Transcript 2011 fractions pick ups

Proportions and Ratios
Workshop
4 out of 3 people have
trouble with fractions
Have a go at the Pirate Problem or
Fraction Hunt on your table while
you are waiting!
Objectives
• Understand the progressive strategy stages of
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
Overview
• Key Teaching Ideas (Stages 2-6)
• Number Framework Progressions
Morning Tea
• Decimals and Stage 7
• Getting into Book 7 and the Planning Units
Lunch
• Modelling Sessions
• Year Overiews – Putting it all together
Play Which Mystery Letter Am I?
1/
3
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
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.
What misconceptions might 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?
Introduce the vocabulary of equal / fair shares with both
regions and sets for halves and then quarters.
Bev Dunbar: ‘Exploring Fractions’
Key Teaching Ideas
Stages 2- 6
Draw two pictures of one
quarter
Fractions Key Teaching Ideas
1. Use sets as well as regions and lengths
from early on and connect different
representations
1 quarter
Shapes/Regions
1
4
Lengths
Sets
3 out of 7
7/3
3 sevenths
7 thirds
Fractions Key Teaching Ideas
1.
Use sets as well as regions from early on and connect different
representations.
2. Use words first then introduce symbols
with care.
How do you explain the top and bottom
numbers?
1
2
The number of parts chosen
The number of parts the whole has been
divided into
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 Symbols
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
Emphasise the ‘ths’ code
1 dog + 2 dogs = 3 dogs
One third
One quarter
Don’t know
1 fifth + 2 fifths = 3 fifths
1/
5
+ 2 /5 = 3 /5
3 fifths + ?/5 = 1
1
-
?/
5
= 3 /5
6
is one third of what number?
This is one quarter of a
shape. What does the
whole look like?
Fractions Key Teaching Ideas
1.
Use sets as well as regions from early on and connect different
representations.
2.
Use words first & introduce symbols with care.
3. Go from part-to-whole as well as whole-to-part
with both shapes and sets.
Which letter shows 5 halves as a number?
A
0
B
C
1
D
E
2
F
3
Fractions Key Teaching Ideas
1.
2.
3.
Use sets as well as regions from early on and connect different
representations.
Use words first & introduce symbols with care.
Go from part-to-whole as well as whole-to-part with both
shapes and sets.
4. Fractions are not always less than 1. Push over 1
early to consolidate the understanding of the top
and bottom numbers.
What is this fraction?
2 fifths,
5/
five lots of halves,
2
tenth, five twoths
How do I write 3 halves?
1
3 1/2
/3
Y7 student responses decile 10
Teaching Ideas
Fraction Circles
Fraction number lines and counting in fractions
0
0
1 half
1/
2
2 halves
2/
2
3 halves
3/
2
0
1/
1
11/2
2
4 halves
4/
2
2
Understanding Fraction Representations
A.
B.
C.
D.
E.
F.
G.
H.
I.
Spin a Whole
• Form groups of 3.
Fraction Dots
• Explore your game.
Happy Families
• Number yourselves 1 – 3
Fraction Circles
• Number 1’s get together…
I have…, Who Has…
• Share your game
Number Mat
Numerators and Denominators
Fraction Bingo
Dominoes
How could these activities be adapted?
e.g. decimal identification
5 children share three chocolate bars
evenly.
How much chocolate does each child
receive?
3÷5
Fractions Key Teaching Ideas
1.
Use sets as well as regions from early on and connect different
representations.
2.
Use words first & introduce symbols with care.
3.
Go from part-to-whole as well as whole-to-part with both
shapes and sets.
4.
Fractions are not always less than 1. Push over 1 early to
consolidate understanding.
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.
e.g. 3 ÷ 5
5 children share three chocolate bars evenly.
How much chocolate does each child receive?
What are these
3÷5
pieces called?
1/
2
1/
2
1/
1/
½
+
2
10
1/
2
=
1/
2
2/
12
1/
2
!!
What do you think they have done?
