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## Pickups 2010 Fractional Thinking

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### Pickups 2010 Fractional Thinking Lisa Heap, Jill Smythe & Alison Howard Numeracy Facilitators – PowerPoint PPT presentation

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Title: Pickups 2010 Fractional Thinking

1
Pickups 2010Fractional Thinking
• Lisa Heap, Jill Smythe Alison Howard
• Numeracy Facilitators

2
Pirate Problem
• While you are waiting
• 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 1/3 of
the treasure.
• The second pirate woke at 3.00am and took 1/3 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?

3
The Rope Activity
4
Objectives
• Identify the progressive strategy stages of
fractions, proportions and ratios.
• Further develop teachers confidence and content
knowledge of fractions.
• Explore key ideas, equipment and activities used
to teach fraction knowledge and strategy.

5
The 4 Stages of the P.D 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
6
Developing Proportional Thinking
• A chance to recap what needs to be taught at the
different stages.
• Decide which strategy stage fits each scenario.
• Highlight all the fractional knowledge across the
stages (pg18-22).

7
Fraction Knowledge Test
• Draw 2 pictures (a) one half (b) one eighth
• Mark 5 halves on a number line from 1-5
• 12 is three fifths of what number?
• What is 3 5?
• Draw a picture of 7 thirds
• Write one half as a ratio.
• The ratio of kidney beans to green is 34. What
fraction of the beans are green?
• Order these fractions
• 2/4, 3/4, 2/5, 7/16, 2/3,
6/49
• Now include these and decimals into your
order
• 30, 75, 0.38, 0.5

8
Morning Tea
• After morning tea we will split again
• Stages 1, 2, 3 with Alison
• Stages 4 8 with Jill and Lisa

9
Body Fractions
10
Ratios
• In the rectangle below, what is the ratio of
green to blue cubes?
• What is the fraction of blue and green cubes?
• Can you make another structure with the same
ratio? What would it look like?
• What confusions may children have here?

11
More on Ratios.
• Divide a rectangle up so that the ratio of its
blue to green parts is 73.
• Think of other ways that you can do it.
• What is the fraction of each colour?
• If I had 60 cubes how many of them will be of
each colour?

12
A Ratio Problem to Solve
• There are 27 pieces of fruit. The ratio of fruit
that I get to the fruit that you get is 27. How
many pieces do I get?
• How many pieces would there have to be for me to
get 8 pieces of fruit?
• What key mathematical knowledge is required here?

13
• Two students are measuring the height of the
plants their class is growing.
• Plant A is 6 counters high.
• Plant B is 9 counters high.
• When they measure the plants using paper clips
they find that Plant A is 4 paper clips high.
• What is the height of Plant B in paper clips ?
• Consider..
• Scott thinks Plant B is 7 paper clips high.
• Wendy thinks Plant B is 6 paper clips high.
• Who is correct?
• What is the possible reasoning behind each of
their
• How would you further support Scotts thinking?

14
Key Idea
• The key to proportional thinking is to be able to
see combinations of factors within numbers.
• Wendy is correct, Plant B is 6 paper clips high.
• Scotts reasoning To find Plant Bs height you
add 3 to the height of Plant A 4 3 7.
• Wendys reasoning
• Plant B is one and a half times taller than Plant
A 4 x 1.5 6.
• The ratio of heights will remain constant. 69 is
equivalent to 46.
• 3 counters are the same height as 2 paper clips.
There are 3 lots of 3 counters in plant B,
therefore 3 x 2 6 paper clips.

15
Exploring Book 7
• Stage 4-5
• Fraction Circles (page 20)
• Stage 5-6
• Birthday Cakes (page 26)
• Stage 6-7
• Hot Shots (page 46)

16
Welcome Back Alisons Group
17
Views of Fractions
• What does this fraction mean?

3 7
3 out of 7
3 7
3 over 7
3 sevenths
18
The Problem with Language
Use words first before using the symbols e.g.
one half not 1 out or 2 How do you explain the
top and bottom numbers? 1 2
The number of parts chosen The number of equal
parts the whole has been divided into
19
Continuous Model
• Models where the object can be divided in any way
that is chosen.
• e.g. ¾ of this line and this square are blue.

