PROBLEM SOLVING - Institute for Student Achievement · Web viewHow do the properties of real numbers apply to or not apply to imaginary numbers. Why? Example Project/Research: Sum

USING THE ISA MATHEMATICS RUBRIC EVERYDAY

Jonathan Katz, Ed.D, ISAJoe Walter, ISA

http://www.studentachievement.org/index.html

Regine Philippeaux-Pierre, Ed.D, NCREST

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We would like to acknowledge the following ISA coaches, teachers, and NCREST researchers for their help in preparing this document:

Marian Mogulescu, ISA CoachDr. Denise Levine, ISA Coach

Marylin Cano, Teacher, Queens High School of TeachingMatthew Sullivan, Teacher, Bushwick School for Social Justice

Maria Peralta, Teacher, Pablo Neruda AcademyPatrice Nichols, NCREST

Dr. Charles Tocci, NCREST

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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 CEOInstitute for Student Achievement

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Using the ISA Mathematics Rubric Everyday.............................................................................................................................................5

Rubrics.........................................................................................................................................................................................................6

Problem Solving..........................................................................................................................................................................................8Teaching Idea #1: Choose the appropriate problem/task..................................................................................................................10Teaching Idea #2: Use problems that can be solved with multiple strategies..................................................................................11Teaching Idea #2a: Selecting and applying an appropriate strategy to find a solution......................................................................13Teaching Idea #3: Value process in addition to the answer..............................................................................................................15Teaching Idea #4: Answer student questions in ways that foster understanding.............................................................................16Teaching Idea #5: Use Error as a tool of inquiry..............................................................................................................................17Teaching Idea #6: Have students create their own problems based on their experience with solving different problems..............18

Reasoning and Proof..................................................................................................................................................................................19Teaching Idea #1: Conjecturing........................................................................................................................................................22Teaching Idea #2: Encouraging the use of evidence and proof in daily problem solving................................................................23Teaching Idea #3: Increasing students’ metacognition.....................................................................................................................24

Communications........................................................................................................................................................................................26Teaching Idea 1a: Writing in mathematics gives students the opportunity to reflect on mathematical concepts and clarify their

ideas through the use of journals........................................................................................................................28Teaching 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...............................................29Teaching Idea 2: Oral communication............................................................................................................................................31

Connections...............................................................................................................................................................................................32Teaching Idea #1: There are common structures (e.g. patterns) that bind together multiple ideas of mathematics.........................35Teaching Idea #2: The history of mathematics helps students make sense of and appreciate mathematics....................................36Teaching Idea #3: Using contextual problems that are meaningful to students...............................................................................37

Representations..........................................................................................................................................................................................38Teaching Idea #1a: Help students see how the concrete is connected to the abstract, such as hoe the concrete leads to

generalizations....................................................................................................................................................40Teaching Idea #1b: Use concrete structures or examples of concrete structures to have students examine mathematical ideas......42

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Teaching Idea #1c: Use additional mathematical modeling and design projects to move students from the concrete to the abstract and develop mathematical understanding...........................................................................................................43

Teaching Idea #2: Moving from the arithmetic to the algebraic......................................................................................................44Teaching Idea #3: Using mathematical representations to investigate, analyze, interpret, explain, and justify..............................45

Appendix A................................................................................................................................................................................................47Appendix B................................................................................................................................................................................................49Appendix C................................................................................................................................................................................................51Appendix D................................................................................................................................................................................................52Appendix E................................................................................................................................................................................................53

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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.6

ISA Mathematics Rubric Student __________________________________ Scorer’s Initials ____________

1NOVICE

2APPRENTICE

3PRACTITIONER

4EXPERT SCORE

PRO

BLE

M S

OLV

ING

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.

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.

A correct strategy is chosen based on the mathematical situation in the task.

Evidence of applying prior knowledge to the problem-solving situation.

Begins to build new mathematical knowledge by extending prior knowledge.

All parts of the solution may be correct.

An efficient strategy is chosen and progress toward a solution is evaluated.

Adjustments in strategy, if necessary, are made along the way, and/or alternative strategies are considered.

Evidence of analyzing the situation in mathematical terms. Constructs new mathematical knowledge by extending prior knowledge is present.

All parts of the solution must be correct.

REA

SON

ING

AN

D P

RO

OF

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.

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.

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.

Evidence is used to justify and support decisions made and conclusions reached. This leads to:

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.

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CO

MM

UN

ICA

TIO

NResponses are not clear, thorough, or detailed.

Little or no communication of ideas is evident.

Responses are somewhat clear, sequenced, thorough, and detailed manner.

Some communication of an approach is evident through verbal/written accounts and explanations, use of diagrams or objects, writing, and using mathematical symbols.

Student uses mostly familiar, everyday language with some formal mathematical language interspersed.

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.

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.

CO

NN

ECTI

ON

S

No attempt to connect mathematical ideas.

