Transcript Otara Lead T Cluster 5

Otara LT Cluster Meeting
Vanitha Govini
Phoebe Fabricius
Numeracy facilitators
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 1/3 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?
Game time!
FRACDICE!
• Learning intention
• Key knowledge
Objectives:
• Explore common misconceptions with fractions in order to
develop teachers content knowledge.
• Explore key ideas, equipment and activities used to teach
fraction knowledge and strategy.
• Exploring the language demands of mathematics
• Scaffolding learning for English Language Learners (ELL)
The ‘big’ ideas??
Brainstorm any key ideas that underpin
fractional understanding
• Comparing
•
•
•
•
•
•
**What’s it got to do with
division?
Ordering
Improper
**Finding a fraction of a set
Mixed
Equivalency
**Representing a fraction
Unit fractions
Why does ‘the whole’ matter?
Write down all the misconceptions you
think exist within Fractions.
The Problem with Language
•Stress the meaning of the numerators and denomina
Use words first before using the symbols
e.g. one fifth not 1/5
How do you explain the top and bottom numbers?
1
The number of parts chosen
2
The number of parts the whole
has been divided into
Ideas for Fraction language
and recording.
Fraction as a Number
• Do
not assume children understand the
symbols.
• Call the fraction by its name:
• It is wise to use words first not symbols
e.g. record 1 half rather than 1/2.
See
Say
Do
Doing some ‘Skemp’ activities to
teach Fractions.
Skemp activity
START
ACTION
RESULT
NAME
Biscuit
Leave this on
as it is
(Put it here)
The whole of a biscuit
Biscuit
Make 2 equal
parts
(Put it here)
These are halves of a
biscuit
Biscuit
Make 3 equal
parts
(Put it here)
These are third-parts of
a biscuit
Biscuit
Make 4 equal
parts
(Put it here)
These are fourth-parts
of a biscuit. Also called
Quarters.
Biscuit
Make 5 equal
parts
(Put it here)
These are fifth-parts of
a biscuit.
What does Fractions mean?
3÷7
3 out of 7
3:7
3
7
3 sevenths
3 over 7
How would each of the views
solve this problem?
3
7
3 out of 7
3÷7
3 
over 7
3:7
3 sevenths
of
42
3
7
of
1
3
Is possible but refers to both numerator
and denominator as being whole numbers.
Need to picture groups of 7 and then three of
each of these.
Encourages additive thinking
3 out of 7
Fraction is actually tenths
1/10 of 42 = 4 so 3 x 4 =, 12 with
some left over
3/7 is actually a ratio of 3:4
Demonstrate with unifix cubes
3:7
Difficult as you need to work out what 3 divided by
7 as a decimal (leads to needing calculator).
Can be worked out by (3x42) / 7.
Generates unnecessary multiplication.
A calculator will give the answer, though this would
be devoid of meaning
3÷7
3
7
3 sevenths
Refers to both numerator
and denominator as being whole
numbers. If you place three over
seven are you actually
working with tenths.
3 over 7
1 seventh of 42 is 6 so 3 x 6 = 18
Children need good Mult and Div basic facts
7 is telling you how many equal parts and the 3 is telling how
many of those parts have been selected
Draw 3 pictures to represent three
quarters.
Discrete model of
Continuous model of
fractions
fractions
0
1
Label your pictures of three quarter
as either continuous (shape/region)
or discrete (sets).
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?
Perception Check:
Model
Continuous
(Region or
length)
Discrete
(sets)
Part - to - Whole
Whole- to Part
Perception Check:
Model
Continuous
(Region or
length)
Discrete
(sets)
Part - to - Whole
Whole- to Part
This is one quarter of How many ways
a shape. What is the can you cut this
shape?
shape into
quarters?
Hemi got two thirds
of the lollies. How
many were there
altogether?
Here are 12
lollies. If you eat
one quarter of
them, how many
do you get?
Emphasise the ‘ths’ code
1 dog + 2 dogs = 3 dogs
1 fifth + 2 fifths = 3 fifths
3
2
1
+
=
5
5
5
3 fifths + ? = 1



1 -
5
?
=
5
3
5
Scenario one
• A group of students are investigating the books they have in
their homes.
1
2
• Steve notices that
1
books, while
of the books in his house are fiction
5
• Andrew finds that
of the books his family owns are fiction.

