Transcript

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Presented by       Tr. Harvey F. Silver Ed.D. Silver Strong & Associates    

Math Tools: Research-Based Practices for Differentiating Instruction and Raising Achievement in Mathematics

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Mathematical Styles and Strategies for Differentiating Instruction and Increasing Student Engagement

Our thoughtful questions…

• Why do some students succeed in mathematics while others do not? Is it a matter of skill or will?

• How can we use research-based teaching tools and strategies to address

the styles of all learners so they succeed in mathematics?

Our workshop is based on the following assumptions… • What teachers do and the instructional decisions they make have a

significant impact on what students learn and how they learn to think.

• Different students approach mathematics using different learning styles and need different things from their teachers to achieve in mathematics.

• Style-based mathematics instruction is more than a way to invite a greater

number of students into the teaching and learning process; it is, plain and simple, good math—balanced, rigorous, and diverse.

In this workshop, you will learn:

• The characteristics of the four basic mathematical learning styles (Mastery, Understanding, Self-Expressive, and Interpersonal).

• How to use a variety of mathematical teaching tools to differentiate instruction and increase student engagement.

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MEET YOUR NEIGHBOR BY THE NUMBERS

Numbers play an important role in our life experiences, from a person’s age to important dates, to birth order, to college GPA, and so on.

1. Select five numbers that are meaningful to you and that will help someone learn a little bit more about you.

2. Write a sentence or question for each number, leaving a blank line where the

number should go (e.g. The number of people in my family is __). Share your numbers with a neighbor. See if your neighbor can match the right number to your sentence.

Sentence Answer

1.

2.

3.

4.

5.

3. Meet with two other pairs (to form a group of six) and write each of your

numbers on a sticky note. Place all of your numbers on your table and see how many groups you can make that share a common characteristic (e.g. 2, 12, 32—numbers that have “2” in the ones column).

4. Visit another table and try to guess the reason for their groupings.

5. Return to your table and discuss how you might use some of the parts of this activity with your students.

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WHO AM I AS A LEARNER OF MATHEMATICS?

The three things I remember most from learning mathematics are… 1. 2. 3.

I learn mathematics best when I can:

Mathematics is… Because…

Which of these terms best describe you as a learner of mathematics? Circle all that apply.

variable cubical spherical

irrational equilateral congruent

infinite finite rational

point acute factorial

parabolic minimum constant

exponential divisible square

maximum obtuse transformation

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MATHEMATICAL ANTICIPATION GUIDE

Complete the first (Before Workshop) and last columns (After Workshop) by placing an “A” for “Agree” or a “D” for “Disagree” in each box. Share your thoughts with your group and give one or two reasons to support your position. Before After Workshop Workshop

1. Most mathematics teachers use a variety of teaching tools and strategies to teach mathematics.

2. Differences in mathematical teaching styles account for 65% of the reason that students are not successful in mathematics.

3. There is an inverse correlation between writing in mathematics and mathematical achievement.

4. Success in mathematics has more to do with feeling than with thinking.

5. Cooperative learning is a highly effective strategy for learning mathematics.

6. Proficiency in mathematical procedures is more important than understanding mathematical concepts (procedure vs. content).

7. In the United States, teachers of mathematics cover more content in a year than their counterparts in other countries whose students score higher on international tests of mathematical achievement.

8. Access to high-level concepts in mathematics is an important equity issue.

9. Mathematical achievement has little to do with career success.

10. Divergent thinking and creativity are more important to learning in the humanities than they are to learning in mathematics.

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WHAT IS MATHEMATICAL LITERACY?

Range Finder Note: A full description of this tool can be found on pages 208‐211 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement. 

Examine the three sets of mathematical problems A, B, and C below. Complete the one set that you feel most comfortable solving in five minutes.

Set A 1) 28 + 32 + 51 2) 3 x 37 3) 225 – 114 Set B 1) 4(20) + 31 2) 3(52)+ 62 3) 10(26 – 15) + 1 Set C 1) Solve for x: 3x – 133 = 200 2) Find the LCM of 3 and 37 3) Evaluate: 2a + 4b + c when a = 50, b = 5, and c = –9.

Reflection Reflect upon the level you chose and answer the questions below. Then meet with a partner and share your responses. Why did you select the set you did? What makes Set B more challenging than Set A? What makes Set C more challenging than Set B? How well do you believe your mathematics education has provided you with the skills you need in life?

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Yes, But Why?

Note: A full description of this tool can be found on pages 114‐116 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement. 

Work with a partner. Write down the last two digits of the year you were born. Add that number to the age you will be on your birthday this calendar year. Compare your answer with your partner’s. What did you discover? Work with your partner to develop an explanation for why you both had the same answer. “Neat Trick”

My Numbers My Partner’s Numbers Last two digits of

the year I was born

Age for this calendar year

Total

Yes, the answers are the same but why? 3-D Approach

Note: A full description of this tool can be found on pages 127‐129 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement. 

Use a 3-D approach to explain your reasoning for the “Neat Trick” above. How would you represent your answer algebraically, graphically, numerically, with a diagram, or through writing?

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Create Your Own How could you modify this “Neat Trick” to make it more remarkable? Use your understanding of this “Neat Trick” to create your own mathematical trick. Explain the steps in your new trick below.

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What Is Mathematical Literacy?

Mastery of procedural and

conceptual knowledge.

_______________________

A language to communicate ideas and

solve real-world problems.

_______________________

Understanding of logical reasoning to

explain and prove a solution.

_______________________

Application of strategies to formulate

and solve problems.

_______________________

What percentage of your classroom practice in mathematics would you estimate you spend in each of these areas? (Write your percentage on the line in each box above) How does your classroom practice compare with the NAEP data?