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 does at least one of the following: 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.

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.

REP

RES

ENTA

TIO

N No attempt is made to construct accurate mathematical representations.

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.

Generalized and accurate mathematical representations are constructed to analyze relationships, extend thinking, and/or interpret thought processes.

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DIMENSION IPROBLEM SOLVING

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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/task2. Use problems that can be solved with multiple strategies

a. Selecting and applying an appropriate strategy to find a solution3. Value process in addition to the answer4. Answer student questions in ways that foster understanding5. Use Error as a tool of inquiry6. 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. 10

Problem SolvingTEACHING IDEA #1: Choosing the appropriate problem

Teaching Idea Questions/Prompts Suggestions

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 students2. The task is mathematically rich: new

concepts and/or procedures are embedded within the problem/tasks.

3. Students will find the task interesting4. The task asks students to construct new

knowledge.

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?

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.

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Problem SolvingTEACHING IDEA #2: Use Problems with multiple strategies


Use Problems that can be solved with multiple strategies

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.

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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?

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Problem SolvingTEACHING IDEA #2a: Selecting an appropriate strategy

Teaching Idea Teaching Questions/Prompts Suggestions

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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 problemb. Pattern recognitionc. Working backwardsd. 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.

To support students in selecting and applying appropriate strategies, have students:

- Restate a problem for understanding.

- Connect the problem to prior knowledge, ideas in math, and skills from prior lessons, subjects, or disciplines.

- Make observations.

- Decide on a strategy.

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?

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

3 Please see problems in Appendix A.15

Assess/test your strategy. Re-evaluate 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

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Problem SolvingTEACHING IDEA #3: Value process and answer


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.

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.

For students:What are you doing?

Why are you doing it? Why did you do it?

Why does this approach make sense to you?

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?

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Problem SolvingTEACHING IDEA #4: Answer student questions to foster understanding


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.

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?

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?

EXAMPLE PROBLEM:

a) How many squares are on an 8 by 8 checkerboard?

b) How many rectangles are on an 8 by 8 checkerboard?

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Problem SolvingTEACHING IDEA #5: Error as a tool for inquiry


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.

For Student:Does anyone disagree/agree with the process and solution?

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?

Let students wonder about their work; do not rush to correct mistakes. Self-correction is more valuable than teacher correction.

When you question students, the purpose should be to have students reflect on their process and thinking.

EXAMPLE PROBLEM:

John solved the equation in this way- 3x = 5x – 4 -3x -3x____________________

2x = -4 x = -2

Do you agree or disagree with John’s method? Why or why not?

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Problem SolvingTEACHING IDEA #6: Student create their own problems


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.

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)

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.20

DIMENSION IIREASONING AND PROOF

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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, 20096 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.

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http://www.merriam-webster.com/dictionary/proof

Teaching Ideas in Reasoning and Proof

1. Conjecturing 2. Encouraging the use of evidence and proof in daily problem solving3. Increasing students’ metacognition

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Reasoning and ProofTEACHING IDEA #1: Conjecturing


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.

One of the most useful ways to develop rules, procedures, and concepts is to give students opportunities to:

- Observe- Look for patterns- Make conjectures

For students:What did you observe or notice?

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?

Observations, pattern hunting, and conjecturing are all part of the inquiry process.

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.

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Reasoning and ProofTEACHING IDEA #2: Evidence and proof


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.

Effective proofs and justifications for conjecture/answer should include:

1. Clear articulation/demonstration of patterns

2. Clear description of reasoning3. Clear examples, possibly addressing

counter-examples4. 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)

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?

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

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Reasoning and ProofTEACHING IDEA #3: Metacognition


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. Am I on the right track?b. Does my thinking make sense?c. Where am I going?d. Let me try another strategye. Is this question connected to another

question I’ve seen before?

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?

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.

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:26

1. Have students try the problem on their own, see how they approach the problem and where they hit blocks2. Teacher intervention point: Determine at which points you will intervene to help facilitate discussion and reasoning3. Elicit strategies from students

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DIMENSION IIICOMMUNICATION

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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 journalsb. Through mathematical research and writing within problems and projects

2. Oral communication

7 For a more thorough explanation of communicating symbolically, please see REPRESENTATIONS in Section V.8 Hiebert, J. et al., (1997). Making sense. Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

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CommunicationsTEACHING IDEA #1a: Writing Communication -Writing in journals


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.

For Student:

Explain the [math concept] that was learned yesterday. Why/How is it important in solving [a particular open-ended 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

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.

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CommunicationsTEACHING IDEA #1b: Writing Communication -Writing in problems and projects


31

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.

Explore the mathematical phenomenon, theory, or concept

- Explain the phenomenon- Use a particular problem to

describe how else or where else the concept, theory or phenomenon may be used.

- Provide examples- Connect the phenomenon, theory

or concept to a concept we have learned. How is it similar? How is it different?

