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Developing Proportional Reasoning Jim Hogan School Support Services, University of Waikato National Numeracy Facilitators Conference 2008 Waipuna

Developing Proportional Reasoning

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Developing Proportional Reasoning. Jim Hogan School Support Services, University of Waikato National Numeracy Facilitators Conference 2008 Waipuna. Bepatientatintersectionsandwaitforagap. With reference to:-. Proportionality and the Development of Pre-Algebra Understandings - PowerPoint PPT Presentation

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Page 1: Developing Proportional Reasoning

Developing Proportional Reasoning

Jim HoganSchool Support Services, University of Waikato

National Numeracy Facilitators Conference 2008

Waipuna

Page 2: Developing Proportional Reasoning

Bepatientatintersectionsandwaitfora gap

Page 3: Developing Proportional Reasoning

With reference to:-• Proportionality and the Development of Pre-Algebra Understandings

Thomas Post, Merlyn Behr, Richard Lesh

The Rational Number Projecthttp://education.umn.edu/rationalnumberproject

Many Vince Wright papers and Figure It OutsAnother year of thinking about all this.

Thanks Vince!

Page 4: Developing Proportional Reasoning

Session Aims• increase our understanding of proportional reasoning (PR) and the links to algebra

• demonstrate a structured set of resources for you to actively teach PR

• recommend several valuable readings and resources for your learning of PR today

• empower you to evaluate resources for PR in an informed and and critical manner.

Page 5: Developing Proportional Reasoning

The Real World• Many aspects of our world operate according to proportional rules.

• Being able to operate and reason to interpret nature is a key competency.

Also powerful and vital for thinking success in upper secondary, tertiary.

Page 6: Developing Proportional Reasoning

Activity• Buddy up and try and define proportional reasoning.

• Give a few examples of problems

• What mathematical prior knowledge and skills do you need?

Page 7: Developing Proportional Reasoning

Comparing in MathematicsComparing in Mathematics

Shall I compare thee to a summer's day? Thou art more lovely and more temperate…

How do we compare things in mathematics?

Shall I compare thee to a summer's day? Thou art more lovely and more temperate…

How do we compare things in mathematics?

Page 8: Developing Proportional Reasoning

Choice!

We can

subtract (add)or divide (multiply/group)

or maybe some measure against a standard.

Page 9: Developing Proportional Reasoning

Adding just don’t work!Adding just don’t work!Start with 1 red block: 3 yellow blocks

What happens if we keep adding one block of each colour?

Start with 1 red block: 3 yellow blocks

What happens if we keep adding one block of each colour?

Page 10: Developing Proportional Reasoning

Keeping the ratio the sameKeeping the ratio the same

Really important multiplicative idea…Groups of groups

Really important multiplicative idea…Groups of groups

Page 11: Developing Proportional Reasoning

May the moose ne’er leave yer girnall wi’ a tear-drap in his e’e.

Robert Burns

Page 12: Developing Proportional Reasoning

Well…so far I think PR involves

PR

Multiplicative

fractions

decimals

literate

experience

Problem solving skills

failure

success

additive

Seeing relationships

Ability to learn

Visual/drawing

Page 13: Developing Proportional Reasoning

Post, Behr and Lesh say…

• PR involves co-variation, multiple comparison, storing and processing several pieces of information.

• PR is concerned with inference, prediction, the qualitative and the quantitative.

Eeek gooble de gook!It’s why the moose left the hoose.

Page 14: Developing Proportional Reasoning

Lets Unpack Post, Behr and Lesh !

• CovariationTwo or more things varying at the same time.

EgNikki jogs 2.4km, or 6 laps in 12 minutes.

EgOne ink cartridge costs $39.75.

Page 15: Developing Proportional Reasoning

More Unpacking• Multiple Comparison

Eg4 cups of sugar to 7 litres3 cups to 5 litres

Which is sweeter?

Page 16: Developing Proportional Reasoning

Yet More Unpacking• Storing and processing several data

EgTwo stroke mix is 1 part oil to 50 parts oil.

Do I add 80ml of oil to 16 litres of petrol to get the correct mix?

Page 17: Developing Proportional Reasoning

More Plus Unpacking• Qualitative thinking

EgNikki ran fewer laps today in more time.

Did she run faster, slower, the same or can’t you tell?

Page 18: Developing Proportional Reasoning

Extra Unpacking• Quantitative thinking

EgInterest is 8.5% pa flat. How much interest do I repay weekly on $120,000?

Eg Discount $18 by 20%

Page 19: Developing Proportional Reasoning

Extra Plus Unpacking• Inference

EgJacko said the ratio of boys to girls in his class on Monday was exactly 4:5. On Tuesday there was one more person in class and the ratio was 5:6.

Is this possible?

Page 20: Developing Proportional Reasoning

Still More Unpacking• Prediction

EgI run 2.4km in 11.2minutes.

About how far can I run in 55 minutes?

Page 21: Developing Proportional Reasoning

What do we need to begin?

