Exploring the Progression of Proportional

Reasoning From K-9tapintoteenminds.com/OTF

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Visual and ConcreteContextual

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Interconnected

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VISION

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CONFIDENCE

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PERCEPTION

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ACHIEVEMENT

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Proportional Reasoning

Ratio and proportional thinking and reasoning abilities are seen as a corner stone of middles

school mathematics and this observation is reflected in current syllabus documents (e.g.,

National Council of Teachers of Mathematics, 2004) and by educators such as (e.g., Nabors, 2002).

Norton, S. (2005). The Construction of Proportional Reasoning. Queensland University of Technology.

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Proportional Reasoning

A number of authors (e.g., Ilany, Keret & Ben-Chaim, 2004; Lo & Watanabe, 1997) have noted that the essence

of such thinking is essentially multiplicative. Ability in such thinking is needed for and understanding of percentages, gradient, trigonometry and algebra.

Norton, S. (2005). The Construction of Proportional Reasoning. Queensland University of Technology.

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Proportional Reasoning

Lamon (1995) noted that proportional reasoning has typically been taught in a single chapter of the mathematical text book, in which symbols are

introduced before sufficient ground work has been laid for students to understand them (p. 167).

Norton, S. (2005). The Construction of Proportional Reasoning. Queensland University of Technology.

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Paying Attention to Mathematics provided an overview of what it would take to help Ontario students make and sustain - gains in their learning and understanding of mathematics. It outlined seven foundational principles for planning and implementing improvements and gave examples of what each principle would involve.

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Focus on mathematics.Coordinate and strengthen mathematics leadership.Build understanding of effective mathematics instruction.Support collaborative professional learning in mathematics.Design a responsive mathematics learning environment.Provide assessment and evaluation in mathematics that supports student learning.Facilitate access to mathematics learning resources.

SEVEN FOUNDATIONAL PRINCIPLESFOR IMPROVEMENT IN MATHEMATICS, K12

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All ability to reason using proportional relationships is a complex process that develops over an extended period of time. It takes many varied physical

experiences to develop an understanding of what a proportional relationship is and then more time to

gain the ability to deal with it abstractly.kylep.ca/patprop

Proportional Reasoning

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when you think of 8 as two fours or four twos rather than thinking of

it as one more than seven.

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What is Proportional Reasoning?

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when you think of 8 as two fours or four twos rather than thinking of

it as one more than seven.

kylep.ca/patprop

What is Proportional Reasoning?

when you think of how a speed of 50 km/h is the same as a speed of

25 km/30 min.

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The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms.

What is Proportional Reasoning?

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Which shape is more purple?

A

BPaying Attention to Proportional ReasoningAdapted from Marian Small (2008, p. 254)

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Which shape is more purple?

A

BPaying Attention to Proportional ReasoningAdapted from Marian Small (2008, p. 254)

24

23

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You invest $100 and it grows to $400.

Consider:

Scenario #1

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Consider:

Scenario #1 Scenario #2

You invest $1,000 and it grows to $1,500.

You invest $100 and it grows to $400.

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Consider:

Scenario #1 Scenario #2

Which Investment Is Better?

You invest $100 and it grows to $400.

You invest $1,000 and it grows to $1,500.

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You invest $1,000 and it grows to $1,500.

Consider:

Scenario #1 Scenario #2

Which Investment Is Better?

You invest $100 and it grows to $400.

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You invest $1,000 and it grows to $1,500.

You invest $100 and it grows to $400.

Consider:

Scenario #1 Scenario #2

Which Investment Is Better?

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Which Investment Is Better?

Scenario #1

Scenario #2

You invest $100 and it grows to $400.

You invest $1,000 and it grows to $1,500.

$100$100$100$100

$100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500

$400

$100

$1,000

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In Absolute Terms

Scenario #1

Scenario #2

You invest $100 and it grows to $400.

You invest $1,000 and it grows to $1,500.

$100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500

$1,000

$1,000 $500

$100$100$100$100

$100

$100 $300

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In Absolute Terms

Scenario #1

Scenario #2

You invest $100 and it grows to $400.

