIMENSION V
REPRESENTATION
38
V. REPRESENTATION
Defining Mathematical Representation
Mathematical representation is any of the myriad ways mathematical ideas may be presented and demonstrated: written, oral and
visual (e.g. pictures, graphs, charts, tables, diagrams, symbols, mathematical expressions and statements, etc.)
Goal(s) of Mathematical Representation
Teachers:
Teachers will help students to think in the abstract through the development of multiple representations.
Students:
Students will be encouraged to use a variety of ways to represent mathematical ideas to model, interpret and extend understanding
of physical, social and mathematical phenomenon.
We will discuss three teaching ideas in Representation. We will also provide examples of problems, teaching prompts, questions, and
suggestions that may help teachers develop and implement these ideas.
Teaching Ideas in Representations
1. Moving from the concrete to the abstract
a. Help students see how the concrete is connected to the abstract, such as how the concrete leads to
generalizations.
b. Use concrete structures or examples of concrete structures to have students examine mathematical ideas
c. Use Additional mathematical modeling and design projects to move students from the concrete to the abstract
and develop mathematical understanding.
2. Moving from the arithmetic to the Algebraic
3. Using mathematical representations to investigate, analyze, interpret, explain, and justify.
39
Representation
TEACHING IDEA #1a: Moving from the concrete to the abstract
Teaching Idea
To support algebraic understanding, one
main goal is to move students to see how
the concrete is connected and leads to the
abstract or generalizations
Questions/Prompts
For students17:
Describe the concrete situation
Suggestions
.
See tasks below
Give other examples of this situation
Pattern recognition is essential if students are
to make connections between the concrete and
the abstract. It is the patterns students
understand in the concrete that will help them
to make algebraic generalizations.
What can you generalize about the
concrete situation?
How does it connect to the abstract
situation?
Example Problem 1:
1. Describe the similarities and differences in these two statements. What do you think the answers will be?
- 3 apples plus 5 apples
- 3 apples plus 5 bananas
2. Explain the similarities and differences between the two statements. What would you conjecture the answers to be?
- 3a +5a
- 3a + 5b
17
Teachers, please insert the proper wording according to the problem you are doing.
40
Example Problem 2
Farmer Jose has a farm in upstate New York with cows and chickens. Jose is forgetful. His worker, who likes to count odd things, told him that
he has a total of 50 cows and chickens that combined have a total of 124 legs. But neither of them were sure how many were chickens and how
many were cows. Your task is to find this out for them.
Teacher: Let kids play with this any way they want. If they are struggling, recommend to them that they organize this into a table with headings
# of cows, # of chickens, and Total # of legs. They can use intelligent guess and check.
# Cows
20
10
12
# Chickens
30
40
38
Question for the teacher: How can we use this to develop an algebraic approach?
Questions for kids: If you chose 20 cows, how did you find out the number of chickens? If you chose 10 cows, how did you…? If you chose “x”
cows, how would you represent the number of chickens? (x and 50-x).
# Cows
20
10
12
# Chickens
30
40
38
Total # legs
140
120
124
Question for Teacher to ask: How did you get the total number of legs?
Here you want kids to talk about multiplying the number of cows by 4 and the chickens by 2.
So the equation could become 4 x + 2(50 – x) = 124.
41
Representation
TEACHING IDEA #1b: Use examples of physical structures
Teaching Idea
Use physical examples to help students
connect the abstract to the concrete.
Many students find it difficult to think
abstractly. One way of helping students enter
into that world is using concrete examples and
models.
Questions/Prompts
For the teacher:
How is the topic I am teaching connected
to the real world? What mathematical
ideas do I see reflected in the world
around me? (see suggestions)
Suggestions
Begin the study of a particular unit by
showing pictures of the phenomena being
studied (e.g. arches, bridges, trajectory of
a ball for quadratic functions, speed limit
signs for inequalities, steps and ramps for
slope)
How can I use the concreteness of
physical structures to help students
understand abstract ideas?
Example Problem using Ratio and Proportion:
Use photographs of architectural structures (buildings, pyramids, domes, etc.) and have students examine, predict, and calculate mathematical
ideas related to these structures. For example, students could be presented with pictures of the Seagram building in Manhattan which is a
rectangular prism. They would be given the height. After the class comes to consensus about the width and depth, students would then use
construction paper to build a replica using different scales.
