Transcript WORKSHOP V - Mathematics resources for learning

I can’t believe
the price of
gas these days!
Oh, has it
gone up?
You mean
you haven’t
noticed?
Nope. It still costs
the same for me.
Five dollars worth.
COURSE NM309
FRACTIONS, DECIMALS &
PERCENTAGES
TEACHING FOR UNDERSTANDING
From Vince
OBJECTIVES
Developing understandings of fractions
and decimals
Discuss the difficulties and misperceptions
Identify strategies at different stages
Discuss learning processes
Discuss teaching strategies
Explore activities to use in the classroom
Which family has more girls?
The Jones Family
The King Family
Before, tree A was 8m tall and tree B was 10m tall.
Now, tree A is 14m tall and tree B is 16m tall.
Which tree grew more?
A
B
Before
A
B
Now
A fishy problem
 Two-thirds
 There
 How
of the goldfish are male
are 24 male goldfish
many goldfish are there
altogether?
Share your strategy
 How
did you do it?
 Discuss
 Who
your method in groups
taught you how to do it this
way?
The Bill Gates question
You have a fish tank containing 200 fish
and 99% of them are guppies. You will
remove guppies until 98% of the
remaining fish are guppies. How many
will you remove?
Report on Numeracy Project 2003

The performance of year 7 and 8 students on
fractions and decimals is well below what would
be wished.
 Integration of fractions with proportional
reasoning would aid understanding of those
topics.
 Decimals are of particular concern
 Decimals need to be taught using the Numeracy
principles: using materials and imaging before
number properties
Why do students have difficulty with
fractions?





Rational number ideas are sophisticated and
different from natural number ideas
Natural numbers can be represented
individually, rational numbers cannot.
Students’ whole number schemes can interfere
with their efforts to learn fractions
Students have to learn new ways to represent,
describe and interpret rational numbers
Rote procedures for manipulating fractions (eg
making equivalent fractions) may not be enough
Initial Fraction Interview: Task 1
This is three-quarters of the lollies I started
with. How many lollies did I start with?
Why did you choose that many lollies?
Initial Fraction Interview: Task 2
2
5
2
5
2
3
1
4
2
8
2
3

Which of these pairs of fractions are
equivalent (have the same value)?

How did you decide?
1
3
Initial Fraction Interview: Task 2
Typical responses:
 One-quarter is equivalent to two-eighths ‘cos ‘1
goes into 4, four times, and 2 goes into 8, four
times.
 If you were to simplify it (2/8) it would go down to
a quarter. You just halve it.
 Double one-quarter to get two-eighths.
All were successful except one student who said
that one third and two thirds were equivalent
because ‘the bottom is the same’
Initial Fraction Interview: Task 3
3
10
=
21
What number do you need to write in the
box so that the fractions are equivalent?
How did you decide?
Initial Fraction Interview: Task 4
0.5 0.25 0.1 0.4
1
2
1
4
1
10
Match each fraction with the equivalent
decimal.
How did you decide?
2
5
Initial Fraction Interview: Task 4
 Most
confidently matched fraction and
decimal equivalents for one-half and onetenth, were less confident with one-quarter
and put two-fifths with 0.4 because it was
‘just the one left’
 Difficulties
arose when students were
asked to choose the larger of two
fractions…
Probing Task 1
3
5
2
3
Which is larger, three-fifths or two-thirds?
How did you decide?
Probing Task 2
3
5
5
8
Which is larger, three-fifths or five-eighths?
How did you decide?
Probing Task 3
3
5
3
4
Which is larger, three-fifths or five-quarters?
How did you decide?
Probing Task 4

Pick one of the tasks where the student was
incorrect. Hand the student one card and a
number line marked 0 to 1.
‘Place this fraction on the number line.’
‘How did you decide?’

