2016 05-25- HPEDSB Making Math Contextual, Visual and Concrete

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

Making Math
Contextual, Visual,
and Concrete

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

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.

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

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
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
=
understanding
or

LEARNING

NEWOLD
PROGRESSIVETRADITIONAL
ACTIVITYLECTURE
INQUIRY BASEDDIRECT INSTRUCTION
TECHNOLOGYPEN & PAPER
MATH WARS

BALANCE
NEW
OLD
PROGRESSIVE
TRADITIONAL
ACTIVITY
LECTURE
INQUIRY BASED
DIRECT INSTRUCTION
TECHNOLOGY
PEN & PAPER

What do you notice?
What do you wonder?

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

AnswerKey:55.15
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?

HOW MANY SMALL VASES WILL IT
TAKE TO FILL THE TALL VASE?

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

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

Visual and Concrete
Contextual
TASKS
CONSOLIDATE INQUIRY
Move Towards
Procedural Fluency
Connect to Prior
Knowledge
CONNECTIONS
Construct Conceptual
Understanding
VISION
@MathletePearcekylep.ca/curiousmath1p

Four Stages
of
Mastery
COMPETENCE
CONSCIOUSNESS
2 3
41

Four Stages
of
Mastery
COMPETENCE
CONSCIOUSNESS
2 3
4
Consciously
Competent
Consciously
Incompetent
1
Unconsciously
Incompetent
Unconsciously
Competent

Consciously Incompetent
The Four Stages
of
Mastery
COMPETENCE
CONSCIOUSNESS
1
2 3
4
Unconsciously Competent
Consciously Competent
Unconsciously Incompetent1
2
3
4

1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
1
1
1
4 - 8 + 6 - 4 + 7
= - 4 + 6 - 4 + 7
= 2 - 4 + 7
= - 2 + 7
= 5

What equation am I?
x
1 1
1 1
=
1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1
1
1
1
1
1

x = 20+ 5
x
1 1
1 1
=
1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1
1
1
1
1
1

x
1 1
1 1
=
1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1
1
1
1
1
1
x = 20+ 5
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
- 5 - 5

x
1 1
1 1
=
1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1
1
1
1
1
1
x = 20+ 5
+ 5 + 5
1
1
1
1
1
1
1
1
1
1

x
1 1
1 1
=
1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1
1
1
1
1
1
x = 20+ 5
+ x + x
x
x

x
1 1
1 1
-x =
1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1
1
1
1
1
1
x = 20+ 5
- x - x
-x

x
=
x = 9- 3
-1
-1
-1
-1
-1
-1
-
-1
-1
-1
-1
-1
-1

x
=
1
1
1
x = 9- 3
-1
-1
-1
-1
-1
-1
+ 3 + 3
-
-1
-1
-1
-1
-1
-1 1
1
1

x
=
x = 9- 3
-1
-1
-1
-1
-1
-1
- 3 - 3
-
-1
-1
-1
-1
-1
-1-1
-1
-1
-1
-1
-1

x
=
x = 9- 3
-1
-1
-1
-1
-1
-1
+ x
-
-1
-1
-1
-1
-1
-1
+ x
x
x

x
-x =
x = 9- 3
-1
-1
-1
-1
-1
-1
- x
-
-1
-1
-1
-1
-1
-1
- x
-x

What equation am I?
x 1
1
-1
=x
x
1 1
1 1
1 1 1
1 1 1 1
1
1
1
-1
-1 -1
-1 -1 1

x 1
1
-1
=x
x
1 1
1 1
1 1 1
1 1 1 1
1
1
1
-1
-1 -1
-1 -1 1
3(x - 2) = 17

1
1
1
3 x 2
“3 groups of 2”

1 1
1
1
1
3 x 2
“3 groups of 2”

