Transcript power point presentation

Fractions
A Staff Tutorial
Workshop Format
This workshop is based around seven
teaching scenarios.
From each of these scenarios will be
drawn:
key ideas about fractions
how to communicate these to students
Scenario One
A group of students are investigating the books they
have in their homes.
1
2
Steve notices that of the books in his house are
1
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 Andrew’s.
Consider…
Is Steve necessarily correct?
Why / Why not?
What action, if any, do you take?
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…
Number of
books
Fraction of books that Number of fiction
are fiction
books
Steve’s family
30
1
2
15
Andrew’s family
100
1
5
20
Andrew’s family has more fiction books than Steve’s.
Number of
books
Steve’s family
40
Andrew’s family
40
Fraction of books that Number of fiction
are fiction
books
1
2
1
5
Steve’s family has more fiction books than Andrew’s.
20
8
Key Idea:
The size of the fractional amount depends on
the size of the whole.
To communicate this key idea to students
you could…
Demonstrate with clear examples, as in the
previous tables.
Use materials or diagrams to represent the
numbers involved (if appropriate).
Question the student about the size of one whole:
Is one half always more than one fifth?
What is the number of books we are finding one fifth
of? How many books is that?
What is the number of books we are finding one half
of? How many books is that?
Scenario Two
You observe the following equation in Emma’s work:
1 + 2
2
3
Is Emma correct?
= 3
5
Consider…
You question Emma about her understanding and
she explains:
“I ate 1 of the 2 sandwiches in my
lunchbox, Kate ate 2 of the 3 sandwiches in
her lunchbox, so together we ate 3 of the 5
sandwiches we had.”
What, if any, is the key understanding Emma needs
to develop in order to solve this problem?
Emma needs to know that the
whole than the 23 .
1
2
relates to a different
If it is clarified that both lunchboxes together
represent one whole, then the correct recording is:
1 + 2 =3
5
5
5
Emma also needs to know that she has written an
incorrect equation to show the addition of fractions.
Key Idea:
When working with fractions, the whole needs to
be clearly identified.
To communicate this key idea to students
you could…
Use materials or diagrams to represent the situation. For
example:
Question the student about their understanding.
The one out of two sandwiches refers to whose lunchbox?
Whose lunchbox does the two out of three sandwiches
represent?
Whose lunchbox does the three out of five sandwiches
represent?
Key Idea:
When adding fractions, the units need to be the
same because the answer can only have one
denominator.
To communicate this idea to students you
could…
Use a diagram or materials to demonstrate
that fractions with different denominators
cannot be added together unless the units
are changed. For example:
Scenario Three
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 answers?
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.
Key Idea:
The key to proportional thinking is being able to
see combinations of factors within numbers.
To communicate this idea to students you
could…
Draw a diagram to show the relationships
between the numbers.
Use ratio tables to identify the
multiplicative relationships between the
numbers involved.
Use double-number lines to help visualise
the relationships between the numbers.
Scenario Four
Anna says
7
is not possible as a fraction.
3
Consider…..
Is 7 possible as a fraction?
3
What action, if any, do you take?
7
3 is possible as a fraction.
It is read as “seven thirds.”
Seven thirds is equivalent to two and one third and can also
be recorded as 2 13 .
7
The shaded area in the diagram represents 3 .
Key Idea:
A fraction can represent more than one whole.
The denominator tells the number of equal parts into which
a whole is divided. The numerator specifies the number of
these parts being counted.
To communicate this idea to students you
could…
Use materials and diagrams to illustrate.
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?
Let’s try
Scenario Five
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?
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.
Key Idea:
To divide the number A by the number B
is to find out how many lots of B are in
A.
For example:
•
There are 4 lots of 2 in 8
•
There are 5 lots of 1/2 in 2 1/2
To communicate this idea to students you
could…
Use meaningful representations for the
problem. For example:
I am making hats. If each hat takes 1/2 a
metre of material, how many hats can I make
from 2 1/2 metres?
Use materials or diagrams to show there
are 5 lots of 1/2 in 2 1/2:
Key Idea:
Division is the opposite of multiplication.
The relationship between multiplication and division can
be used to help simplify the solution to problems involving
the division of fractions.
To communicate this idea to students you
could…
Use contexts that make use of the inverse operation:
A rectangular vegetable garden is 2.5 m2. If one
side of the garden is 1/2 a metre long, what is the
length of the other side?
Half of a skipping rope is 2.5 metres long. How
long is the skipping rope?
Scenario Six
Which shape has
of its area shaded?
Sarah insists that none of the shapes have
shaded.
of their area
Consider:
Do any of the shapes have
of their area shaded?
What action, if any, do you take?
The shape on the right has of it’s area shaded.
is equivalent to , that is it represents the same
quantity. The same amount of each of the circles is
shaded:
Key Idea:
Equivalent fractions have the same value.
To communicate this idea to students you
could…
Use diagrams or materials to show equivalence.
– Paper folding
– Cut up pieces of fruit to show, for example, that one
half is equivalent to two quarters.
– Fraction tiles
Question students about their understanding. For
example, using the fraction tiles you could ask:
How many twelfths take up the same amount of space as
two sixths?
How many sixths take up the same amount of space as
one third?
Can you see any other equivalent fractions in this wall?
Record the equivalent fractions as they are identified.
Scenario Seven
You observe the following equation in Bruce’s work:
Consider:
Is he correct?
After checking that Bruce understands what the
“>” symbol means, what action, if any, do you
take?
No he is not correct.
The correct equation is
because
one sixth is less than one quarter.
Key Idea:
The more pieces a whole is divided into, the
smaller each piece will be.
To communicate this idea to students you
could…
Demonstrate the relative size of fractions
with materials or diagrams.
Question students about the relative size
of each fractional piece:
If we had 2 pizzas and we cut one pizza into six
pieces and the other into 4 pieces, which pieces
would be bigger?
The use of reference points 0, 1/2 and 1 can
be useful for ordering fractions larger than unit
fractions. For example:
Which is larger
8
11
is larger than one half and
8
one half, so 11 is greater than
7
15
is less than
7
.
15
Key Ideas about Fractions
The size of the fractional amount depends on the size
of the whole.
When working with fractions, the whole needs to be
clearly identified.
When adding fractions, the units need to be the same
because the answer can only have one denominator.
The key to proportional thinking is being able to see
combinations of factors within numbers.
A fraction can represent more than one whole.
The denominator tells the number of equal parts
into which a whole is divided. The numerator
specifies the number of these parts being counted.
Division is the opposite of multiplication.
The relationship between multiplication and division
can be used to help simplify the solution to
problems involving the division of fractions.
Equivalent fractions have the same value.
The more pieces a whole is divided into, the
smaller each piece will be.