Introduction
Evil Forces
Role of Analogous Tasks and Examples
Analogous Tasks and the Common Core
Dr. Roger Fischer
Project Facilitator
Exploring Mathematics with Analogous Tasks
Montana State University
October 16, 2015
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
O VERVIEW
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
M Y B ACKGROUND
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B.A. Mathematics: Teaching Secondary
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2004, Central Washington University
M.A.T. Mathematics Teaching
2007, Central Washington University
Ph.D. Mathematics: Mathematics Education Concentration
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2014, Montana State University
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Two years teaching middle school mathematics
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One year teaching developmental math to undergraduates
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Eight years teaching college level mathematics content and
pedagogy
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Currently in my second year of teaching developmental
math to undergraduates
Introduction
Evil Forces
Role of Analogous Tasks and Examples
A NALOGOUS TASKS AND M ATHEMATICAL
P RACTICES
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MP.1: Make sense of problems and persevere in solving
them
Mathematically proficient students start by explaining
the meaning of a problem and look for entry points to
its solution. . . they consider analogous problems, and
try special cases and simpler forms of the original
problem in order to gain insight into its solution.
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Introduce yourself to someone next to you.
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Discuss with this person what an “analogous problem”
might be.
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
E XAMPLE PROBLEM
Warm Up: Write the following expression in simplest form.
Then suggest some difficulties students may encounter as they
work.
2y2 − 4y y2 − 2y − 3
·
2y2 − 2 y2 − 5y + 6
Introduction
Evil Forces
Role of Analogous Tasks and Examples
E VIL F ORCES
One particularly robust misconception students have is
centered on the idea of “cancelling.”
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
E XAMPLE OF AN A NALOGOUS TASK
Simplify the following expressions. Try to find two different
approaches.
12 + 8
12
5 + 40
5
What are some advantages and disadvantages of each
approach?
12 + 6
24 + 18
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A NALOGOUS TASK S EQUENCING
Write each of the following expressions so that the numerator
and denominator have no common factors.
12x + 8
12
5a + 40
5
3m2 + 3m
6m2 + 9m
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A NALOGOUS TASK S EQUENCING
Write each of the following expressions so that the numerator
and denominator have no common factors.
x2 − 25
x2 − 10x + 25
4a2
3a − 6
·
2
3a − 12a + 12
2a
3a3 + 3
6a2 − 6a + 6
Introduction
Evil Forces
Role of Analogous Tasks and Examples
ATTENDING TO P RECISION
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a
=1
a
This fact underlies our strategy for writing rational
expressions in simplest form.
For all non-zero a,
Without consistent use of precise vocabulary, “remove
factors of one” becomes “cross out things that are the
same.”
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
A NOTHER C OMMON M ISCONCEPTION
Everyone’s favorite “identity:”
(a + b)2 = a2 + b2
Why is this misconception so robust?
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
A NALOGOUS TASK
Let’s use a word problem and particular numbers to analyze
the structure of the square of a binomial expression.
Chuck Norris has a vegetable garden that measures 4
feet long and 4 feet wide. One day, Steven Seagal
makes fun of how small Chuck’s garden is. After
Chuck stares at Steven until he explodes, Chuck adds
three feet to both the length and the width. Draw a
picture of Chuck’s old garden, the new one, and find
the area of each.
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
A NALOGOUS TASK
Let’s use a word problem and introduce a variable to further
analyze the structure of the square of a binomial expression.
Chuck Norris has a vegetable garden that measures x
feet long and x feet wide. One day, Steven Seagal
makes fun of how small Chuck’s garden is. After
Chuck stares at Steven until he explodes, Chuck adds
three feet to both the length and the width. Draw a
picture of Chuck’s old garden, the new one, and find
an expression that represents the area of each.
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
A NALOGOUS TASK
Let’s use a word problem and introduce another variable to
further analyze the structure of the square of a binomial
expression.
Chuck Norris has a vegetable garden that measures x
feet long and x feet wide. One day, Steven Seagal
makes fun of how small Chuck’s garden is. After
Chuck stares at Steven until he explodes, Chuck adds
y feet to both the length and the width. Draw a picture
of Chuck’s old garden, the new one, and find an
expression that represents the area of each.
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
A NALOGOUS TASK
Let’s use a word problem and introduce another variable to
further analyze the structure of the square of a binomial
expression.