Compare and contrast the different structures and discuss how the set of building models are similar and different.
42
Representation
TEACHING IDEA #1c: Projects -Modeling
Teaching Idea
Use modeling to help students see the
connections between the concrete and
abstract and develop mathematical
understanding.
Questions/Prompts
For Teachers:
What understandings will students gain
through engagement in a modeling
project?
Suggestions
Make certain that the projects are
mathematically rich and challenging.
They should require students to abstract
by developing mathematical models.
What do you want students to know and
think about at the end of this project?
Does the project lead to the development
by students of mathematical models?
Example 1
Have students model their route from home to school, first by describing the trip with a map, followed by an expression of the trip in words, and
finally by representing train stops, bus stops, walking, etc with variables. This makes the abstract concrete then moves back to the abstract
allowing students to make and evaluate a mathematical expression.
Example 2
Students will create a redesign (floor plan) for their classroom or a room/space where they live to improve the aesthetics and functionality. They
could use the area of an existing room which they can measure and calculate as a basis for their redesign. They will then make a model to scale
of this design showing all internal characteristics.
Example 3: Model Suspension Bridge Project
You are an engineer for Bechtel and charged with designing and building a scale model of a suspension bridge. You will be given dimensions
of a waterway that will be crossed by your bridge. Your task is to build a suspension bridge.18
18
Please see Appendix D for the full project.
43
Representation
TEACHING IDEA #2: Connecting the Arithmetic to the Algebraic
Teaching Idea
It also important that students develop the
ability to go from the arithmetic to the
algebraic.
Teaching Questions/Prompts
Suggestions
.
Example 1
What is the relationship
2 • 2 • 2 • 2 = 2 4 and 5 • 5 • 5 • 5 = 54
to x • x• x • x = x4
Example 2: Distributive Property
You and your friend Shawn are selling boxes of candy on the subway. You begin your speeches by saying, “I’m not going to lie to you. I’m
not selling candy for my team, my church or any other organization. I’m selling this for me, to keep me off the street.” At the end of the
week, your plan is to share the profits equally. For each box of candy you sell, you get $7. You sold 18 boxes and Shawn sold 23 boxes.
a. Shawn chose to calculate amount you will earn by summing the total number of boxes sold and multiplying by 7.
b. You figured out your own earnings before you saw Shawn. Then you calculated Shawn’s earnings and added them
together.
Compare your results with Shawn’s. What do you notice?
Let a stand for the number of boxes you sold, b stand for the number Shawn sold and let c stand for the amount you earn on each box.
a. Write an algebraic expression showing the math that Shawn did.
b. Then write an algebraic expression showing what you did.
c. Write an equation that includes the information from both of your methods.
44
Representation
TEACHING IDEA #3: Investigate, analyze, interpret, explain, and justify
Teaching Idea
Have students represent ideas, thinking,
and/or solutions using graphs, tables,
mathematical statements, pictures,
symbols, animation, skits, poems, prose,
diagrams (i.e. Venn Diagrams), etc.
Students can be asked to create a skit (with
their partner) whose main characters are
two geometric shapes. In the writing of the
skit each partner takes on one shape and
tries to show why his shape is unique.
Teaching Questions/Prompts
How many ways can this [solution,
situation, etc…] be represented?
Suggestions
As much as possible each
concept/lesson/unit/ should have a
written, oral and visual component.
What tool would best represent this idea?
What does this symbol tell us?
Walls should be covered with examples of
the variety of student work demonstrating
representations
How else can you represent this idea?
Is there a picture that could represent this
idea?
Where might you see this outside the
class?
Can you draw an example of this idea?
How would a table show the solutions to
this equation?
How do we make math in the media
meaningful?
What is the appropriate representation?
E.g. Should I use a graph to represent a
linear function instead of a bar graph?
Rooms should have lots of models,
manipulatives and materials for students
to work with in developing mathematical
representations.
Use graphical support as a tool for
inquiry. Use the graphing calculator,
geometer’s Sketchpad, Cad Cam
programs for architectural design and
other mathematical software to have
students discover and represent properties
of geometric figures and other concepts.