NAEP data shows that proficiency in these four areas has developed unevenly. In many classrooms, students are able to mimic rules and procedures demonstrated by their teacher: however, students often acquire these skills with little depth of understanding or the ability to use them to solve complex problems (Kowley & Wearing 2000).

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WHAT KIND OF PROBLEM SOLVER ARE YOU? Math is all about problem solving. But not all students and not all mathematicians solve problems in the same way. In fact, even though your textbook might tell you otherwise, there are many different ways to solve math problems. Your own preferences as a problem solver can tell you a lot about how your mind works and how you learn best. So, how do you go about solving problems in mathematic? Let’s conduct a little experiment to find out. Read “The Canoe Problem” below. When you feel ready, use the workspace to solve “The Canoe Problem.” But here’s the twist: As you are getting ready to solve the problem and as you are doing the work of problem solving, try to look and listen in on your own mind. What is it doing? What is it saying? How is it attempting to solve the problem? The Canoe Problem Nineteen campers are hiking through a state park when they come to a river. The river is moving too rapidly for the campers to swim across. The campers have one canoe, which fits three people. On each trip across the river, one of the three canoe riders must be an adult. There is only one adult among the nineteen campers. How many trips across the river will be needed to get all of the children to the other side of the river?

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How Four Different Students Responded to “The Canoe Problem” Maria Well, the first thing I did was gather up the facts quickly: 19 campers, 1 canoe, 3 people per canoe, etc. Then — don’t think this is crazy — I used a piece of paper to stand for the boat, with one red pen on it to stand for the adult and two blue pens on it to stand for the children.  Using actual objects to simulate the problem really helps me — it makes it easier to grasp the problem. To solve the problem I moved step‐by‐step from beginning to end.  First, I took the facts I gathered up and set them up carefully on paper. Then, I used basic math to get my answer of 17 trips across the river. Finally, I double checked my calculations to make sure I had done my math correctly. 

Giovanni I was very happy when the teacher said we could work with a learning partner. For me, the best way to learn math and solve problems like this one is to talk. I really like it when the teacher comes around and asks me how I’m doing, and I also like when I can work with friends and share my ideas. The best ideas seem to come when people are talking or working together. Anyway, what I really liked about today’s learning partnership with Jody is that we didn’t just get the answer to the problem right and wait around. We also talked about how we solved the problem and what we might do next time to improve as problem solvers. 

Tanisha I find that problems like this one often have hidden questions or little tricks in them that aren’t always so obvious. For example, I bet some people missed the fact that every time 2 children get across the river, that’s 2 trips across — one there and one back. By looking for the hidden question, I saw the pattern to the problem pretty quickly: 2out of 18 children get to the other side for every 2 trips across the river. That means it will take 18 trips to get all 18 children across. But here’s another little trick: On the last trip, they only need to go one way and not back again. So the answer’s actually 17. Anyway, once I figured out the answer, I checked to make sure it made logical sense and that it answered the question posed by the problem. In both cases, it did. 

Al I need to see the problem in my head. I closed my eyes and actually pictured the river and saw the 18 kids and the 1 adult with that 1canoe. Then, I generated possible answers by sort of playing with the numbers, trying different things out. When I do a problem like this, I try out different ways to solve it. Sometimes, I come up with more than one solution. For this problem, I came up with 9 and 17 as possible answers, so I explored each one to see which one worked. That’s how I came up with 17. Sometimes, I like to imagine cool twists or variations that would make the problem more interesting. For example, what if the boat held only a certain amount of weight and all the campers’ weights were given?  Then we would have to find the best way to load the canoe on each trip. 

What we do as problem solvers is closely related to the way we learn. Everyone learns, but we don’t all learn in the same way. The differences in how people learn are called learning styles. You can see your style in the way you talk, the way you think, and the way you solve problems. Some students, like Maria, solve math problems using step-by-step procedures. Others are like Tanisha. These students prefer to find patterns and discover hidden questions. Students like Al are drawn to problems that are unique and love to speculate on the possible solutions. For students like Giovanni, there’s no better way to solve a challenging math problem than by discussing it with friends and fellow students. Which of these students sounds most like you?

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Observations

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Four Styles of Mathematical Learners

Working with a group of four, each takes one of the four learners to analyze. Read their responses to the “canoe problem.” Then answer the following questions:

1. What 3 adjectives would you use to describe the learner?

2. What kind(s) of math problem does the student like?

3. How does the student learn best?

4. What problems or challenges may this student have in becoming a better math student?

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Which of the following best represents you as a learner of mathematics? Explain your choice.

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The T.E.M.P.O. of Style

Mastery (S + T)

Thinking Goal Environment Motivation Process Outcome

Interpersonal (S + F) Thinking Goal Environment Motivation Process Outcome

Understanding (N + T) Thinking Goal Environment Motivation Process Outcome

Self-Expressive (N + F) Thinking Goal Environment Motivation Process Outcome

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TEACHING TO, WITH, AND ABOUT STYLE

Many of the students we are consigning to the dust heaps of our classrooms have the abilities to succeed. It is we, not they, who are failing. We are failing to recognize the variety of thinking and learning styles they bring to the classroom, and teaching them in ways that don’t fit them well.

Robert J. Sternberg

Teaching to is: An individualized approach to suit a particular student’s style who may not be performing to the best of his/her abilities because of a lack of motivation or insufficient mastery of specific knowledge or skills.

