Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden...

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


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!!!!


Barriers: Comfort Zones

)1)(1(12

1)1()12(

12)1(

2

2

22

xxxx

xxxx

xxx


Three Main Applications

• Algebraic Connections (tiles)

• Concrete Functions (cubalinks)

• General Modeling (ex. trig)

Gun building


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


Multiply/Divide/Factor

)1)(1(12

1)1()12(

12)1(

2

2

22

xxxx

xxxx

xxx

4312

4312

1243


Factoring

652 xx )3)(2( xx

Make a Rectangle, and look at the dimensions!


Factoring Practice

352 2 xx )1)(32( xx


Factoring Practice

762 xx )1)(7( xx


From Tiles to Diagrams

652 xx )3)(2( xx

2x x2

x3 6

x

x2

3


Student Work


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

http://www.sirrobertbordenhs.ocdsb.ca/math/math.htm


Completing the Square

622 xx

Make a SQUARE shape + Extras

extras

5)1( 2 x


Practice: Completing the Square

742 xx

extras

3)2( 2 x


More Practice

142 xx

extras

3)2( 2 x

Add ZERO group


Student Work


Looking Forward…


Looking Forward

• Algebra Blocks Calculus


Our Proof of Equivalence


Equivalence / Representations

542 xx

1)2( 2 x

2)3)(1( xx


Link to Transformations


What are First Differences?


First Differences

Linear? Yes / No

Height Difference Between Stacks


More Differences

Linear? Yes / No



What are Second Differences?


Second Differences



First Differences are Linear


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


Building Quadratic Relations


Cubalink Parabola – Second Differences of 1


Cubalinks Insights

• Very rich model

• Explore concavity

• Explore transformations

• Takes some time – give YOURSELF time to play


Actual Students!


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?


Trig Ratios

θ

cos θ = 14/17

What is θ ?


Similar Triangles


Modeling


MATH IS FUN!


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

Manipulatives in Grade 10 Math Robin McAteer Head of Math and Computer Science Sir Robert Borden...

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