SERCC - Making Math Contextual, Visual and Concrete

Making Math Contextual,
Visual, and Concrete in
Grade 7-12
@MathletePearcewww.tapintoteenminds.com

Making Math
Contextual, Visual,
and Concrete in
Grade 7-12

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.

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

Two Groups of Students
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

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

PREDICTIONS

REQUEST
INFORMATION

How many packs of paper
will it take to reach a
height of 273 cm?
height
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?

Making Connections

273cm
24.75cm
=
24.75cm24.75cm

273cm
=
24.75cm
24.75cm
24.75cm

273cm
=
25cm
25cm
25cm
Friendly Numbers

275cm
=
25cm
25cm
25cm
Friendly Numbers

275cm
=
25cm
25cm
25 cm
Friendly Numbers
Height

275cm
=
25cm
25cm
Friendly Numbers
25 cm
25 cm
Height
50 cm

275cm
=
25cm
25cm
Friendly Numbers
25 cm
25 cm
Height
25 cm
75 cm

275cm
=
25cm
25cm
Friendly Numbers
25 cm
25 cm
Height
25 cm
25 cm
100 cm

275cm
=
25cm
25cm
Friendly Numbers
25 cm
25 cm
Height
25 cm
25 cm
200 cm
25 cm
25 cm
25 cm
25 cm

275cm
=
25cm
25cm
Friendly Numbers
100 cm
Height
300 cm
100 cm
100 cm

275cm
=
25cm
25cm
Friendly Numbers
100 cm
Height
275 cm
100 cm
75 cm

275cm
=
25cm
25cm
Friendly Numbers
“Array”

275cm
=
25cm
25cm
Friendly Numbers
Area Model

275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groupsArea Model

275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groups
55 packs
Area Model

5packs
11 groups
275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groups
55 packs
5packs
11 groups
55 packs
Area Model

275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groups
55 packs
5packs
11 groups
55 packs
25cm
11 groups
Area ModelDouble

275cm
=
25cm
25cm
Friendly Numbers
5packs
11 groups
55 packs
5packs
11 groups
55 packs
25cm
11 groups
275 cm
Area ModelDouble

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
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?
How many packs of paper
will it take to reach a
height of 273 cm?
packs of paper
height
273cm
4.95 cm

packs of paper height
Independent Dependent
273cm
4.95 cm

0 0

0 0
1 4.95

0 0
1 4.95
2 9.9

0 0
1 4.95
2 9.9
4 19.8

0 0
1 4.95
2 9.9
4 19.8
40 198

packs of paper height:

packs of paper
height

packs of paper
height
24.75 cm
5 reams
=

24.75 cm
5 reams
=
273 cm
How Many?

5 reams
=
x
24.75 cm 273 cm
x ??

5 reams
=
x24.75 cm
273 cm
x ??

5 reams
=
x24.75 cm
273 cm
x ??
11.03

24.75 cm
5 reams
=
273 cm
11.03x
x

24.75 cm
5 reams
=
273 cm
55.15 reams
11.03
x 11.03
x

VISION
Visual and Concrete
Contextual
TASKS

Visual and Concrete
Contextual
TASKS
INQUIRY
Construct Conceptual
Understanding
VISION

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

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

3 Act Math Tasks
iTunes U
kylep.ca/curiousmath1p

iTunes U

iTunes U
Tap

3 Act Math Tasks
tapintoteenminds.com/3act-math
ﬁshing4tech.com/mtbos.html

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
1
Enactive

Concreteness Fading
1
Enactive
Concrete

Concreteness Fading
1 2
Enactive
Concrete
Iconic

Concreteness Fading
1 2
Enactive
Concrete
Iconic
Visual

Concreteness Fading
1 2 3
Enactive
Concrete
Iconic
Visual
Symbolic

Concreteness Fading
1 2 3
Enactive
Concrete
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

SERCC - Making Math Contextual, Visual and Concrete

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SERCC - Making Math Contextual, Visual and Concrete