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Complete and return to your presenter

PIMSER2011 Keynote 

Name: Position/Title:

Organization:

Address:

Work Phone Preferred e-mail (please print clearly):

Three ideas from our work today:

One thing I would tell a friend about this workshop:

Before today I thought: Now I think:

My conference learning experience is best described as a function that is:

Linear

Polynomial (3rd)

Sinusoidal

Exponential Explain your choice

Complete and return to your presenter

PIMSER2011 Keynote 

Math Tools Breakout

1

Math ToolsMath ToolsGRADES 3‐12

64WAYS TO DIFFERENTIATE INSTRUCTION AND INCREASE STUDENT ENGAGEMENT

Presented by Tr. Harvey F. Silver, EdD

INTRODUCTION

Our Thoughtful Questions…

• Why do some students succeed in mathematics while others do not? Is it a matter of skill or will?

H h b d hi l d

INTRODUCTION

• How can we use research‐based teaching tools and strategies to address the styles of all learners so they succeed in mathematics?

Math Tools Breakout

2

This workshop is based on the following assumptions…

• What teachers do and the instructional decisions they make have a significant impact on what students learn and how they learn to think.

INTRODUCTION

• Different students approach mathematics using different learning styles and need different things from their teachers to achieve in mathematics.

• Style‐based mathematics instruction is more than a way to invite a greater number of students into the teaching and learning process; it is, plain and simple, good math—balanced, rigorous, and diverse.

In this workshop you will learn…

• The characteristics of the four basic mathematical learning styles (Mastery, Understanding, Self‐Expressive, and Interpersonal).

INTRODUCTION

• How to use a variety of mathematical teaching tools to differentiate instruction and increase student engagement.

Meet Your Neighbor by the Numbers1. Select five numbers that are meaningful to you and that will help 

someone learn a little bit more about you. 

2. Write a sentence or question for each number, leaving a blank line where the number should go (e.g., The number of people in my family is __). Share your numbers with a neighbor.  See if your neighbor can match the correct number to each of your lines.

INTRODUCTION

g y

3. Meet with two other pairs (to form a group of six) and write each of your numbers on a sticky note. Place all of your numbers on your table and see how many groups you can make that share a common characteristic (e.g., 2, 12, 32—numbers that have “2” in the ones column).

4. Visit another table and try to guess the reason for their groupings.

5. Return to your table and discuss how you might use this activity with your students.

Math Tools Breakout

3

WHO AM I AS A LEARNER OF MATHEMATICS?

The three things I remember most from learning mathematics are…1.

2.

3.

I learn mathematics best when I can…

WHO AM I AS A LEARNER OF MATHEMATICS?

Mathematics is…

Because…

Which of these terms best describe you as a learner of mathematics? Circle all that apply. variable cubical spherical

irrational equilateral congruent

infinite finite rational

Point acute factorial

parabolic minimum constant

exponential divisible square

maximum obtuse transformation

1. Most mathematics teachers use a variety of teaching tools and strategies to teach mathematics. 

2. Differences in mathematical teaching styles account for 65% of the reason that students are not successful in mathematics.

3. There is an inverse correlation between writing in mathematics and mathematical achievement.

4. Success in mathematics has more to do with feeling than with thinking.

5 C ti l i i hi hl ff ti t t f l i

ANTICIPATION GUIDE

5. Cooperative learning is a highly effective strategy for learning mathematics.

6. Proficiency in mathematical procedures is more important than understanding mathematical concepts (procedure vs. content).

7. In the United States, teachers of mathematics cover more content in a year than their counterparts in other countries whose students score higher on international tests of mathematical achievement.

8. Access to high‐level concepts in mathematics is an important equity issue.

9. Mathematical achievement has little to do with career success.10. Divergent thinking and creativity are more important to learning in 

the humanities than they are to learning in mathematics.

Math Tools Breakout

4

Examine the three sets of mathematical problems A, B, and C below.  Complete the one set that you feel most comfortable solving in five minutes.

RANGE FINDER 

Set A 1) 28 + 32 + 51 2) 3 x 37 3) 225 – 114

Set B 1) 4(20) + 31 2) 3(52) + 62 3) 10(26 – 15) + 1

Set C 1) Solve for x: 3x – 133 = 200 2) Find the LCM of 3 and 373) Evaluate: 2a + 4b + c when a = 50, b = 5, and c = –9.

Note: A full description of this tool can be found on pages 208‐211 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

ReflectionReflect upon the level you chose and answer the questions below.  Then meet with a partner and share your responses.

• Why did you select the set you did?

RANGE FINDER

• What makes Set B more challenging than Set A?          • What makes Set C more challenging than Set B?

• How well do you believe your mathematics education           has provided you with the skills you need in life?

A “Neat Trick”

• Work with a partner.  Write down the last two digits of the year you were born.  

• Add that number to the age you will be on your birthday this calendar year

YES, BUT WHY?

birthday this calendar year.  

• Compare your answer with your partner’s.  

• What did you discover?  Work with your partner to develop an explanation for why you both had the same answer.

Note: A full description of this tool can be found on pages 114‐116 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

Math Tools Breakout

5

Use a 3‐D approach to explain your reasoning for the “Neat Trick.” 

• How would you represent your answer

3‐D APPROACH

How would you represent your answer algebraically, graphically, numerically, with a diagram, or through writing?

Note: A full description of this tool can be found on pages 127‐129 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

How could you modify the “Neat Trick” to make it more remarkable?  Use your understanding of this “Neat Trick” to create your own mathematical trick.  Make sure you clearly explain the steps in your new trick.

CREATE YOUR OWN

Note: A full description of this tool can be found on pages 162‐164 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

MATHEMATICAL LITERACY

Math Tools Breakout

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WHAT IS MATHEMATICAL LITERACY

Literacy in reading _______not only being able to pronounce and decode _______, but also being able to read and ______________ what one reads.

means

words

comprehend

What is mathematical literacy?

______________

Mathematical literacy means the ________thing—having procedural and computational skills as well as _______________ understanding.