• Steve states that his family has more fiction books than
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
1
1
number of books
each
family
owns.
2
5
• For example: of 30 is less than of 100.

• Key is to always refer to the whole. This will be
dependent on the problem!
Scenario Two
• 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
•
•
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.
Chocolate activity
Fraction starter: Fraction Three in a
Row
Dotty Pairs Game: p. 22 Book 7
• One player is dots the other is crosses
• Number line from 0 to 6 or 0 to 10
• Roll 2 dice and form a fraction, place this on
number line (use materials if necessary)
• Aim is to get 3 marks uninterrupted by your
opponent’s marks on the number line.
• If a player chooses a fraction that is
equivalent to a mark that is already there they
lose a turn.
Three in a row:
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
Are fractions always less than
1?
Using the material provided, can you
show 5 halves?
5 halves will
always be 2
and a half as a
number.
How would you mark 5 halves on a
number line?
0
5
Book 7:
• Explore one activity from a strategy stage in your
table groups.
• Focus on :
• What key knowledge is required before beginning this
stage.
• Highlight the important key ideas at this stage.
• The learning intention of the activity.
• Work through the teaching model (materials, imaging,
number properties).
• Possible follow up practice activities.
• The link to the planning units and Figure It Out
support.
• Let’s have a look at another example.
• How much more of green is there than of blue?
• What knowledge do children need to be able to apply
this understanding to realitity?
Ratios:
• 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?
Pipe Music
Using Deci-pipes to teach decimals!
Objectives:
•Identify and Order decimals
•Identify no. of tenths / hundredths in a
number
•Add / Sub decimals
Game: Zap!
Decimals
Winnie uses materials and claims
2•5 + 1•8 = 3•13.
What error has Winnie probably made?
Problem solving
In a game of netball, Irene gets in
43 out of her 50 shots. Sarah
takes 20 shots and gets in 17.
Who is the better shot?
The Connection between
Fractions and Percentages
What does % mean?
• In mathematics, a percentage is a way of expressing a
number as a fraction of 100 (per cent meaning "per
hundred"). It is often denoted using the percent sign, % For
example, 45% (read as "forty-five percent") is equal to 45
hundredths or 0.45.
• What do we need to do to fractions so that it can be read as
a percentage?
• What key mathematical knowledge do children need
to be able to do this?
Teaching Percentages:
• Double Number lines:
Students enter the information they have onto a
double number line, then extend the pattern to
find the information they need. Students are
encouraged to find relationships vertically and
horizontally.
For example: 40% of 70 = ?
4
0
7
0
10
% 4
28
7010 = 7
40%
10010 =
10
70
100
%
Solve these Problems using
the Double Number line:
• Emily’s team won the basket ball game 120-117. Emily shot
60% of the goals. How many goals did Emily get?
• John scored 104 runs in a one day cricket match, that was
40% of the teams total. How many runs did his team score
altogether?
• In a bike race, 30% of cyclists drop out. 42 riders finish the
race. How many cyclists started the race?
Summary of key ideas
•
•
•
•
•
•
•
Fraction language - emphasise the “ths” code
Fraction symbols - use words and symbols with caution
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
Book 7:
• Explore one activity from a strategy stage in your table
groups.
• Focus on :
• What key knowledge is required before beginning this stage.
• Highlight the important key ideas at this stage.
• The learning intention of the activity.
• Work through the teaching model (materials, imaging, number
properties).
• Possible follow up practice activities.
• The link to the planning units and Figure It Out support.
The Connection between
Fractions and Percentages
What does % mean?
• In mathematics, a percentage is a way of expressing a
number as a fraction of 100 (per cent meaning "per
hundred"). It is often denoted using the percent sign, % For
example, 45% (read as "forty-five percent") is equal to 45
hundredths or 0.45.
• What do we need to do to fractions so that it can be read as
a percentage?
• What key mathematical knowledge do children need
to be able to do this?
Lemon activity
What are the key messages? Why?
Share it with a partner.