Download 2010 TNISFractions Master

Intermediate Effective
Numeracy Fractions
Workshop
4 out of 3 people have
trouble with fractions
Have a go at the Fraction Hunt
on your table while you are
waiting!
Objectives
• Understand the progressive strategy stages of
fractions, proportions and ratios
• Understand common misconceptions and key
ideas when teaching fractions and decimals.
• Explore equipment and activities used to teach
fraction knowledge and strategy
Choose your share of chocolate!
4 Stages of the PD Journey
Organisation
Organising routines, resources etc.
Focus on Content
Familiarisation with books, teaching model etc.
Focus on the Student
Move away from what you are doing to noticing what the
student is doing
Reacting to the Student
Interpret and respond to what the student is doing
Assess
Yourself –
Fraction
Strategy and
Knowledge
Stage 1
Stage 2-4 (AC)
Stage 5 (EA)
Unequal Sharing
Equal Sharing
Use of Addition and
known facts e.g.
5 + 5 + 5 = 15
Stage 6 (AA)
Using multiplication
Stage 7 (AM)
Using division
Stage 8 (AP)
What misconceptions may children have
with fractions?
Misconceptions about finding one half when beginning
fractions:
• Share without any attention to equality
• Share appropriate to their perception of size, age etc.
• Measure once halved but ignore any remainder
So what do we need to teach to move to equal
sharing?
Introduce the vocabulary of equal / fair shares with both
regions and sets for halves and then quarters, then other
fractions.
Key Idea 1
Draw two pictures of one
quarter
one quarter
Connect different representations
words - symbols - drawings - number lines
1
4
Sets
(Discrete Models)