20
Discrete Model
• Discrete Made up of individual objects.
• e.g. ¾ of this set is blue

21
Whole to Part
• Most fraction problems are about giving students
the whole and asking them to find parts.
• Show me ¼ of this circle?

22
Part to Whole
• We also need to give them part to whole problems,
like
• ¼ of a number is 5.
• What is the number?

23
Teaching Fractions
• What do you see as some of the confusions
associated with the teaching and understanding
• of fractions?

24
Misconceptions with Fractions
• Charlotte believes that one eighth is bigger than
one half.
• 1/2 ? 1/3 ? 1/4 ? 1/8
• Why do you think Charlotte has this
misunderstanding?
• How would you address this misconception?
• What equipment would you use?

25
Misconceptions with Fractions
• Fiona says the following
• ¼ ¼ ¼ 3/12
• Why do you think Fiona has this misunderstanding?
• How would you address this misconception?
• What equipment would you use?

26
Misconceptions with Fractions
• A group of students are investigating the books
they have in their homes.
• Steve notices that of the books in his house
are fiction books, while Andrew finds that of
the books his family owns are fiction.
• Steve states that his family has more fiction
books than Andrews.
• Consider.
• Is Steve necessarily correct?
• Why/Why not?
• What action, if any, do you take?

27
Key Idea
• The size of the fraction depends on the size of
the whole.
• Steve is not necessarily correct because the
amount of books that each fraction represents is
dependent on the number of books each family
owns.
• For example of 30 is less than of 100.
• Key is to always refer to the whole. This will
be dependent on the problem!

28
Misconceptions with Fractions
• Heather says is not possible as a fraction.
• Consider..
• Is possible as a fraction?
• Why does Heather say this?
• What action, if any, do you take?

29
Key Idea
• A fraction can represent more than one whole.
• Can be illustrated through the use of materials
and diagrams.
• Question students to develop understanding
• Show me 2 thirds, 3, thirds, 4 thirds
• How many thirds in one whole? two wholes?
• How many wholes can we make with 7 thirds?

30
What could be the misconception here?
• 2 chocolate bars shared amongst 5 students
• What does each student get?

31
Reason
• Because the divisor is 5 the natural denominator
is fifths. Each bar is broken into five equal
pieces.
• One way of solving the problem is to give each
student one piece from each bar. Each will have
2 pieces. Compared with one bar each student has
2 fifths of a bar.
• The common error here is for students to think
is 2 out of 10.

32
Misconceptions with Fractions
• You observe the following equation in Bills
work
• Consider..
• Is Bill correct?
• What is the possible reasoning behind his
• What, if any, is the key understanding he needs
to develop in order to solve this problem?

33
Key Idea
• To divide the number A by the number B is to find
out how many lots of B are in A. When dividing
by some unit fractions the answer gets bigger!
• No he is not correct. The correct equation is
• Possible reasoning behind his answer
• 1/2 of 2 1/2 is 1 1/4.
• He is dividing by 2.
• He is multiplying by 1/2.
• He reasons that division makes smaller
therefore the answer must be smaller
• than 2 1/2.

34
Misconceptions with Fractions
• When you multiply by some fractions the answer
gets smaller
• 1/4 x 1/3 1/12
• This is ? of one whole strip.
• If it is cut into quarters, four equivalent
pieces, what will each new piece be called?

35
Fractions Video
• What was the key purpose of the lesson?
• What key mathematical language was being
developed?
• How did materials/equipment support the
childrens learning? What may have happened if
the equipment was not present?
• Why did the teacher use the example 101/4 in the
lesson?
• In terms of the teaching model, where do you
think the children are at?
• What would be you next step with this group of
children?

36
Summary of key ideas
• Fraction language - emphasise the ths code
• Fraction symbols use symbols with caution,