16

p

same 

conceptual

MATHEMATICAL LITERACY

The importance of mathematical literacy and the need to understand and be able to use mathematics in everyday life and in the workplace have never been greater and willworkplace have never been greater and will continue to increase.

National Commission on Mathematics and Science for the 21st Century

17

Jobs requiring mathematical and technical skills are growing the fastest among the eight professional and related occupations.

Sixty percent of all new jobs beginning in the 21st

MATHEMATICAL LITERACY

Sixty percent of all new jobs beginning in the 21st century require skills that are possessed by only 20% of the current workforce.

18

Math Tools Breakout

7

Mastery of procedural and conceptual knowledge

A language to communicate and solve real‐world problems

What percentage of your classroom practice in mathematics would you estimate you spend in each of these areas? 

MATHEMATICAL LITERACY

Understanding of logical reasoning to explain and prove a 

solution

Application of strategies to formulate and solve problems

Mathematics Is…Real‐World Connections 

20

Mathematics Is…Information and Procedures

21

Math Tools Breakout

8

Mathematics Is…Reasoning and Problem Solving

22

Mathematics Is…Reasoning and Problem Solving

23

Mathematics Is…Creative Expression

24

Math Tools Breakout

9

WHAT KIND OF PROBLEM SOLVER ARE YOU?

The Canoe Problem 

Nineteen campers are hiking through a state park when they come to a river. The river is moving too rapidly for the campers to swim across. The campers have one canoe, which fits three people On each trip across the river one

WHAT KIND OF PROBLEM SOLVER ARE YOU?

which fits three people. On each trip across the river, one of the three canoe riders must be an adult. There is only one adult among the nineteen campers. How many trips across the river will be needed to get all of the children to the other side of the river?

Working with your team, have each member analyze one of the four learners. Review your learner’s response to the “canoe problem.”  

Then answer the following questions:• What three adjectives would you use to describe the learner?

WHAT KIND OF PROBLEM SOLVER ARE YOU?

• What kind(s) of mathematics problems does the student like to solve?

• How does the student learn best?

• What problems or challenges may this student face in becoming a better mathematics student?

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How Maria Responded to the Canoe Problem

Well, the first thing I did was gather up the facts quickly: nineteen campers, one canoe, three people per canoe, etc. Then—don’t think this is crazy—I  used a piece of paper to stand for the boat, with one red pen on it to stand for the adult and two blue pens on it to stand for the children. Using 

l bj i l h bl ll h l i

WHAT KIND OF PROBLEM SOLVER ARE YOU?

actual objects to simulate the problem really helps me—it kind of takes the abstraction away and makes it easier to grasp the problem. To solve the problem I moved step‐by‐step from beginning to end. First, I took the facts I gathered up and set them up carefully on paper. Then, I used basic math to get my answer of 17 trips across the river. Finally, I double checked my calculations to make sure I had done my math correctly.

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I find that problems like this one often have hidden questions or little tricks in them that aren’t always so obvious. For example, I bet some people missed the fact that every time 2 children get across the river, that’s two trips across—one there and one back. By looking for the hidden question, I saw the 

WHAT KIND OF PROBLEM SOLVER ARE YOU?

How Tanisha Responded to the Canoe Problem

y g q ,pattern to the problem pretty quickly: 2 out of 18 children get to the other side for every 2 trips across the river. That means it will take 18 trips to get all 18 children across. But—here’s another little trick—on the last trip they only need to go one way and not back again. So the answer’s actually 17. Anyway, once I figured out the answer, I checked to make sure it made logical sense and answered the question posed by the problem. In both cases, it did.

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I was very happy when the teacher said we could work with a learning partner. For me, the best way to learn math and solve problems like this one is to talk. I really like it when the teacher comes around and asks me how I’m doing and I also like when I can work with friends

WHAT KIND OF PROBLEM SOLVER ARE YOU?

How Giovanni Responded to the Canoe Problem

I m doing, and I also like when I can work with friends and share my ideas. The best ideas seem to come when people are talking or working together. Anyway, what I really liked about today’s learning partnership with Jody is that we didn’t just get the answer to the problem right and wait around.  We also talked about how we solved the problem and what we might do next time to improve as problem solvers.

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I need to see the problem in my head. I closed my eyes and actually pictured the river and saw the 18 kids and the one adult with that one canoe. Then, I generated possible answers by sort of playing with the numbers, trying different things out. When I do a problem like this, I try out different ways to solve it. Sometimes, I come up with more than 

WHAT KIND OF PROBLEM SOLVER ARE YOU?

How Al Responded to the Canoe Problem

one solution.

For this problem, I came up with 9 and 17 as possible answers, so I explored each one to see which one worked. That’s how I came up with 17. Sometimes I like to imagine cool twists or variations that would make the problem more interesting. For example, what if the boat held only 400 lbs. and all the campers’ weights were given? Then we would have to find the most efficient way to load the canoe on each trip.

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MATH LEARNING STYLES

MATH LEARNING STYLES

Which of the following objects best represents you as a learner of mathematics?  Explain your choice.

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How are all minds alike?

MATH LEARNING STYLES

Carl Jung Carl Jung asked…asked…

How are all minds different?

MATH LEARNING STYLES

All minds PERCEIVE and PROCESS information, 

Carl Jung answered…Carl Jung answered…

All minds P RC IV and PROC SS information,but differ in how they pay attention.

