understanding multiplication and division of whole and decimal numbers
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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 Modelling and Representing. Aims of Numeracy Professional Learning - PowerPoint PPT PresentationTRANSCRIPT
The Literacy and Numeracy Secretariat Professional Learning Series
Number Sense and Numeration, Grades 4 to 6 (Volumes 1, 3, 4, and 6)
Understanding Multiplication and Division of Whole and Decimal Numbers
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Session AModelling and Representing 1. Aims of Numeracy Professional
Learning2. Learning Goals of the Module3. Book Walk – Tabbing the Volumes4. Warm Up – What Ways Do We Use
Math?5. Modelling and Representing
Multiplication – Problem #1
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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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• 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.
Aims continued
5Teaching Mathematics Through Problem Solving• 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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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 student-generated 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.
Learning Goals of the Module
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Book Walk:Tabbing the Volumes (1, 3, 4, and 6)
Number Sense and Numeration, Grades 4 to 6
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Volume 1: The Big Ideas
Volume 3: Multiplication
Volume 4: Division
Volume 5: Fractions
Volume 6: Decimal Numbers
Volume 2: Addition and Subtraction
Number Sense and Numeration, Grades 4 to 6
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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.
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.
Connecting mathematics to a real world context
Warm Up – What Ways Do We Use Math?
Think-Pair-Share
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Sort your group’s multiplication and division examples. Describe your sorting rule and label each column.
label 1 label 2 label 3 label 4
Examples of Multiplication
and Division in Our Daily Lives
Warm Up – What Ways Do We Use Math?
Connecting situated knowledge, and informal, lived, or embodied mathematics to formal mathematics
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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
12Solving the ProblemThere 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.
13Session BConceptual Development
1. Problem Solving to Develop Conceptual Understanding
2. Warm Up – A Math Congress3. The Concepts of Multiplication –
Problem #2 4. A Gallery Walk
14The 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
15Session CExploring Alternative Algorithms
1. Applying Student-Generated Algorithms and Analysing Standard Algorithms
2. Partitive and Quotative Division3. 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 #3Ben 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 DCommunicating 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
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• Give each group of 4 a set of cards with whole and decimal numbers on the cards.
6.969
tenths
• 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.
Warm Up: “All About Place Value” Game
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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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For a pentagon, I need to divide the copper into 5 equal lengths.
I round 6.9 to 7. I can mentally calculate 7 divided by 5.
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.
Julie’s Julie’s MethodMethod
For a hexagon, I know there would be 6 equal sides.
I can divide 6.9 if I think 6 x ? = 6.9.
I can estimate the value by multiplying by decimals.6 x 1.0 = 66 x 1.1 = 6.66 x 1.2 = 7.2
I can use these numbers to estimate the length.
Sarah’s Sarah’s MethodMethod
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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 OpportunitiesCollaborate 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