Presented by Tr. Harvey F. Silver Ed.D. Silver Strong & Associates Math Tools: Research-Based Practices for Differentiating
Instruction and Raising Achievement in Mathematics
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 1 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.
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 2 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.
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 3 WHO AM I AS A LEARNER OF MATHEMATICS?
The three things I remember most from
learning mathematics are…
I learn mathematics best when I can:
1.
2.
3.
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
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 4 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
Workshop
After
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.
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 5 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?
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 6 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?
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 7 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.
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 8 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).
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 9 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?
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 10 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. 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. 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. 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?
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 11 Observations
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 12 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?
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 13 Which of the following best represents you as a learner of mathematics? Explain
your choice.
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 14 The T.E.M.P.O. of Style
Mastery (S + T)
Interpersonal (S + F)
Thinking Goal
Thinking Goal
Environment
Environment
Motivation
Motivation
Process
Process
Outcome
Outcome
Understanding (N + T)
Self-Expressive (N + F)
Thinking Goal
Thinking Goal
Environment
Environment
Motivation
Motivation
Process
Process
Outcome
Outcome
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 15 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.
Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 16 Copyright ©2011 Thoughtful Education Press. All Rights Reserved. This document cannot be reproduced without written permission. Page 17 Complete and return to your presenter
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:
My conference learning experience is
best described as a function that is:
Linear
Polynomial (3rd)
Now I think:
Sinusoidal
Exponential
Explain your choice
PIMSER2011 Keynote Complete and return to your presenter
PIMSER2011 Keynote Math Tools Breakout
Math Tools
GRADES 3‐12
64 WAYS TO DIFFERENTIATE INSTRUCTION AND INCREASE STUDENT ENGAGEMENT
Presented by Tr. Harvey F. Silver, EdD
INTRODUCTION
INTRODUCTION
Our Thoughtful Questions…
• Why do some students succeed in mathematics while others do not? Is it a matter of skill or will?
• H
How can we use research‐based teaching tools and hb d
hi
l
d
strategies to address the styles of all learners so they succeed in mathematics?
1
Math Tools Breakout
INTRODUCTION
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.
• 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.
INTRODUCTION
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.
INTRODUCTION
Meet Your Neighbor by the Numbers
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 g
y
neighbor can match the correct number to each of your lines.
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.
2
Math Tools Breakout
WHO AM I AS A LEARNER OF MATHEMATICS?
WHO AM I AS A LEARNER OF MATHEMATICS?
The three things I remember most from
learning mathematics are…
1.
I learn mathematics best when I can…
2.
3.
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
Point
acute
parabolic
exponential
maximum
minimum
divisible
obtuse
rational
factorial
constant
square
transformation
ANTICIPATION GUIDE
1.
Most mathematics teachers use a variety of teaching tools and strategies to teach mathematics. 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
5.
C
Cooperative learning is a highly effective strategy for learning ti l
i i hi hl ff ti
t t
f l
i
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.
2.
3
Math Tools Breakout
RANGE FINDER 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.
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.
RANGE FINDER
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?
YES, BUT WHY?
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
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.
4
Math Tools Breakout
3‐D APPROACH
Use a 3‐D approach to explain your reasoning for the “Neat Trick.” How would you represent your answer would you represent your answer
• How
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.
CREATE YOUR OWN
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.
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
5
Math Tools Breakout
WHAT IS MATHEMATICAL LITERACY
What is mathematical literacy?
Literacy in reading _______not only being able to means
pronounce and decode _______, but also being words
______________
able to read and ______________ what one reads.
p
comprehend
same Mathematical literacy means the ________thing—
having procedural and computational skills as well conceptual
as _______________ understanding.
16
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 will
continue to increase.
National Commission on Mathematics and Science for the 21st Century
17
MATHEMATICAL LITERACY
ƒ Jobs requiring mathematical and technical skills are growing the fastest among the eight professional and related occupations.
ƒ Sixty
Sixty percent of all new jobs beginning in the 21st percent of all new jobs beginning in the 21st
century require skills that are possessed by only 20% of the current workforce.
18
6
Math Tools Breakout
MATHEMATICAL LITERACY
What percentage of your classroom practice in mathematics would you estimate you spend in each of these areas? Mastery of procedural and conceptual knowledge
A language to communicate and solve real‐world problems
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
7
Math Tools Breakout
Mathematics Is…
Reasoning and Problem Solving
22
Mathematics Is…
Reasoning and Problem Solving
23
Mathematics Is…
Creative Expression
24
8
Math Tools Breakout
WHAT KIND OF PROBLEM SOLVER ARE YOU?