Hand the student the second card
‘Place this fraction on the number line.’
‘What did you find when you placed your
fractions on the number line?’
Misconceptions 1: ‘gap’ thinking
Interviewer: ‘Which is larger: 3
or 5 ?
5
8
Student 1: ‘Three-fifths is larger because
there is less of a gap between the three
and the five than the five and the eight’.
Misconceptions 2:
‘comparing to a whole’ thinking
Interviewer: ‘Which is larger: 3
or 5 ?
5
8
Student 2: ‘Three-fifths is larger because it is
two numbers away from being a whole
and five-eighths is three away from being
a whole’.
Misconceptions 3:
‘larger is bigger’ thinking
Interviewer: ‘Which is larger 2/3 or 3/5 ?
Student 3:
2
6
12 18
3
9
18 27
3
5
6
10
12
20
18
30
18 is larger than 18 because 30 > 27
30
27
Probing with student A
Chose 3 as larger than 3
4
5
Int: ‘Can you do it another way?’
A: ‘I automatically said it.’
He was given a sheet with empty number
lines
Further probing with student A
A placed 3 close to 1.
4
On the number line he put 3 twice as far away from 1 as 3
5
4
0
0
3/4
3/5
1
1
Probing with student B
Student B said correctly that 2/3 was larger
than 3/5.
His reason was that three-fifths is
‘two numbers away from being a whole and
two-thirds is one number away from being
a whole’
He applied the same reasoning to 3/5 and
5/8 arguing that ‘three-fifths must therefore
be bigger’
Probing with student B
Int: ‘Think about 2/3 and 3/4.’
B: ‘I think they are equal. Not just because
they are one away form being a whole.
This (3/4) is 75% and 2/3 is about 75%.’
He didn’t have any idea of how he could
check how close 2/3 was to 75%
Probing with student B
He was given an empty number line.
He marked the number line in fifths.
On the second number line he marked onehalf, one-quarter and three-quarters by
eye.
From his diagram he concluded that ¾ was
bigger than 3/5
He reiterated that ¾ was 75% and used a
calculator to show that 3/5 was 60%
Probing with student B
0
1
1/5
3/5
0
1
¼
½
¾
Probing with student B
To compare 3/5 and 5/8, B subdivided the
second number line from quarters into
eighths by eye
He then said ‘5/8 is bigger- it is a bit ahead
of 3/5. My old method doesn’t work.’
Int: ‘Consider one-half and four-eighths’
B: ‘ my old method would say that ½ is
bigger but they are the same’
Probing with student C
To compare 3/5 and 2/3,
C said ‘Both go into 15’ and then wrote
2/3 as 10/15, and 3/5 as 9/15.
To compare 3/5 and 5/8, C first said that ‘3/5
is bigger by one’. He then converted both
fractions to the same denominator (24/40
and 25/40) and said that 5/8 was bigger.
Probing with student C
He converted 3/5 and ¾ to 12/20 and 15/20 and
correctly concluded that ¾ is bigger.
Using number lines to compare ¾ and 3/5, he
divided the first number line by eye into quarters
and marked one-half and three-quarters.
He placed one-half on the number line below in a
corresponding position.
He said that ‘three-fifths is smaller than threequarters and marked three-fifths to the right of
one-half and the left of three-quarters on the
number line
Probing with student C
0
1
½
¾
0
1
½
3/5
Probing with student C
He placed 3/5 and ¾ approximately where
we would expect.On a pencil and paper
test his response would be OK…
However it was not clear to the interviewer
why student C had placed the fractions
where he did.
Further probing was required.
Further Probing with student C
Int: ‘Can you place 3/5 on the number line?
Int: ‘Where would 1/5 be?’
C: ‘one-fifth is more than one-half (I think)’
He then placed one-fifth to the right of onehalf.
Int: ‘where would one-third and one-quarter
be on the number line?’
He placed these two fractions in between
one-half and one-quarter.
Further Probing with student C
0
1
½ 1/3
1/4 1/5
Findings
Procedural competence can disguise whole
number thinking about fractions
 eg scaling up to equivalent fractions is a rote
technique and students may relate new
numerators and denominators as discrete whole
numbers
Whole number thinking
 Treats numerators and denominators as discrete
whole numbers (gap thinking and larger is
bigger)
 Treats the ‘gap’ as a whole number not a fraction
Conclusions
To overcome whole number thinking
students need to:
 Make multiple representations of fractions
using discrete and continuous quantities
 Use a number line to represent and
compare fractions
 Check results and estimate answers
 Deal explicitly with whole number thinking
Models for fractions
Discrete models

