Preparing Smart Teachers to Teach with SMART Technology ...€¦ · NCTM 2012 Presentation no...
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NCTM 2012 Presentation no video.notebook Metz, Reilly & Alarcon 1 April 26, 2012 Preparing Smart Teachers to Teach with SMART TM Technology NCTM Annual Conference April 26, 2012 Philadelphia, PA Mary Lou Metz ([email protected]) Edel Reilly Francisco Alarcon Indiana University of PA
Transcript of Preparing Smart Teachers to Teach with SMART Technology ...€¦ · NCTM 2012 Presentation no...
NCTM 2012 Presentation no videonotebook
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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
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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
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Agendabull Example from Geometrybull Example from Algebrabull Example from Number Theorybull Example from Fraction Concepts
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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
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Shapes and Properties - Level 0
Click here to check your answer
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Shapes and Properties - Level 1
Back to Geometry Goals
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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
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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
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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
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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
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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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4 (2x2 + 2) shy (x2 + 2x shy 1)
x2 x2
11
x shyx 1 shy1
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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
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Multiplication BinomialsshyshyUsing Algebra Tiles
x
x 1 1 1
11
x2
x 1
(x + 3)(x + 2)
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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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3 x2 + 6x + 8
x2
x 1
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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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shy5 0 5 10
1
2
x
y
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shy10 shy5 0 5 10
shy20
shy10
10
20
30
x
y
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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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Some Other Number Theory Magicians Diophantus of Alexandria Euclid Fibonacci
Euler Carl Friedrich Gauss
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Onto some magic with numbers
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Was this really a magic trick
Is there a pattern you notice
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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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Making a Video of the Class
Class April 18 2011wmv
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Examples from Fraction Concepts
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Smart Pen background
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Fraction Concepts Task 1
Video of Task 1
= 1 =
=
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Attachments
fractions1pdf
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SMART Notebook
- Page 1 Apr 17-216 PM
- Page 2 Apr 17-216 PM
- Page 3 Apr 17-216 PM
- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
- Page 6 Apr 12-1010 AM
- Page 7 Apr 12-1124 AM
- Page 8 Apr 5-636 PM
- Page 9 Apr 5-636 PM
- Page 10 gridlines
- Page 11 Apr 17-216 PM
- Page 12 Mar 19-540 AM
- Page 13 Mar 31-138 PM
- Page 14 Mar 31-141 PM
- Page 15 Mar 19-606 AM
- Page 16 Mar 31-215 PM
- Page 17 Apr 24-150 PM
- Page 18 Mar 19-637 AM
- Page 19 Apr 24-126 PM
- Page 20 Apr 24-118 PM
- Page 21 Apr 24-144 PM
- Page 22 Apr 17-216 PM
- Page 23 Aug 30-130 AM
- Page 24 Apr 24-101 PM
- Page 25 Apr 24-102 PM
- Page 26 Aug 30-154 AM
- Page 27 Apr 23-1220 PM
- Page 28 Aug 30-154 AM
- Page 29 Aug 30-156 AM
- Page 30 Aug 30-852 AM
- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
- Page 2 Apr 17-216 PM
- Page 3 Apr 17-216 PM
- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
- Page 6 Apr 12-1010 AM
- Page 7 Apr 12-1124 AM
- Page 8 Apr 5-636 PM
- Page 9 Apr 5-636 PM
- Page 10 gridlines
- Page 11 Apr 17-216 PM
- Page 12 Mar 19-540 AM
- Page 13 Mar 31-138 PM
- Page 14 Mar 31-141 PM
- Page 15 Mar 19-606 AM
- Page 16 Mar 31-215 PM
- Page 17 Apr 24-150 PM
- Page 18 Mar 19-637 AM
- Page 19 Apr 24-126 PM
- Page 20 Apr 24-118 PM
- Page 21 Apr 24-144 PM
- Page 22 Apr 17-216 PM
- Page 23 Aug 30-130 AM
- Page 24 Apr 24-101 PM
- Page 25 Apr 24-102 PM
- Page 26 Aug 30-154 AM
- Page 27 Apr 23-1220 PM
- Page 28 Aug 30-154 AM
- Page 29 Aug 30-156 AM
- Page 30 Aug 30-852 AM
- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 10 gridlines
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- Page 33 Apr 17-233 PM
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- Attachments Page 1
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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
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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SMART Notebook
- Page 1 Apr 17-216 PM
- Page 2 Apr 17-216 PM
- Page 3 Apr 17-216 PM
- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
- Page 6 Apr 12-1010 AM
- Page 7 Apr 12-1124 AM
- Page 8 Apr 5-636 PM
- Page 9 Apr 5-636 PM
- Page 10 gridlines
- Page 11 Apr 17-216 PM
- Page 12 Mar 19-540 AM
- Page 13 Mar 31-138 PM
- Page 14 Mar 31-141 PM
- Page 15 Mar 19-606 AM
- Page 16 Mar 31-215 PM
- Page 17 Apr 24-150 PM
- Page 18 Mar 19-637 AM
- Page 19 Apr 24-126 PM
- Page 20 Apr 24-118 PM
- Page 21 Apr 24-144 PM
- Page 22 Apr 17-216 PM
- Page 23 Aug 30-130 AM
- Page 24 Apr 24-101 PM
- Page 25 Apr 24-102 PM
- Page 26 Aug 30-154 AM
- Page 27 Apr 23-1220 PM
- Page 28 Aug 30-154 AM
- Page 29 Aug 30-156 AM
- Page 30 Aug 30-852 AM
- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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
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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
