Transcript Swanson Frac Dec

Proportions & Ratios
Workshop
Lisa Heap and Alison Howard
Mathematics Facilitators
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
Revising the Framework:
• Sort the addition/subtraction framework.
• Align the multiplication/division framework.
• How do they fit?
Solving a Division Problem:
A sheep station
has eight
paddocks and
296 sheep. How
many sheep are
there in each
paddock?
296 ÷ 8
Proportional
296
÷ 8 =:
Adjustment
148 ÷ 4 =
74 ÷ 2 = 37
Reversibilit
Place Value
y
8 x 30 =240
240 ÷ 8 = 30
56 ÷ 8 = 7
8 x 7 = 56
30 + 7 = 37
Rounding and
Tidy Numbers
Compensating
320
÷
8==40
4000
÷8
500
Algorithm
500 - (320 ÷ 8)=
40 - (24 ÷ 8)=
500 - 40 = 460
40 - 3= 37
Multiplication Division Share Back:
Did you try…
– A knowledge check?
– Diagnostic snapshot?
– Recording in your modelling book?
– Some Equal grouping/Sharing?
Importance of Place Value
• What is place value?
• Where does place value
start?
• What place value equipment
have you currently got in your
school?
• Order the equipment from
least abstract to most abstract.
Place Value Hats:
Place Value Ideas:
• 100 Day Party.
• Place Value Hats.
• Large Numbers Roll Over Page 43.
Place Value
Read, Say, Do x2
Write the number as 63
Write the number as sixty-three
Say the numeral one way, 63 is sixty-three
Say the numeral another way, 63 is six tens and
three ones
• Model the number as ones, 63 individual ones
• Model the number in the PV form as 6 tens and
3
ones
•
•
•
•
The Rope Game
Fraction
Rope Game
Find 2/5
Fraction Knowledge Test:
•
Write the symbols for one half, one eighth, one quarter,
one third and one fifth
•
Put those fractions in order (smallest first)
•
Draw 3 pictures to represent one quarter
•
7 is one third of what number?
•
12 is three fifths of what number?
•
What is 3 ÷ 5?
• On a number line from 1 – 5 show where five halves
live.
• Show me one half as a ratio.
5 children share three chocolate bars evenly.
How much chocolate does each child receive?
What are these
3÷5
pieces called?
1/
2
1/
2
1/
1/
2
2+
1/
1/
2
10
1/
2
=
What do you think they have done?
2/
12
!!
A more sophisticated method for 3 ÷ 5
1/
1/ +1/ =3/
+
5
5
5
5
Y7 response: “3 fifteenths!” Why?
Which letter shows 5 halves as a number?
A
0
B
C
1
D
E
2
F
3
Ratios (Introduced at Stage 6)
1:1
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 contexts when ratios are used.
Choose your share of chocolate!
Framework Practice
Match the strategy stages
to their definitions and
assessment task(s) from
GloSS.
What does this mean?
3÷7
3 out of 7
3:7
3
7
3 sevenths
3 over 7
The problem with “out of”
1
2 +
2
3
8
6
2
3
3 = 5
x 24 =
“I ate 1 out of my 2
sandwiches, Kate ate 2 out
of her 3 sandwiches so
together we ate 3 out of the
5 sandwiches”!!!!!
2 out of 3 multiplied by 24!
= 8 out of 6 parts!
The Problem with Language
Use words first before using the symbols
e.g. one fifth not 1/5
How do you explain the top and bottom numbers?
The number of parts chosen
1
2
The number of parts the whole has been
divided into
Models of Fractions:
• Complete the activity on discrete and
continuous models of fractions.
• In your thinking groups discuss the
meaning of continuous and discrete.
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 one quarter of
this circle?
Part to Whole:
• We also need to give them part to
whole problems, like:
• 5 is a quarter of this
number.
What is the number?
Missing Number
Decimal Fraction Bingo
1.4 1.5
2.4
2.9
2.6
3.3
The Strategy Teaching Model
Existing Knowledge
& Strategies
Using
Material
Materials
s
Using Imaging
Using Number Properties
New Knowledge &
Strategies
What equipment do you use to
teach decimals?
Developing Decimal Place Value Understanding
Decipipes, candy bars, or decimats can be used
to demonstrate how tenths and
hundredths arise and what decimal
numbers ‘look like’ and to compare decimals
numbers.
Using Decipipes
Explore the Decipipes. What is each piece called? How
did you know?
•
•
Build 0.365 and 0.37
Which is bigger? Why?
•
Add 0.4 and 0.25 on your decipipes. Discuss what
you did and what you found out.
Book 7: Pipe Music with Decimals, p.38
Using candy bars
(and expressing remainders as decimals)
3÷5
3 chocolate bars shared between 5 children.
30 tenths ÷ 5 =
0 wholes + 6 tenths each = 0.6
Solve this problem…
Standard written
form (algorithm)
Place Value
I had 1.6m of ribbon. I used 0.98m
for my gift wrapping. How much
ribbon do I have left?
Equal Additions
Reversibility
Ratios
• In the rectangle below, what is the ratio of
blue cubes to green 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?
More on Ratios
• Divide a rectangle up so that the
ratio of its blue to green parts is
7:3.
• What is the fraction of blue and
green?
• 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?
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
Final Evaluation
Complete your initial evaluation
and mark on your progress.
Thank you!