1 1
1
1
1
3 x 2
“3 groups of 2”
1 1

1 1
1
1
1
3 x 2
“3 groups of 2”
1 1
1 1

1 1
1
1
1
3 x 2
“3 groups of 2”
1 1
1 1
1 1

1 1
1
1
1
3 x 2
“3 groups of 2”
1 1
1 1
1 1
= 6

1 1
1
1
1
5 x 3 = ?
1
1
1
“5 groups of 3”

1 1
1
1
1
5 x 3
“5 groups of 3”
1 1
1 1
1 1
= ?
1
1
1
1
1
1
1 1
1 1
1
1

1 1
1
1
1
5 x 3
“5 groups of 3”
1 1 1 1
= ?
1
1
1
1 1
1 1 1 11 1
1 1 1 11 1
1 1 1 11 1
1 1 1 11 1

1 1
1
1
1
5 x 3
“5 groups of 3”
1 1 1 1
= ?
1
1
1
1 1
1 1 1 11 1
1 1 1 11 1

1 1
1
1
1
5 x 3
“5 groups of 3”
1 1
= ?
1
1
1
1 1
1 1
1 1

5 x 14 = ?
“5 groups of 14”
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 1

5 x 14 = ?
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 110 4
5
“5 groups of 10 plus 5 groups of 4”
or
?

5 x 14 = ?
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 110 4
5
or
? ?

5 x 14 = 70
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 110 4
5
or
50 20

5 x 14 = 70
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 110 4
5
or
20 50

5 x 14 = 24
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 110 4
5
or
15 9

5 x 14 = 140
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 110 4
5
or
70 70

9 x 12 = ?
1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 1

9 x 12 = ?
“9 groups of 12” 1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 110 2
5
4
“5 groups of 10
plus
4 groups of 2”
or

9 x 12 = ?
1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 110 2
5
4
?
“5 groups of 10
plus
4 groups of 2”
or

9 x 12 = ?
1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 110 2
5
4
? ?
“5 groups of 10
plus
4 groups of 2”
or

9 x 12 = ?
1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 110 2
5
4
?
?
?
“5 groups of 10
plus
4 groups of 2”
or

9 x 12 = ?
1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 110 2
5
4
?
?
?
?
“5 groups of 10
plus
4 groups of 2”
or

9 x 12 = ?
1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 110 2
5
4
50
40
10
8
“5 groups of 10
plus
4 groups of 2”
or

9 x 12 = ?
1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 110 2
5
4
15
14
7
6
“5 groups of 10
plus
4 groups of 2”
or

9 x 12 = ?
1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 110 2
5
4
40
50
20
8
“5 groups of 10
plus
4 groups of 2”
or

9 x 12 = ?
1
1
1
1
1
1
1
1
1
1 1 1 1 1 1 1 1 1 1 1 110 2
5
4
10
8
50
40
“5 groups of 10
plus
4 groups of 2”
or

What equation am I?
x
1
1
-1
=x
x
1 1
1 1
1 1 1 1
1
1
1
-1
1
x
x
x
-1
-1 -1 -1
-1 -1 -1
1

x
1
1
-1
=x
x
1 1
1 1
1 1 1 1
1
1
1
-1
1
x
x
x
-1
-1 -1 -1
-1 -1 -1
1
3(2x - 3) = 15

What’s My Area Equation?
x x x x

x x x x
1
1
1

x x x x
1
1
1
Area = 3( )4x

x x x x
1
1
1
x x x x
x x x x
x x x x
Area = 3( )4x
= 12x units2

1
1
1 1
1 1
1 1 1 1
1
1
1
1
1
x x x x
1
1
1
Area = 3( )4x
= 45
1
1
1
1
1
1
1
1
1 1
1 1
1 1 1 1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
units2

x
x x x x
1
1
1
Area = 3( )4x
= 15x units2
x
x
x
x
x
x
x
x
x
x
x
x
x
x

x x x x
1
1
1
Area = 3( )4x
= 10x + 15 units2
x
x
x
x
x
x
x
x
x
x
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

