Making Math Contextual, Visual and Concrete with Technology

Making Math Contextual, Visual, and Concrete

kylep.ca/curiousmath1p khmath.com

tapintoteenminds.com/uwin

@MathletePearcewww.tapintoteenminds.com

http://khmath.com

https://tapintoteenminds.com/uwin

ABOUT MEBelle River, ONMath Teacher / Instructional Coach at Greater Essex County District School BoardApple Distinguished EducatorApple Education TrainerGoogle Certified Teacher



PROBLEM


ENGAGED YET?


ENGAGED YET?



ENGAGEMENT“There are several types/categories of engagement – academic, cognitive, intellectual, institutional, emotional, behavioural, social, psychological, to name a few.”

Parsons, J., & Taylor, L. (2011). Student Engagement: What do we know and what should we do? University Partners (pp 3). University of Alberta.

[email protected]


Math Class


MathClass

1. Take Up Homework


Math Class


2. Definitions, Formulae, Procedures/Algorithms

1. Take Up Homework

Math Class


3. Examples


1. Take Up Homework

Math Class


4. Homework

3. Examples


1. Take Up Homework

Math Class

Example 1(a) List all combinations of three out of the four colours: red, blue, white, and green.


Two Groups of Students

Result



Good atMath

NotGood at

Math



Not

Good at Memorization

Not Good at Memorization













Struggle withunfamiliarproblems


A VISION FOR MATHEMATICS

EDUCATION

VISION


Contextual

VISION


Visual and ConcreteContextual

VISION


Interconnected


VISION


Interconnected


CONFIDENCE


Interconnected


PERCEPTION


Interconnected


STUDENT SUCCESS

Change


Change


Immune to

http://www.vancouversun.com/life/From+geek+chic/6296707/story.htmlPhotograph by: Maggie Wong


@MathletePearcewww.tapintoteenminds.com https://mathstrategies.wordpress.com/tag/numb3rs/

http://www.wykamath.com/

entertainment

engagement=



algorithm

what’sthis?





algorithm

s

[email protected]

algorithms

[email protected]

algorithms

engagement=


algorithms

engagement=

understandingor


iBooks


iBooks Author


iBooks Author

Available on iTunes and…

iBooks

3 Act Math Tasks

iTunes U

kylep.ca/[email protected]

kylep.ca/curiousmath1p

iTunes U



iTunes U



iTunes U


TapFollow the Math

Twitter Blogosphere


playkh.complaykh.comGo to:

http://playkh.com

http://playkh.com



Tap


Tap


Tap


Tap, then wait.


Tap


QUESTIONS

-Part Math Lesson

How many packs of paper would it take to reach the ceiling?

4. Consolidate3. Connections

2. InquiryOUR FIRST QUESTION:QUESTIONSPREDICTIONS


PREDICTIONS


REQUEST INFORMATION


How many packs of paper would it take to reach the ceiling?

4. Consolidate3. Connections 273

cm

24.7

5 cm

QUESTION


QUESTION


Mr. Pearce stacks reams of paper vertically against a wall. If the height of 5 reams is 24.75 cm and the height of the room is

273 cm, how many reams of paper would he need to reach the ceiling?

Answer Key: 55.15

QUESTION



273 cm, how many reams of paper would he need to reach the ceiling?FUN, RIGHT?

Answer Key: 55.15QUESTION



273 cm, how many reams of paper would he need to reach the ceiling?

Answer Key: 55.15


Extensions


Interconnected


VISION


Interconnected


VISION



Interconnected


VISION

The Next Day…



How tall is the table?

4.95 cm130

.75

cm

?

130

.75

cm


4.95 cm

130

.75

cm

?

packs5

130

.75

cm



4.95 cm

130

.75

cm

?

packs5 x = 24.75 cm

130

.75

cm 24.75 cm



4.95 cm

130

.75

cm

?

packs5 x = 24.75 cm

130.75 cm24.75 cm-



4.95 cm

130

.75

cm

?

packs5 x = 24.75 cm

130.75 cm24.75 cm-

106 cm106 cm



4.95 cm

130

.75

cm packs5 x = 24.75 cm

130.75 cm24.75 cm-

106 cm

106

cm



-Part Math Lesson

4. Consolidate

Minds On

Inquiry2. -Part Math Lesson

Making Connections



4.95 cm

130

.75

cm packs5 x = 24.75 cm

130.75 cm24.75 cm-

106 cm

106

cm

How tall is the table?4.95 cmpacks5 x = 24.75 cm


130

.75

cm

130

.75

4.95 cmx = 24.75 cm

130.75 cm 24.75 cm 106 cm= +

packs5 4.95 cmx = 24.75 cmpacks5

106

cm


x = 24.75 cm

130.75 cm 24.75 cm 106 cm= +

packs5

4.95 cm (

= 24.75 cm

packs5

130.75 cm 24.75 cm 106 cm= +

4.95 cm

)( )