Sets for counting
•counters, blocks, beans
Continuous models
Area for dividing and shading
•circles, triangles, rectangles
Number lines
•rope and paper strips for folding
•double number lines

FRACTION NUMBER SENSE
Developing an understanding of Fraction includes:
Representing
the fraction as an expression of a
relationship between a part and a whole and relationships
among parts and wholes.
Regardless
of the representation used for a fraction and
regardless of the size, shape, colour, arrangement ,
orientation, and the number of equivalent parts, the student
can focus on the relative amount
Recognising
that in the symbolic representation of a
fraction the denominator indicates how many parts the
whole has been divided into, and the numerator indicates
how many parts of the whole have been chosen
FRACTION NUMBER SENSE
Developing an understanding of
Fraction Number Sense includes
five different but interconnected
subconstructs:
(Kieran 1976,1980)
FRACTION NUMBER SENSE
Part-Whole,
e.g. ‘3 parts out of every 4’
CLASSROOM EXAMPLES
Fold a strip of paper into four equal parts (quarters).
What are three of these called?
Fold each quarter into three equal parts.
What are the new parts called?
What are three of these new parts called?
1
WHAT CAN YOU SEE ?
Can You See It?
Can you see 3/5 of something?
Can you see 5/3 of something?
Can you see 2/3 of 3/5?
Can you see 1 divided by 3/5?
Can you see 3/5 divided by 2?
Big Stix Chocolate Bar
Half the candy bar is how many
sticks?
2 sticks is what part of the bar?
If you have half and I have 1/3, who
has more?
How much more?
How much is half and 1/3 together?
What part remains for someone
else?
How much of the candy bar is half
of a third?
How many times will 1/3 fit into ½?
FRACTION NUMBER SENSE
Operator, i.e.
‘3/4 of something’
2
MEANING
3/4 gives a rule that tells how to operate on a unit (or the result of
a previous operation), that is find 3/4 of something.
CLASSROOM EXAMPLES
A photo measures 26cm x 15cm. You want a copy made which has
each side three quarters of its original length. How big will the
copy be?
You have a collection of bubble gum cards. You divide the
collection into 4 equal piles and give your friend three of the piles.
How much of the whole collection do you give them?
Thinking Up And Down
FRACTION NUMBER SENSE
Ratios and Rates, i.e.
‘3 parts of one thing to 4 parts of another’
3
MEANING
3:4 means 3 parts of A to 4 parts of B, where A and
B are of like measure (ratio) or of different
measure (rate)
CLASSROOM EXAMPLES
Sally mixes 12 tins of yellow paint with 9 tins of red paint.
Tane mixes 8 tins of yellow paint with 6 tins of red paint.
Each tin holds the same amount. Whose paint is the darkest
shade of orange? How do you know?
FRACTION NUMBER SENSE
Quotient, i.e.
‘3 divided by 4’
MEANING
3/4 is the amount when each party gets when 3
units are shared equally among four parties.
CLASSROOM EXAMPLES
There are three chocolate bars to share equally
among four people. How much chocolate bar will
each person get?
4
Three pizzas for five children:

How much pizza does each
child eat?

