manipulatives in grade 10 math robin mcateer head of math and computer science sir robert borden...
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Manipulatives in Grade 10 Math
Robin McAteer
Head of Math and Computer Science
Sir Robert Borden High School
Ottawa, Ont
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
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!!!!
Created by Robin McAteer for OAME 2008
Barriers: Comfort Zones
)1)(1(12
1)1()12(
12)1(
2
2
22
xxxx
xxxx
xxx
Created by Robin McAteer for OAME 2008
Three Main Applications
• Algebraic Connections (tiles)
• Concrete Functions (cubalinks)
• General Modeling (ex. trig)
Gun building
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
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
Created by Robin McAteer for OAME 2008
Factoring
652 xx )3)(2( xx
Make a Rectangle, and look at the dimensions!
Created by Robin McAteer for OAME 2008
Factoring Practice
352 2 xx )1)(32( xx
Created by Robin McAteer for OAME 2008
Factoring Practice
762 xx )1)(7( xx
Created by Robin McAteer for OAME 2008
From Tiles to Diagrams
652 xx )3)(2( xx
2x x2
x3 6
x
x2
3
Created by Robin McAteer for OAME 2008
Student Work
Created by Robin McAteer for OAME 2008
Student Work
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
Created by Robin McAteer for OAME 2008
Completing the Square
622 xx
Make a SQUARE shape + Extras
extras
5)1( 2 x
Created by Robin McAteer for OAME 2008
Practice: Completing the Square
742 xx
extras
3)2( 2 x
Created by Robin McAteer for OAME 2008
More Practice
142 xx
extras
3)2( 2 x
Add ZERO group
Created by Robin McAteer for OAME 2008
Student Work
Created by Robin McAteer for OAME 2008
Looking Forward…
Created by Robin McAteer for OAME 2008
Looking Forward
• Algebra Blocks Calculus
Created by Robin McAteer for OAME 2008
Our Proof of Equivalence
Created by Robin McAteer for OAME 2008
Equivalence / Representations
542 xx
1)2( 2 x
2)3)(1( xx
Created by Robin McAteer for OAME 2008
Link to Transformations
Created by Robin McAteer for OAME 2008
What are First Differences?
Created by Robin McAteer for OAME 2008
First Differences
Linear? Yes / No
Height Difference Between Stacks
Created by Robin McAteer for OAME 2008
More Differences
Linear? Yes / No
Height Difference Between Stacks
Created by Robin McAteer for OAME 2008
What are Second Differences?
Created by Robin McAteer for OAME 2008
Second Differences
Height Difference Between Stacks
Height Difference Between Stacks
First Differences are Linear
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
Created by Robin McAteer for OAME 2008
Building Quadratic Relations
Created by Robin McAteer for OAME 2008
Cubalink Parabola – Second Differences of 1
Created by Robin McAteer for OAME 2008
Cubalinks Insights
• Very rich model
• Explore concavity
• Explore transformations
• Takes some time – give YOURSELF time to play
Created by Robin McAteer for OAME 2008
Actual Students!
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?
Created by Robin McAteer for OAME 2008
Trig Ratios
θ
cos θ = 14/17
What is θ ?
Created by Robin McAteer for OAME 2008
Similar Triangles
Created by Robin McAteer for OAME 2008
Modeling
Created by Robin McAteer for OAME 2008
MATH IS FUN!
Created by Robin McAteer for OAME 2008
Contact Info
• E-mail: robin.mcateer@ocdsb.ca
- for more information
• My Website: mrsmac.pbwiki.com
- my courses, links, sequencing, handouts
• School Site: www.srbhs.ca- manipulatives videos
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