Making Math Contextual,
Visual, and Concrete in
Grade 7-12
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Making Math
Contextual, Visual,
and Concrete in
Grade 7-12
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PROBLEM
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ENGAGED YET?
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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.
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Math Class
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Math
Class
1. Take Up Homework
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Math Class
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2. Definitions, Formulae, Procedures/Algorithms
1. Take Up Homework
Math Class
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3. Examples
2. Definitions, Formulae, Procedures/Algorithms
1. Take Up Homework
Math Class
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4. Homework
3. Examples
2. Definitions, Formulae, Procedures/Algorithms
1. Take Up Homework
Math Class
Example 1
(a) List all combinations of three out of
the four colours: red, blue, white, and
green.
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My Result:
Successful
in Math
Unsuccessful
in Math
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A Perception Develops
Good at
Math
Poor at
Math
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Confident in
Math Class
Doubtful in
Math Class
Students Feel…
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Enjoying
Mathematics
Disliking
Mathematics
This May Lead To…
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Willingness
to Engage In
Math Class
Unwillingness
to Engage In
Math Class
This May Lead To…
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Moving Ahead Falling Behind
This May Lead To…
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Two Groups of Students
Good at
Math
Poor at
Math
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Two Groups of Students
Good at
Math
Poor at
Math
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Two Groups of Students
Good at
Math
Poor at
Math
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Two Groups of Students
Struggle with
unfamiliar
problems
Good at
Math
Poor at
Math
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A VISION FOR
MATHEMATICS
EDUCATION
VISION
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Contextual
VISION
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Visual and Concrete
Contextual
VISION
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Interconnected
Visual and Concrete
Contextual
VISION
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Interconnected
Visual and Concrete
Contextual
CONFIDENCE
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Interconnected
Visual and Concrete
Contextual
PERCEPTION
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Interconnected
Visual and Concrete
Contextual
STUDENT
SUCCESS
Change
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Change
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Change
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Immune to
http://www.vancouversun.com/life/From+geek+chic/6296707/story.html
Photograph by: Maggie Wong
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http://www.wykamath.com/
entertainment
engagement
=
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algorithm
what’s
this?
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algorithm
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algorithms
engagement
=
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algorithms
engagement
=
understanding
or
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Notice? Wonder?
How many packs of paper would it
take to reach the ceiling?
OUR FIRST QUESTION:
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PREDICTIONS
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REQUEST
INFORMATION
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How many packs of paper
will it take to reach a
height of 273 cm?
height
273cm
24.75cm
QUESTION
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QUESTION
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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
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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?FUN, RIGHT?
AnswerKey:55.15
QUESTION
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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?
Making Connections
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@MathletePearcewww.tapintoteenminds.com
How many packs of paper
will it take to reach a
height of 273 cm?
height
273cm
24.75cm
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273cm
24.75cm
=
24.75cm24.75cm
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273cm
=
24.75cm
24.75cm
24.75cm
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273cm
=
24.75cm
24.75cm
24.75cm
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273cm
=
25cm
25cm
25cm
Friendly Numbers
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275cm
=
25cm
25cm
25cm
Friendly Numbers
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275cm
=
25cm
25cm
25 cm
Friendly Numbers
Height
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275cm
=
25cm
25cm
Friendly Numbers
25 cm
25 cm
Height
50 cm
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275cm
=
25cm
25cm
Friendly Numbers
25 cm
25 cm
Height
25 cm
75 cm
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275cm
=
25cm
25cm
Friendly Numbers
25 cm
25 cm
Height
25 cm
25 cm
100 cm
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275cm
=
25cm
25cm
Friendly Numbers
25 cm
25 cm
Height
25 cm
25 cm
200 cm
25 cm
25 cm
25 cm
25 cm
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275cm
=
25cm
25cm
Friendly Numbers
100 cm
Height
300 cm
100 cm
100 cm
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275cm
=
25cm
25cm
Friendly Numbers
100 cm
Height
275 cm
100 cm
75 cm
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275cm
=
25cm
25cm
Friendly Numbers
100 cm
Height
275 cm
100 cm
75 cm
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275cm
=
25cm
25cm
Friendly Numbers
“Array”
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275cm
=
25cm
25cm
Friendly Numbers
Area Model
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275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groupsArea Model
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275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groups
55 packs
Area Model
5packs
11 groups
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275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groups
55 packs
5packs
11 groups
55 packs
Area Model
