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Understanding Relationships Between Fractions, Decimals, Ratios, Rates, and Percents Number Sense and Numeration, Grades 4 to 6 (with reference to Volumes 1, 5, and 6) The Literacy and Numeracy Secretariat Professional Learning Series Session A – Activating Mathematical Knowledge 1. 2. 3. 4. Aims of Numeracy Professional Learning Learning Goals of the Module Warm Up We Are Fractions! Overview of Number Sense and Numeration, Grades 4 to 6 a) Scavenger Hunt – Volume 1: The Big Ideas b) Book Walk – Volume 5: Fractions 2 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 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. 3 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. 4 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 5 Learning Goals of the Module 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. 6 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! 7 Scavenger Hunt – Volume 1: The Big Ideas • 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. Book Walk – Volume 5: Fractions • The Mathematical Processes • Characteristics of Junior Learners • Learning About Fractions in the Junior Grades Number Sense and Numeration, Grades 4 to 6 8 Session 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 9 Warm Up – Anticipation Guide Talk to your table partners. Come up with a table answer for the following statements: What do you think? True Untrue 3 4 represents a fraction, but 0.67 and 25% do not. A discount of 0.75 means I will pay 1 4 of the price. The following are in ascending order: 21 , 0.37, 0.93, 54 . Mathematical processes: Reasoning and proving, connecting, communicating 10 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 13 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 11 Solving the Problem Patrik sees white shirts on sale. A sign in the window shows a 25% discount. Another sign shows different white shirts with off. A 13 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 12 A Mini-Gallery Walk 13 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 Session C – Conceptual Development 1. Warm Up – A KWL Chart Know, Wonder, Learned 2. Quilting – Problem #2 3. A Gallery Walk 14 Warm Up KWL Chart Know, Wonder, Learned What we think we know about fractions and decimals What we still wonder about fractions and decimals 15 What we learned about fractions and decimals Reasoning and proving, connecting, communicating 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. 4 They have finished 10 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 grid 2) two stacked number lines Connections to Number Sense and Numeration, Grades 4 to 6, Volume 6, pages 14 and 19 16 A Gallery Walk 17 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. Sharing strategy: Gallery Walk Reflecting, communicating Session D – Alternative Algorithms 1. 2. 3. 4. 5. Warm-Up – The Meaning of Ratio Best Buy on Juice – Problem #3 Bansho Engaging in Rich Problems Professional Learning Opportunities 18 Warm 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 19 Best Buy on Juice – Problem #3 Sandro 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? Connections to Number Sense and Numeration, Grades 4 to 6, Volume 1, page 41 Problem solving, reasoning and proving, selecting tools and computational strategies, representing, communicating 20 Reflecting/Connecting 21 Bansho: sorting and classifying the details in the solutions presented by participants Bansho 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. Sharing strategy: Bansho Reflecting, connecting, communicating Engaging 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. 22 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) 23