A more sophisticated method for 3 ÷ 5
1/
1/ +1/ =3/
+
5
5
5
5
Y7 response: “3 fifteenths!” Why?
Put a peg where you think 3/5 will be.
1
0
Put a peg where you think 3/5 of 100 will be.
Fractions Key Teaching Ideas
1.
Use sets as well as regions from early on and connect different
representations.
2.
Use words first & introduce symbols with care.
3.
Go from part-to-whole as well as whole-to-part with both
shapes and sets.
4.
Fractions are not always less than 1. Push over 1 early to
consolidate understanding.
5.
Division is the most common context for fractions, e.g. 3 ÷ 5
6. Fractions are numbers as well as operators
3/ is a number between 0 and 1 (number)
5
Three fifths of 100 is 60
(operator)
Teaching Ideas
Using double number lines
x3
0
20
60
100
0
1
5
3
5
1
Connecting sets with regions and lengths
¼
¼
of 12
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
Fractions Key Teaching Ideas
1.
Use sets as well as regions from early on and connect different
representations.
2.
Use words first & introduce symbols with care.
3.
Go from part-to-whole as well as whole-to-part with both
shapes and sets.
4.
Fractions are not always less than 1. Push over 1 early to
consolidate understanding.
5.
Division is the most common context for fractions, e.g. 3 ÷ 5.
6.
Fractions are numbers as well as operators.
7. Fractions are always relative to the whole.
Continually ask “What is 1?”
What
Whatisisthe
B? whole? (Trains Book 7, p32)
A
A
B
B
B
B
C
D
D
D
D
D
D
D
D
Fractions Key Teaching Ideas
1.
2.
3.
4.
Use sets as well as regions from early on and connect different
representations.
Use words first & introduce symbols with care.
Go from part-to-whole as well as whole-to-part with both
shapes and sets.
Fractions are not always less than 1. Go over 1
5. Division is the most common context for fractions, e.g. 3 ÷ 5.
6.
Fractions are numbers as well as operators
7.
Fractions are always relative to the whole.
8. Fractions are really a context for applying
add/sub and mult/div strategies
Connect their division strategies to finding a
fraction of a number, i.e. finding 1 third of a
number is the same as dividing a number by 3.
Framework Practice
Match the strategy stages
to their definitions and
assessment task(s) from
GloSS.
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
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.
What strategy would be used to find 1 third
of 27 when the division fact is unknown?
Stage 2- 4: Equal sharing by ones
Stage 2- 4:
Stage 5:
Stage 5:
Anticipate the result of equal sharing using
repeated addition or skip counting,
Stage 6:
e.g. 9 + 9 + 9
or
9, 18, 27
Stage 6:
Use multiplication, e.g. 3 x ? = 27
Can easily extend to finding 2 thirds of 27.
Ratios (Introduced at Stage 6)
Write 1/2 as a ratio
1:1
3: 4 is the ratio of red to blue beans.
What fraction of the beans are red?
3/
7
Think of some contexts when ratios are used.
Ratios
How are ratios and fractions connected?
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
Perception
Check
What have you
remembered
about these
important key
teaching ideas?
Choose your share of chocolate!
Pirate Problem
• Three pirates have some treasure to share.
They decide to sleep and share it equally in
the morning.
• One pirate got up at at 1.00am and took 1
third of the treasure.
• The second pirate woke at 3.00am and took 1
third of the treasure.
• The last pirate got up at 7.00am and took the
rest of the treasure.
Do they each get an equal share of the
treasure? If not, how much do they each get?
Pirate Problem
• One pirate got up at at 1.00am and took 1 third of the treasure.
• The second pirate woke at 3.00am and took 1 third of the treasure.
• The last pirate got up at 7.00am and took the rest of the treasure.
1st pirate = 1 third
2nd pirate =1/3 x 2/3
= 2 ninths
3rd
pirate = the rest
= 1 - 5 ninths
= 4 ninths
Stage 7
Decimals
Decimals are special cases of equivalent
fractions where the denominator is always a
power of ten.
