### Understanding Relationships Between Fractions

```Understanding Relationships
Between Fractions, Decimals,
Ratios, Rates, and Percents
Number Sense and Numeration,
(with reference to Volumes 1, 5, and 6)
The Literacy and Numeracy Secretariat Professional Learning Series
Session A – Activating
Mathematical Knowledge
1.
2.
3.
4.
Aims of Numeracy Professional Learning
Learning Goals of the Module
Warm Up  We Are Fractions!
Overview of Number Sense and
a) Scavenger Hunt – Volume 1: The Big Ideas
b) Book Walk – Volume 5: Fractions
2
Aims of Numeracy
Professional Learning
• Promote the belief that all students have learned
some mathematics through their lived experiences in
the world and that the math classroom is one where
students bring that thinking to their work.
• Build teachers’ expertise at setting classroom
conditions where students can move from their
informal math understandings to forming concepts,
making sense of procedures and becoming
comfortable with formal mathematical
representations.
• Assist educators working with teachers of students in
the junior division to implement student-focused
instructional methods to improve student achievement
– as referenced in Number Sense and Numeration,
3
Aims continued
• Have teachers experience mathematical problem
solving as a model of what effective math instruction
entails by:
– collectively solving problems relevant to students’
lives that reflect the expectations in the Ontario
mathematics curriculum;
– viewing and discussing the thinking and strategies
in the solutions;
– sorting and classifying the responses to a problem
to provide a visual image of the range of
experience and understanding of the mathematics;
and
– analysing the visual continuum of thinking to
determine starting points for instruction.
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Teaching Mathematics Through
Problem Solving
•
•
•
•
•
•
•
•
Sharing thinking
Listening to and considering ideas of others
Understanding and analysing solutions
Comparing and contrasting different solutions
Discussing
Generalizing
Communicating
5
Learning Goals of the Module
During these sessions, participants will:
• develop an understanding of the conceptual
models of fractions, decimals, ratios, rates, and
percents;
• explore conceptual and algorithmic models of
fractions and decimals through problem solving;
• analyse and discuss the role of
student-generated strategies and standard
algorithms in teaching the concepts and
relationships of fractions, decimals, ratios, rates,
and percents; and
• identify, reflect on, and connect strategies that
form a major component of an effective
mathematics classroom.
6
Warm Up  We Are Fractions!
Introduce yourself to anyone at your
table you do not know.
In your group, make a list of the
following:
3 or 4 four things that might be
true of nearly all of us
3 or 4 four things that might be
true of nearly half of us
3 or 4 four things that might be
true of nearly none of us
Connecting
mathematics
to a real
world context
Be
prepare
d to
share!
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Scavenger Hunt – Volume 1:
The Big Ideas
• what the big ideas are;
• the importance of learning big ideas;
• characteristics of student learning as students
relate to big ideas; and
• instructional strategies related to big ideas.
Book Walk – Volume 5:
Fractions
• The Mathematical Processes
• Characteristics of Junior Learners
Number Sense and Numeration, Grades 4 to 6
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Session B –
Modelling and Representing
1. Warm Up  Anticipation Guide
2. What Does It Mean to Model and
Represent Mathematical Thinking?
3. Save, Save, Save – Problem #1
4. A Mini-Gallery Walk
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Warm Up – Anticipation Guide
Talk to your table partners. Come up with a table
What do you think?
True
Untrue
3
4
represents a fraction, but 0.67 and
25% do not.
A discount of 0.75 means I will pay
1
4 of the price.
The following are in ascending
order: 21 , 0.37, 0.93, 54 .
Mathematical processes: Reasoning and
proving, connecting, communicating
10
Save, Save, Save – Problem #1
shows a 25% discount. Another sign shows different
white shirts with 13 off. A third sign shows discounted
prices that are 0.45 less than the original price on
white shirts.
Show Patrik which discount he should ask for in order
to save the most money on a white shirt.
Connections to Number Sense and Numeration,
Grades 4 to 6, Volume 5, page 58
Problem solving, reasoning and proving,
selecting tools and computational
strategies, representing, communicating
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Solving the Problem
Patrik sees white shirts on
shows a 25% discount.
