A Focus on Proportional Reasoning, Grades 4 - 8 February, 2015 - - PowerPoint PPT Presentation

A Focus on Proportional Reasoning,

Grades 4 - 8

February, 2015 Marian Small

• Dr. Marian Small, University of New

Brunswick February 2015

Agenda

• What does/can proportional

reasoning look like in Grades 4 – 8?

• Dr. Marian Small, University of New

Brunswick February 2015

Agenda

• What have we seen Ontario

students do when confronted with proportional reasoning problems?

• Dr. Marian Small, University of New

Brunswick February 2015

Agenda

• What manipulatives are useful to

evoke proportional reasoning?

• Dr. Marian Small, University of New

Brunswick February 2015

Agenda

• Creating rich proportional

reasoning problems

• Dr. Marian Small, University of New

Brunswick February 2015

• Proportional reasoning involves the

use of multiplicative relationships to compare quantities and to predict the value of one quantity based on the values of another.

Proportional reasoning

• Dr. Marian Small, University of New

Brunswick February 2015

• But it’s not about actually seeing

multiplication signs.

Proportional reasoning

• Dr. Marian Small, University of New

Brunswick February 2015

• For example, if I ask a child what 8

cookies should cost, s/he is thinking proportionally whether adding 89 + 89 or thinking 2 x 89.

Proportional reasoning

89¢

• Dr. Marian Small, University of New

Brunswick February 2015

In the curriculum

Most obvious spots

• Grades 4 – 8: sections under number

entitled Proportional Relationships

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more Examples

• Grade 4 on: any measuring activity

using units

• If I ask you to predict how many

metres long a room is if I let you see one or two metre sticks against the wall.

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

How long is this room?

• Dr. Marian Small, University of New

Brunswick February 2015

So what do you think?

Which is your choice? Type in the chat box.

1. Measuring is always about proportional reasoning. 2. When you use a partial measurement to predict a full one, that’s when you use proportional reasoning.

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: exchanging coins
• If I ask you to show me 60¢ with

fewer coins and you exchange 2 dimes and a nickel for a quarter, you are changing to bigger units to get fewer coins.

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: multiplication/division
• If I ask you how many children are

in a class if there are 6 tables, each with 4 children at the table…. I am changing from a unit of table (6 units) to a unit of child (24 units)

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: fractions
• Any work with fractions involves

the multiplicative relationship between numerator and denominator; for example

• Fractions are equal to 1/2 if the

denominator is twice the numerator

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: fractions
• How far apart numerators and

denominators are tells me nothing about relative size.

• For example, 2/3 > 3/5 (1 apart vs

2) but 1/2 < 7/9 (1 apart vs 2)

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: graphs with scales
• Using a many-to-one

correspondence on a graph (e.g.

• ne icon represents 4 people) is

proportional thinking--- thinking of a number as groups of, e.g. 4

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

A circle graph or any other graph that shows fractions

• r percents of people in categories involves a

multiplicative comparison between the part and the whole. Make a multiplicative comparison about this graph in the chat box.

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: probability
• All probability work involves

comparing, fractionally, the desired events to the total number of events

• e.g. the probability of rolling 1 on a

die is 1 out of 6

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: solving problems

relating to magnitudes of 1000, etc.

• These problems normally involve
• unitizing. For example, if there are

200 sheets of paper in a pack costing $1.20, how much would 1000 sheets cost?

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: area/volume formulas
• The area of a rectangle describes

the number of squares that form equal rows of squares.

3 units of 5 squares

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: area/volume formulas
• The volume of a prism is about the

number of cubes that form equal layers of cubes.

3 layers of the area of the base

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: unit conversions
• Determining the number of metres

for 430 cm involves unit changes. There will be 1/100 as many units.

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 4 on: linear patterns
• Determining what the 100th term of

5, 10, 15, 20,…. is involves thinking

• f 100 units of 5.
• Determining what the 100th term of

4, 9, 14, 19, … is involves thinking

• f 100 units of 5, less 1.
• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 5 on: mean of a set of data
• Calculating the mean is about

replacing n pieces of data with n identical units; the mean is the size

• f that unit
• e.g. the mean of 3, 4, 5 is 4
• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 6 on: percent work
• Any percent is a multiplicative

comparison between a number and 100. Thinking of 50% of 32 is about relating the relationship between 50 and 100 to a number and 32.