130

.75

cm

130

.75

106

cm


x = 24.75 cm

130.75 cm 24.75 cm 106 cm= +

packs5 = 24.75 cm

130.75 cm

24.75 cm

106 cm= +

4.95 cm

4.95 cm ( packs5 )( )

130

.75

cm

130

.75

106

cm


x = 24.75 cm

130.75 cm 24.75 cm 106 cm= +

packs5 = 24.75 cm

130.75 cm

24.75 cm

106 cm= +

height initial height of table= +number

of packs

4.95 cm

height of a pack )

130

.75

cm

130

.75

106

cm ( )(

4.95 cm ( packs5 )( )


x = 24.75 cm

130.75 cm 24.75 cm 106 cm= +

packs5 = 24.75 cm

130.75 cm

24.75 cm

106 cm= +

height initial height of table= +

4.95 cm13

0.7

5 c

m13

0.7

5

106

cm number

of packsheight of

a pack )( )(

4.95 cm ( packs5 )( )


x = 24.75 cm

130.75 cm 24.75 cm 106 cm= +

5 = 24.75 cm

130.75 cm

24.75 cm

106 cm= +


4.95 cm13

0.7

5 c

m13

0.7

5

106

cm

packs

number of packs

height of a pack )( )(

4.95 cm ( packs5 )( )


x = 24.75 cm

130.75 cm 24.75 cm 106 cm= +

5 = 24.75 cm

130.75 cm

24.75 cm

106 cm= +


4.95 cm13

0.7

5 c

m13

0.7

5

106

cm

packs

number of packs


4.95 cm ( packs5 )( )


x = 24.75 cm

130.75 cm 24.75 cm 106 cm= +

5 = 24.75 cm

130.75 cm

24.75 cm

106 cm= +


4.95 cm13

0.7

5 c

m13

0.7

5

106

cm

packs

number of packs


4.95 cm ( packs5 )( )


x = 24.75 cm

130.75 cm 24.75 cm 106 cm= +

5 = 24.75 cm

130.75 cm

24.75 cm

106 cm= +


4.95 cm13

0.7

5 c

m13

0.7

5

106

cm

packs

y b= +m x

number of packs


4.95 cm ( packs5 )( )

