manipulatives in grade 10 math robin mcateer head of math and computer science sir robert borden...
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![Page 1: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/1.jpg)
Manipulatives in Grade 10 Math
Robin McAteer
Head of Math and Computer Science
Sir Robert Borden High School
Ottawa, Ont
![Page 2: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/2.jpg)
Created by Robin McAteer for OAME 2008
WHY? Mathematical Processes• Problem Solving : conduct investigations to help
deepen their mathematical understanding• Selecting Tools : select and use a variety of
concrete, visual, and electronic learning tools…. to investigate mathematical ideas…
• Representing: create a variety of representations of mathematical ideas
• Commmunicate mathematical ideas using… a variety of appropriate representations and, and observing mathematical connections
![Page 3: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/3.jpg)
Created by Robin McAteer for OAME 2008
Why: Instructional Effectiveness
• Differentiated Instruction– Appeals to kinesthetic learners
– Provides alternate access to concepts
• Cooperative Learning– Group investigations
• Enduring Understandings– Students forget “rules”, but REMEMBER concepts
• FUN!!!!
![Page 4: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/4.jpg)
Created by Robin McAteer for OAME 2008
Barriers: Comfort Zones
)1)(1(12
1)1()12(
12)1(
2
2
22
xxxx
xxxx
xxx
![Page 5: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/5.jpg)
Created by Robin McAteer for OAME 2008
Three Main Applications
• Algebraic Connections (tiles)
• Concrete Functions (cubalinks)
• General Modeling (ex. trig)
Gun building
![Page 6: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/6.jpg)
Created by Robin McAteer for OAME 2008
Algebraic ConnectionsArea Model
Unit tile
( 1 x 1)
Area=1
X tile
( X x 1)
Area=X
X2 tile
( X x X)
Area= X2
Negative Tiles
![Page 7: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/7.jpg)
Created by Robin McAteer for OAME 2008
Multiply/Divide/Factor
)1)(1(12
1)1()12(
12)1(
2
2
22
xxxx
xxxx
xxx
4312
4312
1243
![Page 8: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/8.jpg)
Created by Robin McAteer for OAME 2008
Factoring
652 xx )3)(2( xx
Make a Rectangle, and look at the dimensions!
![Page 9: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/9.jpg)
Created by Robin McAteer for OAME 2008
Factoring Practice
352 2 xx )1)(32( xx
![Page 10: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/10.jpg)
Created by Robin McAteer for OAME 2008
Factoring Practice
762 xx )1)(7( xx
![Page 11: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/11.jpg)
Created by Robin McAteer for OAME 2008
From Tiles to Diagrams
652 xx )3)(2( xx
2x x2
x3 6
x
x2
3
![Page 12: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/12.jpg)
Created by Robin McAteer for OAME 2008
Student Work
![Page 13: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/13.jpg)
Created by Robin McAteer for OAME 2008
Student Work
![Page 14: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/14.jpg)
Created by Robin McAteer for OAME 2008
Looking Back
• Grade 7/8/9
– concept of x as a variable
- integer concepts
- subtracting is adding the opposite
- negative + positive pair makes zero
- Evaluating expressions with x
- Video 1 Evaluating Linear Expressions
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Created by Robin McAteer for OAME 2008
Completing the Square
622 xx
Make a SQUARE shape + Extras
extras
5)1( 2 x
![Page 16: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/16.jpg)
Created by Robin McAteer for OAME 2008
Practice: Completing the Square
742 xx
extras
3)2( 2 x
![Page 17: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/17.jpg)
Created by Robin McAteer for OAME 2008
More Practice
142 xx
extras
3)2( 2 x
Add ZERO group
![Page 18: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/18.jpg)
Created by Robin McAteer for OAME 2008
Student Work
![Page 19: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/19.jpg)
Created by Robin McAteer for OAME 2008
Looking Forward…
![Page 20: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/20.jpg)
Created by Robin McAteer for OAME 2008
Looking Forward
• Algebra Blocks Calculus
![Page 21: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/21.jpg)
Created by Robin McAteer for OAME 2008
Our Proof of Equivalence
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Created by Robin McAteer for OAME 2008
Equivalence / Representations
542 xx
1)2( 2 x
2)3)(1( xx
![Page 23: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/23.jpg)
Created by Robin McAteer for OAME 2008
Link to Transformations
![Page 24: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/24.jpg)
Created by Robin McAteer for OAME 2008
What are First Differences?
![Page 25: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/25.jpg)
Created by Robin McAteer for OAME 2008
First Differences
Linear? Yes / No
Height Difference Between Stacks
![Page 26: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/26.jpg)
Created by Robin McAteer for OAME 2008
More Differences
Linear? Yes / No
Height Difference Between Stacks
![Page 27: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/27.jpg)
Created by Robin McAteer for OAME 2008
What are Second Differences?
![Page 28: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/28.jpg)
Created by Robin McAteer for OAME 2008
Second Differences
Height Difference Between Stacks
Height Difference Between Stacks
First Differences are Linear
![Page 29: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/29.jpg)
Created by Robin McAteer for OAME 2008
Exercise
Exercise: In your groups, build a quadratic relation with a second difference of 1.
DO NOT USE A TABLE OF VALUES!
Hint: Could start like this
![Page 30: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/30.jpg)
Created by Robin McAteer for OAME 2008
Building Quadratic Relations
![Page 31: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/31.jpg)
Created by Robin McAteer for OAME 2008
Cubalink Parabola – Second Differences of 1
![Page 32: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/32.jpg)
Created by Robin McAteer for OAME 2008
Cubalinks Insights
• Very rich model
• Explore concavity
• Explore transformations
• Takes some time – give YOURSELF time to play
![Page 33: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/33.jpg)
Created by Robin McAteer for OAME 2008
Actual Students!
![Page 34: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/34.jpg)
Created by Robin McAteer for OAME 2008
General Modeling
θ
What is the tan ratio for θ ?
What angle gives a tan ratio of 1?
What angle gives a ratio of 1/8?
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Created by Robin McAteer for OAME 2008
Trig Ratios
θ
cos θ = 14/17
What is θ ?
![Page 36: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/36.jpg)
Created by Robin McAteer for OAME 2008
Similar Triangles
![Page 37: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/37.jpg)
Created by Robin McAteer for OAME 2008
Modeling
![Page 38: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/38.jpg)
Created by Robin McAteer for OAME 2008
MATH IS FUN!
![Page 39: Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden High School Ottawa, Ont](https://reader035.vdocuments.net/reader035/viewer/2022081512/551790475503463e368b5609/html5/thumbnails/39.jpg)
Created by Robin McAteer for OAME 2008
Contact Info
• E-mail: [email protected]
- for more information
• My Website: mrsmac.pbwiki.com
- my courses, links, sequencing, handouts
• School Site: www.srbhs.ca- manipulatives videos