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
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?
WHAT KIND OF PROBLEM SOLVER ARE YOU?
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(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?
9
Math Tools Breakout
WHAT KIND OF PROBLEM SOLVER ARE YOU?
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 actual objects to simulate the problem really helps me—it l bj
i l
h
bl
ll h l
i
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.
28
WHAT KIND OF PROBLEM SOLVER ARE YOU?
How Tanisha Responded to the Canoe Problem
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 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.
29
WHAT KIND OF PROBLEM SOLVER ARE YOU?
How Giovanni Responded to the Canoe Problem
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 II’m
m doing, and I also like when I can work with friends 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.
30
10
Math Tools Breakout
WHAT KIND OF PROBLEM SOLVER ARE YOU?
How Al Responded to the Canoe Problem
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 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.
31
MATH LEARNING STYLES
MATH LEARNING STYLES
Which of the following objects best represents you as a learner of mathematics? Explain your choice.
11
Math Tools Breakout
MATH LEARNING STYLES
Carl Jung asked…
Carl Jung asked…
How are all minds alike?
How are all minds different?
MATH LEARNING STYLES
Carl Jung answered…
All minds P RC IV and PROC SS information,
All minds PERCEIVE and PROCESS information, but differ in how they pay attention.
The Four Functions of Style
SENSING
Physical
Facts
Details
Here & Now
Objective
Perspiration
Analyze
Logic
Likes/Dislikes
Truth
Procedures
Subjective
Harmonize
Tact
Inspiration
People
Past & Future
Ideas
Possibilities
Patterns
INTUITION
36
12
Math Tools Breakout
From Function to Style:
S+T
S+F
Mastery
Interpersonal
Sensing
Fe
eeling
Intuition
Understanding
Self-Expressive
N+T
N+F
37
MATH LEARNING STYLES
Mastery Learner (ST):
• Thinking Goal:
REMEMBERING
• Environment:
CLARITY & CONSISTENCY
• Motivation:
SUCCESS
• Process:
STEP‐BY‐STEP, EXERCISE, & PRACTICE
• Outcome:
WHAT? CORRECT ANSWERS
MATH LEARNING STYLES
Understanding Learner (NT):
• Thinking Goal:
REASONING
• Environment:
CRITICAL THINKING & CHALLENGE
• Motivation:
CURIOSITY
• Process:
DOUBT‐BY‐DOUBT, EXPLAIN, & PROVE
• Outcome:
WHY? ARGUMENTS
13
Math Tools Breakout
MATH LEARNING STYLES
Self‐Expressive Learner (NF):
• Thinking Goal:
REORGANIZING
• Environment:
COLORFUL & CHOICE
• Motivation:
ORIGINALITY
• Process:
DREAM‐BY‐DREAM, EXPLORE POSSIBILITIES
• Outcome:
WHAT IF? CREATIVE ALTERNATIVES
MATH LEARNING STYLES
Interpersonal Learner (SF):
• Thinking Goal:
RELATE PERSONALLY
• Environment:
COOPERATIVE & CONVERSATION
• Motivation:
RELATIONSHIPS
• Process:
FRIEND‐BY‐FRIEND, EXPERIENCE, & PERSONALIZE
• Outcome:
SO WHAT? CURRENT & CONNECTED
MATH LEARNING STYLES
Percentages of Learning Style Preferences 35%
35%
S+T
MASTERY
S+F
INTERPERSONAL
12% 65%
1% 22%
UNDERSTANDING
N+T
10%
SELF‐EXPRESSIVE
N+F
20%
14
Math Tools Breakout
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 Study
Five different ways for teaching mathematics
y
pp
p
g
• A memory‐based approach emphasizing identification and recall of facts 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 Tools Breakout
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
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Style and NCTM Standards
(The Mathematics of Sensing‐Thinkers)
MASTERY STYLE
NCTM 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 Communication
Abilities 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
•To notice, control, and make use of emotional responses in mathematical work
(The Mathematics of Intuitive‐Thinkers)
UNDERSTANDING STYLE
NCTM Focus: Mathematics as Reasoning
Abilities 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
THE RIGHT LEARNING SITUATION?