The Four Functions of Style

SENSINGPhysical

Facts

Details

Here & Now

PerspirationObjective

AnalyzeSubjective

Harmonize

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INTUITIONPatterns

Possibilities

Ideas

Past & Future

Inspiration

Logic

Truth

Procedures

Likes/Dislikes

Tact

People

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From Function to Style:

Sensing

Fe

S + T

Mastery

S + F

Interpersonal

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Intuition

ee

ling

Self-Expressive

N + F

Understanding

N + T

MATH LEARNING STYLES

• Thinking Goal:

• Environment:

REMEMBERING

CLARITY & CONSISTENCY

Mastery Learner (ST):

• Motivation:

• Process:

• Outcome:

SUCCESS

STEP‐BY‐STEP, EXERCISE, & PRACTICE

WHAT? CORRECT ANSWERS

MATH LEARNING STYLES

• Thinking Goal:

• Environment:

REASONING

CRITICAL THINKING & CHALLENGE

Understanding Learner (NT):

• Motivation:

• Process:

• Outcome:

CURIOSITY

DOUBT‐BY‐DOUBT, EXPLAIN, & PROVE

WHY? ARGUMENTS

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MATH LEARNING STYLES

• Thinking Goal:

• Environment:

REORGANIZING

COLORFUL & CHOICE

Self‐Expressive Learner (NF):

• Motivation:

• Process:

• Outcome:

ORIGINALITY

DREAM‐BY‐DREAM, EXPLORE POSSIBILITIES

WHAT IF? CREATIVE ALTERNATIVES

MATH LEARNING STYLES

• Thinking Goal:

• Environment:

RELATE PERSONALLY

COOPERATIVE & CONVERSATION

Interpersonal Learner (SF):

• Motivation:

• Process:

• Outcome:

RELATIONSHIPS

FRIEND‐BY‐FRIEND, EXPERIENCE,              & PERSONALIZE

SO WHAT? CURRENT & CONNECTED

MATH LEARNING STYLES

Percentages of Learning Style Preferences 

S+TMASTERY

S+FINTERPERSONAL

35% 35%

UNDERSTANDINGN+T

SELF‐EXPRESSIVEN+F

10% 20%

12%1% 22%

65%

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MATH LEARNING STYLES

Dr. Robert J. Sternberg, is Provost of Oklahoma State University, former professor of psychology at Yale University, and past President of the American Psychological Association.

Learning Style Research StudyFive different ways for teaching mathematics

• A memory‐based approach emphasizing identification and recall of y pp p gfacts and concepts

• An analytical approach emphasizing critical thinking, evaluation, and comparative analysis

• A creative approach emphasizing imagination and invention

• A practical approach emphasizing the application of concepts to real‐world contexts and situations

• A diverse approach that incorporates all the approaches

MATH LEARNING STYLES

Sternberg and his colleagues drew two conclusions.

First, whenever students were taught in a way that matched their own style preferences those students outperformed students who were mismatched. 

Second, students who were taught using a diversity of approaches outperformed all other students on both performance assessments and on multiple‐choice memory tests.

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MATH LEARNING STYLES

Many of the students we are consigning to the dust heaps of our classrooms have the abilities to succeed. It is we, not they, who are failing. We are failing to recognize the variety of 

thinking and learning styles they bring to the classroom, and teaching them in ways that don’t fit them well.

Dr. Robert J. Sternberg

Style and NCTM Standards(The Mathematics of Sensing‐Thinkers)

MASTERY STYLENCTM Focus: Mathematics as Computation

Abilities Needed:•To remember and store mathematical procedures•To select a procedure appropriate to a particular, well‐defined problem•To check for accuracy•To take notes and maintain organized records

(The Mathematics of Sensing‐Feelers)INTERPERSONAL STYLE

NCTM Focus: Mathematics as CommunicationAbilities Needed:•To reflect on and write about mathematical practice and feelings•To tolerate ambiguity, complexity, and lack of closure•To make choices•To work together in cooperative groups•To notice control and make use of emotional

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•To notice, control, and make use of emotional responses in mathematical work

(The Mathematics of Intuitive‐Thinkers)UNDERSTANDING  STYLE

NCTM Focus: Mathematics as ReasoningAbilities Needed:•To perceive mathematical patterns•To spot anomalies and discrepancies•To generalize•To discriminate•To use evidence•To use deduction as a tool of discovery

(The Mathematics of Intuitive‐Feelers)SELF‐EXPRESSIVE  STYLE

NCTM Focus: Mathematics as Connections and Problem Solving

Abilities Needed:•To identify similarities and differences among problems•To adjust and adapt procedures to new situations•To generate a wide variety of possible ideas•To use and form criteria to select among options•To connect mathematical ideas to a wide variety of mathematical and non‐mathematical situations

HOW DO I SELECT THE RIGHT TOOL FOR HOW DO I SELECT THE RIGHT TOOL FOR THE RIGHT LEARNING SITUATION?

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Vital Statistic 1: Title and Flash Summary

HOW DO I SELECT THE RIGHT TOOL FOR THE RIGHT LEARNING SITUATION?

Knowledge Cards—Students create “flash cards” to visualize Knowledge Cards—Students create flash cards to visualize and remember complex terms and concepts.

Summary of Knowledge Cards

Vital Statistic 2: NCTM Process Standards

HOW DO I SELECT THE RIGHT TOOL FOR THE RIGHT LEARNING SITUATION?

NCTM Process Standards for Knowledge Cards

Vital Statistic 3: Educational Research

HOW DO I SELECT THE RIGHT TOOL FOR THE RIGHT LEARNING SITUATION?

Educational Research Base for Knowledge Cards

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Vital Statistic 4: Instructional Objectives

HOW DO I SELECT THE RIGHT TOOL FOR THE RIGHT LEARNING SITUATION?

Instructional Objectives for Knowledge Cards

MATH TOOLS

1. Try one out2. Use tools to help you meet a particular 

standard or objective3. Individualize instruction4. Differentiate instruction for the entire class

FIVE WAYS TO USE MATH TOOLS

5. Design more powerful lessons, assessments, and units

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PurposeMental Math War provides a quick, engaging, and interactive way for students to practice working with variables, equations, and their solutions.

