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 Learning Goals of the Module Book Walk – Tabbing the Volumes Warm Up – What Ways Do We Use Math? 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.

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.

Teaching 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

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 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. Share the session goals with the participants.

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

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. 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 Think-Pair-Share

Warm Up – What Ways Do We Use Math? Sort your group’s multiplication and division examples. Describe your sorting rule and label each column. Examples of Multiplication and Division in Our Daily Lives Connecting situated knowledge, and informal, lived, or embodied mathematics to formal mathematics 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

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.

Session B Conceptual Development Problem Solving to Develop Conceptual Understanding Warm Up – A Math Congress The Concepts of Multiplication – Problem #2 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. Fac Guide: Problem 1 – what makes it the problem is the fact the we ask for sevreal solutiaons – otherwise just a word problem Problem #2 – the concept we are aiming for is the distributive property which is inherent in the problem. The 150 contraint makes the problem not as obvious. Connections to Number Sense and Numeration, Grades 4 to 6, Volume 5: page 23

Session C Exploring Alternative Algorithms Applying Student-Generated Algorithms and Analysing Standard Algorithms Partitive and Quotative Division Student-Generated and Standard Algorithms for Division – Problem #3 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

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. Remind teachers to adapt problems to suit their students’ needs – for example reduicng extraneous words to differentiate instruction,

Session D Communicating Mathematical Thinking Estimating Decimal Division Warm Up – “All About Place Value” Game Making the Strategies and Math Talk Explicit – Problem #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 6.9 69 tenths 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.

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?

Julie’s Method Sarah’s Method 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 = 6 6 x 1.1 = 6.6 6 x 1.2 = 7.2 I can use these numbers to estimate the length. Sarah’s Method 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.

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?

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 <in facilitator’s guide go to eWorkshop Success Factors>