Calgary Regional Consortium Webinar - Making Math Contextual, Visual and Interconnected with Technology

Making Math
Contextual, Visual,
and Concrete With
Technology
@MathletePearcewww.tapintoteenminds.com

ABOUT ME
Belle River, ON
Math Teacher / Instructional Coach at
Greater Essex County District School
Board
Apple Distinguished Educator
Apple Education Trainer
Google Certiﬁed Teacher

Norway
AND
Canada:
An international
partnership pursuing
equity in mathematics
education
COMM-212 2015-03

PROBLEM

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.

ENGAGED YET?

Transformation
Enhancement

Math
Class

4. Homework
3. Examples
2. Deﬁnitions, Formulae, Procedures/Algorithms
1. Take Up Homework
Math Class

(b) Considering the colour combinations
as arrangements with indistinguishable
elements, check the number of
combinations of three out of four items.

My Result:
Successful
in Math
Unsuccessful
in Math

A Perception Develops
Good at
Math
Poor at
Math

Conﬁdent in
Math Class
Doubtful in
Math Class
Students Feel…

Enjoying
Mathematics
Disliking
Mathematics
This May Lead To…

Willingness
to Engage In
Math Class
Unwillingness
to Engage In
Math Class
This May Lead To…

Moving Ahead Falling Behind
This May Lead To…

Two Groups of Students
Good at
Math
Poor at
Math

Struggle with
unfamiliar
problems
Good at
Math
Poor at
Math

A VISION FOR
MATHEMATICS
EDUCATION
VISION

Contextual
VISION

Visual and Concrete
Contextual
VISION

Interconnected
Visual and Concrete
Contextual
VISION

Interconnected
Visual and Concrete
Contextual
CONFIDENCE

Interconnected
Visual and Concrete
Contextual
PERCEPTION

Interconnected
Visual and Concrete
Contextual
STUDENT
SUCCESS

Change

Change
Immune to

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

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

entertainment
engagement
=

algorithm

algorithms
engagement
=

algorithms
engagement
=
understanding
or

3 Act Math Tasks
iTunes U
kylep.ca/curiousmath1p

playkh.complaykh.comGo to:

wiz623

Tap
wiz623

Tap, then wait.
wiz623

How many packs of paper would it take to
reach the ceiling?
OUR FIRST QUESTION:
wiz623

How many packs of paper would
it take to reach the ceiling?
273cm
24.75cm
wiz623

3 Act Math Tasks
kylep.ca/curiousmath1p
iTunes U
tapintoteenminds.com

How tall is the table?
4.95 cm
130.75cm
?
130.75cm
wiz623

4.95 cm
130.75cm
?
packs5
130.75cm How tall is the table?

4.95 cm
130.75cm
?
packs5 x = 24.75 cm
130.75cm
24.75 cm

4.95 cm
130.75cm
?
packs5 x = 24.75 cm
130.75 cm
24.75 cm-

4.95 cm
130.75cm
?
packs5 x = 24.75 cm
130.75 cm
24.75 cm-
106 cm106 cm

4.95 cm
130.75cm
packs5 x = 24.75 cm
130.75 cm
24.75 cm-
106 cm
106cm

Making Connections

4.95 cm
130.75cm
packs5 x = 24.75 cm
130.75 cm
24.75 cm-
106 cm
106cm
4.95 cmpacks5 x = 24.75 cm

130.75cm130.75
4.95 cmx = 24.75 cm
130.75 cm 24.75 cm 106 cm= +
packs5 4.95 cmx = 24.75 cmpacks5
106cm

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
)( )
130.75cm130.75
106cm

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 )( )
130.75cm130.75
106cm

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
)
130.75cm130.75
106cm ( )(
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= +
of table
= +
4.95 cm
130.75cm130.75
106cm 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= +
of table
= +
4.95 cm
130.75cm130.75
106cm
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= +
of table
= +
4.95 cm
130.75cm130.75
106cm
packs
y b= +mx
number
of packs
height of
a pack
)( )(
4.95 cm ( packs5 )( )

130.75cm
packs of paper height
Independent Dependent
5
packs130.75 5
4.95 cm

How tall is the table?130.75cm
5
packs130.75 5
4.95 cm
Slope

5
packs
130.755
4.95 cm
Slope

5
packs
130.755
4.95 cm
Slope
)x y
(

130.75cm packs of
paper
height
5 packs
130.7554.95 cm
Slope
)x y
( ,
InitialValue
y-intercept

130.75cm packs of
paper
height
130.7554.95 cm
Slope
)x y
( ,
y b= +mx
InitialValue
y-intercept
5 packs5
130.7554.95
130.75