How much of the pizza does
each child eat?
FRACTION NUMBER SENSE
Measure, i.e.
‘3 measures of 1/4’
5
MEANING
3/4 means the putting together of three 1/4 units
CLASSROOM EXAMPLES
There are 7 otters and 5 sea-lions in Marineland.
At feed time each otter gets one-quarter of an eel,
and each sea-lion gets one third of an eel. Which
hgroup of animals takes the most eels to feed?
EARLY STAGES
COUNTING FROM ONE / IMAGING TO ADVANCED COUNTING
- Common language
- Initially focus on unit fractions with 1 as numerator, but it is
also important to introduce non unit fractions like ¾
-Use continuous models and discrete
-Use whole number strategies to anticipate result of equal
sharing
PAGES 23 & 25 BOOK 3, PAGE 2 BOOK 7
STAGES 4 TO 5
ADVANCED COUNTING TO EARLY ADDITIVE
Students must realize that the symbols for fractions tell how
many parts the whole has been divided into (denominator),
and how many of those parts have been chosen
(numerator). The terminology is not as important as the
understanding.
Students need to appreciate that fractions are both
numbers and operators. It is vital to develop an
understanding of the home of fractions among the whole
numbers
PAGE 27 BOOK 3, PAGE 5 BOOK 7
STAGES 5 TO 6
EARLY ADDITIVE TO ADVANCED ADDITIVE
Early additive students are progressing towards
multiplicative thinking. Fractions involve a significant mental
jump for students because units of one, which are the basis
for whole number counting, need to be split up
(partitioned), and repackaged (re-unitised).
PAGE 30 BOOK 3, PAGE 14 BOOK 7
STAGES 6 TO 7
ADVANCED ADDITIVE TO ADVANCED MULTIPLICATIVE
Use a diverse range of strategies involving multiplication and
division with whole numbers including:
Compensation from tidy numbers 28 x 7 as 30 x 7 - 2 x 7
Place value 64 x 8 as 60 x 8 + 4 x 8
Reversibility and commutativity e.g., 84 ÷ 7 as 7 x = 84 or
2.37 x 6 as 6 x 2.37
Proportional adjustment e.g., doubling and halving
Changing numbers e.g., 201 ÷ 3 as (99 ÷ 3) + (99 ÷ 3) + (3 ÷ 3)
STAGES 6 TO 7 (cont)
ADVANCED ADDITIVE TO ADVANCED MULTIPLICATIVE
•Solve division problems with remainders and express answers
in fractional, decimal and whole number form
•Use written working forms or calculators where the numbers
are difficult and / or untidy
BOOK 3 Page 33 / BOOK 7 Pages 21 & 22
STAGES 7 TO 8
ADVANCED MULTIPLICATIVE TO ADVANCED PROPORTIONAL
To become strong proportional thinkers, the students need to
be able to find multiplicative relationships in a variety of
situations involving fractions, decimals, ratios and proportions.
It is also important that they see the relationships between the
three views of fractional numbers: proportions, ratios and
fractional/decimal operators. (Examples page 31, Book 7)
BOOK 3 Page 35 / BOOK 7 Pages 30 & 31
Operations with fractions
Experience with operations on fractions using
manipulatives in contextual problem solving settings
Select
appropriate visual models, methods and tools for
computing with fractions, decimals and percents
Explain
methods for solving problems by developing and
analyzing algorithms for computing with fractions, decimals and
percentages
Understand
the meaning and effects of arithmetic operations
with fractions, decimals and percents
Division with fractions

People seem to have different approaches to solving
problems involving division with fractions.

How do you solve a problem like this one?
3 1
1  
4 2
Division with fractions

Imagine you are teaching division with fractions. To
make this meaningful for kids, something that many
teachers try to do is relate mathematics
to other things.
3 1
 
Sometimes they try to come up1 4with
2 real-world situations
or story problems to show the application of some
particular piece of content.