MENTAL MATH WAR

Note: A full description of this tool can be found on pages 47‐49 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

Steps1. Review the concept of variables and their function in an equation.2. Review the traditional War card game and explain the different 

rules and procedures for Mental Math War.3. Group students into pairs.4. Provide each pair of students with a deck of playing cards to be 

divided equally.

MENTAL MATH WAR

q y5. From each pair, assign one student to play cards for the x variable 

and one student to play cards for the y variable6. Remind students that cards must be kept face down until played, 

and that when played, the cards must be in full view of both students.

7. Start the game by writing or revealing an equation that includes the variables x, y, and z.

8. Pause the game and introduce a new equation (still using the variables x, y, and z) that will guide the next series of play.

MENTAL MATH WAR

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MENTAL MATH WAR

PurposeAlways‐Sometimes‐Never (ASN) is a reasoning activity that focuses students’ thinking around the important, and often subtle, facts and details associated with mathematical concepts.

ALWAYS-SOMETIMES-NEVER (ASN)

concepts.

Note: A full description of this tool can be found on pages 66‐69 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

Steps1. Provide students with a list of statements about a 

recently discussed or familiar mathematical concept or topic.

2 Allow students enough time to read and consider all of

ALWAYS-SOMETIMES-NEVER (ASN)

2. Allow students enough time to read and consider all of the statements carefully.

3. Have students think about each of the statements and decide whether each is always true, sometimes true, or never true.

4. Make sure that students explain the reasoning behind their choices.

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Arithmetic: Addition and Subtraction1. The sum of two 3‐digit numbers is a 3‐digit number. 2. The sum of two even numbers is an odd number. 3. The difference of two odd numbers is an even number. 4. The sum of additive inverses is zero. 

h ff f h b b

ALWAYS‐SOMETIMES‐NEVER (ASN)

(Sometimes)(Never)

(Always)(Always)

(Al )5. The difference of three odd numbers is an odd number. 6. The sum of three even numbers is zero. 7. The sum of three odd numbers is zero. 8. The sum of two counting numbers is greater than the difference         

of the same numbers. 

(Always)

(Always)

(Sometimes)

(Never)

Statistics: Mean, Median, Mode1. A list of numbers has a mean. 2. A list of numbers has a median. 3. A list of numbers has a mode. 4. The mean of a set of numbers is one of the numbers                                            

of that set. 

ALWAYS-SOMETIMES-NEVER (ASN)

(Sometimes)(Always)

(Always)

(Sometimes)5. The median of ten consecutive integers is one of                                         

those integers. 6. If the mode of a set of numbers is 14, then 14 is one of the 

numbers of that set.7. The mean of a set of numbers is greater than the median of that 

set of numbers. 8. The mode of a set of numbers, without repeated values, can be 

found by arranging the numbers in increasing order and then calculating the mean of the middle two numbers.  (Never)

(Always)

(Never)

(Sometimes)

Trigonometry: Graph Analysis1. The graph of a trigonometric function is periodic. 2. Doubling the amplitude of a trigonometric function doubles the period of 

the function. 3. The graph of a cosecant function has an infinite number of asymptotes.4. The period of y = sin bx + h is equivalent to the period                                                

of y = sec bx + k.                                                  5. A cosine function has both a maximum and a minimum value.

ALWAYS-SOMETIMES-NEVER (ASN)

(Always)

(Never)(Always)

(Always; both)(Always)5. A cosine function has both a maximum and a minimum value. 

6. For numbers a and b, the graph of y = cos x on the interval a < 0 < b is an increasing function. 

7. Stretching the graph of a trigonometric function changes the period of the function. 

8. Applying a phase shift on a secant graph changes the location of vertical asymptotes. 

9. For any value of a, the graph of y = tan ax will have the y‐axis as an asymptote. 

10. Secant graphs have horizontal asymptotes.

(Never)

(Sometimes; true for horizontal stretches, false for vertical stretches.)

(Sometimes)

(Never)

(Never)

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PurposeResearch shows that teaching students how to identify similarities and differences is the single most effective way to increase understanding and raise achievement levels (Marzano, Pickering, & Pollock, 2001). Setting two concepts against one another and using each as a frame of reference for examining the other allows students to see deeply into

COMPARE AND CONTRAST

reference for examining the other allows students to see deeply into the content they are studying and fuels new insights about mathematics. 

Note: A full description of this tool can be found on pages 98‐105 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

Steps1. Select two (or more) related concepts or mathematics 

problems.2. Specify criteria for comparison.3. Provide (or teach students how to create) graphic 

COMPARE AND CONTRAST

organizers for describing items and comparing them.4. Guide students through the four phases of comparison:

• Description• Comparison• Conclusion• Application

Problem 1Haylee sells kitchen equipment. As part of her salary, Haylee receives 12% commission on her sales. In February, Haylee sold $7,250 worth of kitchen equipment. How much commission should she 

COMPARE AND CONTRAST

q preceive?

Problem 2For March, Haylee forgot to check her total sales figures. She received a commission of $825 for the month. How much did Haylee sell?

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COMPARE AND CONTRAST

Description Phase

COMPARE AND CONTRAST

Comparison Phase

Conclusion Phase• Are these two problems more alike or

more different?

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COMPARE AND CONTRAST

• How did these differences affect the way you solved each problem?

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Application PhaseAfter the conclusion phase, in which students decide if the two problems are more alike or more different and discuss how the differences affect the problem-solving procedure, you might ask them to apply their new understanding with a task like the following.