You invest $1,000 and it grows to $1,500.

$100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500

$1,000

$1,000 $500

$100$100$100$100

$100

$100 $300

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$100$100$100$100

$100

$100

In Absolute Terms

Scenario #1

Scenario #2

You invest $100 and it grows to $400.

You invest $1,000 and it grows to $1,500.

$100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500

$1,000

$1,000

$300

$500

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In Relative Terms

Scenario #1

Scenario #2

You invest $100 and it grows to $400.

You invest $1,000 and it grows to $1,500.

$100$100$100$100

$100

$100

$100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500

$1,000

$1,000 $1,000

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In Relative Terms

Scenario #1

Scenario #2

You invest $100 and it grows to $400.

You invest $1,000 and it grows to $1,500.

$100$100$100$100

$100

$100

$100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500

$1,000

$1,000 $1,000

x 121

x 4

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In Relative Terms

Scenario #1

Scenario #2

You invest $100 and it grows to $400.

You invest $1,000 and it grows to $1,500.

$100$100$100$100

$100

$100

$100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500

$1,000

$1,000 $1,000

x 121

x 4

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In Relative Terms

Scenario #1

Scenario #2

You invest $100 and it grows to $400.

You invest $1,000 and it grows to $1,500.

$100$100$100$100

$100

$100

$100$100$100$100 $100$100$100$100 $100$100 $100$100$100$100 $100$1,500

$1,000

$1,000 $1,000

x 121

x 4

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Unitizing

There are 4 fish bowls with 5 fish in each. How many fish in total are there?

Multiplication:

There are 4 fish bowls with 5 fish in each. How many fish in total are there?

Unitizing Multiplication:

There are 4 fish bowls with 5 fish in each. How many fish in total are there?

Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously.

Unitizing Multiplication:

You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need?

Unitizing Division:

You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need?

Unitizing Division:

You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need?

Unitizing Division:

You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need?

Unitizing Division:

You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need?

Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously.

Unitizing Division:

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There are 4 fish bowls with 5 fish in each. How many

fish in total are there?

Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously.

Unitizing

MultiplicationYou buy 15 goldfish. You are

going to put 3 fish in each bowl. How many bowls will you need?

Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously.

Division

There are 4 fish bowls with 5 fish in each. How many

fish in total are there?

Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously.

Unitizing

MultiplicationYou buy 15 goldfish. You are

going to put 3 fish in each bowl. How many bowls will you need?

Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously.

Division

5 fish to 1 bowl

5:13 fish to 1 bowl

3:1

There are 4 fish bowls with 5 fish in each. How many

fish in total are there?

MultiplicationYou buy 15 goldfish. You are

going to put 3 fish in each bowl. How many bowls will you need?

Division

Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously.

Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously.

5 fish to 1 bowl

5:13 fish to 1 bowl

3:1

There are 4 fish bowls with 5 fish in each. How many

fish in total are there?

MultiplicationYou buy 15 goldfish. You are

going to put 3 fish in each bowl. How many bowls will you need?

Division

Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously.

Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously.

1 bowl to 5 fish

1:51 bowl to 3 fish

1:3

There are 4 fish bowls with 5 fish in each. How many

fish in total are there?

MultiplicationYou buy 15 goldfish. You are

going to put 3 fish in each bowl. How many bowls will you need?

Division

1 bowl to 5 fish

1:51 bowl to 3 fish

1:3

Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

2 3:

Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

1 3

Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

11

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Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

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Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

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Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

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Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

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Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

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Ratios and Rates

For every 2 red candies in a package, there are 3 green candies. How many red candies would

there be if you have 12 green candies?

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many red candies would there be if you have 12 green candies?

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More RepresentationsFor every 2 red candies in a package, there are 3 green candies. How

many red candies would there be if you have 12 green candies?