Trips to art museums (for example to see
Islamic art, Frank Lloyd Wright, rooms
from other cultures at the Met) to observe
and discover mathematical
representations in other cultures.
Students should have the experience of
using math representations to teach a
concept/idea to each other, teachers,
45
family members and younger students.
Example 1
Using mathematical statements in the media to interpret the physical, social, and political world. Example: A newspaper says that 8 million in
Africa have AIDS. What is the mathematical comparison to people with AIDS in America?
Example 2:
Have different groups choose a way to represent a solution in different ways. For example, group 1 would graph, group 2 would draw a diagram,
etc.
Example 3
From an article about a political, social, environmental issue or event, have students ask mathematical questions that arise from the material.
Example 4
Allow students to discover the advantages/disadvantages of describing a procedure using flow charts or step/step instructions.
Example 5
Develop a timeline of important mathematical developments.
46
APPENDIX A
Examples of procedural, conceptual, and real-world problems
PROCEDURAL PROBLEMS
CONCEPTUAL PROBLEMS
REAL-WORLD PROBLEMS
Example Problem 1:
Example Problem 1:
Example Problem 1:
Given what you know about equality and
equations, find a way to solve 3x - 6 = 14
(Your students have discussed the notion of
equality but have not yet worked with
equations other than one step equations.)
Ask your students to discover the relationship
between two tangents and a circle using tools
of measurement.
Sue Flay opened a McDonalds on White
Plains Road and Cassa Role opened a Burger
King across the street. Both had to borrow
money to open their fast food franchises.
After 500 customers, Sue was still $4000 in
debt. By the time she had served 3000
customers, she was ahead $1000.
A. Write the equation that expresses income
in terms of the number of customers
served.
After 2000 customers, Cassa Role still owed
$6000 to the bank. However, after 4500
customers, she was ahead by $1500.
A. Write the equation that expresses income
in terms of the number of customers
served.
B. At this point, which restaurant would you
rather own? Why?
C. Plot both equations on the same set of
axes.
D. How many customers would Sue Flay
have to serve to break even? (Income
cancels the debt.).
47
E. Repeat E for Cassa Role.
F. At what point do the two lines intersect?
What does that point mean in the real
world of the two restaurants and their
owners?
G. How much did Sue Flay have to borrow
from the bank to open the McDonalds?
H. How much did Cassa Role have to borrow
to open Burger King?
I. If each of the restaurants serves 50,000
customers, which would you rather own?
Explain. Does this go along with your
thinking in Question C?
PROCEDURAL
Example Problem 2:
CONCEPTUSL
Example Problem 2:
Using your calculator, find the results of the
following
20
100
50
2-1
10-1
5-1
2-2
10-2
5-2
Jose and Tanisha were arguing about a
problem posed by the teacher. They were
asked to talk about the difference between x +
x and x•x.
Jose said that
x +x = 2x2 and x•x = x2
Tanisha said that
x +x = 2x and x•x = x
Write down all your observations. What
conjectures can you make? How can you prove
your conjectures are true?
Who is correct and why? Justify your
reasoning.
48
Appendix B
Brief Explanations of Socratic Seminars and Cooperative Learning
Socratic Seminar
The Socratic method of teaching is based on Socrates' theory that it is more important to enable students to think for themselves than
to transmit "right" answers. Therefore, he regularly engaged his pupils in dialogues by responding to their questions with questions,
rather than answers. This process encourages divergent thinking.
Students are given opportunities to examine a text, in any form. After "reading" the common text open-ended questions are posed.
Open-ended questions allow students to think critically, analyze multiple meanings in text, and express ideas with clarity and
confidence19.
Mathematic Socratic Seminar
 Begin with a text. For the mathematics class, the text can be a rich problem, a graph, table or diagram.

19
Students Should:
- Question the “text.” An example of an opening question is “What are we trying to solve?”
- Discuss what the “text” is really asking or describing
- Brainstorm strategies for solving the problem or analyze diagrams, graphs, or tables given in the problem.
