WERKLUND SCHOOL OF EDUCATIONGALILEO EDUCATIONAL NETWORK

LEARNING IN MATHEMATICS (K-6)A CBE RESEARCH & PROFESSIONAL LEARNING SERIES

In partnership with Werklund School of Education, University of CalgarySession 3: Friday November 17, 1:00-4:00 p.m.

The agenda, working documents and other materials for each session can be accessed on Galileo’s Professional Learning website http://galileo.org/pl

Focus: Concepts related to number division (partitive and quotative) [multiplicative reasoning, connection to number systems]

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which it is useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know.

Learning Outcome: Recognize division as the inverse of multiplication. Multiplicative reasoning is required to be successful with division.

Participant Agenda

Time Activity Learning Intentions/ Outcomes

1:00

1:10

Welcome & Overview Document sharing established

for data collectionApplied Learning Individual reflection to prepare

for sharing (page 1) In triads, review the evidence

of learning Record a synopsis of the

conversation in the Google form goo.gl/8FRLDq

Intention: Identify multiplicative reasoning in teachers’ contexts and examine errors students are making.Outcome: Recognize multiplicative reasoning begins in K. Analyzing errors can support differentiated learning designs.

1:50 Discuss key insights Large group discussion

Intention: Building collective and expansive understanding of multiplicative reasoning using an array representation.Outcome: Deepen teachers’ awareness of multiplicative reasoning.

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2:00 Multiplicative Reasoning: Problem

Who is it?

Intention:  Demonstrate that children’s everyday activities (enactive) contain multiplicative reasoning.Outcome:  Use the enactive to bridge to iconic and symbolic.

2:10 Break

2:20 What is division? Generate word cloud

www.menti.com code 70 45 2Different meanings of division Early experiences with division Meanings represented in text

books Division of different kinds of

numbers (fractions, decimals)

Intention: Demonstrate that partitive meaning is overrepresented in textbooks.Recognize the importance of balancing the meaning of division to support learners' longitudinal development (division involving fraction, algebraic thinking).Distinguish discrete and continuous numbers within the context of division.Outcome: Identify critical features of division starting in early years.

2:50 Create a problem Using the chart paper, create a

story problem which models 1 ¾ ½

Construct a representation to show your thinking

Revisit: What is division?www.menti.com code 70 45 2

Outcome: Create an accurate division problem.

3:15 Multiplication Matrix Division and multiplication are

inverse relationships

Intention: Explore patterns inherent in the multiplication matrix.  Explore the ways in which the matrix makes the inverse relationships visible.Outcome:  Multiplication and division are inverse relationships

3:45 For next session:1. Introduce students to ideas of division using measurement. Bring

your task to the next session with a sample of student work. Look for student misconceptions.

2. Use the Principled Practice document to reflect on your experience as you introduce your students to the ideas of division. Identify places in the table where you could provide examples to illustrate

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your experience.

3:504:00

Feedback Survey: link goo.gl/JWdqKAAdjourn

Individual Reflection

Please take a moment to photograph the sample(s) of student work you brought to share and upload the image(s) to your folder.

1. Describe how you introduced multiplication arrays to your students. What did you do? What were the students doing?

2. What errors in reasoning did your students encounter as they worked through the activity?

4

Multiplicative Reasoning: Who is it?

There are 8 people standing in a circle. As you start to count, every third person will be eliminated. Where should you stand so that you will be it?

5

Division Situations1. There are 50 cards. Divide the cards equally for 4 players to play a

game.

2. Each lesson is 45 minutes. How many lessons can we have, if a day is 6 hours long?

3. In a university dormitory, 3 students share a room. There are 178 new students. How many rooms do we need to accommodate the new students?

4. 3 children received 6 apples. How many apples will each child receive?

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Different Meanings of Division

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Different Meanings of Division

Different Meanings of Division

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Action Known Unknown

Quotative Measure an object with a

given unit

Unit Measurement

Quotative(Discrete)

Make groups of a given quantity of

each

Cardinality of the groups

Number of groups

Partitive Separating an object into a

given number of equal parts

Number of the parts

Size of the parts(How

much/many)

Partitive(Discrete)

Split a collection of objects into a given number of equal groups

Number of groups Cardinality of the group (How

many)

Create a Problem

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Working as a group, use the chart paper provided to do the following:

1. Calculate 1¾ ½

2. Make up a story problem which could be solved using this calculation.

3. Construct a representation to show your thinking.

Multiplication MatrixComplete the multiplication table. Fill in each square with the appropriate product.

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(Eg. 7 x 8 = 56)What are some of the patterns you notice? How does the matrix make the inverse relationship visible?

X 0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

9

10

11

12

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Multiplication Matrix

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Principled PracticeGuiding

Principles

Domains of Practice

Attending to the integrity

of the mathematics

Committing to the

learning & achievement

of each student

Establishing and

managing a productive learning

environment

Learning from &

systematically improving

practice

Leading a whole class discussion

Representing mathematical ideas

Assessing students’ knowledge, skill, and dispositions

Planning math lessons