Stage 7 (AM) Level 4 Key Ideas
Fractions
• Rename improper fractions as mixed numbers, e.g. 17/3 = 52/3
• Find equivalent fractions using multiplicative thinking,, e.g. 2/6 = how many twelfths?
• Order fractions using equivalence and benchmarks like 1 half , e.g. 2/5 < 11/16
• Add and subtract related fractions, e.g. 2/4 + 5/8
• Find fractions of whole numbers using mult’n and div’n e.g.2/3 of 36 and 2/3 of ? = 24
• Multiply fractions by other factions e.g.2/3 x ¼
• Solve measurement problems with related fractions, e.g. 1½ ÷ 1/6 = 9/6 ÷ 1/6 =9
Decimals
• Order decimals to 3dp
• Round whole numbers and decimals to the nearest whole or tenth
• Solve division problems expressing remainders as decimals, e.g. 8 ÷ 3 = 22/3 or 2.66
• Convert common fractions, i.e. halves, quarters etc. to decimals and percentages
• Add and subtract decimals, e.g. 3.6 + 2.89
Percentages
• Estimate and solve percentage type problems like ‘What % is 35 out of 60?’, and ‘What
is 46% of 90?’ using benchmark amounts like 10% & 5%
Ratios and Rates
•
Find equivalent ratios using multiplication and express them as equivalent fractions, e.g.
16:8 as 8:4 as 4:2 as 2:1 = 2/3
•
Begin to compare ratios by finding equivalent fractions, building equivalent ratios or
mapping onto 1).
•
Solve simple rate problems using multiplication, e.g. Picking 7 boxes of apples in ½ hour
is equivalent to 21 boxes in 1½ hours.
Misconceptions with Decimal Place Value:
How do these children view decimals?
1. Bernie says that 0.657 is bigger than 0.7
2. Sam thinks that 0.27 is bigger than 0.395
3. James thinks that 0 is bigger than 0.5
4. Adey thinks that 0.2 is bigger than 0.4
5. Claire thinks that 10 x 4.5 is 4.50
Developing understanding of
decimal place value
The CANON law in our place value system is that 1
unit must be split into TEN of the next smallest
unit AND NO OTHER!
Developing Decimal Place Value Understanding
1. Use decipipes, candy bars, or decimats to
understand how tenths and hundredths
arise and what decimal numbers ‘look like’
2. Make and compare decimal numbers, e.g.
Which is bigger? 0.6 and 0.47
3. How much more make.. e.g. 0.47 + ? =
Using Decipipes
1. establish the whole, half, quarter rods then tenths
2. 1 half = ? Tenths, why is it 0.5 as a decimal?
3. 1 quarter = ? tenths +
4. 1 eighth = ? tenths? +
View children’s response to this task
Now compare:
0.4 0.38 0.275
Using candy bars
(and expressing remainders as decimals)
3÷5
3 chocolate bars shared between 5 children.
30 tenths ÷ 5 =
0 wholes + 6 tenths each = 0.6
Using decimats and arrow cards
1. Read and Make (stage 6)
2. Compare and Order (stage 6-7)
•
•
Which is bigger: 0.6 or 0.43?
How much more make…
3. Add and Subtract (stage 7)
Rank these questions in order of difficulty.
a)0.8 + 0.3,
Exchanging ten for 1
b)0.6 + 0.23
Mixed decimal values
c)0.06 + 0.23,
Same decimal values
4. Multiply and Divide (stage 8)
Add and subtract decimals (Stage 7)
using decipipes or candy bars
Place Value
Tidy Numbers
1.6 - 0.98
Equal Additions
Reversibility
Standard written
form (algorithm)
Decimal Games and Activities
• Digital learning Objects:
http://digistore.tki.org.nz/ec/viewMetadata.action?id=L1079
1. Decimal Sort
2. First to the Draw
3. Four in a Row Decimals
4. Beat the Basics
5. Decimal Keyboard
6. Target Time FIO N3:2,16
‘Target Time’
(from FIO Number L3 Book 2 page 16)
Target Number is 6
+
=
• Roll a dice and place the number thrown.