What would you say would be a good story of model for
3 1
1  
4 2
?
DECIMALS



Develop sound understanding of fractions
first.
Develop understanding of decimals as special
fractions
Develop understanding of percentage once
decimal understanding is sound
Highest Number: 0._ _ _
Play in pairs
Aim is to make the largest 3 digit number
1.
2.
3.
Take turns to roll the die
Place card in column of choice on your
sheet
If you get a repeat, ignore and roll again
Highest Number
What is a good strategy?
If you rolled a 5 where would you put it?
Why?
How does your strategy change if
 the winner is the closest to 0.5?
 You use a 10 sided die with numbers 0 to 9?
 You play with 4 place value positions?
Why do students have difficulties with
decimals?
•The relative size of decimals.
Ordering 0.4, 0.23, 0.164.
•Decimal place value.
•Multiplication and division ideas extended from
whole numbers.
‘When you multiply the numbers get bigger
When you divide the numbers get smaller’
1. Whole number thinking
0.25 is larger than 0.6
…because 25 is larger than 6
So 1.5 is smaller than1.45
but 1.50 is bigger than1.45
The decimal point is seen as a separator of two
number systems. No understanding of place
value to the right of the decimal point.
Work with money tends to reinforce this
misperception as do ‘tricks’ like adding zeros
2. Denominator focused thinking
0.4 is larger than 0.51
… because 4 is for tenths and the 1 is for
hundredths and tenths are bigger than
hundredths
The longer the decimal number, the smaller it is
3. Reciprocal thinking
0.2 is bigger than 0.3
… because you have one of two pieces (halves)
versus one of three pieces (thirds)
Numbers after the decimal point indicate how
many pieces there are, of which you have one.
Fraction / decimal equivalents confusion so 1/5
the same as 0.5
4. Negative number thinking
5.3 is bigger than 5.4
… because 5.3 is 3 units away from 5 and 5.4 is
4 units away.
Thinks of decimals as measuring a distance
from the whole number. The distance is seen as
diminishing the whole number.
5. Money thinking
2.45 is bigger than 2.452
… because decimals beyond 2 decimal places
are not really real – they are either ignored or
they are regarded as diminishing the earlier
number.
These students may not be noticed in class as
they are successful dealing with decimals up to
hundredths.
6. Place value thinking
4.08 is bigger than 4.7
8.0527 is smaller than 8.54
… you ignore the zero because zero is nothing.
Confused about concept of zero as a
placeholder.
7. Zero on number line
0.03 is bigger than 0.00
…But 0.03 is smaller than 0.0
… and 0.0 is not the same as 0.00
Unsure of relationship between zero, one and
decimal numbers
8. Task Experts
 Get
most tasks correct but make a few
errors with no pattern
9. Not consistent
 No
clear way of dealing with decimals but
know that whole number thinking is not
correct.
UNDERSTANDING DECIMALS
•Decimal place value
•Ordering decimals
•Relationship between fractions and decimals
•Placing decimals on a number line
•Operating with decimals:
Addition and subtraction as extensions of whole
number operations
Multiplication and division as extension of
operations with fractions
EXTENDING UNDERSTANDING
Write down a number that is:
Bigger
than 3.9 and smaller than 4
Bigger
than 6 and smaller than 6.1
Bigger
than 0.52 and smaller than 0.53
Bigger
than 8.9 and smaller than 8.15
Zero insertion task
Which is bigger?
2.35
or 2.350?
2.305
or 2.35?
2.035
or 2.35?
QUESTIONS
•How important is context?
•Is money an appropriate context for development of
decimal understanding?
•What language should we use for decimals?
•Should we consider the decimal point as a marker for
the ones place?
•Is Multiplicative thinking needed for understanding of
fractions and decimals?
SUGGESTIONS
•Promote connections between decimals, fractions and other
mathematical contexts such as metric measures and
percentages
•Decimal place value language 0.1 is ‘1 tenth’, 0.01 is said as ‘1
hundredth’, 3.14 as ‘3 and fourteen hundredths’ or ‘3 and 1 tenth
and 4 hundredths’
•Multiplicative thinking is essential for understanding of
equivalent fractions and ratios. The equivalent fractions concept
is essential for understanding decimals and percentages
•Consider the decimal point as a marker for the ones place
rather than a barrier that separates the wholes from the fractions
Course NUM309
Developing understandings of fractions
and decimals
Discuss difficulties and misperceptions
Identify strategies at different stages
Discuss learning processes
Discuss teaching strategies
Explore activities to use in the classroom
Oh, one more thing.
Cut that pizza into six slices.
I can’t eat eight.
Thanks for your contribution today