Transcript Developing Proportional Reasoning

Developing Proportional
Reasoning
Jim Hogan
School Support Services, University of Waikato
National Numeracy Facilitators
Conference 2008
Waipuna
Bepatientatintersectionsandwaitfora gap
With reference to:• Proportionality and the Development of PreAlgebra Understandings
Thomas Post, Merlyn Behr, Richard Lesh
The Rational Number Project
http://education.umn.edu/rationalnumberproject
Many Vince Wright papers and Figure It Outs
Another year of thinking about all this.
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 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.
Activity
• Buddy up and try and define
proportional reasoning.
• Give a few examples of problems
• What mathematical prior knowledge and
skills do you need?
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
May the moose ne’er leave yer girnall wi’
a tear-drap in his e’e.
Robert Burns
Well…so far I think PR involves
additive
Visual/drawing
experience
Multiplicative
literate
failure
fractions
PR
Ability to learn
Problem solving skills
Seeing relationships
su
cc
es
s
decimals
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 2.4km, 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 80ml 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 2.4km in 11.2minutes.
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?
The red group of 2, is 2 thirds the size of, the yellow group
of 3.
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
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 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.
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?
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 x3 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 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?
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
3km per 2 litres.
Y = mx
• Start with +ve
quadrant
problems and
y=mx
6 km per 4 litres
is the same as
3km 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 100km using 12.5L 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 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?
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 = 6.7 laps.
To Cross-Multiply
• Nicole ran 4 laps in 6 minutes. How far did she run in
10 minutes?
4
___
6
=
hence
d
___
10
6d = 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
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, 101a to 101g, .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, cos, 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 and
critical manner.
Thank you
This powerpoint is available from my
website
http:schools.reap.org.nz/advisor
Along with many other resources.