NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 1
April 26 2012
Preparing Smart Teachers to Teach with SMARTTM Technology
NCTM Annual ConferenceApril 26 2012Philadelphia PA
Mary Lou Metz (mlmetziupedu)Edel Reilly
Francisco Alarcon
Indiana University of PA
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 2
April 26 2012
Description of PresentationAppropriate use of technology is an essential component of preparing preshyservice teachers to teach mathematics We will share smart technology lessons activities strategies and tools we have used in our work with preshyservice mathematics teachers at the elementary middle school and secondary levels
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 3
April 26 2012
Agendabull Example from Geometrybull Example from Algebrabull Example from Number Theorybull Example from Fraction Concepts
NCTM 2012 Presentation no videonotebook
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April 26 2012
Examples from Geometrybull Methods Course for PreK shy 4 Education Majorsbull Lesson Teaching Geometry concepts shy Four NCTM content goals for geometry
Shapes and propertiesTransformationsLocationVisualization
shy Five van Hiele levels of geometric thought
0 Visualization1 Analysis2 Informal Deduction3 Deduction4 Rigor
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 5
April 26 2012
Shapes and Properties - Level 0
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 6
April 26 2012
NCTM 2012 Presentation no videonotebook
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April 26 2012
Shapes and Properties - Level 1
Back to Geometry Goals
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 8
April 26 2012
Transformations - Level 1
Back to Geometry Goals
bull Reflect the shape across the x- axisbull Reflect the shape across the y-axis
Click here to check your answer
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 9
April 26 2012
Visualization - Level 0Pentominoes
Using 5 squares make as many DIFFERENT shapes as you can Each square must share a side with at least one other square
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
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April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
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April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
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April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 2
April 26 2012
Description of PresentationAppropriate use of technology is an essential component of preparing preshyservice teachers to teach mathematics We will share smart technology lessons activities strategies and tools we have used in our work with preshyservice mathematics teachers at the elementary middle school and secondary levels
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 3
April 26 2012
Agendabull Example from Geometrybull Example from Algebrabull Example from Number Theorybull Example from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 4
April 26 2012
Examples from Geometrybull Methods Course for PreK shy 4 Education Majorsbull Lesson Teaching Geometry concepts shy Four NCTM content goals for geometry
Shapes and propertiesTransformationsLocationVisualization
shy Five van Hiele levels of geometric thought
0 Visualization1 Analysis2 Informal Deduction3 Deduction4 Rigor
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 5
April 26 2012
Shapes and Properties - Level 0
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 6
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 7
April 26 2012
Shapes and Properties - Level 1
Back to Geometry Goals
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 8
April 26 2012
Transformations - Level 1
Back to Geometry Goals
bull Reflect the shape across the x- axisbull Reflect the shape across the y-axis
Click here to check your answer
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 9
April 26 2012
Visualization - Level 0Pentominoes
Using 5 squares make as many DIFFERENT shapes as you can Each square must share a side with at least one other square
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 3
April 26 2012
Agendabull Example from Geometrybull Example from Algebrabull Example from Number Theorybull Example from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 4
April 26 2012
Examples from Geometrybull Methods Course for PreK shy 4 Education Majorsbull Lesson Teaching Geometry concepts shy Four NCTM content goals for geometry
Shapes and propertiesTransformationsLocationVisualization
shy Five van Hiele levels of geometric thought
0 Visualization1 Analysis2 Informal Deduction3 Deduction4 Rigor
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 5
April 26 2012
Shapes and Properties - Level 0
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 6
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 7
April 26 2012
Shapes and Properties - Level 1
Back to Geometry Goals
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 8
April 26 2012
Transformations - Level 1
Back to Geometry Goals
bull Reflect the shape across the x- axisbull Reflect the shape across the y-axis
Click here to check your answer
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 9
April 26 2012
Visualization - Level 0Pentominoes
Using 5 squares make as many DIFFERENT shapes as you can Each square must share a side with at least one other square
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 4
April 26 2012
Examples from Geometrybull Methods Course for PreK shy 4 Education Majorsbull Lesson Teaching Geometry concepts shy Four NCTM content goals for geometry
Shapes and propertiesTransformationsLocationVisualization
shy Five van Hiele levels of geometric thought
0 Visualization1 Analysis2 Informal Deduction3 Deduction4 Rigor
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 5
April 26 2012
Shapes and Properties - Level 0
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 6
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 7
April 26 2012
Shapes and Properties - Level 1
Back to Geometry Goals
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 8
April 26 2012
Transformations - Level 1
Back to Geometry Goals
bull Reflect the shape across the x- axisbull Reflect the shape across the y-axis
Click here to check your answer
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 9
April 26 2012
Visualization - Level 0Pentominoes
Using 5 squares make as many DIFFERENT shapes as you can Each square must share a side with at least one other square
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
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April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
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April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 5
April 26 2012
Shapes and Properties - Level 0
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 6
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 7
April 26 2012
Shapes and Properties - Level 1
Back to Geometry Goals
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 8
April 26 2012
Transformations - Level 1
Back to Geometry Goals
bull Reflect the shape across the x- axisbull Reflect the shape across the y-axis
Click here to check your answer
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 9
April 26 2012
Visualization - Level 0Pentominoes
Using 5 squares make as many DIFFERENT shapes as you can Each square must share a side with at least one other square
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 6
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 7
April 26 2012
Shapes and Properties - Level 1
Back to Geometry Goals
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 8
April 26 2012
Transformations - Level 1
Back to Geometry Goals
bull Reflect the shape across the x- axisbull Reflect the shape across the y-axis
Click here to check your answer
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 9
April 26 2012
Visualization - Level 0Pentominoes
Using 5 squares make as many DIFFERENT shapes as you can Each square must share a side with at least one other square
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
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Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
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Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
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Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
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Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
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Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
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Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
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Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
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Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
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Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
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Metz Reilly amp Alarcon 35
April 26 2012
Attachments
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 7
April 26 2012
Shapes and Properties - Level 1
Back to Geometry Goals
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 8
April 26 2012
Transformations - Level 1
Back to Geometry Goals
bull Reflect the shape across the x- axisbull Reflect the shape across the y-axis
Click here to check your answer
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 9
April 26 2012
Visualization - Level 0Pentominoes
Using 5 squares make as many DIFFERENT shapes as you can Each square must share a side with at least one other square
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 8
April 26 2012
Transformations - Level 1
Back to Geometry Goals
bull Reflect the shape across the x- axisbull Reflect the shape across the y-axis
Click here to check your answer
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 9
April 26 2012
Visualization - Level 0Pentominoes
Using 5 squares make as many DIFFERENT shapes as you can Each square must share a side with at least one other square
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
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Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
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Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
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Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 9
April 26 2012
Visualization - Level 0Pentominoes
Using 5 squares make as many DIFFERENT shapes as you can Each square must share a side with at least one other square
Click here to check your answer
