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.
2
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
3
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?
6
Different Meanings of Division
7
Different Meanings of Division
Different Meanings of Division
8
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
9
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.
10
(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
11
Multiplication Matrix
12
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