130.75cm packs of
paper
height
130.7554.95 cm
Slope
)x y
( ,
y b= +mx
InitialValue
y-intercept
5 packs5
130.75
5
130.75
b+
4.95

130.75cm packs of
paper
height
130.7554.95 cm
Slope
)x y
( ,
y b= +m x
InitialValue
y-intercept
5 packs5
130.75
5
130.75 = )(
)( b+
4.95

130.75cm packs of
paper
height
130.7554.95 cm
Slope
)x y
( ,
y b= +m x
InitialValue
y-intercept
5 packs
5130.75 5130.75 =
)(
)(
)( b+
55)(4.95

130.75cm packs of
paper
height
130.7554.95 cm
Slope
)x y
( ,
y = m
InitialValue
y-intercept
5 packs
130.75130.75 =
)(
b+130.75 = b+
x
55 )(
)(
55)(4.95
b+ b+
54.95

54.95
130.75cm packs of
paper
height
130.7554.95 cm
Slope
)x y
( ,
y = m
InitialValue
y-intercept
5 packs
130.75130.75 =
)(
130.75 = b+24.75
b+ b+
x
55 )(
)(
55)(4.95
b+ b+

130.75cm packs of
paper
height
130.7554.95 cm
Slope
)x y
( ,
y = m
InitialValue
y-intercept
5 packs
130.75130.75 =
)(
130.75 = b+
b+ b+
x
55 )(
)(
55)(4.95
b+ b+
54.9524.75
24.75-

130.75cm packs of
paper
height
130.7554.95 cm
Slope
)x y
( ,
y = m
InitialValue
y-intercept
5 packs
130.75130.75 =
)(
130.75 = b+
b+ b+
x
55 )(
)(
55)(4.95
b+ b+
54.9524.75
24.75-24.75-
b+

130.75cm
packs of
paper
height
130.755
4.95 cm
Slope
)x y
( ,
y = m
InitialValue
y-intercept
5 packs
130.75130.75 =
)(
130.75 = b+
b+ b+
x
55 )(
)(
55)(4.95
b+ b+
54.9524.75
24.75-24.75-
106 = b0 +

In the past…

Example
Consolidation
Task #1:

5
packs
130.755( )x y4.95 cm
Slope
2 1515
130.75cm

packs
( )x y
2 15151

wiz623
What does the point (1, 5) represent in
the context of the Stacking Paper task?
packs
( )x y
2 15151

wiz623
What does the slope of 2 represent in
the context of the Stacking Paper task?
packs
( )x y
2 15151

wiz623
packs
( )x y
2 15151

packs
( )x y
2 1515
1
5
1
52
y = m x + b
2
( )
1 pack

packs
( )x y
2 15151 55
2 2( )
1
1y = m + b( )
5
1 pack

packs
( )x y
2 15151
5
5
2 2( )
1
1y = m + b( )5
1 pack

1pack
packs
( )x y
2 15151
5
5
2 2()
1
1y = + b()5 2

1pack
packs
( )x y
2 15151
5
5
2()
1
1y = + b()5 2
-2-2
2

1pack
packs
( )x y
2 15151
5
5
2()
1
1y = +b()5 2-2-22
3
3

1pack
packs
( )x y
2 15151
5
5
2()
1
1y = +b()5 2-2-2
2
3
3
y = 2x + 3

Poor at
Math
Struggle with
unfamiliar
problems
Good at
Math

Digging Deeper…

142.75cm
100.75cm
?
142.75
100.75?
wiz623

142.75cm
100.75cm
?
142.75 100.75
?
-

142.75cm
100.75cm
?
142.75 100.75
?
- = 42 cm42 cm

142.75cm
100.75cm
?
142.75 100.75
?
- = 42 cm
42cm

142.75cm
100.75cm
?
142.75 100.75
?
- = 42 cm
42cm
12packs
5
packs
12packs
5
packs

142.75cm
100.75cm
?
142.75 100.75
?
- = 42 cm
42cm
12packs
5
packs
12 packs 5 packs-

142.75cm
100.75cm
?
142.75 100.75
?
- = 42 cm
42cm
12packs
5
packs
12 packs 5 packs- = 7 packs7 packs

142.75cm
100.75cm
?
142.75 100.75
?
- = 42 cm
42cm
12packs
5
packs
12 packs 5 packs- = 7 packs
7packs

142.75cm
100.75cm
?
142.75 100.75
?
-
=
42 cm
42cm
12packs
5
packs
12 packs 5 packs-
=
7 packs
7packs

