Dates:Tuesdays, Jan 7 – Feb 11 Time: 5:30 pm to 8:30 pm Location: Victor Scott School

Tentative Course Timeline Week/Content Topic Dates Pedagogy topic Week 1 Operational Strategies Jan 7 Introductory information, survey, pre-assessment Teaching through Problem Solving Framework Week 2 Operational Strategies & Fractions Jan 14 Planning a Three-Part Lesson Week 3 Fractions Jan 21 Week 4 Fractions & Algebra Jan 28 Formative Assessment Teachers present planned lessons Week 5 Algebra Feb 4 Questioning Week 6 Feb 11 Post Test & Exit survey 60-40 Content through pedagogy Switch hats between student and teacher.

Math and Science Institute Exploring Numbers and Algebraic Thinking Today’s Agenda Recap on Fraction Concepts Engaging in Operational Fractions Tasks Planning an Effective Math Lesson Plan a lesson for your year level 5:30 pm 5:45 pm 6:45 pm 7:00 pm Facilitator: Rebeka Matthews Sousa – rsousa@moed.bm Content Specialist Teacher for Mathematics

Learning Objectives In this session, Mathematicians will: Have a deeper conceptual understanding of fractions Investigate various ways to represent, compare, and operate with fractions

Exploring Fraction Concepts –Fraction Equivalents Find fraction names for each shaded region. Explain how you saw each name you find. p.229 of JVdW 3-5 Dot-Paper Fraction Equivalence

Exploring Fraction Concepts –Fraction Equivalents Using set models for equivalent fractions. Set out 24 counters – 16 red (apples) and 8 yellow (bananas) Your task: Group the counters into different fractional parts of the whole and use the parts to create fraction names for the fractions that are apples and the fractions that are bananas. p.119 of JVdW 6-8 Act 8.12 Insert fig 8.8 Ask: If we make groups of four, what part of the set is red?” Notice that the phrase reducing fractions is not used – this alone suggests that the amount is smaller – fractions are simplified not reduced Do not tell students their answer is incorrect if not in simpliest or lowest form

Exploring Fraction Concepts –Fraction Equivalents Using set models for equivalent fractions. Set out 24 counters – 16 red (apples) and 8 yellow (bananas) Your task: Group the counters into different fractional parts of the whole and use the parts to create fraction names for the fractions that are apples and the fractions that are bananas. p.119 of JVdW 6-8 Act 8.12 Insert fig 8.8 Ask: If we make groups of four, what part of the set is red?” Notice that the phrase reducing fractions is not used – this alone suggests that the amount is smaller – fractions are simplified not reduced Do not tell students their answer is incorrect if not in simpliest or lowest form

Exploring Fraction Operations –Adding Fractions Where and Why is it Wrong? A student adds these two fractions: 1 2 + 1 3 = 2 5 The answer is incorrect. Explain why it is incorrect. Is the answer reasonable? Using a strategy, other than using “like denominators” and following a rule for adding fractions, solve the problem. Explain using words or diagrams, how you would come to the correct answer. Introduce fractions strips

Using Fraction Strips to show addition of fractions: 1 2 + 1 3 = ? ?

Models for Adding Fractions Have students develop algorithms rather than just memorizing them. Make sense out of what the algorithm means. Some may develop the understanding on their own from the models. p.124 Fig 8.10 p.125 Fig 8.11 models for adding fractions

Models for Adding Fractions Have students develop algorithms rather than just memorizing them. Make sense out of what the algorithm means. Some may develop the understanding on their own from the models. p.124 Fig 8.10 p.125 Fig 8.11 models for adding fractions

Models for Adding Fractions Have students develop algorithms rather than just memorizing them. Make sense out of what the algorithm means. Some may develop the understanding on their own from the models. p.124 Fig 8.10 p.125 Fig 8.11 models for adding fractions

Models for Adding Fractions Have students develop algorithms rather than just memorizing them. Make sense out of what the algorithm means. Some may develop the understanding on their own from the models. p.124 Fig 8.10 p.125 Fig 8.11 models for adding fractions

Models for Adding Fractions Have students develop algorithms rather than just memorizing them. Make sense out of what the algorithm means. Some may develop the understanding on their own from the models. p.124 Fig 8.10 p.125 Fig 8.11 models for adding fractions

Models for Adding Fractions Have students develop algorithms rather than just memorizing them. Make sense out of what the algorithm means. Some may develop the understanding on their own from the models. p.124 Fig 8.10 p.125 Fig 8.11 models for adding fractions

Exploring Fraction Operations –Adding/Subtracting Fractions Can You Make it True? There are two missing values (in the numerators, or denominators, or one of each). Can you determine the digits that will make each problem true? You may not use digits already in the problem Use fraction benchmarks (0, ½, 1) to support thinking. These require reasoning and thinking about the relative size of fractions Which one is impossible? Why? p.123 Act 8.15 JVdW 6-8 6 + 8 =1 5 − 3 =1 9 + 8 =3 6 𝑛 + 5 𝑛 = 1 2 1 𝑛 − 5 𝑛 =2 𝑛 7 − 5 𝑛 = 1 2

Exploring Fraction Operations –Adding/Subtracting Fractions Jumps on the Ruler Using the ruler, find the results of these three problems without applying the common denominator algorithm, 3 4 + 1 2 = 2 1 2 −1 1 4 = 1 1 8 +1 1 2 = p.125 Act 8.16 JVdW 6-8 May need to discuss how imperial divisions work Give ruler

