Developing Proportional Reasoning Jim Hogan School Support Services

Developing Proportional Reasoning Jim Hogan School Support Services, University of Waikato National Numeracy Facilitators Conference 2007 Waipuna

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With reference to: • Proportionality and the Development of Pre. Algebra Understandings Thomas Post, Merlyn Behr, Richard Lesh The Rational Number Project http: //education. umn. edu/rationalnumberproject Many Vince Wright papers and Figure It Outs. Thanks Vince!

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 critical manner.

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.

Comparing 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?

Choice! We can subtract (add) or divide (multiply/group) or maybe some measure against a standard.

Adding just don’t work! Start with 1 red block: 3 yellow blocks What happens if we keep adding one block of each colour?

Keeping the ratio the same Really important multiplicative idea… Groups of groups

Activity • Buddy up and try and define proportional reasoning. • Give a few examples of problems • What mathematical prior knowledge and skills do you need?

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

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.

Lets Unpack Post, Behr and Lesh ! • Covariation Two or more things varying at the same time. Eg Nikki jogs 2. 4 km, or 6 laps in 12 minutes. Eg One ink cartridge costs $39. 75.

More Unpacking • Multiple Comparison Eg 4 cups of sugar to 7 litres 3 cups to 5 litres Which is sweeter?

Yet More Unpacking • Storing and processing several data Eg Two stroke mix is 1 part oil to 50 parts oil. Do I add 80 ml of oil to 16 litres of petrol to get the correct mix?

More Plus Unpacking • Qualitative thinking Eg Nikki ran fewer laps today in more time. Did she run faster, slower, the same or can’t you tell?

Extra Unpacking • Quantitative thinking Eg Interest is 8. 5% pa flat. How much interest do I repay weekly on $120, 000? Eg Discount $18 by 20%

Extra Plus Unpacking • Inference Eg Jacko 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?

Still More Unpacking • Prediction Eg I run 2. 4 km in 11. 2 minutes. About how far can I run in 55 minutes?

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

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.

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

An Issue…revealed Ratio is commonly written 2: 3 = 2/3 … what does this mean? Where is the 2 thirds in this group of five? of 3. The red group of 2, is 2 thirds the size of, the yellow group Did we just suddenly rename one?

Check Point Charlie Session 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 critical manner.

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

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.

Teaching Proportion 101 • The Unit Rate Method This 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.

Teaching Proportion • The Unit Rate Method Extending 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.

Teaching Proportion • The Unit Rate Method Combining 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.

Teaching Proportion • The Unit Rate Method We 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.

The Rate and the Reciprocal • There always two unit rates. My truck travels 750 km on 60 L of diesel. How many km per L? How many L per km? This idea needs developing.

The Rate and the Reciprocal • There 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?

Factor of Change Method • One quantity is a multiple of the other. Eg Bananas are 5 for $3. 50. How much did Sally pay for 15 bananas? Numbersense allows the simple x 3 multiple to be identified. Many problems can be solved this way.

Factor of Change Method • Not so useful if the numbers are obscure! Eg Bananas are 5 for $3. 50. How much did Sally pay for 12 bananas? Did you multiply by two and two fifths?

On to harder problems • Numerical comparison is the next stage to develop in students. Quantitative comparison. Eg Billy bought 4 apples for $2. 40 and Joe 7 for $4. 20. Who got the best deal?

On to harder problems • Numerical comparison is the next stage to develop in students. Qualitative comparison. Eg This 3 L tin of pink paint is mixed 1 red and 2 white. This 5 L tin is 2 red and 3 white. Which is more pink? What if they were both 3 L tins?

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!

Equivalent rates 6 km per 4 litres is the same as 3 km per 2 litres.

Y = mx • Start with +ve quadrant problems and y=mx 6 km per 4 litres is the same as 3 km per 2 litres.

Y = mx • Which fraction is bigger? 2 thirds or 5 sevenths?

Y = mx • Which ratio is bigger? 3: 4 or 5: 7?

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?

Reciprocal meaning What is represented by 5/7? What is represented by 7/5? apples dollars

Rate • Compares two quantities of different units. Eg I travel 100 km using 12. 5 L of petrol. How many km/L? How many L/km? Which rate is more meaningful?

Ratio • Compares two quantities of same units. Eg I am 180 cm tall and my daughter is 160 cm tall. This is a ratio of 180: 160 or simplified 9: 8. What would 8: 9 mean? What does to simplify mean?

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.

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 = 15 laps.

To Cross-Multiply • Nicole ran 4 laps in 6 minutes. How far did she run in 10 minutes? 4 ___ 6 = hence d ___ 10 6 d = 40 And so on to the answer.

Check Point Charlie Session 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 critical manner.

• May you always cast a shadow.

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, 101 a to 101 g, . ppts are posted on my website http: //schools. reap. org. nz/advisor

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

Developing 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

…extend to • And so to – non linear, inverse, square • awkward numbers – density, solubility » applications…. physics, economics etc

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, cosec, sec – Other proportionalities • Inverse square, log, exponential, cube – Applications of rational numbers • Probability, Conditional probability,

How do texts do it? Take a look in these commonly available books to see how others teach this.

Check Point Charlie Session 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 critical manner.

Thank you This powerpoint is available from my website http: schools. reap. org. nz/advisor email jimhogan@clear. net. nz Along with many other resources.