Minds On


-Part Math Lesson13

0.7

5 c

m

packs of paper heightIndependent Dependent

5packs13

0.7

55

4.95 cm



How tall is the table?13

0.7

5 c

m


5packs13

0.7

55

4.95 cmSlope



0.7

5 c

m


5packs

130.755

4.95 cmSlope



0.7

5 c

m


5packs

130.755

4.95 cmSlope )x y(



0.7

5 c

mpacks of

paper height5 packs130.7554.95 cm

Slope

)x y

( ,In

itia

l Val

ue

y -i

nter

cep

t



0.7

5 c

mpacks of

paper height

130.7554.95 cmSlope

)x y

( ,

y b= +mxInit

ial V

alue

y -i

nter

cep

t

5 packs5130.7554.95

130

.75



0.7

5 c

mpacks of

paper height

130.7554.95 cmSlope

)x y

( ,

y b= +mxInit

ial V

alue

y -i

nter

cep

t

5 packs5

130.75

5

130.75

b+

4.95



0.7

5 c

mpacks of

paper height

130.7554.95 cmSlope

)x y

( ,

y b= +m xInit

ial V

alue

y -i

nter

cep

t

5 packs5

130.75

5

130.75 = )()( b+

4.95



0.7

5 c

mpacks of

paper height

130.7554.95 cmSlope

)x y

( ,

y b= +m xInit

ial V

alue

y -i

nter

cep

t

5 packs

5130.75 5130.75 =)(

)()( b+

55)(4.95



0.7

5 c

mpacks of

paper height

130.7554.95 cmSlope

)x y

( ,

y = mInit

ial V

alue

y -i

nter

cep

t

5 packs

130.75130.75 =)(

b+130.75 = b+x55 )(

)(55)(4.95

b+ b+54.95

54.95



0.7

5 c

mpacks of

paper height

130.7554.95 cmSlope

)x y

( ,

y = mInit

ial V

alue

y -i

nter

cep

t

5 packs

130.75130.75 =)(

130.75 = b+24.75b+ b+

x55 )(

)(55)(4.95

b+ b+



0.7

5 c

mpacks of

paper height

130.7554.95 cmSlope

)x y

( ,

y = mInit

ial V

alue

y -i

nter

cep

t

5 packs

130.75130.75 =)(

130.75 = b+b+ b+

x55 )(

)(55)(4.95

b+ b+

54.9524.7524.75-



0.7

5 c

mpacks of

paper height

130.7554.95 cmSlope

)x y

( ,

y = mInit

ial V

alue

y -i

nter

cep

t

5 packs

130.75130.75 =)(

130.75 = b+b+ b+

x55 )(

)(55)(4.95

b+ b+

54.9524.7524.75-24.75-

b+



0.7

5 c

mpacks of

paper height

130.7554.95 cm

Slope )x y

( ,

y = mIn

itia

l Val

ue

y -i

nter

cep

t

5 packs

130.75130.75 =)(

130.75 = b+b+ b+

x55 )(

)(55)(4.95

b+ b+

54.9524.7524.75-24.75-

106 = b0 +

In the past…



4. Consolidate


ExampleConsolidation

ExampleConsolidation


Task #1:

Minds On




5packs

130.755( )x y4.95 cmSlope

2 1 51 513

0.7

5 c

m

Minds On



packs of paper heightIndependent Dependentpacks

( )x ySlope

2 1 51 51


What does the point (1, 5) represent in the context of the Stacking Paper task?


( )x ySlope

2 1 51 51


What does the slope of 2 represent in the context of the Stacking Paper task?


( )x ySlope

2 1 51 51



( )x y1


2 1 51 51



( )x y

2 1 51 5

1

5

1

52

y = m x + b

2

( )1 pack



( )x y

2 1 51 51 55

2 2( )

1

1y = m + b( )

51 pack



( )x y

2 1 51 51

5

5

2 2( )

1

1y = m + b( )51 pack

1pack



( )x y

2 1 51 51

5

5

2 2()

1

1y = + b()5 2

1pack



( )x y

2 1 51 51

5

5

2()

1

1y = + b()5 2-2-2

2

1pack



( )x y

2 1 51 51

5

5

2()

1

1y = +b()5 2-2-22

3

3

1pack



( )x y

2 1 51 51

5

5

2()

1

1y = +b()5 2-2-22

3

3y = 2x + 3








Two Groups of StudentsStruggle with

unfamiliarproblems

Good at Math

Not Good at Math



Good at Math

Not Good at Math



Struggle withunfamiliarproblemsDigging Deeper…


142

.75

cm

100

.75

cm


?142

.75

100

.75

?


142

.75

cm

100

.75

cm


?

142.75 100.75

?

-


142

.75

cm

100

.75

cm


?

142.75 100.75

?

- = 42 cm42 cm


142

.75

cm

100

.75

cm


?

142.75 100.75

?

- = 42 cm42

cm


142

.75

cm

100

.75

cm


?

142.75 100.75

?

- = 42 cm42

cm

12p

ack

s

5p

ack

s

12p

ack

s

5p

ack

s


142

.75

cm

100

.75

cm


?

142.75 100.75

?

- = 42 cm42

cm

12p

ack

s

5p

ack

s12 packs 5 packs-


142

.75

cm

100

.75

cm


?

142.75 100.75

?

- = 42 cm42

cm

12p

ack

s

5p

ack

s12 packs 5 packs- = 7 packs7 packs


142

.75

cm

100

.75

cm


?

142.75 100.75

?

- = 42 cm42

cm

12p

ack

s

5p

ack

s12 packs 5 packs- = 7 packs

7p

ack

s


142

.75

cm

100

.75

cm


?

142.75 100.75

?

- = 42 cm42

cm

12p

ack

s

5p

ack

s12 packs 5 packs- = 7 packs

7p

ack

s


142

.75

cm

100

.75

cm


?

142.75 100.75

?

-=

42 cm42

cm

12p

ack

s

5p

ack

s12 packs 5 packs-

=7 packs

7p

ack

s


142

.75

cm

100

.75

cm


?

142.75 100.75

?

-=

42 cm42

cm

12p

ack

s

5p

ack

s12 packs 5 packs-

=

7 packs7

pa

cks

6 cm

1 packs=

6 cm


142

.75

cm

100

.75

cm


?

142.75 100.75

?