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

0 50 100 0 ?? 32

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 6: rotation work with

patterns

• Asking what the 50th term of the

pattern below looks like requires you to think of 50 as groups of 4.

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 7 on: Two shapes are

similar if the proportions relating their side lengths are maintained.

1 5 0.75 3.75

• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 7 on: solving linear

equations by, for example, multiplying both sides by the same amount

• You can multiply both sides of an

equation by 3 since if one item equals another, 3 of them match 3

• f the other.
• Dr. Marian Small, University of New

Brunswick February 2015

But there are SO MANY more

• Grade 7 on: linear relationships
• When students look at

relationships between two variables and see a line, through (0,0), they are recognizing that one variable’s value is always the same multiple of the other’s

• Dr. Marian Small, University of New

Brunswick February 2015

For example

(1,7) (2,14)

Days vs. Weeks

• Dr. Marian Small, University of New

Brunswick February 2015

Your turn

• How might one of these curriculum

expectations be about proportional reasoning?

• Gr 4: demonstrate an understandng of

place value…

• Gr 6: identify composite and prime

numbers

• Gr 8:measure circumference and area
• f circles
• Dr. Marian Small, University of New

Brunswick February 2015

Big Ideas of PR

• It is often useful to think of one amount

as groups of another amount. e.g.

• one loonie as 4 quarters
• 14 days as 2 weeks
• 20 eggs as 1 2/3 dozen
• 25 as ¼ of 100
• Dr. Marian Small, University of New

Brunswick February 2015

In fact…

• Any number can be compared to

any other number multiplicatively, e.g. 8 can be compared to 2 by thinking of it as 4 twos. And 2 can be compared to 8 as 2/8 (or ¼) of an 8.

• Dr. Marian Small, University of New

Brunswick February 2015

In fact…

And 9 can be compared to 2 by

thinking of it as 4 ½ twos. And 2 can be compared to 9 as 2/9

• f a 9.
• Dr. Marian Small, University of New

Brunswick February 2015

Which price changed the most?

Comparing changes

$46 945.00 to $44 999.00 $5.99 to $2.99

• Dr. Marian Small, University of New

Brunswick February 2015

• Dr. Marian Small, University of New

Brunswick February 2015

Related important ideas

• If you use a bigger unit, you need

fewer of them.

• If units are related, you can use

that relationship to predict how many of one unit if you know how many of the other.

• Dr. Marian Small, University of New

Brunswick February 2015

Related important ideas

• How far apart numbers are

additively has nothing to do with how far apart they are multiplicatively.

• For example, 2 and 2000 are far

apart both ways.

• But 1000 and 2000 are only far

apart additively.

• Dr. Marian Small, University of New

Brunswick February 2015

Related important ideas

• Using a fraction, decimal or

percent is a way of comparing numbers multiplicatively.

• For example, 2/3 tells us that 2 is
• nly 2/3 of a 3.
• 0.4 is a way to compare 4 to 10
• 35% is a way to compare 35 to

100

• Dr. Marian Small, University of New

Brunswick February 2015

What does it look like?

• What sorts of problems involve

proportional reasoning?

• Dr. Marian Small, University of New

Brunswick February 2015

Dogs

• 1 out of every 3 Canadian

households has a dog.

• About how many dogs would you

predict for the students in your class?

• How would you envision a

Grade 4 solving this?

• Dr. Marian Small, University of New

Brunswick February 2015

Or..

• On average, Canadians consume

18% of their daily calories at breakfast.

• Is that true in your class?
• Dr. Marian Small, University of New

Brunswick February 2015

Probability

• You are pulling out a counter from

each bag.

• Which bag gives you the best chance
• f pulling out a red?
• Dr. Marian Small, University of New

Brunswick February 2015

Speeds

• A car goes 280 km in 3 hours.
• How far, at that speed, will they go in another

1.5 hours?