• Certainly to be developing or have multiplicative ability

• Experience of real world situations

• Problem solving ability• Time to allow the growth…then it is a matter of application

Page 22: Developing Proportional Reasoning

So…Is it any wonder we find a long wait in our Year 10 data?

This is quite hard to learn and needs time to develop.

Page 23: Developing Proportional Reasoning

An Issue…An Issue…

Where is the 2 thirds in this group of five?

The red group of 2, is 2 thirds the size of, the yellow group of 3.

Did we just suddenly rename one?

Where is the 2 thirds in this group of five?

The red group of 2, is 2 thirds the size of, the yellow group of 3.

Did we just suddenly rename one?

Ratio is commonly written 2:3 = 2/3…what does this mean?

Ratio is commonly written 2:3 = 2/3…what does this mean?

Page 24: Developing Proportional Reasoning

An Issue…revealedAn Issue…revealed

Where is the 2 thirds in this group of five?

The red group of 2, is 2 thirds the size of, the yellow group of 3.

Did we just suddenly rename one?

Where is the 2 thirds in this group of five?

The red group of 2, is 2 thirds the size of, the yellow group of 3.

Did we just suddenly rename one?

Ratio is commonly written 2:3 = 2/3 …what does this mean?

Ratio is commonly written 2:3 = 2/3 …what does this mean?

Page 25: Developing Proportional Reasoning

Check Point CharlieSession Aims• increase our understanding of proportional

reasoning (PR) and links to algebra

• demonstrate a structured path (and resources) to actively teach PR

• recommend several valuable readings and resources for your learning of PR today

• empower you to evaluate resources for PR in an informed and and critical manner.

Page 26: Developing Proportional Reasoning

May yer lum continue reekin’ ‘til ye’re auld enough tae d’e.

R Burns

Page 27: Developing Proportional Reasoning

A PR Progression…Post et al

Pretest of this- Unit rate method and inverse, x- Two step, more complicated, /- Reciprocal use and meaning- factor of change, related numbers- quantitative, qualitative comparison- graphical interpretation- missing , rate, ratio- discover cross multiply and of course always generalising the problems

and post test if needed.

Page 28: Developing Proportional Reasoning

Teaching Proportion 101

• The Unit Rate MethodThis is by far the most intuitive and the obvious first step.

Disks cost 90 cents each (unit rate). How much for 15 disks?

Children have experience of this transaction. One step multiplication problem. Start with what they know.

Page 29: Developing Proportional Reasoning

Teaching Proportion• The Unit Rate MethodExtending this to using the inverse idea.

Sally bought 15 disks for $9.00. How much did each disk cost?

Children have experience of this transaction. One step division problem.

Page 30: Developing Proportional Reasoning

Teaching Proportion• The Unit Rate MethodCombining these skills.

Sally bought 15 disks for $9.00. How much would she pay for 10 disks?

Two step problem needing both divison and multiplication.

Page 31: Developing Proportional Reasoning

Teaching Proportion• The Unit Rate MethodWe can make the problem more obscure by adjusting the numbers.

Sally bought 15 disks for $7.50. How much would she pay for 11 disks?

Two step problem needing both divison and multiplication and number sense. A missing value problem.

Page 32: Developing Proportional Reasoning

The Rate and the Reciprocal

• There are always two unit rates.

My truck travels 750km on 60L of diesel.

How many km per L?How many L per km?

This idea needs developing.

Page 33: Developing Proportional Reasoning

The Rate and the Reciprocal

• There are always two unit rates.

My truck travels “b” km on “a” L of diesel.

How many km per L?How many L per km?

This is pre-algebra. Which rate is more useful?

Page 34: Developing Proportional Reasoning

Factor of Change Method

• One quantity is a multiple of the other.

EgBananas are 5 for $3.50. How much did Sally pay for 15 bananas?

Numbersense allows the simple x3 multiple to be identified. Many problems can be solved this way.

Page 35: Developing Proportional Reasoning

Factor of Change Method

• Not so useful if the numbers are obscure!

EgBananas are 5 for $3.50. How much did Sally pay for 12 bananas?

Did you multiply by two and two fifths?

Page 36: Developing Proportional Reasoning

On to harder problems• Numerical comparison is the next stage to develop in students. Quantitative comparison.

EgBilly bought 4 apples for $2.40 and Joe 7 for $4.20.

Who got the best deal?

Page 37: Developing Proportional Reasoning

On to harder problems• Numerical comparison is the next stage to develop in students. Qualitative comparison.

EgThis 3L tin of pink paint is mixed 1 red and 2 white. This 5L tin is 2 red and 3 white. Which is more pink?

What if they were both 3L tins?

Page 38: Developing Proportional Reasoning

Y = mx• Equivalent rates, ratios and rational numbers can be represented graphically as a gradient on the (x,y) plane.

Caution!• (0,0) usually has meaning• The x and y axes do not. They remain uninterpretable. – Involves ‘derision’ by zero!

Page 39: Developing Proportional Reasoning

Equivalent rates

6 km per 4 litresis the same as

3km per 2 litres.