- Discuss applications of the concept
http://www.studyguide.org/socratic_seminar.htm, retrieved December 23, 2008
49
Cooperative Learning
In cooperative groups students work in groups or pairs to solve a problem. They share the result of their work with the class in a
formal or informal presentation. Other groups ask the presenting group at least one question and/or provide feedback.
Teachers must be sure that each member of the group plays a specific role. Examples of possible roles are:
 Recorder
 Reporter
 Responder
50
Appendix C
Mini Project: Creating a Quadratic Situation
You will begin this project with a partner. Your task is to create a situation that you can model with a quadratic function. You
will use that function to answer questions about your situation that you develop. There are two ways you can approach this. You can
use situations similar to ones we’ve studied. For these, you must alter the story and the data, coming up with a new equation. Or you
can select a situation (realistic or not) that interests you and that you want to develop into a quadratic situation.
You can look at situations that represent parabolas in the physical world, such as the Gateway Arch problem and the crumpled
paper toss. You might create a roller coaster, a skateboard half pipe (which you can make parabolic), a parabolic ski slope, any kind
of projectile motion (throwing any object which is acted upon by gravity), water fountains, etc.
Or you can look at relations between variables that are not physical, such as the pizza problem or the bathtub problem, where
you are looking at water volume and time, where the rate of change is quadratic. Other examples might include the relationship of the
area to the perimeter of a rectangle. Another example that could be made quadratic is gain and loss of weight over time. Find
something that interests you and try to create a story that you will represent with a quadratic equation.
[Teacher: Let kids create a story around anything that interests them. However, you will need to meet with them to help them think
of variables to go with their story. Once the scenario is decided upon, you should encourage them to work backwards. They might
make an in and out table that shows a quadratic relationship starting from a key point that might be the vertex. They can use data from
this table for their problem and be assured that the equation will involve reasonable coefficients.]
Your written piece must include;
A. The story with its data that allows the reader to find the equation.
B. A set of at least five questions that requires the reader to understand the equation and/or its graph for the real world
implications.
C. A detailed explanation of the answer to each question, showing the mathematical process using mathematical terms that
you have learned. This explanation must be connected to the real world. (e.g., what the y-intercept, the vertex, x
intercepts, etc. represent)
51
Appendix D
Model Suspension Bridge Project
You are an engineer for the City of New York and charged with designing and building a scale model of a suspension bridge. You
will be given dimensions of a waterway that will be crossed by your bridge. Your task is:
1. Decide where to put the towers over the water. This will enable you to calculate the span.
2. Decide how high the roadway will be above the water.
3. Calculate how tall the towers must be above the roadway. (The height of the tower above the roadway must be 10% of the
length of the span.)
4. Before you can build your model, you must make an architectural design on a coordinate plane. Decide on a scale based
on the distance across the water so that your whole bridge will fit on the paper. The span between the towers is parabolic
(quadratic). The x-axis represents the roadway. The y axis runs through the right side of the left tower (the beginning
point of the quadratic curve.) Remember, there are two kinds of cables: the big suspension cable and the vertical
suspenders that hold up the road. To complete your design you must do the following:
a. In addition to the length of the span and the height of the towers above the roadway, you must decide how high you
want the center of the span (at its lowest point in the middle) to be above the roadway.
b. With this information, you should be able to find three sets of coordinate pairs which you will use to find the
equation of the parabolic span.
c. Decide the equal spacing you will use between your vertical suspenders.
d. Enter the necessary information in the graphing calculator to find your equation.
e. Scroll through the window to find enough points on your suspension cable so you can draw it neatly on the
coordinate plane.
f. Assume that the suspension cable that connects to the anchors on both ends of the bridge is linear.
g. Find two points on each of these parts of the suspension cable to find the two linear equations that describe them.
h. Finally, on the coordinate axes, draw your bridge with the towers, roadway, suspension cables, and vertical
suspension cables.
5. Build your model using the dimensions on your architectural drawing.
52
Appendix E
Vignette 2: Creating Definition (Geometry Groups)
You will be given ten geometric figures. In your group classify them in any way that you want. Why did you put them together in
these groups? You may put a figure into more than one group.
Chart your findings. Be sure to make clear your explanations as to your groupings.
Each group will present their findings and other groups will ask questions.
Through these classifications can we create a definition for each of the different groups?
53
54