• Why was it smart to

ask about 1.5?

• To use 280 and not 270?
• Dr. Marian Small, University of New

Brunswick February 2015

Length

• How long is a line of 1 000 000

pennies?

19 cm

• Dr. Marian Small, University of New

Brunswick February 2015

How much faster?

• You normally drive 90 km/h on a certain
• road. How much faster would you have

to go to save 15 minutes on a 400 km trip on that road?

• Dr. Marian Small, University of New

Brunswick February 2015

A Fermi problem, e.g.

Estimate the number of square centimetres of pizza that all of the students in Toronto eat in one week.

Estimation

• Dr. Marian Small, University of New

Brunswick February 2015

An EQAO video

• http://www.youtube.com/watch?v=LPkQvN3r8js
• Dr. Marian Small, University of New

Brunswick February 2015

Let’s look at the types of problems that involve proportional reasoning that students around the province have been solving.

• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• Which sequence gets past 1000

first? 15, 25, 35, 45, 55,…. 500, 502, 504, 506, 508,… Why is this about proportional reasoning?

• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• You have linking cubes to build a

rectangle.

• The perimeter has to be three times as

much as the length.

• What do you know about the length and

width?

• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• A yellow pattern block is worth A.
• Build a design worth B.
• Choice 1: A is 6 and B is 20
• Choice 2: A is 5.1 and B is 17
• Choice 3: A is ½ and B is 1 2/3
• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• A light green Cuisenaire rod is worth 9 (or 15).
• What should the other rods be worth?
• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• Make a rectangle. Figure out its perimeter.
• Then make a rectangle with half the area.
• Figure out that perimeter.

P = 18 P = 12

• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• What fraction of the big perimeter

is the small one?

• Try more times. What fractions are

possible and which are not?

• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• You model a number with base ten

blocks.

• There are twice as many rods as

flats.

• There are 3 times as many unit

blocks as rods.

• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• What could the number be?
• Think of as many numbers as you

can that are less than 1000.

• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• Make a design with pattern blocks

that is half yellow.

• Dr. Marian Small, University of New

Brunswick February 2015

Maybe

• Dr. Marian Small, University of New

Brunswick February 2015

Maybe

• Dr. Marian Small, University of New

Brunswick February 2015

Maybe

• Dr. Marian Small, University of New

Brunswick February 2015

Problems we’ve tried

• Make a design that is 2/3 red and

1/3 green.

• Dr. Marian Small, University of New

Brunswick February 2015

Maybe

• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Pattern blocks

• The ___ block is worth ___. What are the other

blocks worth? 12

• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Pattern blocks

• The ___ block is worth ___. Make a design

worth ____. 12 44

• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Pattern blocks

• Make a design where ¾ of the area is yellow.
• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Cuisenaire rods

• Find rods that are ½ (or 2/3 or 5/6) as long as
• ther rods.
• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Cuisenaire rods

• One rod is 2 ½ times as long as another. What

rods could they be?

• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Cuisenaire rods

• A line of 8 of one colour rod matches a line of 5
• f another colour rod. What rods could you use?
• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Cuisenaire rods

• What single colour rods can make a line as

long as 4 orange rods?

• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Cuisenaire rods

• You measure something with orange rods. It

takes 4 orange rods. How many yellow would it take? How many pink?

• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Square tiles

• What does 3 x 4 look like?
• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Square tiles

• Why did 8 x 3 have to be the same as 4 x 6?
• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Square tiles

• Build a rectangle with a width of 3. How does

the area relate to the length? Could the perimeter be a multiple of the length?

• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Square tiles

• Build a shape with 3 times as many blue

squares as yellow ones, but 2 times as many red squares as yellow ones.

• Dr. Marian Small, University of New

Brunswick February 2015

Useful manipulatives

Square tiles

• Make a design that is 2/3 red and 1/4 green.
• Dr. Marian Small, University of New

Brunswick February 2015

Base Ten Blocks

• Show any 2-digit number with base ten blocks.
• Now show a number 10 times as big.
• Dr. Marian Small, University of New

Brunswick February 2015

Your turn

• Have any of you used other

manipulatives in a valuable way for proportional reasoning ?