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 10
April 26 2012
NCTM 2012 Presentation no videonotebook
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April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
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April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
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April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
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April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
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April 26 2012
3 x2 + 6x + 8
x2
x 1
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April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
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April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
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April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
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April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
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April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
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April 26 2012
NCTM 2012 Presentation no videonotebook
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April 26 2012
NCTM 2012 Presentation no videonotebook
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April 26 2012
Onto some magic with numbers
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Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
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April 26 2012
Was this really a magic trick
Is there a pattern you notice
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April 26 2012
NCTM 2012 Presentation no videonotebook
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Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
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April 26 2012
Making a Video of the Class
Class April 18 2011wmv
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Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
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Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
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Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
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Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 11
April 26 2012
Examples from Algebra
bull Grades 4 through 8 Algebra Course
bull Modeling the simplification of expressionsDistributive propertyAdding and subtracting polynomialsMultiplying and factoring polynomials
bull Graphing equations
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 12
April 26 2012
Distributive PropertyUse the same concept that was applied with multiplication of integers think of the first factor as the counter
The same rules apply 3(x + 2)
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
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Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
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Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
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Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
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Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
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Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 13
April 26 2012
Addition and Subtraction of Polynomials
1 (x2 + 2x + 1) + (x2 shy x shy 4)
x2
x2
xx
shyx
1
shy1 shy1
shy1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 14
April 26 2012
4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
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Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
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Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
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Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 15
April 26 2012
SubstitutionAlgebra tiles can be used to model substitution Represent original expression with tiles Then replace each rectangle with the appropriate tile value Combine like terms
3 + 2x let x = 4
1x
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 16
April 26 2012
Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
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Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
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Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
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Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
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Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
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Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
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Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
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Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 17
April 26 2012
Make a collection of 2 big squares 7 rectangles and 6 ones How many ways can they be arranged Which set looks aesthetically pleasing (in a rectangle) What are the dimensions of the rectangle This will lead to factoring
x2x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
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Metz Reilly amp Alarcon 18
April 26 2012
3 x2 + 6x + 8
x2
x 1
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Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
2
x
y
f(x) =p(x)q(x)
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Metz Reilly amp Alarcon 20
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shy5 0 5 10
1
2
x
y
NCTM 2012 Presentation no videonotebook
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shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
NCTM 2012 Presentation no videonotebook
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Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
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Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
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Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
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Metz Reilly amp Alarcon 27
April 26 2012
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Metz Reilly amp Alarcon 28
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Was this really a magic trick
Is there a pattern you notice
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Metz Reilly amp Alarcon 29
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
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Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
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Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
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Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
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Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
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April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 19
April 26 2012
A rational function is a function f that is a quotient of two polynomials that is
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial The domain of f consists of all inputs x for which q(x)ne 0
shy10 shy5 0 5 10
shy2
shy1
1
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x
y
f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
1
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y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
shy20
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
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NCTM 2012 Presentation no videonotebook
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Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
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y
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 22
April 26 2012
Example from Number Theorybull Introduction to Number Theory for secondary mathematics education majorsbull Lesson introduction to divisibility and proof techniques
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 23
April 26 2012
Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
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Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
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NCTM 2012 Presentation no videonotebook
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Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
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Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
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Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
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Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
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Metz Reilly amp Alarcon 35
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Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 24
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 25
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 26
April 26 2012
Onto some magic with numbers
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 27
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 28
April 26 2012
Was this really a magic trick
Is there a pattern you notice
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 29
April 26 2012
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 30
April 26 2012
Theorem An integer is divisible by 9 if and only if the sum of its digits is divisible by 9
Definition An integer b is divisible by an integer ane0 if and only if b = a c for some integer c
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 31
April 26 2012
Making a Video of the Class
Class April 18 2011wmv
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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audiotime=47213172162 sessionstart=47213172162 sessionend=47213481922gtltsessionannot audiotime=47213481178 sessionstart=47213595683 sessionend=47213596003gtltsessionsgtltcontentgtltpagegtltuserdatagtltdocumentgtltimlgt
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 32
April 26 2012
Examples from Fraction Concepts
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
=
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 33
April 26 2012
Smart Pen background
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
= 1 =
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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