Fractions Workshop Marie Hirst

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Fractions WorkshopMarie Hirst

• Have a go at the Fraction Hunt
• on your table while you are waiting!

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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

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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
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The Number Frameworks
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Assess Your Fraction Strategies and Fraction
Knowledge
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Assigning a strategy stage for proportions and
ratios
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Fraction Snapshots

Here are 12 jelly beans to spread on the cake.
If you ate one third of the cake how many jelly
beans will you eat?
Stage 1 Stage 2-4 (AC) Stage 5 (EA)
Unequal Sharing Equal Sharing Use of Addition and known facts e.g. 4 4 4 12
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Fraction Snapshots (contd)
Stage 6 (AA) Using multiplication
What is 3/4 of 80?
16 is four ninths of what number?
Stage 7 (AM) Using division
To make 8 aprons it takes 6 metres of cloth. How
many metres would you need to make 20 aprons?
Stage 8 (AP)
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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.

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• Draw two pictures of one quarter

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Discrete and continuous models
One Quarter
Continuous Discrete
(regions/lengths) (sets) Label
your drawings as discrete or continuous
models. Children need experience with both models
from the very start.
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Key Idea 1

• Work with both shapes and sets of fractions
  from early on.

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Linking regions/shapes and sets
Find one quarter
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The Strategy Teaching Model
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Using Materials - fraction regions
Find one quarter
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Using Materials - fraction regions
Find one quarter of 12
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The Strategy Teaching Model
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Using Imaging
Find one quarter of 12
Key idea quarters means you need 4 equal groups.
One quarter is the number in one of those groups.

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The Strategy Teaching Model
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Using Number Properties
Find one quarter of 40, 400, 4000
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Develop early additive thinking by using addition
facts
Find one quarter of 12
?
?
?
?
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Using Materials - cubes
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?
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Key Idea 2

• 3 sevenths 3 out of 7 7/3
  7 thirds

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5 views of fractions
3 7
3 out of 7

• 3 over 7

3 7
3 sevenths
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The problem with out of
I ate 1 out of the 2 sandwiches in my lunchbox,
Kate ate 2 out of the 3 sandwiches in her
lunchbox, so together we ate 3 out of the 5
sandwiches
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Fraction Language
Use words before and use symbols with care. e.g.
one fifth not 1/5 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
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Fractional vocabulary

• One half
• One third
• One quarter
• Dont know

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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

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1 - ?/20 3/20
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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.

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Key Idea 3

• 6 is one third of what number?

This is one quarter of a shape. What does the
whole look like?
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18
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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.

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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 Whole-to-Part
Continuous (region or length)
Discrete (sets)
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Write 3 more questions to fit the other parts of
the table.
Model Part - to - Whole Whole - to - Part
Continuous (Region or length)
Discrete (sets) Hemi got two thirds of the lollies. How many were there altogether?
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Extending the idea of going from part-to-whole
with non-unit fractions

• Hemi got three fifths of the lollies and got 12.
  How many lollies were there altogether?
• i.e. 12 is three fifths of what number?
• Draw a diagram/use equipment to help your
  thinking.

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12 is three fifths of what number?
20
12
8
4 4
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5 children share three chocolate bars evenly. How
much chocolate does each child receive?
Key Idea 4
3 5
Discuss in groups what you think children would
do and then how you would solve this problem.
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Division
3 5
1/51/51/5 3/5
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Key Idea 4
Division is the most common context for
fractions when units of one are not accurate
enough for measuring and sharing problems.
e.g. 5 3
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Which letter shows 5 halves as a number?
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Key Idea 5Fractions are not always less than
1.Push over 1 early to consolidate the
understanding of the top and bottom numbers.
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Using fraction number lines to consolidate
understanding of denominator and numerator
Push over 1
0 1/2 2/2
3/2 4/2
0 1/2 1 11/2
2
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Fraction Circles
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.
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Three in a row (use two dice or numeral cards)A
game to practice using improper fractions as
numbers
0 1 2 3 4 5 6
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Key Idea 6Fractions are numbers as well as
operators
1/2 is a number between 0 and 1 (number)
Find one half of 12 (operator)
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Using Double Number Lines
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
3/5 of 100. (Fractions as an operator). How will
you work it out?
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Key Idea 7
Sam had one half of a cake, Julie had one quarter
of a cake, so Sam had most. True or False
or Maybe
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Key Idea 7

