148_Understanding Multiplication and Division of Whole and Decimal Numbers

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

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

Session A

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.

• 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

• Understanding and analysing solutions

• Comparing and contrasting different solutions

• Discussing

• Generalizing

• Communicating

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

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 mathematic

s to a real world

context

Warm Up – What Ways

Do We Use Math?

Think-Pair-Share

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

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

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.

2 Warm Up – A Math Congress

3 The Concepts of Multiplication –

Problem #2

4 A Gallery Walk

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

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

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

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.

2 Warm Up – “All About

Place Value” Game

3 Making the Strategies

and Math Talk Explicit –

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

“ 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

• Give each group of 4 a set of

cards with whole and decimal

numbers on the cards

6.9 69

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

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?

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

I can divide 6.9 if I think 6 x ? = 6.9.

I can estimate the value by multiplying

• 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?

• Coaching Videos on Demand www.curriculum.org

• Deborah Ball webcast www.curriculum.org

• E-workshop www.eworkshop.on.ca