1111

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Interconnected

Visual and ConcreteContextual

VISION

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playkh.complaykh.comGo to:

http://playkh.comhttp://playkh.com

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wiz623

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Tap

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Tap

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Tap, then wait.

wiz623

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Tap

wiz623

Notice? Wonder?What do you

BAG?How many hot are in a

PREDICTIONMake a

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HOT

Hot to Not Ratio

1:6

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HOT

Hot to Not Ratio

1:6

Assuming the "Hot to Not" ratio remains 1:6 for every

bag of Doritos Roulette, how many "hot" chips

should you expect in a bag with 36 "not hot" chips?

@MathletePearcewww.tapintoteenminds.com

HOT

Hot to Not Ratio

1:6

Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

HOT

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HOT

Hot to Not Ratio

1:6

Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

HOT

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HOT

Hot to Not Ratio

1:6

Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

HOT

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HOT

Hot to Not Ratio

1:6

Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

HOT

@MathletePearcewww.tapintoteenminds.com

HOT

Hot to Not Ratio

1:6

Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

HOT 6

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Hot to Not Ratio

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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@MathletePearcewww.tapintoteenminds.com

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Hot to Not Ratio

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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Hot to Not Ratio

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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Hot to Not Ratio

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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Hot to Not Ratio

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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Hot to Not Ratio

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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Assuming the "Hot to Not" ratio remains 1:6 for every bag of Doritos Roulette, how many "hot" chips should you expect in a bag with 36 "not hot" chips?

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@MathletePearcewww.tapintoteenminds.com

HOT

New Hot to Not Ratio

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Assume Doritos changes the "Hot to Not" ratio to 2:5 for

every bag of Doritos Roulette. How many not hot"

chips should you expect in a bag with 12 "hot" chips?

HOT

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HOT

2:5

Assume Doritos changes the "Hot to Not" ratio to 2:5 for

every bag of Doritos Roulette. How many not hot"

chips should you expect in a bag with 12 "hot" chips?

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2:5Assume Doritos changes the "Hot to Not" ratio to 2:5

for every bag of Doritos Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

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111111

111111

6 6

2 5

12 30

If there are ___ hot there are ___ not

2 54 106 158 201012

2530

0

0

2

5

4

10

6

15

8

20

10

25

12

30

14

35

+2

+5

+2

+5

+2

+5

+2

+5

+2

+5

+2

+5

HOT

1 3 5 7 9 11 13

@MathletePearcewww.tapintoteenminds.com

HOT

HOT

HOT

HOT

HOTHOT

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

111111

111111

6 6

2 5

12 30

@MathletePearcewww.tapintoteenminds.com

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

111111

111111

HOT

HOT

HOT

HOTHOT

HOT

HOT

@MathletePearcewww.tapintoteenminds.com

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

HOT

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

?

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

?

2 5

12

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

?

2 5

12

111111

111111

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

?

2 5

12

111111

111111

x

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

?

2 5

12

111111

111111

x x

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

?

2 5

12

111111

111111

x x

6 6

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

30

2 5

12

111111

111111

x x

6 6 30

2 5

126 6

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

30

2 5

12

111111

111111

x x

6 6 3025126 6

=

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

30

2 5

12

111111

111111

x x

6 63025

126 6

=

x x

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

30

2 5

12

111111

111111

x x

6 63025

12

6

6

=

x

x

@MathletePearcewww.tapintoteenminds.com

HOT

ConsolidationAssume Doritos changes the "Hot to Not" ratio to 2:5 for every bag of Doritos

Roulette. How many not hot" chips should you expect in a bag with 12 "hot" chips?

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOT

HOTHOT

HOT

111111

111111

x x

6 63025

12

6

6

=

x

x

12

2 5

30

@MathletePearcewww.tapintoteenminds.com

Your Turn

The ratio of boys to girls in a class is 3 to 7. If there are 12 boys total in the class, how many

girls are there?

@MathletePearcewww.tapintoteenminds.com

Your TurnThe ratio of boys to girls in a class is 3 to 7.

If there are 12 boys total in the class, how many girls are there?

x37

12=12

3 7

x

@MathletePearcewww.tapintoteenminds.com

Your TurnThe ratio of boys to girls in a class is 3 to 7.