• Fractions are always relative to the whole.
• Halves are not always bigger than quarters, it
  depends on what the whole is.

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What is the whole?
A A A A A A A A
B B B B B B B B
C C C C C C C C
D D D D D D D D
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Key Idea 8 - Ratios!
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Write 1/2 as a ratio 3 4 is the ratio of red to
blue beans. What fraction of the beans are red?
3/7
Think of some real life contexts when ratios are
used.
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Key Idea 8

• There is a 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 parts
  with the whole, e.g.
• The paint mixture above is 2/5 blue

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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)
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Exploring simple ratios at Stage 6
2 green beans 3 red beans How many green and
red beans in 6 packets?
green
red
I have 22 green beans, how many red will I have?
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Summary of Fractions Key Ideas(Stages 2 - 6)

• Use sets as well as shapes/regions from early on
• Fraction Language - use words first and introduce
  symbols carefully
• Go from Part-to-Whole as well as Whole-to-Part
• Division is the most common context for
  fractions.
• Fractions are not always less than 1, push over 1
  early.
• Fractions are numbers as well as operators.
• Fractions are always relative to the whole.
• Be careful of the relationship between ratios and
  fractions
• Fractions are a context for add/sub and mult/div
  strategies

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Choose your share of chocolate!
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Getting into book 7

• Explore an activity in book 7.
• Focus on the key ideas we have discussed whilst
  exploring the activity.

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Fractions, Ratios and Decimals

• "My life is all arithmetic
• the young businesswoman
• explains.
• "I try to add to my income,
• subtract from my weight,
• divide my time, and avoid
• multiplying..."

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Little League Video Clip
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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.
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Numerical Reasoning Testas used for the NZ
Police Recruitment
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• ½ is to 0.5 as 1/5 is to
• a. 0.15
• b. 0.1
• c. 0.2
• d. 0.5

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• 1.24 is to 0.62 as 0.54 is to
• a. 1.08
• b. 1.8
• c. 0.27
• d. 0.48

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• 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

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• If a man weighing 80kg increased his weight by
  20, what would his weight be now?
• a. 96kg
• b. 89kg
• c. 88kg
• d. 100kg

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Objectives

• Consolidate understanding of key ideas when
  teaching fractions, decimals and percentages
• Understand common misconceptions with ratios and
  decimals.
• Explore equipment and activities used to teach
  key ideas within these higher stages.

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Decimals

At what stage are decimals introduced? (knowledge
and strategy)
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Teaching Decimal Knowledge using Book 4

• Decimal Number Lines (Bk 4 15) MM 4-31
• Squeeze / Number Line Flips Bk 4 (15)
• Using Decimats (Bk 4 8,9), MM 4-21

What did these activities practice?
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How are your decimals?

• Order these decimals from smallest to largest
  . 3.48 3.6
  3.067
• Write one eighth as a decimal
• What is the answer to 5 4
• What is the answer to 3 - 1.95
• What is 0.3 x 0.4
• Order these fractions decimals and percentages
  . 2/3 7/16 30 0.61
  2/5 75 0.38

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Staceys Homework

• Continue these sequences
• 0.7, 0.8, 0.9, 0.10, 0.11, 0.12
• 2.97, 2.98, 2.99 2.100, 2.101, 2.102
• Write down which is the smallest number
• 0.8, 0.5, 0.1 0.1
• 2.3, 2.191, 2.161 2.3
• 3.856, 3.29, 3.4 3.4
• What do you think Stacey is doing?