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275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groups
55 packs
5packs
11 groups
55 packs
25cm
11 groups
Area ModelDouble
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275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groups
55 packs
5packs
11 groups
55 packs
25cm
11 groups
275 cm
Area ModelDouble
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275cm
25cm
Friendly Numbers
Double Number Line
Friendly Numbers
Double Number Line
275cm
25cm
0
0
Friendly Numbers
Double Number Line
275cm
25cm
0
0
Number of Packs
Total Height
Friendly Numbers
Double Number Line
275cm
25cm
0
0
5
25
Number of Packs
Total Height
Friendly Numbers
Double Number Line
275cm
25cm
0
0
5
25 50
10
Number of Packs
Total Height
Friendly Numbers
Double Number Line
275cm
25cm
0
0
5
25 50
10
75
15
250
50
Number of Packs
Total Height
20
100 125
25
150
30 35
175 200
40
225
45 55
275
Friendly Numbers
Double Number Line
275cm
25cm
0
0
5
25 50
10
Number of Packs
Total Height
Friendly Numbers
Double Number Line
275cm
25cm
0
0
5
25 50
10
Number of Packs
Total Height
20
100
Friendly Numbers
Double Number Line
275cm
25cm
0
0
5
25 50
10
Number of Packs
Total Height
20
100 200
40
Friendly Numbers
Double Number Line
275cm
25cm
0
0
5
25 50
10
Number of Packs
Total Height
20
100 200
40
250
50
225
45 55
275
Friendly Numbers
Double Number Line
275cm
25cm
0
0
5
25 50
10
Number of Packs
Total Height
20
100 200
40
250
50
225
45 55
275275
5
25
Friendly Numbers
Double Number Line
275cm
25cm
0
0
Number of Packs
Total Height
275
x
275
5
25
5
25
Friendly Numbers
Double Number Line
275cm
25cm
0
0
Number of Packs
Total Height
275
x5
25
5
?x
27525
Friendly Numbers
Double Number Line
275cm
25cm
0
0
Number of Packs
Total Height
275
x
275
5
25
5
25
?x
= 11111111
Friendly Numbers
Double Number Line
275cm
25cm
0
0
Number of Packs
Total Height
275
x
275
5
25
5
25
x
= 1111
11
11
Friendly Numbers
Double Number Line
275cm
25cm
0
0
Number of Packs
Total Height
275
x
275
5
25
5
25
x
= 11
11
11
x 11
Friendly Numbers
Double Number Line
275cm
25cm
0
0
Number of Packs
Total Height
275
x
275
5
25
5
25
x
= 11
11
11
x
11x = 5555
Friendly Numbers
Double Number Line
275cm
25cm
0
0
Number of Packs
Total Height
275
275
5
25
5
25
x
= 11
11
11
x
11x = 55
55
How many packs of paper would it
take to reach the ceiling?
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How many packs of paper
will it take to reach a
height of 273 cm?
packs of paper
height
273cm
4.95 cm
How many packs of paper would it
take to reach the ceiling?
@MathletePearcewww.tapintoteenminds.com
How many packs of paper
will it take to reach a
height of 273 cm?
packs of paper
height
273cm
4.95 cm
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packs of paper height
Independent Dependent
273cm
4.95 cm
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0 0
packs of paper height
Independent Dependent
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0 0
1 4.95
packs of paper height
Independent Dependent
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0 0
1 4.95
2 9.9
packs of paper height
Independent Dependent
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0 0
1 4.95
2 9.9
packs of paper height
Independent Dependent
4 19.8
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0 0
1 4.95
2 9.9
packs of paper height
Independent Dependent
4 19.8
40 198
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0 0
1 4.95
2 9.9
packs of paper height
Independent Dependent
4 19.8
40 198
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packs of paper height:
@MathletePearcewww.tapintoteenminds.com
packs of paper
height
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packs of paper
height
24.75 cm
5 reams
=
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24.75 cm
5 reams
=
273 cm
How Many?
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5 reams
=
x
24.75 cm 273 cm
x ??
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5 reams
=
x
24.75 cm 273 cm
x ??
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5 reams
=
x24.75 cm
273 cm
x ??
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5 reams
=
x24.75 cm
273 cm
x ??
11.03
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24.75 cm
5 reams
=
273 cm
11.03x
x
@MathletePearcewww.tapintoteenminds.com
24.75 cm
5 reams
=
273 cm
55.15 reams
11.03
x 11.03
x
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VISION
Visual and Concrete
Contextual
TASKS
Visual and Concrete
Contextual
TASKS
INQUIRY
Construct Conceptual
Understanding
VISION
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Connect to Prior
Knowledge
Visual and Concrete
Contextual
TASKS
CONNECTIONS
INQUIRY
Construct Conceptual
Understanding
VISION
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Visual and Concrete
Contextual
TASKS
CONSOLIDATE INQUIRY
Move Towards
Procedural Fluency
Connect to Prior
Knowledge
CONNECTIONS
Construct Conceptual
Understanding
VISION
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3 Act Math Tasks
iTunes U
kylep.ca/curiousmath1p
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kylep.ca/curiousmath1p
iTunes U
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kylep.ca/curiousmath1p
iTunes U
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kylep.ca/curiousmath1p
iTunes U
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Tap
3 Act Math Tasks
tapintoteenminds.com/3act-math
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fishing4tech.com/mtbos.html
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The CN Tower has a height of 553 m and
your stack of 5 packages of paper have a
height of approximately 0.25 m.
Approximately how many packages of
paper would you need to stack up to the
height of the CN Tower?
24.75cm
Concreteness Fading
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1
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1 2
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Concrete
Iconic
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1 2
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Concrete
Iconic
Visual
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1 2 3
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Concrete
Iconic
Visual
Symbolic
Concreteness Fading
1 2 3
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Concrete
Iconic
Visual
Symbolic
Abstract
Concreteness Fading
1 2 3Enactive
Concrete
Iconic
Visual
Symbolic
Abstract
Concreteness Fading
1 2 3Enactive
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Iconic
Visual
Symbolic
Abstract
Concreteness Fading
1 2 3Enactive
Concrete
Iconic
Visual
Symbolic
Abstract
Concreteness Fading
1 2 3Enactive
Concrete
Iconic
Visual
Symbolic
Abstract
2
2x
2
6
1
044
21 3
+
275
Making Math Contextual,
Visual, and Concrete
@MathletePearcewww.tapintoteenminds.com

SERCC - Making Math Contextual, Visual and Concrete