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Math Tools Breakout
HOW DO I SELECT THE RIGHT TOOL FOR
THE RIGHT LEARNING SITUATION?
Vital Statistic 1: Title and Flash Summary
Knowledge Cards—Students create “flash
flash cards
cards” to visualize
and remember complex terms and concepts.
Summary of Knowledge Cards
HOW DO I SELECT THE RIGHT TOOL FOR
THE RIGHT LEARNING SITUATION?
Vital Statistic 2: NCTM Process Standards
NCTM Process Standards for Knowledge Cards
HOW DO I SELECT THE RIGHT TOOL FOR
THE RIGHT LEARNING SITUATION?
Vital Statistic 3: Educational Research
Educational Research Base for Knowledge Cards
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Math Tools Breakout
HOW DO I SELECT THE RIGHT TOOL FOR
THE RIGHT LEARNING SITUATION?
Vital Statistic 4: Instructional Objectives
Instructional Objectives for Knowledge Cards
MATH TOOLS
FIVE WAYS TO USE MATH TOOLS
1. Try one out
2. Use tools to help you meet a particular standard or objective
3. Individualize instruction
4. Differentiate instruction for the entire class
5. Design more powerful lessons, assessments, and units
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Math Tools Breakout
MENTAL MATH WAR
Purpose
Mental Math War provides a quick, engaging, and interactive way for students to practice working with variables, equations, and their solutions.
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.
MENTAL MATH WAR
Steps
1. 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.
q y
5. From each pair, assign one student to play cards for the x variable and one student to play cards for the y variable
6. 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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Math Tools Breakout
MENTAL MATH WAR
ALWAYS-SOMETIMES-NEVER (ASN)
Purpose
Always‐Sometimes‐Never (ASN) is a reasoning activity that focuses students’ thinking around the important, and often subtle, facts and details associated with mathematical 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.
ALWAYS-SOMETIMES-NEVER (ASN)
Steps
1. 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 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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Math Tools Breakout
ALWAYS‐SOMETIMES‐NEVER (ASN)
Arithmetic: Addition and Subtraction
1. The sum of two 3‐digit numbers is a 3‐digit number. (Sometimes)
2. The sum of two even numbers is an odd number. (Never)
3. The difference of two odd numbers is an even number. (Always)
4. The sum of additive inverses is zero. (Always)
(Al
)
5. The difference of three odd numbers is an odd number. h ff
f h
b
b (Always)
6. The sum of three even numbers is zero. (Sometimes)
7. The sum of three odd numbers is zero. (Never)
8. The sum of two counting numbers is greater than the difference of the same numbers. (Always)
ALWAYS-SOMETIMES-NEVER (ASN)
Statistics: Mean, Median, Mode
1.
2.
3.
4.
5.
6.
7.
8.
A list of numbers has a mean. (Always)
A list of numbers has a median. (Always)
A list of numbers has a mode. (Sometimes)
The mean of a set of numbers is one of the numbers of that set. (Sometimes)
The median of ten consecutive integers is one of those integers. (Never)
If the mode of a set of numbers is 14, then 14 is one of the numbers of that set. (Always)
The mean of a set of numbers is greater than the median of that set of numbers. (Sometimes)
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-SOMETIMES-NEVER (ASN)
Trigonometry: Graph Analysis
1. The graph of a trigonometric function is periodic. (Always)
2. Doubling the amplitude of a trigonometric function doubles the period of the function. (Never)
3. The graph of a cosecant function has an infinite number of asymptotes. (Always)
4. The period of y = sin bx + h is equivalent to the period of y = sec bx + k. (Always; both)
5. A cosine function has both a maximum and a minimum value. A cosine function has both a maximum and a minimum value. (Always)
6.
For numbers a and b, the graph of y = cos x on the interval a < 0 < b is an increasing function. (Never)
7. Stretching the graph of a trigonometric function changes the period of the function. (Sometimes; true for horizontal stretches, false for vertical stretches.)
8. Applying a phase shift on a secant graph changes the location of vertical asymptotes. (Sometimes)
9. For any value of a, the graph of y = tan ax will have the y‐axis as an asymptote. (Never)
10. Secant graphs have horizontal asymptotes. (Never)
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Math Tools Breakout
COMPARE AND CONTRAST
Purpose
Research 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
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.