• Dr. Marian Small, University of New

Brunswick February 2015

Creating good PR problems

• The purpose of the problem should

be to draw out proportional reasoning ideas.

• Here are a number of examples.
• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• You can arrange a batch of

ABOUT 50 counters into equal

• groups. How many groups and of

what size might they be?

• Dr. Marian Small, University of New

Brunswick February 2015

Follow up by asking

• Why did nobody have 100 groups?
• What was the biggest group size

anyone had? Why?

• When did someone have a lot of

groups?

• When did someone have a big group

size?

• When could there be 2 groups?
• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

How many marbles do you think the big container could hold?

Choice 1:

Choice 2:

10 10

• Dr. Marian Small, University of New

Brunswick February 2015

Common questions:

• Are there more than 10 marbles in the big

container? How do you know?

• Do you think there are more than 20

marbles? Why or why not?

• Did it matter how wide the dark blue container

(with 10 marbles) was?

• How?
• Dr. Marian Small, University of New

Brunswick February 2015

Common questions:

• Did it matter how high the dark blue

container of 10 was?

• How?
• How did you decide how many marbles?
• What if there had only been 5 marbles in

the small can? How would your answer change?

• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• How many ears would I draw if I

draw 8 cows?

• How many legs?
• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• How many numbers would I need to

write (say) to continue this way to get to 50? 12, 14, 16, 18, 20,….

• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• You can show an amount of cookies

exactly using groups of 6 cookies.

• How do you know that you can also show

it exactly using groups of 3 cookies?

• What about using groups of 4 cookies?
• Dr. Marian Small, University of New

Brunswick February 2015

You could:

Regularly use multiplicative language such as:

• Twice as much
• Four times as big
• Half as many
• Two thirds as heavy
• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• My brother has 2.5 times as many

games as I have. How many might we each have?

• Do you think I have 9 games?
• Dr. Marian Small, University of New

Brunswick February 2015

• I spin a spinner.
• I am twice as likely to get red as blue.
• I am half as likely to get blue as

green.

• What could the probability of green

be?

A Colourful Spinner

• Dr. Marian Small, University of New

Brunswick February 2015

Blue Blue Red Red Green Green Red Red Green Green

Possibilities

Blue Red Red Green Green Yellow Blue Red Red Green Green

• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• A sentence has 40 letters in it. What

number of words do you think it probably has? Why?

• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• About how many ceiling tiles are

there in the whole school?

• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• You draw a scale diagram and a

___ m distance is represented as ____ cm.

• Choose values for the blanks.
• Then describe how a 17 m and 3.2

m distance would be represented.

• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• The perimeter of one square is 1/3

as long as the perimeter of

• another. What do you know about

the side lengths?

• How could you represent this?
• Dr. Marian Small, University of New

Brunswick February 2015

You could ask:

• A jacket price is reduced by 40%.
• A shirt price is reduced by 20%.
• They end up costing the same

amount (on sale).

• How were the original prices

related?

• Dr. Marian Small, University of New

Brunswick February 2015

Ministry resources

• http://www.edu.gov.on.ca/eng/teachers/

studentsuccess/ProportionReason.pdf

• http://www.edugains.ca/resources/

LearningMaterials/ ContinuumConnection/ BigIdeasQuestioning_ProportionalReas

• ning.pdf
• Dr. Marian Small, University of New

Brunswick February 2015

Ministry resources

• Math camppp materials on proportional

reasoning

• http://gains-camppp.wikispaces.com/

CAMPPP+2010

• http://gains-camppp.wikispaces.com/

CAMPPP+2011+Home

• Dr. Marian Small, University of New

Brunswick February 2015

Message

• Proportional reasoning is about

unitizing, grouping and counting groups, thinking of comparisons multiplicatively.

• Proportional reasoning comes out

if you model it, talk about it, present tasks that allow for it, and encourage it.

• Dr. Marian Small, University of New

Brunswick February 2015