The Literacy and Numeracy Secretariat Professional Learning Series

Number Sense and Numeration, Grades 4 to 6 (with reference to Volumes 1, 5, and 6)

Understanding Relationships Between Fractions, Decimals, Ratios, Rates, and Percents

Understanding Relationships Between Fractions, Decimals, Ratios, Rates, and Percents

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Session A – Activating Mathematical KnowledgeSession A – Activating Mathematical Knowledge

1. Aims of Numeracy Professional Learning

2. Learning Goals of the Module

3. Warm Up We Are Fractions!

4. Overview of Number Sense and Numeration, Grades 4 to 6

a) Scavenger Hunt – Volume 1: The Big Ideas

b) Book Walk – Volume 5: Fractions

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Aims of Numeracy Professional LearningAims 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 is one where students bring that thinking to their work.

• Build teachers’ expertise at setting classroom conditions where students can move from their informal math understandings to forming concepts, making sense of procedures and becoming comfortable with 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 4-6.

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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 continuedAims continued

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Teaching Mathematics Through Problem SolvingTeaching 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 these sessions, participants will:• develop an understanding of the conceptual

models of fractions, decimals, ratios, rates, and percents;

• explore conceptual and algorithmic models of fractions and decimals through problem solving;

• analyse and discuss the role of student-generated strategies and standard algorithms in teaching the concepts and relationships of fractions, decimals, ratios, rates, and percents; and

• identify, reflect on, and connect strategies that form a major component of an effective mathematics classroom.

Learning Goals of the ModuleLearning Goals of the Module

7Warm Up We Are Fractions!Warm Up We Are Fractions!Introduce yourself to anyone at your table you do not know.

In your group, make a list of the following:

3 or 4 four things that might be true of nearly all of us

3 or 4 four things that might be true of nearly half of us

3 or 4 four things that might be true of nearly none of us

Connecting mathematics to a real world context

Be prepare

d to share!

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Scavenger Hunt – Volume 1: The Big IdeasScavenger Hunt – Volume 1: The Big Ideas

Book Walk – Volume 5: FractionsBook Walk – Volume 5: Fractions

Number Sense and Numeration, Grades 4 to 6Number Sense and Numeration, Grades 4 to 6

• what the big ideas are;• the importance of learning big ideas;• characteristics of student learning as students relate to big ideas; and• instructional strategies related to big ideas.

• The Mathematical Processes• Characteristics of Junior Learners• Learning About Fractions in the Junior Grades

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Session B – Modelling and RepresentingSession B – Modelling and Representing

1. Warm Up Anticipation Guide

2. What Does It Mean to Model and Represent Mathematical Thinking?

3. Save, Save, Save – Problem #1

4. A Mini-Gallery Walk

10Warm Up – Anticipation GuideWarm Up – Anticipation GuideTalk to your table partners. Come up with a table answer for the following statements:

What do you think? True Untrue

represents a fraction, but 0.67 and 25% do not.

A discount of 0.75 means I will pay of the price.

The following are in ascending order: , 0.37, 0.93, .

Mathematical processes: Reasoning and proving, connecting, communicating

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11Save, Save, Save – Problem #1 Save, Save, Save – Problem #1 Patrik sees white shirts on sale. A sign in the window shows a 25% discount. Another sign shows different white shirts with off. A third sign shows discounted prices that are 0.45 less than the original price on white shirts.

Show Patrik which discount he should ask for in order to save the most money on a white shirt.

Show more than one way to solve this problem.

Connections to Number Sense and Numeration, Grades 4 to 6, Volume 5, page 58

Problem solving, reasoning and proving, selecting tools and computational strategies, representing, communicating

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12Solving the ProblemSolving the ProblemPatrik sees white shirts on sale. A sign in the window shows a 25% discount. Another sign shows different white shirts with off. A third sign shows white shirts that cost 0.45 less than the original price.

Show Patrik which discount he should ask for in order to save the most money on a white shirt.