-=

42 cm

42

cm

12p

ack

s

5p

ack

s12 packs 5 packs-

=

7 packs7

pa

cks

6 cm

1 packs

= 6 cmSlope m,


142

.75

cm

100

.75

cm

??

42

cm

12p

ack

s

5p

ack

s7

pa

cks

= 6 cmSlope m,packs of

paper height

)x y

( ,

100

.75

5

100.75


142

.75

cm

100

.75

cm

??

42

cm

12p

ack

s

5p

ack

s7

pa

cks


paper height

5 )x y

( ,5 100.75

100

.75

5

6

5

100.75


142

.75

cm

100

.75

cm

??

42

cm

12p

ack

s

5p

ack

s7

pa

cks


paper height

5 )x y

( ,5

y b= +mx

100.75

100

.75

5

6

b+

100.75


142

.75

cm

100

.75

cm

??

42

cm

12p

ack

s

5p

ack

s7

pa

cks


paper height

5 )x y

( ,

5

y b= +mx100.75 6100.75 = )( )( 5 b+

100.75


142

.75

cm

100

.75

cm

??

42

cm

12p

ack

s

5p

ack

s7

pa

cks


paper height

5 )x y

( ,

5

y b= +mx100.75 6100.75 = )( )( 5 b+

5100.75 6100.75 = b+30

100.75


142

.75

cm

100

.75

cm

??

42

cm

12p

ack

s

5p

ack

s7

pa

cks


paper height

5 )x y

( ,

5

y b= +mx100.75 6100.75 = )( )( 5 b+

5100.75 6100.75 = b+3030-

100.75


142

.75

cm

100

.75

cm

??

42

cm

12p

ack

s

5p

ack

s7

pa

cks


paper height

5 )x y

( ,

5

y b= +mx100.75 6100.75 = )( )( 5 b+

5100.75 6100.75 = b+3030-30-

100.75


142

.75

cm

100

.75

cm

??

42

cm

12p

ack

s

5p

ack

s7

pa

cks


paper height

5 )x y

( ,

5

y b= +mx100.75 6100.75 = )( )( 5 b+

5100.75 6100.75 = b+3030-30-

70.75 = b0 +


142

.75

cm

100

.75

cm

?142

.75

100

.75

?

packs of paper height

)x y

( ,packs of

paper height

)x y

( ,12

pa

cks

5p

ack

s

142

.75

100

.75

12 512 5


142

.75

cm

100

.75

cm

?142

.75

100.75

?


)x y

( ,packs of

paper height

)x y

( ,12

pa

cks 5

pa

cks

142

.75

100.7512

512 5


142

.75

cm

100

.75

cm

?142

.75

100.75

?


)x y

( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142

.75

100.7512

512 5

POINT 1


142

.75

cm

100

.75

cm

?

142.75100.75

?


)x y

( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 125 12

5

POINT 1


142

.75

cm

100

.75

cm

?

142.75100.75

?


)( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 125 12

5

POINT 1 POINT 2

x2 y2

142

.75

100

.75

12 5

142.75100.755 12x1 y1 x2 y2


142

.75

cm

100

.75

cm

?

142.75100.75

?


)( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 125 12

5

POINT 1 POINT 2

x2 y212 5

142.75100.755 12x1 y1 x2 y2

142.75 100.75- = 42 cm

pa

cks

pa

cks


142

.75

cm

100

.75

cm

?

142.75100.75

?


)( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 12

512

5

POINT 1 POINT 2

x2 y2142.75100.75

x1 y1 x2 y2

142.75 100.75- = 42 cm12 packs 5 packs- = 7 packs


142

.75

cm

100

.75

cm

?

142.75100.75

?


)( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 12

512

5

POINT 1 POINT 2

x2 y2142.75100.75

x1 y1 x2 y2

142.75 100.75-=

42 cm12 packs 5 packs-

=7 packs


142

.75

cm

100

.75

cm

?

142.75100.75

?


)( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 12

512

5

POINT 1 POINT 2

x2 y2142.75100.75

x1

y1

x2

y2

142.75 100.75- 42 cm12 packs 5 packs- 7 packs

==


142

.75

cm

100

.75

cm

?

142.75100.75

?


)( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 12

512

5

POINT 1 POINT 2

x2 y2142.75100.75

x1

y1

x2

y2

142.75 100.75- 42 cm12 packs 5 packs- 7 packs

--

==


142

.75

cm

100

.75

cm

?

142.75100.75

?


)( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 12

512

5

POINT 1 POINT 2

x2 y2142.75100.75

x1

y1

x2

y2

142.75 100.75- 42 cm

12 packs 5 packs- 7 packs

--

Slope m, =

= ==6 cm

1 packs==


142

.75

cm

100

.75

cm

?

142.75100.75

?


)( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 12

512

5

POINT 1 POINT 2

x2 y2142.75100.75

x1

y1

x2

y2

142.75 100.75- 42 cm


--

Slope m, =

= ==6 cm

1 packs==


142

.75

cm

100

.75

cm

?

142.75100.75

?


)( ,packs of

paper height

)x1 y1

( ,12

pa

cks 5

pa

cks

142.75100.75 12

512

5

POINT 1 POINT 2

x2 y2142.75100.75

x1

y1

x2

y2

142.75 100.75- 42 cm


--

Slope m, =

= ==6 cm

1 packs==


142

.75

cm

100

.75

cm

?

142.75100.75

?

)( ,)x1 y1

( ,12

pa

cks

5

pa

cks

142.75100.75 12

5

POINT 1 POINT 2

x2 y2142.75100.75

Slope m, = 6 cm

y b= +mx b+


142

.75

cm

100

.75

cm

?

142.75100.75

?

)( ,)x1 y1

( ,12

pa

cks

5

pa

cks

142.75100.75 12

5

POINT 1 POINT 2

x2 y2142.75100.75

Slope m, = 6 cm

y b= +mx b+NOW, SUBSTITUTE

SLOPE (m) and EITHER POINT TO FIND INITIAL VALUE/Y-INTERCEPT!

b+


142

.75

cm

100

.75

cm

142.75100.75 )( ,)x1 y1

( ,12

pa

cks

5

pa

cks

142.75100.75 12

5

POINT 1 POINT 2

x2 y2142.75100.75

Slope m, = 6 cm

y b= +mx b+

b

12 142.756


142

.75

cm

100

.75

cm

142.75100.75 )( ,)x1 y1

( ,12

pa

cks

5

pa

cks

142.75100.75 12

5

POINT 1 POINT 2

x2 y2142.75100.75

Slope m, = 6 cm

y b= +mx b+

b12142.75 = 6 )( )( b+


142

.75

cm

100

.75

cm

142.75100.75 )( ,)x1 y1

( ,12

pa

cks

5

pa

cks

142.75100.75 12

5

POINT 1 POINT 2

x2 y2142.75100.75

Slope m, = 6 cm

y b= +mx b+

b12142.75 = 6 )( )( b+

12142.75 = 6 b+72


142

.75

cm

100

.75

cm

142.75100.75 )( ,)x1 y1

( ,12

pa

cks

5

pa

cks

142.75100.75 12

5

POINT 1 POINT 2

x2 y2142.75100.75

Slope m, = 6 cm

y b= +mx b+

b12142.75 = 6 )( )( b+

12142.75 = 6 b+7272-

72

6


142

.75

cm

100

.75

cm

142.75100.75 )( ,)x1 y1

( ,12

pa

cks

5

pa

cks

142.75100.75 12

5

POINT 1 POINT 2

x2 y2142.75100.75

Slope m, = 6 cm

y b= +mx b+

b12142.75 = 6 )( )( b+

12142.75 = 6 b+7272-72-

70.75 = b0 +

6


142

.75

cm

100

.75

cm

142.75100.75 )( ,)x1 y1

( ,12

pa

cks

5

pa

cks

142.75100.75 12

5

POINT 1 POINT 2

x2 y2142.75100.75

Slope m, = 6 cm

y b= +mx b+b 70.75b6

Therefore, the equation of the line is:

Interconnected


VISION

@MathletePearcekylep.ca/curiousmath1p

Interconnected

Visual and ConcreteContextualTASKS

CONNECTIONS

VISION


Interconnected


INQUIRY

CONNECTIONS

VISION


Interconnected


CONNECTIONS

INQUIRYCONSOLIDATE VISION


Interconnected


CONNECTIONS

INQUIRYCONSOLIDATE


VISION


“He’s Crazy” thing.every


“He’s Crazy” thing.every


one thing.


step.one


one vision.

thing.


one


AMAZINGthing .s

NEXT


?

BIGGERand

better

[email protected]

thingNEXT

what will be your

AMAZING

BIGGERbetter

#MTBoS

The Best Teacher PD is FREE on Twitter

Follow the Math Twitter Blogosphere


EXIT SURVEY



Thank you for your feedback!

and session resources


Making Math Contextual, Visual, and Concrete

kylep.ca/curiousmath1p khmath.com



http://khmath.com