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Hemis Homework

• Write down which is the smallest number
• 0.8 0.5 0.1 0.1
• 2.3 2.191 2.16 2.191
• 3.856 3.29 3.4 3.856
• What do you think Hemi is doing?
• Discuss what other common misconceptions you
  think children may have about decimals.

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Decimal Misconceptions

• 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.
• 1/2 is 0.2 and 1/4 is 0.4, e.g. 0.4 is
  smaller than 0.2
• When you multiply decimals the number always gets
  bigger.
• When you multiply a decimal number by 10, just
  add a zero, e.g. 4.5 x 10 4.50

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Equivalent Fractions
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.
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Converting Fractions to Decimals
Using Decipipes Bk 7 p.22 (or Decimats) Start
with tenths, fifths, halves, quarters, and then
eighths,
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Operating with decimals

• Using Candy Bars (book 5) 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

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Using Advanced Additive strategies for decimals
Solve 3.6 - 2.98
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Multiplying Decimals0.3 x 0.4
0
1











Ww w

1
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Multiplying Decimals0.3 x 0.4 0.12
0
1
0.3











0.4
Ww w

1
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How are your decimals?

• Order these decimals from smallest to largest
  . 3.48 3.6
  3.067
• Write one eighth as a decimal
• What is the answer to 5 4
• What is the answer to 3 - 1.95
• What is 0.3 x 0.4
• Order these fractions decimals and percentages
  . 2/3 7/16 30 0.61
  2/5 75 0.38

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Why calculate percentages?

• It is a method of comparing fractions by giving
  both fractions a common denominator -
  hundredths.
• So it is useful to view percentages as
  hundredths.

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Percentages

At what stage are percentages introduced? (knowled
ge and strategy)
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Percentages

• AM (Stage 7 NC Level 4)
• Solve fraction ? decimal ? percentage conversions
  for common fractions e.g. halves, thirds,
  quarters, fifths, and tenths
• AP(Stage 8 NC Level 5)
• Estimate and solve problems using a variety of
  strategies including using common factors,
  re-unitising of fractions, decimals and
  percentages, and finding relationships between
  and within ratios and simple rates.

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Applying Percentages

• Types of Percentage Calculations

• Finding percentages of amounts, e.g. 25 of 80
• Expressing quantities as a percentage (for easy
  comparison), e.g. 18 out of 24
  ?
• Increase and decrease quantities by given
  percentages, including mark up, discount and GST
  e.g. A watch cost 20 after a 33 discount. -
  What was its original price?

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Mini Teaching Session 1

• Estimate and find percentages of whole number
  amounts.
• E.g. Find 25 of 80 (easy!)
• 25 1/4 so 25 1/4 of 80 20
• E.g. Find 35 of 80 (harder!)
• Pondering Percentages NSAT 3-4.1(12-13)

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Find 35 of 80
100
80








80
85
Find 35 of 80
100
80










80
86
Find 35 of 80
100
80










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Find 35 of 80
100
80
35 28










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• Now try this
• 46 of 90

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46 of 90
46 of 90
100 10 40 5 1 6 46
90 9 36 4.50 0.90 5.40 41.40
Is there an easier way to find 46?
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Estimating Percentages
Using Number Properties Explain how you would
estimate 61 of a number?
16 of 3961 TVs are found to be faulty at the
factory and need repairs before they are sent for
sale. About how many sets is that? (book 8 p 26
- Number Sense)
About 600
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What now?

• Use fraction snapshots if you think it would
  be useful to regroup children.(On wikispace)
• Review fraction long term planning units.
• Teach fraction knowledge and proportions
  ratios strategies in your classroom with your
  groups.
• This is our last pick up session -Thank you
  all for coming.

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Thought for the day

• There are three things to remember when teaching
• Know your stuff,
• Know whom you are stuffing,
• And stuff them elegantly.
• Lola May