If there are 12 boys total in the class, how many girls are there?

x37

12=12

3 7

x

@MathletePearcewww.tapintoteenminds.com

Your TurnThe ratio of boys to girls in a class is 3 to 7.

If there are 12 boys total in the class, how many girls are there?

111111

111111

x x

? ?x37

12

?

?

=

x

x

12

3 7

x

@MathletePearcewww.tapintoteenminds.com

Your TurnThe ratio of boys to girls in a class is 3 to 7.

If there are 12 boys total in the class, how many girls are there?

111111

111111

x x

4 4x37

12

4

4

=

x

x

12

3 7

x

@MathletePearcewww.tapintoteenminds.com

Your TurnThe ratio of boys to girls in a class is 3 to 7.

If there are 12 boys total in the class, how many girls are there?

111111

111111

x x

4 42837

12

4

4

=

x

x

12

3 7

28

BAG?How many hot are in a

BAG?How many hot are in a

BAG?How many doritos are in a

50 g

=

50 g50 g50 g50 g

?? 235 g

21 chips

x chips

50 g

??

21

x

50

235

BAG?50 g

=

50 g50 g50 g50 g

?? 235 g

21 chips

x chips

50 g

?? x

50

23521

How many doritos are in a

BAG?50 g

=

50 g50 g50 g50 g

?? 235 g

21 chips

x chips

50 g

?? x 23521

50

How many doritos are in a

BAG?50 g

=

50 g50 g50 g50 g

?? 235 g

21 chips

x chips

50 g

?? 23521

50

x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips =

50 g50 g50 g50 g50 g

?? ??21

50

x

235

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g?

?

x

x

50 g50 g50 g50 g

??

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g1

50 gx

x

50 g50 g50 g?

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g

11

50 gx

x

50 g50 g?

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g

111

50 gx

x

50 g?

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

1111

50 gx

x

?

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

11111 50 g

x

x

?

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

11111 50 g

x

x

?

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

11111 50 g

21 chips21 chips21 chips21 chips21 chips

11111

x

x

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

11111 50 g

21 chips21 chips21 chips21 chips21 chips

11111

4 84 chips

x

x

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

11111 50 g

21 chips21 chips21 chips21 chips21 chips

11111

4 84 chips 4 200 g

x

x

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

11111 35 g

21 chips21 chips21 chips21 chips21 chips

11111

4 84 chips 4 200 g

x

x

?

?

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

1111

21 chips21 chips21 chips21 chips21 chips

11111

4 84 chips 4 200 g

0.7 35 g

4.7

x

x

?x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

1111

21 chips21 chips21 chips21 chips

1111

4 84 chips 4 200 g

0.7 35 g0.7

x

x 4.7

4.7x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

1111

21 chips21 chips21 chips21 chips

1111

4 84 chips 4 200 g

0.7 35 g0.7 14.7 chips

4.7

4.7

x

x

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x=

50 g50 g50 g50 g

1111

21 chips21 chips21 chips21 chips

1111

84 chips 200 g

0.7 35 g0.7 14.7 chips

4.7 4.7

4.7

4.7

x

x

x x

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

x

4.7

4.7

=

x

x

50 g50 g50 g50 g

1111

21 chips21 chips21 chips21 chips

1111

0.7 35 g0.7 14.7 chips

4.7 4.798.7 chips 235 g

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

98.7

4.7

4.7

=

x

x

50 g50 g50 g50 g

1111

21 chips21 chips21 chips21 chips

1111

0.7 35 g0.7 14.7 chips

4.7 4.798.7 chips 235 g

How many doritos are in a

BAG?

235 g

50 g21 chips

x chips 235

21

50

98.7

4.7

4.7

=

x

x

50 g50 g50 g50 g

1111

21 chips21 chips21 chips21 chips

1111

0.7 35 g0.7 14.7 chips

4.7 4.798.7 chips 235 g

How many doritos are in a

BAG?

?

=

x

x

How many hot are in aHOT

1:6

99 doritos per bag

?

x chips

21 chips21 chips21 chips21 chips

1111

0.7 14.7 chips

99 Doritos

1 hot : 6 notTotal of 7 Doritos

BAG?

