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

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

PROBLEM

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.

Math Class

Math
Class

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

Two Groups of Students
Good at
Math
Not
Good at
Math

Good at
Memorization
Not Good at
Memorization

Good at
Memorization
Not Good at
Memorization
Struggle with
unfamiliar
problems

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

iBooks
iBooks Author

iBooks Author
Available on iTunes and…
iBooks

3 Act Math Tasks
iTunes U
kylep.ca/curiousmath1p

iTunes U

iTunes U
Tap

playkh.complaykh.comGo to:

Tap

Tap, then wait.

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

PREDICTIONS

REQUEST
INFORMATION

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

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?

QUESTION
he need to reach the ceiling?FUN, RIGHT?

AnswerKey:55.15
QUESTION
he need to reach the ceiling?

Extensions

The Next Day…

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

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

Example
Consolidation
Task #1:

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

packs
( )x y
2 15151

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

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

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

Struggle with
unfamiliar
problems
Good at
Math
Not Good
at Math

Good at
Math
Not Good
at Math
Struggle with
unfamiliar
problems

Digging Deeper…

142.75cm
100.75cm
?
142.75
100.75?

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
@MathletePearcekylep.ca/curiousmath1p

Interconnected
Visual and Concrete
Contextual
TASKS
CONNECTIONS
VISION

Interconnected
Visual and Concrete
Contextual
TASKS
INQUIRY
CONNECTIONS
VISION

Interconnected
Visual and Concrete
Contextual
TASKS
CONNECTIONS
INQUIRYCONSOLIDATE
VISION

thing.every

one thing.

step.one

one vision.

thing.
one

AMAZING
thing .s

BIGGERand
better

?
thing
NEXT
what will be your
AMAZING

#MTBoS
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Making Math Contextual, Visual and Concrete with Technology

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