Transcript ENW 5 Fractions West

Pickups 2010
Fractional Thinking
Lisa Heap, Jill Smythe & Alison Howard
Numeracy Facilitators
Pirate Problem
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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?
The Rope Activity:
Objectives:
• Identify the progressive strategy stages of
fractions, proportions and ratios.
• Further develop teacher’s confidence and content
knowledge of fractions.
• Explore key ideas, equipment and activities used
to teach fraction knowledge and strategy.
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
Developing Proportional
Thinking:
A chance to recap what needs to be taught at the
different stages.
• Decide which strategy stage fits each scenario.
• Use the number framework to help you.
• Highlight all the fractional knowledge across the
stages (pg18-22).
Fraction Knowledge Test:
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Draw 2 pictures: (a) one half (b) one eighth
Mark 5 halves on a number line from 1-5
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12 is three fifths of what number?
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What is 3 ÷ 5?
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Draw a picture of 7 thirds
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Write one half as a ratio.
• The ratio of kidney beans to green is 3:4. What fraction of the beans
are green?
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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
Morning Tea
• After morning tea we will split again;
Stages 1, 2, 3 – with Alison
Stages 4 – 8 – with Jill and Lisa
Body Fractions
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?
More on Ratios….
• Divide a rectangle up so that the ratio of its blue
to green parts is 7:3.
• 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?
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 2:7. 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?
What about this?
• Two students are measuring the height of the plants their
class is growing.
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Plant A is 6 counters high.
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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
answers?
• How would you further support Scott’s thinking?
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.
• Scott’s reasoning: To find Plant B’s height you add 3 to the
height of Plant A; 4 + 3 = 7.
• Wendy’s 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. 6:9 is equivalent
to 4:6.
– 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.
Exploring Book 7:
Stage 4-5:
Fraction Circles (page 20)
Stage 5-6:
Birthday Cakes (page 26)
Stage 6-7
Hot Shots (page 46)
Welcome
Back
Alison’s
Group
Views of Fractions:
• What does this fraction mean?
3÷7
3 out of 7
3:7
3
7
3 sevenths
3 over 7
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
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.
0
1
Discrete Model:
• Discrete: Made up of individual objects.
• e.g. ¾ of this set is blue
Whole to Part:
• Most fraction problems are about giving students
the whole and asking them to find parts.
• Show me ¼ of this circle?
Part to Whole:
• We also need to give them part to whole
problems, like:
• ¼ of a number is 5.
What is the number?
Teaching Fractions:
• What do you see as some of the
confusions associated with the
teaching and understanding
of fractions?
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?
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?
Misconceptions with Fractions:
• A group of students are investigating the books they
have in their homes.
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• Steve notices that 2 of the books in his1 house are
fiction books, while Andrew finds that 5 of the books
his family owns are fiction.
• Steve states that his family has more fiction books than
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Andrew’s.
Consider….
Is Steve necessarily correct?
Why/Why not?
What action, if any, do you take?
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.
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• For example: 2 of 30 is less than 5 of 100.
• Key is to always refer to the whole. This will be
dependent on the problem! 
Misconceptions with Fractions:
• Heather says 7 is not possible as a fraction.
3
Consider…..
• Is 7
3
possible as a fraction?
• Why does Heather say this?
• What action, if any, do you take?
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?
What could be the misconception
here?
2 chocolate bars shared amongst 5 students:
What does each student get?
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 the
answer is 2/10 because they think the answer is 2
out of 10.
Misconceptions with Fractions:
• You observe the following equation in Bill’s work:
Consider…..
• Is Bill correct?
• What is the possible reasoning behind his answer?
• What, if any, is the key understanding he needs to
develop in order to solve this problem?
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
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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.
Misconceptions with Fractions
• When you multiply by some fractions the answer
gets smaller
• 1/4 x 1/3 = 1/12
• This is ⅓ of one whole strip.
⅓
• If it is cut into quarters, four equivalent pieces,
what will each new piece be called?
1
12
Fractions Video:
• What was the key purpose of the lesson?
• What key mathematical language was being
developed?
• How did materials/equipment support the
children’s 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?
Summary of key ideas:
• Fraction language - emphasise the “ths” code
• Fraction symbols – use symbols with caution, start
with words
• Continuous and discrete models - use both
• Go from Part-to-Whole as well as Whole-to-Part
• Fractions are numbers and operators
• Fractions are a context for add/sub and mult/div
strategies.
• Fractions are always relative to the whole and the
whole can be bigger than one.
Thought for the day:
• Smart people believe only half of what they
hear.
• Smarter people know which half to believe.