?

=

x

x

How many hot are in aHOT

1:6

99 doritos per bag

7 Doritos?

x chips

21 chips21 chips21 chips21 chips

1111

0.7 14.7 chips

99 Doritos

BAG?

x chips

?

?

=

x

x

21 chips21 chips21 chips21 chips

1111

0.7 14.7 chips

How many hot are in aHOT

1:6

99 doritos per bag

1 hot : 6 notTotal of 7 Doritos

7 Doritos

? 99 Doritos

BAG?

x chips

?

?

=

x

x

21 chips21 chips21 chips21 chips

1111

0.7 14.7 chips

How many hot are in aHOT

1:6

99 doritos per bag

1 hot : 6 notTotal of 7 Doritos

7 Doritos

?

1 hot

x hot? 99 Doritos

BAG?

x chips

?

?

=

x

x

21 chips21 chips21 chips21 chips

1111

0.7 14.7 chips

How many hot are in aHOT

1:6

99 doritos per bag

1 hot : 6 notTotal of 7 Doritos

7 Doritos

?

1 hot

x hot? 997

1 x

99 Doritos

BAG?

x chips

?

14.1

=

x

x

21 chips21 chips21 chips21 chips

1111

0.7 14.7 chips

How many hot are in aHOT

1:6

99 doritos per bag

1 hot : 6 notTotal of 7 Doritos

7 Doritos

14.1

1 hot

x hot? 997

1 x

99 Doritos

BAG?

x chips

14.1

14.1

=

x

x

21 chips21 chips21 chips21 chips

1111

0.7 14.7 chips

How many hot are in aHOT

1:6

99 doritos per bag

1 hot : 6 notTotal of 7 Doritos

7 Doritos

14.1

1 hot

x hot 997

1 x

14.1 99 Doritos

BAG?

x chips

14.1

14.1

=

x

x

21 chips21 chips21 chips21 chips

1111

0.7 14.7 chips

99 Doritos

How many hot are in aHOT

1:6

99 doritos per bag

1 hot : 6 notTotal of 7 Doritos

7 Doritos

14.1

1 hot

14.1 hot 997

1 14.1

14.1

Visual and ConcreteContextualTASKS

VISION

@MathletePearcewww.tapintoteenminds.com

Visual and ConcreteContextualTASKS

VISION

@MathletePearcewww.tapintoteenminds.com

INQUIRYConstruct Conceptual

Understanding

Visual and ConcreteContextualTASKS

VISION

@MathletePearcewww.tapintoteenminds.com

INQUIRYConstruct Conceptual

Understanding

Connect to Prior Knowledge

CONNECTIONS

Visual and ConcreteContextualTASKS

VISION

@MathletePearcewww.tapintoteenminds.com

INQUIRYConstruct Conceptual

Understanding

Connect to Prior Knowledge

CONNECTIONS

CONSOLIDATEMove Towards

Procedural Fluency

Two Groups of StudentsStruggle with

unfamiliarproblems

Good at Math

Not Good at Math

@MathletePearcewww.tapintoteenminds.com

Two Groups of Students

Good at Math

Not Good at Math

Struggle withunfamiliarproblems

@MathletePearcewww.tapintoteenminds.com

3 Act Math Tasks

iTunes U

kylep.ca/curiousmath1p@MathletePearcewww.tapintoteenminds.com

3 Act Math Tasks

tapintoteenminds.com/3act-math@MathletePearcewww.tapintoteenminds.com

On The Web

http://tapintoteenminds.com/3act-math

Doritos Roulette

http://kylep.ca/doritosmath@MathletePearcewww.tapintoteenminds.com

http://kylep.ca/doritosmath

@MathletePearcewww.tapintoteenminds.com

Visual and ConcreteContextualTASKS

CONSOLIDATE INQUIRYMove Towards

Procedural Fluency

Connect to Prior Knowledge

CONNECTIONS

Construct Conceptual Understanding

VISION

Exploring the Progression of Proportional

Reasoning From K-9tapintoteenminds.com/OTF

Ontario Teachers Federation

CONNECT