To show what you’ve learned, create three new sales-commission problems. One problem should be like Problem 1 and the other should be like Problem 2 Then since we noticed that Problem 1 asks you to

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COMPARE AND CONTRAST

be like Problem 2. Then, since we noticed that Problem 1 asks you to solve for total commission and Problem 2 asks you to solve for total sales, create a new problem that asks you to figure out the rate (or percentage) of commission.

As we have already noted, Compare and Contrast works as well with mathematical concepts as it does with problem-solving processes. The tool is highly flexible and can be used at varying levels of depth. Students may simply be asked to generate as many similarities and differences as possible and quickly draw conclusions. Here are a few examples of this “down and dirty” approach to Compare and Contrast.

PurposeThree‐Way Tie gives students the opportunity to focus their attention on hidden mathematical relationships. Students identify the relationship between pairs of critical concepts or terms and then distill their

THREE-WAY TIE

critical concepts or terms and then distill their understanding of the relationship into a single sentence.

Note: A full description of this tool can be found on pages 108‐111 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

Steps1. Identify an important mathematical concept.2. Graphically “triangulate” the concept with three related 

terms or concepts. Alternatively, you can have the students generate the three terms themselves by selecting 

THREE-WAY TIE

the three most important ideas in a reading or unit.3. Along each side of the triangle, have students write a 

sentence that clearly relates two of the terms. 4. Have students use their three sentences to develop a brief 

summary of the concept.5. Allow students time to share and explain what they wrote 

on their organizers.

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THREE-WAY TIE

Purpose

While poetry seems out of place in most mathematics classrooms, 

cinquains are powerful learning tools that give students a creative way 

to distill their understanding of important ideas and concepts. The 

concise, five‐line format of a cinquain forces students to “think 

CINQUAINS

economically,” to cut away all of the non‐essential details and include 

only the most critical information.

Note: A full description of this tool can be found on pages 135‐136 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

Steps1. Introduce (or review) the concept of a cinquain and the format it 

follows.2. Model and provide an example of a completed cinquain. (Depending on 

your class’s comfort or familiarity with cinquains, modeling multiple examples may be helpful for students. Make sure models focus on the critical attributes of the concept.)

CINQUAINS

3. Select a word for students to focus on or allow students to pick their own words from a textbook, glossary, or Word Wall.

4. Allow students time to review what they know about the concepts for their cinquains.

5. Have students create a cinquain in class, either individually or within a small group. (Cinquains also make for excellent homework activities.)

6. Encourage students to share and discuss their cinquains, how they were made, and the mathematical concepts they used.

7. Display exemplary cinquains for other students to see and use as models.

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CINQUAINS

Cinquain Format and Successful Cinquain for Algebra

PurposeThe effort that students put into their work is greatly influenced by the three A’s of learning:

•Attention, or their ability to focus on the tasks at hand.•Attitude, or their ability to remain positive and persevere when learning becomes difficult

BRING YOUR “A” GAME

learning becomes difficult.•Aspiration, or their ability to set meaningful learning goals and strive toward excellence.

Bring Your “A” Game gives teachers and students of mathematics a trio of “brain boosters,” or tools that help students put forth their best efforts in the classroom. Each tool builds one of the three A’s of learning.

Note: A full description of this tool can be found on pages 170‐179 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

BRING YOUR “A” GAME

Attention MonitorSteps1. Ask students, “Have you ever stopped paying attention in class?” Kindle and 

extend the discussion by asking, “What happens when you lose your attention?”

2. Have students concentrate on a shaded black dot (4 inches in diameter) for 2 minutes. As students focus on the shape, have them take notes that describe when their attention slipped and how they regained their pp y gattention.

3. Introduce the Four C’s to students for quick ways to correct a lapse in attention. The Four C’s technique works like this:• Change Your Posture• Cut Away the Distractions• Compose or Create• Connect

4. To reinforce the Four C’s, ask students to create an icon for each C in the technique.

5. Provide students with an Attention Monitor so they can reflect upon the lesson and their patterns of attention. Attention Monitors can be created by the teacher or by the students.

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Attention Monitor

BRING YOUR “A” GAME

Teacher’s Attention Monitor

Student’s Attention Monitor Using Emoticons

BRING YOUR “A” GAME

Attitude Catcher

Steps1. Distribute an “All You Need Is Attitude” organizer to 

students. 2. Have students read the statements about attitude. Give2. Have students read the statements about attitude. Give 

students time to explain why they agree or disagree with each statement.

3. Discuss with students the impact that their attitudes can have on their success, both inside and outside of the mathematics classroom.

4. Provide students with an “Attitude Catcher” organizer. 

BRING YOUR “A” GAME

Attitude Catcher

Steps (continued)5. Give students time to reflect and complete the organizer by 

catching attitudes, rating the strengths of each attitude, describing causes, and considering ways to improve or sustain 

i dattitudes. 6. Explain to students that optimists do better in school than 

pessimists because optimists have “inner cheerleaders” who say things like, “This isn’t going to get me down” or “If I work at it I can get it.” Remind students that talking to themselves is productive and positive.

7. Have students review the “Ten Mathitude Adjusters.”8. Encourage students to repeat one or more of the “Mathitude 

Adjusters” to themselves whenever it is productive and positive.

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BRING YOUR “A” GAME

Student’s Completed “All You Need Is Attitude” Organizer

BRING YOUR “A” GAME

Student’s “Attitude Catcher”for Order of Operations

Aspiration Goal PlannerSteps1. Have students reflect on their aspirations by writing a short 

response to the following questions:What would you like to achieve in mathematics class this week? What will you do in order to achieve this goal?What do you plan on doing after you get out of school? What

BRING YOUR “A” GAME

What do you plan on doing after you get out of school? What mathematics do you need to accomplish your goals?What grade in mathematics do you aspire to get this year? How can I help you reach that goal?