Shapes/Regions(Co
ntinous models)
Recapping Key Idea 1
Work with shapes, lengths and sets
of fractions from early on.
Key Idea 2
3 sevenths
3 out of 7
7/3
7 thirds
5 views of fractions
3÷7
3 out of 7
3:7
3
7
3 sevenths
3 over 7
The problem with “out of”
1
2 +
2
3
8
6
2
3
3 = 5
x 24 =
“I ate 1 out of my 2
sandwiches, Kate ate 2 out
of her 3 sandwiches so
together we ate 3 out of the
5 sandwiches”!!!!!
2 out of 3 multiplied by 24!
= 8 out of 6 parts!
Fraction Language
Use words first then introduce symbols with care.
e.g. ‘one fifth’ not 1/5
How do you explain the top and bottom numbers?
The number of parts chosen
1
2
The number of parts the whole has been
divided into
Fractions
In 2001 42% of year 7 & 8 students who sat the
initial NUMPA could not name these symbols
1
4
1
2
1
3
Fractional vocabulary
One half
One third
One quarter
Don’t know
Emphasise the ‘ths’ code
1 dog + 2 dogs = 3 dogs
1 fifth + 2 fifths = 3 fifths
1/
5
+ 2/5 = 3/5
3 fifths + ?/5 = 1
1
-
?/
5
= 3/5
Recapping Key Idea 2
Fraction language is confusing. Emphasise the
‘ths’ code.
Use words before symbols. Introduce symbols
with care.
The bottom number tells how many parts
the whole has been split into, the top
number tells how many of those parts have
been chosen.
Key Idea 3
6
is one third of what number?
This is one quarter of a
shape. What does the
whole look like?
Recapping Key Idea 3
Go from part-to-whole as well as
whole-to-part with both shapes and
sets.
Children need experience in both
reconstructing the whole as well as
dividing a whole.
Perception check on two key ideas
Where in the table does this question fit?
Hemi got two thirds of the lollies. How many were
there altogether?
Part-to-Whole
Continuous
(region or
length)
Discrete
(sets)
Whole-to-Part
Key Idea 4 :
Anticipate the result of equal sharing
by grouping (using addition or skip
counting) to solve fraction problems
rather than equal sharing by
ones(Early Additive Stage 5)
The Strategy Teaching Model
Existing Knowledge
& Strategies
Using
Material
Materials
s
Using Imaging
Using Number Properties
New Knowledge &
Strategies
Developing the use of addition facts
(to find one quarter of 12)
? + ? + ? + ? = 12
3?
?3
?3
?3
Find one quarter of 12 (imaging)
Key idea: quarters means you need 4 equal
groups. One quarter is the number in one of
those groups.
3
Using Number Properties
Find one quarter of 40, 400, 4000
Hungry Birds (Book 7, page 22)
Four birds found a worm in the ground 20 smarties long.
What proportion of the worm do they each get?
How many smarties will each bird get?
5 + 5 + 5 + 5 = 20
Division
1/
5+
1/
5+
1/
5
3÷5
= 3 /5
Y7 response: “3 fifteenths!” Why?
Key Idea 5
Division is the most common context for
fractions when units of one are not
accurate enough for measuring and
sharing problems.
Initially this is done by halving and
halving again but harder examples
require more sophisticated methods
e.g. 3 ÷ 5 = 3 fifths
Key Idea 6
Order these fractions:
¼
6 quarters,
3/
4
nine quarters,
2/
4
Key Idea 6
Fractions are not always less than 1.
Push over 1 early to consolidate the
understanding of the top and bottom
numbers.
6 quarters
1
12/4
5
Year 7 student responses (decile 10)
What is this fraction?
2 fifths,
5/
2
five lots of halves,
How do I write 3 halves?
3 1/2
1
/3
tenth, five twoths
Using fraction number lines to consolidate
understanding of denominator and numerator
Push over 1
0
1 half
0
1/
2
0
1/
2
2 halves
2/
2
1
3 halves
3/
2
11/2
4 halves
4/
2
2
Three in a row (use two dice or numeral cards)
A game to practice using improper fractions as numbers
0
X
1
X
2
3
4
5
e.g. Roll a 3 and a 5
Mark a cross on either 3 fifths or 5 thirds.
The winner is the first person to get three
crosses in a row.
6
Perception Check – Discuss these key
ideas. (Stage 4 and 5).
1. Use sets as well as shapes/regions from early on
2. Fractions are a context for add/sub and mult/div
strategies
3. Fraction Language - use words first and
introduce symbols carefully
4. Go from Part-to-Whole as well as Whole-to-Part.
5. Division is the main context for using fractions
6. Fractions are not always less than 1, push over 1
early to consolidate meaning of fraction symbols.
Watch Vince in action
Moving to Stage 6 (AA)
• What key knowledge is required before
moving on to Stage 6?
• Identify symbols for halves, quarters, thirds and
fifths and tenths
• Order unit fractions (¼ ½,) and fractions with
like denominators (3/4, 1/4 )
• Identifies symbols for improper fractions, e.g. 5/4
Which number games/activities could you
adapt to practice fraction knowledge?
Fraction /
Number Mat
Fraction
Bingo
Fraction
Fraction
Snakes and
Chances
Dominoes
Fraction
Fraction
Memory/
Happy
Families
I have, Who
Has
Key Idea 6.
Which letter shows 5 halves as a number?
A
0
B
C
1
D
E
2
F
3
Key Idea 6 (Stage 6)
Fractions are numbers as well as operators
3/ is a number between 0 and 1 (number)
4
Find three quarters of 80
(operator)
Key Idea 7 (Stage 6 AA)
The distance between Masterton and
Wellington is 80 kilometres. Hemi has
travelled 3/4 of the trip. How many kilometres
is that?
3/4 of 80
80
20
20
20
3/4 of 80 = 3 x 20 = 60
Recapping Key Idea 7 (Stage 6)
Apply understanding of links between
addition and multiplication to solve
fraction problems.
(Birthday Cakes, Book 7 ,p.26)
Using Double Number Lines
0
20
60
100
0
1
5
3
5
1
Put a peg on where you think 3/5 will be.
(Fractions as a number). How will you work it out?
Use a bead string and double number line to find
3/ of 100. (Fractions as an operator). How will you
5
work it out?
Key Idea 8 (stage 6)
Sam had one half of a cake, Julie had one
quarter of a cake, so Sam had most.
True or False or Maybe
Julie
Sam
What
is B?