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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SMART Notebook
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- Attachments Page 1
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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
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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
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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
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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SMART Notebook
- Page 1 Apr 17-216 PM
- Page 2 Apr 17-216 PM
- Page 3 Apr 17-216 PM
- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
- Page 6 Apr 12-1010 AM
- Page 7 Apr 12-1124 AM
- Page 8 Apr 5-636 PM
- Page 9 Apr 5-636 PM
- Page 10 gridlines
- Page 11 Apr 17-216 PM
- Page 12 Mar 19-540 AM
- Page 13 Mar 31-138 PM
- Page 14 Mar 31-141 PM
- Page 15 Mar 19-606 AM
- Page 16 Mar 31-215 PM
- Page 17 Apr 24-150 PM
- Page 18 Mar 19-637 AM
- Page 19 Apr 24-126 PM
- Page 20 Apr 24-118 PM
- Page 21 Apr 24-144 PM
- Page 22 Apr 17-216 PM
- Page 23 Aug 30-130 AM
- Page 24 Apr 24-101 PM
- Page 25 Apr 24-102 PM
- Page 26 Aug 30-154 AM
- Page 27 Apr 23-1220 PM
- Page 28 Aug 30-154 AM
- Page 29 Aug 30-156 AM
- Page 30 Aug 30-852 AM
- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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
SMART Notebook
- Page 1 Apr 17-216 PM
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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
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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
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
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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
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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
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
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 10 gridlines
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- Page 12 Mar 19-540 AM
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- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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
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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
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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Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
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- Page 10 gridlines
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- Page 12 Mar 19-540 AM
- Page 13 Mar 31-138 PM
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- Page 15 Mar 19-606 AM
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- Page 17 Apr 24-150 PM
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- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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
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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 10 gridlines
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- Page 12 Mar 19-540 AM
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- Page 33 Apr 17-233 PM
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- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 5 Apr 5-608 PM
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- Page 10 gridlines
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- Page 33 Apr 17-233 PM
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- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 10 gridlines
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- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
- Page 2 Apr 17-216 PM
- Page 3 Apr 17-216 PM
- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
- Page 6 Apr 12-1010 AM
- Page 7 Apr 12-1124 AM
- Page 8 Apr 5-636 PM
- Page 9 Apr 5-636 PM
- Page 10 gridlines
- Page 11 Apr 17-216 PM
- Page 12 Mar 19-540 AM
- Page 13 Mar 31-138 PM
- Page 14 Mar 31-141 PM
- Page 15 Mar 19-606 AM
- Page 16 Mar 31-215 PM
- Page 17 Apr 24-150 PM
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- Page 19 Apr 24-126 PM
- Page 20 Apr 24-118 PM
- Page 21 Apr 24-144 PM
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- Page 23 Aug 30-130 AM
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- Page 25 Apr 24-102 PM
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- Page 27 Apr 23-1220 PM
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- Page 29 Aug 30-156 AM
- Page 30 Aug 30-852 AM
- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 3 Apr 17-216 PM
- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
- Page 6 Apr 12-1010 AM
- Page 7 Apr 12-1124 AM
- Page 8 Apr 5-636 PM
- Page 9 Apr 5-636 PM
- Page 10 gridlines
- Page 11 Apr 17-216 PM
- Page 12 Mar 19-540 AM
- Page 13 Mar 31-138 PM
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- Page 15 Mar 19-606 AM
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- Page 17 Apr 24-150 PM
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- Page 20 Apr 24-118 PM
- Page 21 Apr 24-144 PM
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- Page 23 Aug 30-130 AM
- Page 24 Apr 24-101 PM
- Page 25 Apr 24-102 PM
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- Page 27 Apr 23-1220 PM
- Page 28 Aug 30-154 AM
- Page 29 Aug 30-156 AM
- Page 30 Aug 30-852 AM
- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 3 Apr 17-216 PM
- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
- Page 6 Apr 12-1010 AM
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- Page 8 Apr 5-636 PM
- Page 9 Apr 5-636 PM
- Page 10 gridlines
- Page 11 Apr 17-216 PM
- Page 12 Mar 19-540 AM
- Page 13 Mar 31-138 PM
- Page 14 Mar 31-141 PM
- Page 15 Mar 19-606 AM
- Page 16 Mar 31-215 PM
- Page 17 Apr 24-150 PM
- Page 18 Mar 19-637 AM
- Page 19 Apr 24-126 PM
- Page 20 Apr 24-118 PM
- Page 21 Apr 24-144 PM
- Page 22 Apr 17-216 PM
- Page 23 Aug 30-130 AM
- Page 24 Apr 24-101 PM
- Page 25 Apr 24-102 PM
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- Page 27 Apr 23-1220 PM
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- Page 29 Aug 30-156 AM
- Page 30 Aug 30-852 AM
- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
- Page 6 Apr 12-1010 AM
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- Page 9 Apr 5-636 PM
- Page 10 gridlines
- Page 11 Apr 17-216 PM
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- Page 17 Apr 24-150 PM