2. Have students write out three things they did yesterday. Then have students choose one item from their list and identify the goal they were trying to achieve.

3. Have students use the Aspiration Goal Planner to help them plot a goal they want to accomplish in their mathematics class. 

4. Meet with students regularly to discuss goals, challenges, and learning progress.

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Aspiration Goal Planner

Mastery

What skills, procedures, and key terms do I want students to master?

Interpersonal

How will students make personal connections or discover the social 

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TASK ROTATION

relevance of mathematics?

Understanding

What core concepts, patterns, or principles do I want students to 

understand deeply?

Self‐Expressive

How will students explore,            visualize, experiment, or apply         

new concepts and skills?

Note: A full description of this tool can be found on pages 222‐238 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement.

Note: A full description of this tool can be found on pages 222‐238 of Math Tools, Grades 3–12: 64 Ways to Differentiate Instructionand Increase Student Engagement.

MASTERY

MATHEMATICAL SUMMARIESThe class uses the average weight data to create a bar graph showing the range of weights associated with their favorite animals.  The students then create a sentence that describes the data.

INTERPERSONAL

WHAT’S YOUR FAVORITE?Each student brings to class a picture of their favorite animal and its average weight.  The students then sit in a circle, identify the animal, share their data, and tell the group why this animal is their favorite. (The teacher records types of animals and facts on chart paper for later use.)

Task RotationWhat an Animal

p p )

UNDERSTANDING

COMPARE AND CONTRAST Students are asked to figure out which animals could make good pets and which should live in the wild.  Students should consider animal weight when making their decisions and record their evaluations in a chart.

SELF‐EXPRESSIVE

CREATE YOUR OWNStudents are to create and solve a mathematical problem using the animal information gathered, graphed, and charted.  Students might create problems that ask:  What’s the difference in weight between your favorite animal and you?  How many of the smallest favorite animal would it take to equal the weight of the heaviest favorite animal?

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MASTERY

MEMORY BOXThink about the different ways you have learned how to find the Greatest Common Factor (GCF) of two numbers.  Without using your notebook or textbook, make a Memory Box for how to find the GCF.

INTERPERSONAL

WHO’S RIGHT?What is the Greatest Common Factor of 72 and 540? Alex says that the GCF of 72 and 540 is 12.  She came up with this answer since the two numbers factor into 23•32 and 33•22•5, so for her the GCF is 3•22 or 12. 

Natasha thinks that the GCF of 72 and 540 is 6, but she didn’t write down any work. 

Who’s right?  

Task RotationGreatest Common Factor (GCF)

Greatest Common Factor

g

Explain any errors that either (or both) of the girls made.  What is the GCF of 72 and 540?

UNDERSTANDING

THREE‐WAY TIELook at the triangle below.  Write a sentence along each side of the triangle that connects the word or phrase at each angle of the triangle.

SELF‐EXPRESSIVE

M + M: MATH AND METAPHORSIs finding the Greatest Common Factor (GCF) of two numbers like building a house?

Like putting together a jigsaw puzzle?  

Or perhaps like panning for gold? Describe one similarity between finding the GCF and each of the actions above.1.2.3.

MASTERY

WHAT’S WRONG?Look at the student work to the right.                                             Find and correct any errors.                                                             What is the correct answer?

INTERPERSONAL

PAIRED LEARNINGWork with a partner to create a set of hints that could be used to help coach someone working on the following problem:

3 − 5(2x − 7)   = 5 − (4 − 2x)Each partner should solve the equation on his or her own before working together to generate a list of helpful hints.

Task RotationFirst Degree Equations

UNDERSTANDING

LEARNING FROM CLUESBelow you will find all of the work required to solve a linear equation. However, the work is in pieces. You must piece the clues together and write the work in the correct order.  What is the original equation? What is the correct answer?

SELF‐EXPRESSIVE

CREATE YOUR OWNKeep in mind that the solution is x = 4.  Create two different first degree equations that each fulfill the following characteristics:1. x =  4 is the only solution.2. Variables are on each side of the equals sign.3. Solution steps to the equation use the distributive  

property.4. The equation contains at least two sets of parentheses. 

Show all of the work and steps used to generate each equation.

MASTERY

KNOWLEDGE CARDSCreate Knowledge Cards for each term on the list below.  Write one term on the front of each card.  On the back of each card, include the value of the first or second derivative for the function for each condition to occur. 

• Inflection Point• Relative maximum •Relative minimum •Concave up 

INTERPERSONAL

COOPERATIVE STRUCTURESWork together in groups. Check over each person’s Knowledge Cards, graphs, and explanations of which types of graphs do not belong. All of the papers from each group will be collected; however, only one of each activity will be graded.  The activity to be graded will be selected at random, and any one student’s work will not be graded more than once. Homework credit will be awarded if and only if all three activities selected at random are completely 

Task RotationCalculus

p•Concave down• Linear functions

y p ycorrect.

UNDERSTANDING

WHICH ONE DOESN’T BELONG?Two of the following types of graphs do not fit with the others. Which ones do not belong? Explain your choices.

1. linear functions2. absolute value functions3. parabolic functions4. sine functions5. step functions6. exponential functions

SELF‐EXPRESSIVE

PICTURE = 1,000 WORDSDraw a graph based on the following information:A continuous function that is concave up, but decreasing from x = −10 to x = −5, where there is an inflection point. 

It is concave down between x = −5 and −3.

There is a relative minimum at x = −3.

A second inflection point occurs at x = 0, and the graph then decreases linearly from x = 1 until x = 5.

The graph is non‐differentiable at x = 5, and slowly increases exponentially thereafter.


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