Transcript Slide 1

Understanding Multiplication
and Division of Whole and
Decimal Numbers
Number Sense and Numeration,
Grades 4 to 6
(Volumes 1, 3, 4, and 6)
The Literacy and Numeracy Secretariat Professional Learning Series
Session A
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Modelling and Representing
1. Aims of Numeracy Professional
Learning
2. Learning Goals of the Module
3. Book Walk – Tabbing the Volumes
4. Warm Up – What Ways Do We Use
Math?
5. Modelling and Representing
Multiplication – Problem #1
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 should be a
place where students bring that thinking to work.
• Build teachers’ expertise in setting classroom
conditions in which students can move from their
informal math understandings to generalizations and
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,
Grades 4 to 6.
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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
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Sharing thinking
Listening to and considering ideas of others
Adapting thoughts
Understanding and analysing solutions
Comparing and contrasting different solutions
Discussing
Generalizing
Communicating
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Learning Goals of the Module
During this session, participants will:
• develop an understanding of the conceptual
models of whole numbers and decimals;
• explore conceptual and algorithmic models of
whole number and decimal multiplication through
problem solving;
• analyse and discuss the role of studentgenerated strategies and standard algorithms in
the teaching of multiplication and division with
whole and decimal numbers; and
• identify the components of an effective
mathematics classroom.
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Book Walk:
Tabbing the Volumes
(1, 3, 4, and 6)
Number Sense
and Numeration,
Grades 4 to 6
Number Sense and
Numeration, Grades 4 to 6
Volume 1: The Big Ideas
Volume 2:
Addition and
Subtraction
Volume 6:
Decimal
Numbers
Volume 3:
Multiplication
Volume 5:
Fractions
Volume 4:
Division
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Warm Up – What Ways
Do We Use Math?
Think of the different ways you have used multiplication
and division in your daily life over the past week.
Record one way per sticky note.
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Connecting
mathematics
to a real world
context
Pair up with your elbow partner and talk about one or two
of the notes you wrote.
Share by introducing yourself to anyone at your table
you do not know. Put your sticky notes onto a piece of
chart paper and report what they say about the different
ways you have used multiplication and division in your
daily life over the past week.
Think-Pair-Share
Warm Up – What Ways
Do We Use Math?
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Sort your group’s multiplication and division examples.
Describe your sorting rule and label each column.
Connecting situated
knowledge, and
informal, lived, or
embodied
mathematics to
formal mathematics
Examples of Multiplication
and Division in Our Daily Lives
label 1
label 2
label 3
label 4
Modelling and Representing
Multiplication – Problem #1
There are 29 students going to a museum.
The museum trip costs $23.00 per student.
The fee includes transportation, a ticket to
the museum, and a lunch.
How much will it cost for 29 students to go
on the field trip?
Connections to Number Sense and Numeration,
Grades 4 to 5, Volume 3: page 47
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Solving the Problem
There are 29 students going to
a museum. The museum trip
costs $23.00 per student. The
fee includes transportation, a
ticket to the museum, and a
lunch.
How much will it cost for 29
students to go on the field trip?
Show more than one way to
solve the problem.
Polya’s Problem-Solving Process
Understand the problem.
Communicate – talk to
understand the problem.
Make a plan.
Communicate – discuss ideas
with others to identify and clarify
strategies.
Carry out the plan.
Communicate – record your
thinking using manipulatives,
pictures, words, numbers, and
symbols.
Look back at the solution.
Communicate – check
reasonableness, review methods,
summarize, generalize.
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Session B
Conceptual Development
1. Problem Solving to Develop
Conceptual Understanding
2. Warm Up – A Math Congress
3. The Concepts of Multiplication –
Problem #2
4. A Gallery Walk
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The Concepts of Multiplication –
Problem #2
Julie can run 100 m in 12.4 seconds.
How long would it take Julie to run 400 m
at that speed?
Show your thinking using a variety of
mathematics – different strategies, tools,
and algorithms.
Connections to Number Sense and Numeration,
Grades 4 to 6, Volume 5: page 23
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Session C
Exploring Alternative Algorithms
1. Applying Student-Generated
Algorithms and Analysing Standard
Algorithms
2. Partitive and Quotative Division
3. Student-Generated and Standard
Algorithms for Division – Problem #3
4. Organizing to See a Range of
Student Thinking – Bansho
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Warm Up – Partitive and
Quotative Division
• Partitive Division
(unknown # of items in each group)
A grocer has 30 apples. He puts the apples in 5 bags.
How many apples will the grocer put in each bag?
• Quotative Division
(unknown # of groups)
A grocer has 30 apples. She wants to put them into
bags, with 5 apples in each bag. How many bags will
the grocer need?
Connections to Number Sense and Numeration,
Grades 4 to 6: Volume 4: page 17
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Student-Generated and
Standard Algorithms for
Division – Problem #3
Ben and his family are planning a charity bike-a-thon.
The total distance is 96 km. They want to have stations
for refreshments about one-fourth of the way, half-way,
and three-fourths of the way.
About how many kilometres should there be between the
starting point, the three stations, and the end point?
Show more than one way to solve the problem.
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Session D
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Communicating Mathematical Thinking
1. Estimating Decimal
Division
2. Warm Up – “All About
Place Value” Game
3. Making the Strategies
and Math Talk Explicit
– Problem #4
4. Professional Learning
Opportunities
“. . . teaching the standard
algorithm for multiplication
should not be the ultimate
teaching goal for students
in the junior grades.
Students need to learn the
importance of looking at
the numbers in the
problem, and then making
decisions about which
strategies are appropriate
and efficient in given
situations.” Volume 3
Warm Up: “All About Place
Value” Game
• Give each group of 4 a set of
cards with whole and decimal
numbers on the cards.
• Players lay the cards face up on the table. They
take turns matching pairs of cards with numbers of
equal value, such as 6.9 and 69 tenths.
• When one player finds a match, he or she takes the
two cards from the array and sets them aside,
scoring 1 point for each pair. Players pass if they
see no matches.
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Making the Strategies and
the Math Explicit –
Problem #4
An artist is creating garden ornaments
out of a strip of copper 6.9 m in length.
She will form either a regular pentagon
or a hexagon as part of the design.
What will be the length of each side of
each polygon?
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Julie’s
Method
Sarah’s
Method
For a pentagon, I need to divide
the copper into 5 equal lengths.
For a hexagon, I know there would be
6 equal sides.
I round 6.9 to 7. I can mentally
calculate 7 divided by 5.
I can divide 6.9 if I think 6 x ? = 6.9.
I know there is one 5 in 7 and 2
ones left over.
I know 2 ones is the same as
20 tenths.
I can divide 20 tenths by 5.
I can estimate the value by multiplying
by decimals.
6 x 1.0 = 6
6 x 1.1 = 6.6
6 x 1.2 = 7.2
I can use these numbers to
estimate the length.
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Problem Solving –Thinking
• Complete each student’s estimate and show
your work.
Reflecting
• Why do you think Sarah stopped multiplying
decimal numbers by 6 after she multiplied
6 x 1.2?
• How would you show another way to estimate
the length of each side of the pentagon and
the hexagon?
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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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