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- Page 20 Apr 24-118 PM
- Page 21 Apr 24-144 PM
- Page 22 Apr 17-216 PM
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- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 6 Apr 12-1010 AM
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- Page 10 gridlines
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- Page 33 Apr 17-233 PM
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- Page 35 Apr 26-911 AM
- Attachments Page 1
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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
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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
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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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SMART Notebook
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- Attachments Page 1
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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
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f(x) =p(x)q(x)
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
April 26 2012
shy5 0 5 10
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
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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 =
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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SMART Notebook
- Page 1 Apr 17-216 PM
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- Attachments Page 1
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 20
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NCTM 2012 Presentation no videonotebook
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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
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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SMART Notebook
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- Attachments Page 1
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 21
April 26 2012
shy10 shy5 0 5 10
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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
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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
NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 34
April 26 2012
Fraction Concepts Task 1
Video of Task 1
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 4 Apr 17-216 PM
- Page 5 Apr 5-608 PM
- Page 6 Apr 12-1010 AM
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- Page 8 Apr 5-636 PM
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- Page 10 gridlines
- Page 11 Apr 17-216 PM
- Page 12 Mar 19-540 AM
- Page 13 Mar 31-138 PM
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- Page 17 Apr 24-150 PM
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- Page 31 Apr 17-216 PM
- Page 32 Apr 17-216 PM
- Page 33 Apr 17-233 PM
- Page 34 Apr 17-229 PM
- Page 35 Apr 26-911 AM
- Attachments Page 1
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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
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 10 gridlines
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- Page 33 Apr 17-233 PM
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- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 10 gridlines
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- Attachments Page 1
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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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SMART Notebook
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- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 10 gridlines
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- Page 33 Apr 17-233 PM
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- Page 35 Apr 26-911 AM
- Attachments Page 1
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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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SMART Notebook
- Page 1 Apr 17-216 PM
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- Page 10 gridlines
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- Page 33 Apr 17-233 PM
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- Attachments Page 1
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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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SMART Notebook
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- Attachments Page 1
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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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SMART Notebook
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- Attachments Page 1
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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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SMART Notebook
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- Attachments Page 1
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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
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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SMART Notebook
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- Attachments Page 1
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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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SMART Notebook
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- Attachments Page 1
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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
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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SMART Notebook
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- Attachments Page 1
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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
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NCTM 2012 Presentation no videonotebook
Metz Reilly amp Alarcon 35
April 26 2012
Attachments
fractions1pdf
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SMART Notebook
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- Attachments Page 1
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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 =
=
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
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