142.75cm
100.75cm
?
142.75 100.75
?
-
=
42 cm
42cm
12packs
5
packs
12 packs 5 packs-
=
7 packs
7packs
6 cm
1 packs
=
6 cm

142.75cm
100.75cm
?
142.75 100.75
?
-
=
42 cm
42cm
12packs
5
packs
12 packs 5 packs-
=
7 packs
7packs
6 cm
1 packs
= 6 cmSlope m,

142.75cm
100.75cm??
42cm
12packs
5
packs7packs
= 6 cmSlope m,
packs of
paper
height
)x y
( ,
100.75
5

100.75
142.75cm
100.75cm??
42cm
12packs
5
packs7packs
= 6 cmSlope m,
packs of
paper
height
5 )x y
( ,5 100.75
100.75
5
6
5

100.75
142.75cm
100.75cm??
42cm
12packs
5
packs7packs
= 6 cmSlope m,
packs of
paper
height
5 )x y
( ,5
y b= +mx
100.75
100.75
5
6
b+

100.75
142.75cm
100.75cm??
42cm
12packs
5
packs7packs
= 6 cmSlope m,
packs of
paper
height
5 )x y
( ,
5
y b= +mx
100.75 6100.75 = )( )(5 b+

100.75
142.75cm
100.75cm??
42cm
12packs
5
packs7packs
= 6 cmSlope m,
packs of
paper
height
5 )x y
( ,
5
y b= +mx
100.75 6100.75 = )( )(5 b+
5100.75 6100.75 = b+30

100.75
142.75cm
100.75cm??
42cm
12packs
5
packs7packs
= 6 cmSlope m,
packs of
paper
height
5 )x y
( ,
5
y b= +mx
100.75 6100.75 = )( )(5 b+
5100.75 6100.75 = b+30
30-

100.75
142.75cm
100.75cm??
42cm
12packs
5
packs7packs
= 6 cmSlope m,
packs of
paper
height
5 )x y
( ,
5
y b= +mx
100.75 6100.75 = )( )(5 b+
5100.75 6100.75 = b+30
30-30-

100.75
142.75cm
100.75cm??
42cm
12packs
5
packs7packs
= 6 cmSlope m,
packs of
paper
height
5 )x y
( ,
5
y b= +mx
100.75 6100.75 = )( )(5 b+
5100.75 6100.75 = b+30
30-30-
70.75 = b0 +

142.75cm
100.75cm?
142.75
100.75?
packs of
paper
height
)x y
( ,
packs of
paper
height
)x y
( ,12packs
5
packs
142.75
100.75
12
5
12
5

142.75cm
100.75cm?
142.75
100.75
?
packs of
paper
height
)x y
( ,
packs of
paper
height
)x y
( ,12packs
5
packs
142.75
100.75
12
5
12
5

142.75cm
100.75cm?
142.75
100.75
?
packs of
paper
height
)x y
( ,
packs of
paper
height
)x1 y1
( ,12packs
5
packs
142.75
100.75
12
5
12
5
POINT 1

142.75cm
100.75cm?
142.75100.75
?
packs of
paper
height
)x y
( ,
packs of
paper
height
)x1 y1
( ,12packs
5
packs
142.75100.75 125 12
5
POINT 1

142.75cm
100.75cm?
142.75100.75
?
packs of
paper
height
)( ,
packs of
paper
height
)x1 y1
( ,12packs
5
packs
142.75100.75 125 12
5
POINT 1 POINT 2
x2 y2
142.75
100.75
12
5
142.75100.755 12
x1 y1 x2 y2

142.75cm
100.75cm?
142.75100.75
?
packs of
paper
height
)( ,
packs of
paper
height
)x1 y1
( ,12packs
5
packs
142.75100.75 125 12
5
POINT 1 POINT 2
x2 y212
5
142.75100.755 12
x1 y1 x2 y2
142.75 100.75- = 42 cm
packs
packs

142.75cm
100.75cm?
142.75100.75
?
packs of
paper
height
)( ,
packs of
paper
height
)x1 y1
( ,12packs
5
packs
142.75100.75 12
512
5
POINT 1 POINT 2
x2 y2
142.75100.75
x1 y1 x2 y2
142.75 100.75- = 42 cm
12 packs 5 packs- = 7 packs

142.75cm
100.75cm?
142.75100.75
?
packs of
paper
height
)( ,
packs of
paper
height
)x1 y1
( ,12packs
5
packs
142.75100.75 12
512
5
POINT 1 POINT 2
x2 y2
142.75100.75
x1 y1 x2 y2
142.75 100.75-
=
42 cm
12 packs 5 packs-
=
7 packs