Chuck Norris has a vegetable garden that measures x
feet long and y feet wide. One day, Steven Seagal
makes fun of how small Chuck’s garden is. After
Chuck stares at Steven until he explodes, Chuck adds
a feet to the length and b feet to the width. Draw a
picture of Chuck’s old garden, the new one, and find
an expression that represents the area of each.
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A N INVITATION
If you found this presentation to be useful and would like to
learn more about using analogous tasks in your classroom, you
can join the Exploring Mathematics with Analogous Tasks
family!
I
You can get paid to attend professional development
sessions much like this one!
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A N INVITATION
If you found this presentation to be useful and would like to
learn more about using analogous tasks in your classroom, you
can join the Exploring Mathematics with Analogous Tasks
family!
I
You can get paid to attend professional development
sessions much like this one!
I
You can attend online sessions or face to face sessions
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A N INVITATION
If you found this presentation to be useful and would like to
learn more about using analogous tasks in your classroom, you
can join the Exploring Mathematics with Analogous Tasks
family!
I
You can get paid to attend professional development
sessions much like this one!
I
You can attend online sessions or face to face sessions
I
You can earn OPI renewal credits.
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A N INVITATION
If you found this presentation to be useful and would like to
learn more about using analogous tasks in your classroom, you
can join the Exploring Mathematics with Analogous Tasks
family!
I
You can get paid to attend professional development
sessions much like this one!
I
You can attend online sessions or face to face sessions
I
You can earn OPI renewal credits.
I
You can even get a tuition reimbursement for a 3 credit
graduate level course.
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A N INVITATION
If you found this presentation to be useful and would like to
learn more about using analogous tasks in your classroom, you
can join the Exploring Mathematics with Analogous Tasks
family!
I
You can get paid to attend professional development
sessions much like this one!
I
You can attend online sessions or face to face sessions
I
You can earn OPI renewal credits.
I
You can even get a tuition reimbursement for a 3 credit
graduate level course.
I
You can get paid to present your analogous tasks to the
rest of the EMAT team.
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A N INVITATION
If you found this presentation to be useful and would like to
learn more about using analogous tasks in your classroom, you
can join the Exploring Mathematics with Analogous Tasks
family!
I
You can get paid to attend professional development
sessions much like this one!
I
You can attend online sessions or face to face sessions
I
You can earn OPI renewal credits.
I
You can even get a tuition reimbursement for a 3 credit
graduate level course.
I
You can get paid to present your analogous tasks to the
rest of the EMAT team.
I
Interested? Please provide your contact information on the
sheet being passed around.
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
A N INVITATION
If you found this presentation to be useful and would like to
learn more about using analogous tasks in your classroom, you
can join the Exploring Mathematics with Analogous Tasks
family!
I
You can get paid to attend professional development
sessions much like this one!
I
You can attend online sessions or face to face sessions
I
You can earn OPI renewal credits.
I
You can even get a tuition reimbursement for a 3 credit
graduate level course.
I
You can get paid to present your analogous tasks to the
rest of the EMAT team.
I
Interested? Please provide your contact information on the
sheet being passed around.
I
Please take a flyer and contact us with any questions.
Introduction
Evil Forces
Role of Analogous Tasks and Examples
C ONCLUSION
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Choose a replacement value of the variable to turn an
intricate algebraic expression into a more concrete
analogous task
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
C ONCLUSION
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Choose a replacement value of the variable to turn an
intricate algebraic expression into a more concrete
analogous task
ATTEND TO PRECISION!
Conclusion
Introduction
Evil Forces
Role of Analogous Tasks and Examples
Conclusion
C ONCLUSION
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Choose a replacement value of the variable to turn an
intricate algebraic expression into a more concrete
analogous task
ATTEND TO PRECISION!
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Students can learn to use precise vocabulary. It begins with
you modeling it in the classroom!
Robust misconceptions will require long-term and
consistent efforts to overcome. Don’t be discouraged!
Introduction
Evil Forces
Role of Analogous Tasks and Examples
T HANKS FOR COMING !
Dr. Roger Fischer
Project Facilitator
Exploring Mathematics with Analogous Tasks
Montana State University
118 Reid Hall
Bozeman, MT 59717
roger.fischer@montana.edu
Conclusion