- What are your lingering questions about this concept?

Some suggestions for students writing within projects 9 :

Describe the problem to be solvedProvide an explanation of how you will approach the problemProvide a thorough explanation of how you arrived at a solution. Describe your problem solving process.

- What was your thinking process?

- What helped you come to a solution?

- What obstacles did you come across?

- 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

Check for spelling, grammar, punctuation, and mathematical mistakes.

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.

Resource: Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth: Heinemann.32

http://edisk.fandm.edu/annalisa.crannell/writing_in_math/guide.html

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?

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CommunicationsTEACHING IDEA #2: Oral Communication


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.

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 ____?

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

11 Please see Appendix B12 Please see Appendix B

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DIMENSION IVCONNECTIONS

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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 Standard36

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 mathematics3. Using contextual problems that are meaningful to students

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ConnectionsTEACHING IDEA #1: There are common structures (e.g. patterns) that bind together the multiple ideas of mathematics


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.

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?

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.

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ConnectionsTEACHING IDEA #2: The history of mathematics help students make sense of and appreciate mathematics


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.

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.

To teacher:

How did numbers come to be?

How did the development of man’s mind lead to the development of the Hindu-Arabic Number System?

How can I use the story of pi to fascinate students about this amazing number?

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

14 A full unit on the history of numbers exists for those who are interested. Please see your ISA math coach.39

ConnectionsTEACHING IDEA # 3: Using contextual problems that are meaningful to students


We have often been told that we have to make mathematics relevant to our students’ lives. Yet, what does that really mean? We can find in many textbooks “real world” problems that have no meaning for students and are not “real” to them. What matters most is that we create tasks that are contextually interesting for students even if they are not part of students’ everyday experiences (Please see Nicol & Crespo, 200515).

For Teachers:

Will the task/activity be interesting to my students? Why?

Does the task have multiple entry points?

How will this task be meaningful to students?

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)

15 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.

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DIMENSION VREPRESENTATION

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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 abstracta. 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 ideasc. 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 Algebraic3. Using mathematical representations to investigate, analyze, interpret, explain, and justify.

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RepresentationTEACHING IDEA #1a: Moving from the concrete to the abstract

Teaching Idea Questions/Prompts SuggestionsTo support algebraic understanding, one main goal is to move students to see how the concrete is connected and leads to the abstract or generalizations

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.

For students17:Describe the concrete situation

Give other examples of this situation

What can you generalize about the concrete situation?

How does it connect to the abstract situation?

.See tasks below

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

Example Problem 2

17 Teachers, please insert the proper wording according to the problem you are doing.43

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 # Chickens20 3010 4012 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 # Chickens Total # legs20 30 14010 40 12012 38 124

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.

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RepresentationTEACHING IDEA #1b: Use examples of physical structures


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.

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)

How can I use the concreteness of physical structures to help students understand abstract ideas?

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)

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.

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RepresentationTEACHING IDEA #1c: Projects -Modeling


Use modeling to help students see the connections between the concrete and abstract and develop mathematical understanding.

For Teachers:What understandings will students gain through engagement in a modeling project?

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?

Make certain that the projects are mathematically rich and challenging. They should require students to abstract by developing 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.46

RepresentationTEACHING IDEA #2: Connecting the Arithmetic to the Algebraic

Teaching Idea Teaching Questions/Prompts SuggestionsIt also important that students develop the ability to go from the arithmetic to the algebraic.

.

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.

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RepresentationTEACHING IDEA #3: Investigate, analyze, interpret, explain, and justify


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.

How many ways can this [solution, situation, etc…] be represented?

What tool would best represent this idea?

What does this symbol tell us?

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?

As much as possible each concept/lesson/unit/ should have a written, oral and visual component.

Walls should be covered with examples of the variety of student work demonstrating representations

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,

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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.

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APPENDIX A

Examples of procedural, conceptual, and real-world problems

PROCEDURAL PROBLEMS CONCEPTUAL PROBLEMS REAL-WORLD PROBLEMS

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.)

Example Problem 1:

Ask your students to discover the relationship between two tangents and a circle using tools of measurement.

Example Problem 1:

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

50

cancels the debt.).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?

PROCEDURALExample Problem 2:

Using your calculator, find the results of the following20 100 50

2-1 10-1 5-1

2-2 10-2 5-2

Write down all your observations. What conjectures can you make? How can you prove your conjectures are true?

CONCEPTUSLExample Problem 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

Who is correct and why? Justify your reasoning.

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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.

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

19 http://www.studyguide.org/socratic_seminar.htm, retrieved December 23, 200852

http://www.studyguide.org/socratic_seminar.htm

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

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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)

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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.

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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?

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Documents

PROBLEM SOLVING - Institute for Student Achievement · Web viewHow do the properties of real numbers apply to or not apply to imaginary numbers. Why? Example Project/Research: Sum