• Try and make the number sentence as close to
the target number as possible.
• Score = the difference between your total and
the target number.
The Strategy Teaching Model
Existing Knowledge
& Strategies
Using
Material
Materials
s
Using Imaging
Using Number Properties
New Knowledge &
Strategies
Long Term Planning Units
NZ Curriculum
strategy
knowledge
Plan a lesson using the
Planning Units and Book 7
Stage 2- 5
Stage 6-7
Tanya, Jacinda,
Alison and William:
Fair Shares
Trains (Stage 6) or
Hot Shots (Stage 7)
Amanda, Jessica, Nicole
Hamish and Marg
Hungry Birds
Seed Packets (Stage 6) or
Mixing Colours (Stage 7)
Emma and Ellie:
Therie and Judy
Fraction Circles
Birthday Cakes
Nikki and Cameron:
Animals or Wafers
Finding Fractions
Throw 2 dice and make a
fraction,
e.g. 4 and 5 could be 4 fifths
of 5 quarters.
Try and make a true
statement each time the dice
is thrown.
Throw dice 10 times, Miss a
go if you cannot place a
fraction.
Modelling Stage 5
• Wafers
• Animals
Putting it all together
Y1-4: 60-80%
How Much Number?
Y5-6: 50-70%
Y7-8:40-60%
Exploring www.nzmaths.co.nz
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
4 Stages of the PD Journey
Organisation
Orgnising 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
Evaluation
Thought for the day
A DECIMAL POINT
When you rearrange the letters becomes
I'M A DOT IN PLACE
Additional Slides
Equivalent
Fractions
Circle the bigger fraction of each pair. What did you
do to order them?
A
½ or ¼
1/ or 1/
5
9
5/ or 2/
9
9
B
6/ or 3/
4
5
7/ or 9/
8
7
7/ or 4/
3
6
C
7/
3/
or
16
8
2/
5/
or
3
9
5/
3/
or
4
2
D
7/
6/
or
10
8
7/
6/
or
8
9
5/
7/
or
7
9
unit
fractions
More or less
than 1
related
fractions
unrelated
fractions
Key Idea
Ordering using equivalence and benchmarks
Example of Stage 8 fraction knowledge
2/
3
3/
4
2/
5
5/
8
3/
8
How could you communicate this
idea of equivalence to students?
Fraction Circles
Multiplicative
thinking
Paper Folding
1/
x2
4
=
x2
Fraction Tiles / Strips
?/
8
Equivalence Games
•
•
•
•
•
•
Fraction circles and dice game
Fraction Wall Tile game (Norma)
Fraction domino pictures then words
The Equivalence Game: PR3+ p.18-19
Fraction Feud
Fraction Board
3
• Collect the chosen
denominators
• Select how many
denominators are
needed.
• Make the fraction
4
• Compare the fraction
(to ½ 1…)
• Make another
equivalent fraction
Once you understand equivalence
you can……
1.Compare and order fractions
2.Add and Subtract fractions
3.Understand decimals, as decimals are
special cases of equivalent fractions
where the denominator is always a
power of ten.
Comparing Fractions - Which is bigger? (Bk 8)
4/5
12/
15
or
2/3
10/
15
Adding Related Fractions: Create 3 (MM7-9) Each player
chooses a
fraction to place
their counter on
Take turns to
move your
counter along
the lines to
another fraction
Add the new
fraction to your
total.
The first player
to make exactly
three wins.
Go over three
and you lose.
A Fractional Thought for the day
Smart people believe only half
of what they hear.
Smarter people know which
half to believe.
Fraction Circles (book 7 p.20)
Which is bigger 3/4 or 9/8?
Play the fraction circle game.
Put the circle pieces in the “bank”.
Take turns to roll the die and collect what ever you
roll from the bank.
You may need to swap and exchange as necessary.
The winner is the person who has made the most
‘wholes’ when the bank has run out of fraction
pieces.
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
Thinkboard Practice
five thirds
or
7÷4