Another sign shows different
white shirts with off. A 13
third sign shows
white shirts that cost 0.45
less than the original price.
Show Patrik which discount
in order to save the most
money on a white shirt.
solve this problem.
Polya’s Problem Solving Process
Understand the problem.
Communicate – talk to understand
the problem
Make a plan.
Communicate – discuss ideas with
others to clarify strategies
Carry out the plan.
using manipulatives, pictures, words,
numbers and symbols
Look back at the solution.
Communicate – verify,
summarize/generalize, validate and
explain
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A Mini-Gallery Walk
13
 Find a partner group.
 Share your group’s solutions with your
partner group. Designate a reporter who
will describe the different ways in which
you solved the problem.
 Listen as the other group’s reporter
describes its solutions.
 Compare the two groups’ solutions. How
are they similar? How are they different?
Sharing strategy: Mini-Gallery Walk
Reflecting, connecting, communicating
Session C –
Conceptual Development
1. Warm Up – A KWL Chart
Know, Wonder, Learned
2. Quilting – Problem #2
3. A Gallery Walk
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Warm Up  KWL Chart
Know, Wonder, Learned
What we think we know
decimals
What we still wonder
decimals
15
fractions and decimals
Reasoning and proving,
connecting, communicating
Quilting – Problem #2
Ahmed and Tamara are sewing a quilt
together. The finished quilt will be square
and have 10 squares on each side. So
far they have finished 0.56 of their quilt.
Their friends, Soumia and Carlos, are
working on another quilt of the same size.
4
They have finished 10 of their quilt.
The friends want to know who has
finished more.
Show the solution of this problem by using:
1) a 10 x 10 grid
2) two stacked number lines
Connections to Number Sense and Numeration,
Grades 4 to 6, Volume 6, pages 14 and 19
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A Gallery Walk
17
 With your group, take a gallery walk to
view the other groups’ solutions.
 Using the strategies you gleaned, edit
your solution. Be prepared to explain how
Sharing strategy: Gallery
Walk
Reflecting, communicating
Session D –
Alternative Algorithms
1.
2.
3.
4.
5.
Warm-Up – The Meaning of Ratio
Best Buy on Juice – Problem #3
Bansho
Engaging in Rich Problems
Professional Learning Opportunities
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Warm Up  The Meaning of
Ratio
Take 3 stick-on notes.
Ask 3 people (from different tables) to share what
“ratio” means to them. Use words, symbols,
pictures, or numbers.
Write or draw what you hear about “ratio.”
Collectively, write a definition and or representation
of “ratio.”
Reasoning and proving, connecting,
reflecting, communicating
19
Problem #3
Sandro and Julia need to buy boxes of
juice for their camping trip. At one
store, the cost is \$27.60 for 24 boxes.
At another store, 18 boxes cost \$19.80.
Their mother told them not to spend
more than \$1.12 per box.
a) Which is the better buy?
b) Where should they buy the juice?
Connections to Number Sense and Numeration,
Grades 4 to 6, Volume 1, page 41
Problem solving, reasoning and proving, selecting tools and
computational strategies, representing, communicating
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Reflecting/Connecting
21
Bansho: sorting and classifying the details in the
solutions presented by participants
Bansho helps students:
see what they need to do and think about;
 see connections between parts of the lesson,
concepts, and solutions;
 organize their thinking; and
 discover new ideas.
Sharing strategy: Bansho
Reflecting, connecting,
communicating
Engaging in Rich Problems
Rich problems:
can be represented with a variety of mathematics;
are grounded in a context meaningful to students;
inherently contain the mathematics that the teacher
wants the students to learn;
have several entry points and are conducive to
extensions, allowing for differentiated instruction;
and
require students to use high-level thinking skills.
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Professional Learning Opportunities
Collaborate with other teachers through:
• Co-teaching
• Coaching
• Teacher inquiry/study groups
View:
• Coaching Videos on Demand (www.curriculum.org)
• Deborah Ball webcast (www.curriculum.org)
• E-workshop (www.eworkshop.on.ca)
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