Show more than one way to solve this problem.

Polya’s Problem Solving Process

Understand the problem.Communicate – talk to understand the problem

Make a plan.Communicate – discuss ideas with others to clarify strategies

Carry out the plan.Communicate – record your thinking using manipulatives, pictures, words, numbers and symbols

Look back at the solution.Communicate – verify, summarize/generalize, validate and explain

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13A Mini-Gallery WalkA Mini-Gallery Walk Find a partner group. Share your group’s solutions with your

partner group. Designate a reporter who will describe the different ways in which you solved the problem.

Listen as the other group’s reporter describes its solutions.

Compare the two groups’ solutions. How are they similar? How are they different?

Sharing strategy: Mini-Gallery Walk

Reflecting, connecting, communicating

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Session C – Conceptual Development Session C – Conceptual Development

1. Warm Up – A KWL Chart Know, Wonder, Learned

2. Quilting – Problem #2

3. A Gallery Walk

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Warm Up KWL ChartKnow, Wonder, LearnedWarm Up KWL ChartKnow, Wonder, Learned

What we think we know about fractions and decimals

What we still wonder about fractions and decimals

What we learned about fractions and decimals

Reasoning and proving, connecting, communicating

16Quilting – Problem #2 Quilting – Problem #2 Ahmed and Tamara are sewing a quilt together. The finished quilt will be square and have 10 squares on each side. So far they have finished 0.56 of their quilt.

Their friends, Soumia and Carlos, are working on another quilt of the same size. They have finished of their quilt.

The friends want to know who has finished more.

Show the solution of this problem by using:1) a 10 x 10 grid2) two stacked number lines

Connections to Number Sense and Numeration, Grades 4 to 6, Volume 6, pages 14 and 19

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17A Gallery WalkA Gallery Walk

Sharing strategy: Gallery Walk

Reflecting, communicating

Post your group’s work. With your group, take a gallery walk to

view the other groups’ solutions. Using the strategies you gleaned, edit

your solution. Be prepared to explain how your solution changed.

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Session D – Alternative Algorithms Session D – Alternative Algorithms

1. Warm-Up – The Meaning of Ratio

2. Best Buy on Juice – Problem #3

3. Bansho

4. Engaging in Rich Problems

5. Professional Learning Opportunities

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Warm Up The Meaning of RatioWarm Up The Meaning of Ratio

Take 3 stick-on notes.

Ask 3 people (from different tables) to share what “ratio” means to them. Use words, symbols, pictures, or numbers.

Write or draw what you hear about “ratio.”

Return to your table.

With your table group, look at all of the comments.

Collectively, write a definition and or representation of “ratio.”

Reasoning and proving, connecting, reflecting, communicating

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Best Buy on Juice – Problem #3Best Buy on Juice – Problem #3Sandro and Julia need to buy boxes of juice for their camping trip. At one store, the cost is $27.60 for 24 boxes. At another store, 18 boxes cost $19.80. Their mother told them not to spend more than $1.12 per box.

a) Which is the better buy?

b) Where should they buy the juice?

Problem solving, reasoning and proving, selecting tools and computational strategies, representing, communicating

Connections to Number Sense and Numeration, Grades 4 to 6, Volume 1, page 41

21Reflecting/ConnectingReflecting/ConnectingBansho: sorting and classifying the details in the solutions presented by participantsBansho helps students:

see what they need to do and think about;

see connections between parts of the lesson, concepts, and solutions;

organize their thinking; and

discover new ideas. Reflecting, connecting,

communicating

Sharing strategy: Bansho

22Engaging in Rich ProblemsEngaging in Rich Problems

Rich problems:

can be represented with a variety of mathematics;

are grounded in a context meaningful to students;

inherently contain the mathematics that the teacher wants the students to learn;

have several entry points and are conducive to extensions, allowing for differentiated instruction; and

require students to use high-level thinking skills.

23Professional Learning OpportunitiesProfessional 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)