What
is the
whole? (Trains Book 7, p32)
A
A
B
B
B
B
C
D
D
D
D
D
D
D
D
Recapping Key Idea 8 (stage 6)
Fractions are always relative to the
whole.
Halves are not always bigger than
quarters, it depends on what the whole
is.
Key Idea 9 (stage 6) - Ratios!
Write 1/2 as a ratio
1:1
3: 4 is the ratio of red to blue beans. 3/
7
What fraction of the beans are red?
Think of some real life contexts
when ratios are used.
Key Idea 9 (Stage 6) Ratios
What is the link between ratios and fractions?
Ratios describe a part-to-part relationship e.g.
2 parts blue paint : 3 parts red paint
But fractions compare the relationships of one of
the parts with the whole, e.g.
The paint mixture above is 2/5 blue
Ratios and Rates
What is the difference between a ratio and a rate?
Both are multiplicative relationships.
A ratio is a relationship between two things that are
measured by the same unit,
e.g. 4 shovels of sand to 1 shovel of cement.
A rate involves different measurement units,
e.g. 60 kilometres in 1 hour (60 km/hr)
Summary of Fractions Key Ideas
1. Use sets as well as shapes/regions from early on
2. Fraction Language - use words first and introduce symbols
carefully.
3. Go from Part-to-Whole as well as Whole-to-Part
4. Division is the most common context for fractions.
5. Fractions are not always less than 1, push over 1 early.
6. Fractions are numbers as well as operators.
7. Fractions are always relative to the whole.
8. Consider the relationship between ratios and fractions
9. Use addition/skip counting to find fractions of sets then
develop and apply multiplicative thinking –
Fractions are really a context for add/sub and
mult/div strategies
Moving to Stage 7 AM
Matiu started with a whole box of chocolates and ate
fife-ninths. That left only 16 chocolates. How many
chocolates were in the box to begin with?
What is the number sentence for this problem?
4/
9
of ? = 16
Draw a diagram to help to solve this problem.
4/
of ? = 16
16 is four ninths of what number? 36
9
16
4 4 4 4
8
4
4
At Stage 7, students should be using a range of
multiplication and division strategies to solve
problems with fractions, proportions and ratios.
Developing Proportional thinking
Fewer than half the adult population can
be viewed as proportional thinkers
And unfortunately…. We do not acquire
the habits and skills of proportional
reasoning simply by getting older.
Summary of Stage 7 Ideas
• Find simple equivalent fractions using multiplicative
thinking.
• Convert common fractions as decimal fractions and
percentages and vice versa.
• Add/subtract related fractions, e.g. 2/4+5/8 and
decimals, e.g.3.6+ 2.89 (multiplication and division of
fractions and decimals is developed further at Stage 8)
• Solve division problems expressing remainders as
fractions or decimals e.g. 8 ÷ 3 = 2 2/3 or 2.66
• Estimate and solve percentage type problems such as
What % is 35 out of 60, and 46% of 90
• Find equivalent ratios and compare by converting to
equivalent fractions, equalising the total number of
parts or mapping onto 1.
• Solve simple rate problems
Equivalence
• A key idea at this stage is equivalence.
• Which equivalent fractions should students be
able to recall and order at this stage?
halves, thirds, quarters, fifths, tenths
Order these fractions:
2/
3
3/
4
2/
5
5/
8
3/
8
What did you do to help order them?
How could you communicate this
idea of equivalence to students?
Paper Folding
Fraction Tiles / Strips/ Fraction circles
Differentiating whole class warm ups
Circle the biggest fraction
A
½ or ¼
1/ or 1/
5
9
D
6/
4
7/
8
B
or 3/5
or 9/7
Adding and Subtracting
fractions with related
denominators
C
4/
3/
or
5
8
7/ or 6/
8
9
Renaming fractions as decimals
You need to understand equivalent
fractions before understanding
decimals, as decimals are special cases
of equivalent fractions where the
denominator is always a power of ten.
What is 3/8 as a decimal? (decipipes)
A Fractional Thought for the day
Smart people believe only half
of what they hear.
Smarter people know which
half to believe.
A sample of numerical
reasoning test questions
as used for the NZ Police
recruitment
½ is to 0.5 as 1/5 is to
a.
0.15
b.
0.1
c.
0.2
d.
0.5
1.24 is to 0.62 as 0.54 is to
a.
b.
c.
d.
1.08
1.8
0.27
0.48
Travelling constantly at 20kmph, how long
will it take to travel 50 kilometres?
a. 1 hour 30 mins
b. 2 hours
c. 2 hours 30 mins
d. 3 hours
If a man weighing 80kg increased his
weight by 20%, what would his weight
be now?
a.
96kg
b.
89kg
c.
88kg
d.
100kg
Misconceptions with Decimal Place Value
What number is three tenths less than 2?
Why might children have get this wrong?
Does the student understand that 1
whole has an infinite number of fraction
names.
In this case, it would be most sensible to
assume ten tenths equals one whole.
Misconceptions with Decimal Place Value:
Bernie says that 0.657 is bigger than 0.7
Why do you think he thinks that?
Does the student mistakenly regard the
digits following the decimal point as
representing whole numbers?
Misconceptions with Decimal Place Value:
Sam thinks that 0.27 is bigger than 0.395
Why do you think he thinks that?
Does the student mistakenly think the
more decimal places a number has, the
smaller the number is because the last
place value digit is very small.
3.9 plus 4 tenths
• Jane thinks it is 3.13.
• Bill thinks it is 7.9.
• Melinda thinks it is 4.3.
Who is correct?
What misconceptions do they have?
How would you respond to this?
Misconceptions with Decimal Place Value:
Martika and Karl are having an argument.
Martika says 1.5  10 = 1.50, while Karl
says its 10.50
Why do you think he thinks that?
What would you do to respond to this?
Decimal Misconceptions Summary