142.75cm
100.75cm?
142.75100.75
?
packs of
paper
height
)( ,
packs of
paper
height
)x1 y1
( ,12packs
5
packs
142.75100.75 12
512
5
POINT 1 POINT 2
x2 y2
142.75100.75
x1
y1
x2
y2
142.75 100.75- 42 cm
12 packs 5 packs- 7 packs
==

142.75cm
100.75cm?
142.75100.75
?
packs of
paper
height
)( ,
packs of
paper
height
)x1 y1
( ,12packs
5
packs
142.75100.75 12
512
5
POINT 1 POINT 2
x2 y2
142.75100.75
x1
y1
x2
y2
142.75 100.75- 42 cm
-
-
==

142.75cm
100.75cm?
142.75100.75
?
packs of
paper
height
)( ,
packs of
paper
height
)x1 y1
( ,12packs
5
packs
142.75100.75 12
512
5
POINT 1 POINT 2
x2 y2
142.75100.75
x1
y1
x2
y2
142.75 100.75- 42 cm
-
-
Slope m, =
= ==
6 cm
1 packs
==

142.75cm
100.75cm?
142.75100.75
?
)( ,)x1 y1
( ,
12packs
5
packs
142.75100.75 12
5
POINT 1 POINT 2
x2 y2
142.75100.75
Slope m, = 6 cm
y b= +mx b+

142.75cm
100.75cm?
142.75100.75
?
)( ,)x1 y1
( ,
12packs
5
packs
142.75100.75 12
5
POINT 1 POINT 2
x2 y2
142.75100.75
Slope m, = 6 cm
y b= +mx b+
NOW, SUBSTITUTE
SLOPE (m) and EITHER
POINT TO FIND INITIAL
VALUE/Y-INTERCEPT!
b+

142.75cm
100.75cm
142.75100.75 )( ,)x1 y1
( ,
12packs
5
packs
142.75100.75 12
5
POINT 1 POINT 2
x2 y2
142.75100.75
Slope m, = 6 cm
y b= +mx b+
b
12 142.75
6

142.75cm
100.75cm
142.75100.75 )( ,)x1 y1
( ,
12packs
5
packs
142.75100.75 12
5
POINT 1 POINT 2
x2 y2
142.75100.75
Slope m, = 6 cm
y b= +mx b+
b
12142.75 = 6 )( )( b+

142.75cm
100.75cm
142.75100.75 )( ,)x1 y1
( ,
12packs
5
packs
142.75100.75 12
5
POINT 1 POINT 2
x2 y2
142.75100.75
Slope m, = 6 cm
y b= +mx b+
b
12142.75 = 6 )( )( b+
12142.75 = 6 b+72

142.75cm
100.75cm
142.75100.75 )( ,)x1 y1
( ,
12packs
5
packs
142.75100.75 12
5
POINT 1 POINT 2
x2 y2
142.75100.75
Slope m, = 6 cm
y b= +mx b+
b
12142.75 = 6 )( )( b+
12142.75 = 6 b+72
72-
6

142.75cm
100.75cm
142.75100.75 )( ,)x1 y1
( ,
12packs
5
packs
142.75100.75 12
5
POINT 1 POINT 2
x2 y2
142.75100.75
Slope m, = 6 cm
y b= +mx b+
b
12142.75 = 6 )( )( b+
12142.75 = 6 b+72
72-72-
70.75 = b0 +
6

142.75cm
100.75cm
142.75100.75 )( ,)x1 y1
( ,
12packs
5
packs
142.75100.75 12
5
POINT 1 POINT 2
x2 y2
142.75100.75
Slope m, = 6 cm
y b= +mx b+b 70.75b6
Therefore, the
equation of the line is:

Interconnected
Visual and Concrete
Contextual
VISION

Interconnected
VISION
Visual and Concrete
Contextual
TASKS

Visual and Concrete
Contextual
TASKS
VISION
INQUIRY
Construct Conceptual
Understanding
Interconnected

Connect to Prior
Knowledge
Visual and Concrete
Contextual
TASKS
CONNECTIONS
VISION
INQUIRY
Understanding

Visual and Concrete
Contextual
TASKS
CONSOLIDATE
VISION
INQUIRY
Move Towards
Procedural Fluency
Connect to Prior
Knowledge
CONNECTIONS
Understanding

#MTBoS
The Best Teacher PD is
FREE on Twitter
Follow the Math
Twitter Blogosphere

Making Math Contextual,
Visual, and Concrete
kylep.ca/curiousmath1p khmath.com

Calgary Regional Consortium Webinar - Making Math Contextual, Visual and Interconnected with Technology

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Calgary Regional Consortium Webinar - Making Math Contextual, Visual and Interconnected with Technology