Page 40: Developing Proportional Reasoning

Y = mx• Start with +ve quadrant problems and y=mx

6 km per 4 litresis the same as3km per 2 litres.

Page 41: Developing Proportional Reasoning

Y = mx• Which fraction is bigger?

2 thirdsor 5 sevenths?

Page 42: Developing Proportional Reasoning

Y = mx• Which ratio is bigger?

3:4or

5:7?

Page 43: Developing Proportional Reasoning

Missing Value problems• 7 apples for $5

• How many for $3

• How much for 2 apples?

• How many for $18?

• How much for 52 apples?

Page 44: Developing Proportional Reasoning

Reciprocal meaningWhat is represented by 5/7?

What is represented by 7/5?

apples

dollars

Page 45: Developing Proportional Reasoning

Rate• Compares two quantities of different units.

Eg I travel 100km using 12.5L of petrol.

How many km/L?How many L/km?

Which rate is more meaningful?

Page 46: Developing Proportional Reasoning

Ratio• Compares two quantities of same units.

Eg I am 180cm tall and my daughter is 160cm tall. This is a ratio of 180:160 or simplified 9:8.

What would 8:9 mean?What does to simplify mean?

Page 47: Developing Proportional Reasoning

At Last!• Allow students to discover the cross multiply principle slowly. There is a lot of foundation work that is needed first.

a/b = c/d => ad = bc

• This is no more than using unit rate Without this understanding it is a

meaningless statement.

Page 48: Developing Proportional Reasoning

To Cross-Multiply• Nicole ran 4 laps in 6 minutes. How far did she run in 10 minutes?

Unit rate is 4/6 laps per minute

Distance is 10 x 4/6 = 6.7 laps.

Page 49: Developing Proportional Reasoning

To Cross-Multiply• Nicole ran 4 laps in 6 minutes. How far did she run in 10 minutes?

4

6

___ =10

___d

hence 6d = 40

And so on to the answer.

Page 50: Developing Proportional Reasoning

Check Point CharlieSession Aims• increase our understanding of proportional

reasoning and the links to algebra

• demonstrate a structured path (and resources) for explicit teaching of PR

• recommend several valuable readings and resources for your learning of PR today

• empower you to evaluate resources for PR in an informed and and critical manner.

Page 51: Developing Proportional Reasoning

• May you always cast a shadow.

Page 52: Developing Proportional Reasoning

Readings• Proportional Reasoning - Some rational Thoughts -Vince Wright, MAV 2005

• Proportionality and the Development of Prealgebra Understandings- Lesh, Post and Behr 1999

And resources…not sure wheer from!Pre, Post, 101a to 101g, .ppts are

posted on my website http://schools.reap.org.nz/advisor

Page 53: Developing Proportional Reasoning

Why emphasise proportional reasoning?

Why emphasise proportional reasoning?

Fewer than half the adult population are proportional thinkers.

We do not acquire the habits and skills of proportional reasoning simply by getting older.

Instruction in proportional reasoning is a must.(Lamon, 1999).

From Teaching Student Centred Mathematics.

Van De Walle

Fewer than half the adult population are proportional thinkers.

We do not acquire the habits and skills of proportional reasoning simply by getting older.

Instruction in proportional reasoning is a must.(Lamon, 1999).

From Teaching Student Centred Mathematics.

Van De Walle

Page 54: Developing Proportional Reasoning

Developing reasoning firstDeveloping reasoning first

… instruction can be effective if rules are delayed.

Premature use of rules encourages students to apply rules without thinking and the ablity to reason proportionally often does not develop.

From Teaching Student Centred Mathematics.

Van De Walle

… instruction can be effective if rules are delayed.

Premature use of rules encourages students to apply rules without thinking and the ablity to reason proportionally often does not develop.

From Teaching Student Centred Mathematics.

Van De Walle

Page 55: Developing Proportional Reasoning

…extend to

• And so to – non linear, inverse, square

•awkward numbers– density, solubility

» applications….physics, economics etc

Page 56: Developing Proportional Reasoning

Senior School• Needs

– Ratio and rate ideas, with inverses– Handling proportional equations

• In geometry, eg similar triangles, circle geometry chord and tangent,

• In trigonometry, sin, cos, cosec, sec

– Other proportionalities• Inverse square, log, exponential, cube

– Applications of rational numbers• Probability, Conditional probability,

Page 57: Developing Proportional Reasoning

How do texts do it?

Take a look in these commonly available books to see how others teach this.

Page 58: Developing Proportional Reasoning

Check Point CharlieSession Aims• increase our understanding of proportional reasoning

(PR) and the links to algebra

• demonstrate a structured path (and resources) to actively teach PR

• recommend several valuable readings and resources for your learning of PR today

• empower you to evaluate resources for PR in an informed and and critical manner.

Page 59: Developing Proportional Reasoning

Thank you

This powerpoint is available from my website

http:schools.reap.org.nz/advisor

Along with many other resources.