Find the Area of The Pool
x x x
1
1
1
Area = 5( )3x
1
1

x x x

x x x
xx

x
x x x
x

x
x x x
x
Area = 2x groups of 3x

x2x2x
x x x
x
x2
Area = 2x( )3x

x
x x x
x
x2 x2 x2
x2 x2 x2
Area = 2x( )3x

x
x x x
x
x2 x2 x2
x2 x2 x2
6x2Area = 2x ( )3x =

x
x x x
x
x2 x2 x2
x2 x2 x2
If the total area
is 96 units2, solve
for the value of x.
6x2Area = 2x( )3x =

x
x x x
x
x2 x2 x2
x2 x2 x2
96 = 6x2
If the total area
is 96 units2, solve
for the value of x.

x x x 1 1

x x x 1 1
1
1
1
1

Area =
x x x 1 1
1
1
1
1
4 groups

Area =
x x x 1 1
1
1
1
1
4 ofgroups 3x + 2

Area =
x x x 1 1
1
1
1
1
4 )( 3x + 2

1 1
1
1
1
x x x
1
1 1x x x
1 1x x x
1 1x x x
1 1x x x
Area = 4 )( 3x + 2
= 12x + 8

1 1
1
1
1
x x x
1
1 1x x x
1 1x x x
1 1x x x
1 1x x x
Area = 4 )( 3x + 2
= 12x + 8
If the total area is 80 units2, solve for the value of x.

1 1
1
1
1
x x x
1
1 1x x x
1 1x x x
1 1x x x
1 1x x x
= 12x + 8
If the total area is 80 units2, solve for the value of x.
80

x
1 1x x x
x

x
1 1x x x
x
Area = 2x groups of 3x + 2

x
1 1x x x
x
Area = 2x( )3x + 2

x2
x
1 1x x x
x
Area = 2x( )3x + 2

x2x2
x
1 1x x x
x
Area = 2x( )3x + 2

x2
x
x
x2x2
x
1 1x x x
x
Area = 2x( )3x + 2

x
1 1x x x
x2 x2 x2
x
x
x
x2 x2 x2
x
x
Area = 2x( )3x + 2

x
1 1x x x
x2 x2 x2
x
x
x
x2 x2 x2
x
x
Area = 2x( )3x + 2 = 6x2 + 4x

x
1 1
1
1
x
1
x x

Area = (x + 3)(3x + 2)
x
1 1
1
1
x
1
x x

x
1 1
1
1
x
1
x x
x2 x2 x2
x x x
x x x
x x x
x
x
1
1
1
1
1
1
Area = (x + 3)(3x + 2)

x
1 1
1
1
x
1
x x
x2 x2 x2
x x x
x x x
x x x
x
x
1
1
1
1
1
1
Area = (x + 3)(3x + 2) = 3x2 + 11x + 6

…NOTICE? …WONDER?
What
do you…

Visual and Concrete
Contextual
TASKS
VISION

Visual and Concrete
Contextual
TASKS
VISION INQUIRY

Interconnected
Visual and Concrete
Contextual
TASKS
CONNECTIONS
INQUIRY
VISION

Interconnected
Visual and Concrete
Contextual
TASKS
CONNECTIONS
INQUIRY
VISIONCONSOLIDATE

@MathletePearce@MrOrr_Geek
Clap as fast as you can

Notice? Wonder?
INQUIRY

Will he beat the record?
INQUIRY

Act 3

CONSOLIDATE

Stage 1 Stage 2
Stage 3 Stage 4

Stage 1 Stage 2
Stage 3 Stage 4
Initial Value:

Stage 1 Stage 2
Stage 3 Stage 4Initial Value:
Stage 0
No
Squares

Stage 1 Stage 2
Stage 0
No
Squares
0

Stage 1 Stage 2
Stage 0
No
Squares
0
Rate of Change:

Stage 1 Stage 2
Stage 0
No
Squares
0
Rate of Change:
4

Stage 1 Stage 2
Stage 0
No
Squares
0
Rate of Change:
4
Equation:

Stage 1 Stage 2
Stage 0
No
Squares
0
Rate of Change:
4
Equation:
y =

Stage 1 Stage 2
Stage 0
No
Squares
0
Rate of Change:
4
Equation:
y = 0

Stage 1 Stage 2
Stage 0
No
Squares
0
Rate of Change:
4
Equation:
y = 0 + 4x

Stage 1 Stage 2
Stage 0
No
Squares
0
Rate of Change:
Equation:
y = 0

Stage 1 Stage 2
Stage 0
No
Squares
0
Rate of Change:
Equation:
y = 0
4
+ 4x

Stage 1 Stage 2
Stage 0
No
Squares
0
Rate of Change:
Equation:
y =
4
4x

Stage 1 Stage 2
Stage 5 Stage 6
Stage 3 Stage 4
? ?

Stage 1 Stage 2
Stage 5 Stage 6
Stage 3 Stage 4
?

Stage 1 Stage 2
Stage 5 Stage 6
Stage 3 Stage 4

130.75cm130.75cm

130.75cm130.75cm
Notice? Wonder?

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

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
How tall is the table?

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

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
151512 15151

packs
( )x y
151512 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

How Many Are In The Bag?

13 13

b6

b6 =

b

b 3

b 3936

b
3
9
3
6

6
b6

b
6 6

b3

b3 =

b3 b b bb b b 6

3 bb b b 6 =+ + + b b b+ + ++

3 bbbb 6 =+ b b b+ + ++3

3 bbbb 6 =+ bbb4+3

3 bbbb 6 =+ bbb4+3
3-

bbbb 6 =+ bbb4+3

bbbb 6 =+ bbb4+3
3-

bbbb 3 =+ bbb4+3

bbbb 3 =+ bbb4+3
3b-

b3 =+
bbb4+

b3 = bbb4+
3b-

b3 = bbb

DOUBLE CLOTHESLINE
17
2x + 5
0
0x

DOUBLE CLOTHESLINE
17
2x + 5
0
0x 2x

DOUBLE CLOTHESLINE
17
2x + 5
0
0x 2x
12

DOUBLE CLOTHESLINE
17
2x + 5
0
0x 2x
12
1x
?

DOUBLE CLOTHESLINE
17
2x + 5
0
0x 2x
12
1x
6

Digging Deeper…

142.75cm
100.75cm
?142.75
100.75
?

142.75cm
100.75cm
?
142.75
100.75?

142.75cm
100.75cm
Equations Given 2 Points

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:

2x - 4 = 8
Solve for x:

2x - 4 = 8
Solve for x:
+ 4 + 4

2x - 4 = 8
Solve for x:
+ 4 + 4
2x = 12

2x - 4 = 8
Solve for x:
+ 4 + 4
2x = 12
2 2

2x - 4 = 8Solve for x:
+ 4 + 4
2x = 12
2 2
x = 6

GAMIFICATION

0
0

0
0
-
+

0
0
-
+
-
+

0
0
0
0

0
0
0 = 0
xx

0
0
x
0 = 0x+
-aa-
a
-
+

0
0
x
-a
+a
= 00 x a a+ -
b
b
+
-
+

0
0
x
-a
+a
-b
+b
=0 x a a+ - + 0 b b+ -
c
d
+
+

0
0
-a
+a
-b
+b
x
+c
+d
0 x a a = 0 b b c d+ - + + - + +

-a
+a
-b
+b
x
x a a = b b c d- + - + +
+c
+d
+0 +

+c
+d
-a+a
-b
+b
x
x aa = b b c d-+ - + +
0
0

+c
+d
-a+a
-b+b
x
x aa = bb c d-+ - + +
0
0
0
0

+c
+d
x
xaa = bbc d- - ++0 0

x = bbc d+

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at Math

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