COMPARE AND CONTRAST
Steps
1. Select two (or more) related concepts or mathematics problems.
2. Specify criteria for comparison.
3. Provide (or teach students how to create) graphic organizers for describing items and comparing them.
4. Guide students through the four phases of comparison:
• Description
• Comparison
• Conclusion
• Application
COMPARE AND CONTRAST
Problem 1
Haylee 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 q p
receive?
Problem 2
For 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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Math Tools Breakout
COMPARE AND CONTRAST
Description Phase
COMPARE AND CONTRAST
Comparison Phase
69
COMPARE AND CONTRAST
Conclusion Phase
• Are these two problems more alike or
more different?
• How did these differences affect the way
you solved each problem?
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Math Tools Breakout
70
COMPARE AND CONTRAST
Application Phase
After 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
2. Then,
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.
THREE-WAY TIE
Purpose
Three‐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
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.
THREE-WAY TIE
Steps
1. 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 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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Math Tools Breakout
THREE-WAY TIE
CINQUAINS
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 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.
CINQUAINS
Steps
1. 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.)
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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Math Tools Breakout
CINQUAINS
Cinquain Format and Successful Cinquain for Algebra
BRING YOUR “A” GAME
Purpose
The 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
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 Monitor
Steps
1. 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 g
attention.
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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Math Tools Breakout
BRING YOUR “A” GAME
Attention Monitor
Teacher’s Attention Monitor
Student’s Attention Monitor Using Emoticons
BRING YOUR “A” GAME
Attitude Catcher
Steps
1. Distribute an “All You Need Is Attitude” organizer to students. 2. Have students read the statements about attitude. Give 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.
6.
7.
8.
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 attitudes. i d
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.
Have students review the “Ten Mathitude Adjusters.”
Encourage students to repeat one or more of the “Mathitude Adjusters” to themselves whenever it is productive and positive.
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Math Tools Breakout
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
BRING YOUR “A” GAME
Aspiration Goal Planner
Steps
1. Have students reflect on their aspirations by writing a short response to the following questions:
x What would you like to achieve in mathematics class this week? What will you do in order to achieve this goal?
x What do you plan on doing after you get out of school? What What do you plan on doing after you get out of school? What
mathematics do you need to accomplish your goals?
x 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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Math Tools Breakout
Aspiration Goal Planner
86
TASK ROTATION
Mastery
Interpersonal
What skills, procedures, and key terms do I want students to master?
How will students make personal connections or discover the social 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.
Task Rotation
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.
What an Animal
MASTERY
INTERPERSONAL
MATHEMATICAL SUMMARIES
The 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.
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.)
p p
)
UNDERSTANDING
SELF‐EXPRESSIVE
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.
CREATE YOUR OWN
Students 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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Math Tools Breakout
Task Rotation
Greatest Common Factor (GCF)
MASTERY
INTERPERSONAL
MEMORY BOX
Think 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.
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. Greatest Common Factor
Natasha thinks that the GCF of 72 and 540 is 6, but she didn’t write down any work. Who’s right? g
Explain any errors that either (or both) of the girls made. What is the GCF of 72 and 540?
UNDERSTANDING
SELF‐EXPRESSIVE
THREE‐WAY TIE
M + M: MATH AND METAPHORS
Look at the triangle below. Write a sentence along each side Is finding the Greatest Common Factor (GCF) of two of the triangle that connects the word or phrase at each numbers like building a house?
angle of the triangle.
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.
Task Rotation
First Degree Equations
MASTERY
INTERPERSONAL
PAIRED LEARNING
WHAT’S WRONG?
Look at the student work to the right. Work with a partner to create a set of hints that could be Find and correct any errors. used to help coach someone working on the following problem:
What is the correct answer?
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.
UNDERSTANDING
SELF‐EXPRESSIVE
LEARNING FROM CLUES
Below 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?
CREATE YOUR OWN
Keep 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.
Task Rotation
Calculus
MASTERY
INTERPERSONAL
KNOWLEDGE CARDS
Create 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 p
•Concave down
• Linear functions
COOPERATIVE STRUCTURES
Work 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 y
p
y
correct.
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 functions
2. absolute value functions
3. parabolic functions
4. sine functions
5. step functions
6. exponential functions
SELF‐EXPRESSIVE
PICTURE = 1,000 WORDS
Draw 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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