Decimals are two independent sets of whole numbers
separated by a decimal point,
e.g. 3.71 is bigger than 3.8
and
1.8 + 2.4 = 3.12

The more decimal places a number has, the smaller the
number is because the last place value digit is very small.
E.g. 2.765 is smaller than 2.4

Decimals are negative numbers.

1/

When you multiply decimals the number always gets
bigger.

When you multiply a decimal number by 10,
just add a zero, e.g. 1.5 x 10 = 1.50
2
is 0.2 and 1/4 is 0.4, so therefore 0.4 is smaller than 0.2
The moral of the story…
• Decimals need explicit teaching – don’t
assume students understand them.
• Introduce them once students understand how
whole numbers work, and have the concept of
a fraction.
• Spend time on decimal place value
• “See, Say, Make”, e.g.0.65
Teaching Decimal Place Value
Decimal Fraction Mats
Book 4:8-9, (MM 4-21)
Decimal Number Lines
Book 4:15 (MM 4-31)
Candy Bars (Book7)
Operating with decimals
Understanding tenths and hundredths using
candy bars:

Pose division problems using the equipment to find the
number of wholes, tenths and hundredths;
e.g. 6 ÷ 5, 4 ÷ 5,

then
5 ÷ 4, 3 ÷ 4, 13÷ 4
Operate with the decimals using addition/subtraction and
multiplication to consolidate understanding requiring
exchanging across the decimal point, e.g.
3.6 - 1.95,
3.4 + 1.8,
4.3 - 2.7,
7 x 0.4,
1.25 x 6
Your Choice! – Further PD with…
• Stage 7
Decimals (Add/sub, Mult/Div)
Percentages
Ratios and rates
• Stages 1 – 5
Beginning Fractions
What misconceptions may young children
have when beginning fractions?
Misconceptions about finding one half when beginning
fractions:
• Share without any attention to equality
• Share appropriate to their perception of size, age etc.
• Measure once halved but ignore any remainder
So what do we need to teach to move to equal
sharing?
Introduce the vocabulary of equal / fair shares with both
regions and sets for halves and then quarters.
Plan a lesson from Book 7
Focus on
the key
ideas we
have
discussed
today
What now?
Use your data from IKAN and GloSS (Re-GloSS
fractions if necessary) to identify class needs.
Use long-term planning units for Fractions
Teach fraction knowledge and proportions &
ratios strategies with your groups/whole class.
In-Class Modelling visit
Evaluation
Little League Video Clip
Thought for the day
A DECIMAL POINT
When you rearrange the letters becomes
I'M A DOT IN PLACE
There are three
things
to day
Thought
for
the
remember when teaching;
Know your stuff,
Know whom you are stuffing,
And stuff them elegantly.
Lola May