Addressing Misconceptions for Adding/Subtracting Fractions Assess students’ understanding and keep an eye on common misconceptions. Adding Both Numerators and Denominators Failing to Find Common Denominators Difficulty Finding Common Multiples Difficulty with Mixed Numbers Give a scenario: Ms. Simmons baked a pan of brownies for the bake sale and cut the brownies into 8 equal size parts. In the morning, three brownies were sold; in the afternoon, two more were sold. What fractional part of the brownies had been sold. Students give a their answers 5/8 or 5/16 half half – model it correctly Students don’t understand the meaning of denominator and different sized portioned. Use number line to indicate different sized pieces Using flash cards LCM Use models: students will try to take smaller fraction from greater fraction P.127

Commercial Break When working with whole numbers we would say 3x5 means 3 sets of 5 (equal sets) or would draw an array. Different visuals are used when working with fractions. Multiplication of fractions is really about scaling. Example: multiply by 1 means that the numbers remain unchanged. Multiply by ½ means that something is half as big Start with fractions of whole numbers

Exploring Fraction Operations –Multiplying Fractions Water, Walking, and Wheels How would you work out the following problems, using a manipulative or drawing to figure out the answers to these three tasks. The walk from school to the public library takes 15 minutes. When Tatiana asked her mom how far they had gone, her mom said that they had gone 2 3 of the way. How many minutes had they walked? (Assume constant walking rate) There are 15 cars in Tiago’s matchbox car collection. Two-thirds of the cars are red. How many red cars does Michael have? Marie filled 15 glasses with 2 3 cup of milk in each row. How much milk did Wilma use? p.130 Act. 8.18 from JVdW 6-8

Modeling Multiplication Problems How would you explain each solution? Fig 8.13 p.129 JVdW 6-8

Exploring Fraction Operations –Multiplying Fractions How much? How much is 3 4 of 2 3 How would you work this problem without using standard algorithm? How much is 3 5 of 3 4 How would you work this problem without using standard algorithm? How can you use this problem to explain the standard algorithm? p.131 Fig 8.14 from JVdW 6-8 p.132 Fig 8.15 from JVdW 6-8

Exploring Fraction Operations –Multiplying Fractions Possible solutions p.131 Fig 8.14 from JVdW 6-8

Exploring Fraction Operations –Multiplying Fractions Possible solutions p.132 Fig 8.15 from JVdW 6-8

Addressing Misconceptions for Multiplying Fractions Assess students’ understanding and keep an eye on common misconceptions. Treating the Denominator the Same as in Addition and Subtraction Problems Inability to Estimate the Approximate Size to the Answer Matching Multiplication Situations with Multiplication (and Not Division) Use models (rectangle, circle, number line to see the conceptual difference) Students often think multiplication makes numbers bigger. Practice estimation Example 1/3 of 24 some may multiply by 1/3 or divide by 3 but some may confuse and divide by 1/3. Estimation again helps P.134

Exploring Fraction Operations –Multiplying Fractions Divided up Determine your answers using a non-standard algorithm 1 ¼ hours to do three chores. How much time for each? 5 3 ÷ 1 2

Exploring Fraction Operations –DividingFractions How would you explain these two models of divisions? p.137 Fig 8.19 from JVdW 6-8 p.138 Fig 8.20 from JVdW 6-8

Exploring Fraction Operations –DividingFractions How would you explain these two models of divisions? p.137 Fig 8.19 from JVdW 6-8 p.138 Fig 8.20 from JVdW 6-8

Addressing Misconceptions for Dividing Fractions Assess students’ understanding and keep an eye on common misconceptions. Thinking the Answer Should Be Smaller Connecting the Illustration with the Answer Writing Remainders Based on whole-number, students think the answer should be smaller – true for greater than 1 Understanding the meaning 1 ½ divided by ¼ means How many fourths in 1 ½ - counting in fourths Example 3 3/8 div ¼ 12 fourths and 2/8 but not sure what to do with extra 1/8 P.139

Effective Teaching Process for Fraction Operations p.120-121 of J. Van de Walle et al. Understanding Fraction Operations “Students must be able to compute with fractions flexibly and accurately. Success with fractions, in particular computation, is closely related to success in Algebra. If students enter formal algebra with a weak understanding of fraction computation (in other words, they have memorized the four procedures but do not understand them), they are at risk for struggling, which in turn can limit college and career opportunities. Deeper understanding and flexibility take time! This is recognized by research…” Read p.121-122 Effective Teaching Process Read p.140 Teaching Considerations for Teaching Fractions

Planning a Lesson Keep in mind the planning criteria and rubric for teaching The lesson should be a condensed lesson (20 mins) And a lesson appropriate for the level that you teach. P3 P4 P5 P6 Middle Denise Angela Kamal Janine Uneaka Chantal Vivienne Desmond Chantel Renee

How will you know that your students know it? Key to planning How will you know that your students know it?

Planning Learning Tasks Asking yourself the following questions will help you plan effective learning tasks: What are the concepts I want my students to learn from the task I plan? How will I determine my students’ prior knowledge? How ill I design lesson (learning tasks) to help student explore and learn these concepts? How will I assess student learning?

…The Three-Part Lesson

Criteria for Effective Mathematics Tasks A good instructional task captures students’ interest and imagination and satisfies the following criteria: The solution is not immediately obvious The problem provides a learning situation related to a key concept or big idea The task is aligned with the Cambridge Objective(s) The context of the problem is meaningful to students. There may be more than one solution. The problem promotes the use of one or more problem solving strategies The situation requires decision making above and beyond the choosing of a mathematical operation. The solution time is reasonable. The situation may encourage collaboration in seeking solutions.

Checklist for Planning Effective Mathematics Tasks The Lesson Has a balance of skills: mental math, conceptual understanding, problem solving, and computational skills May include the Three-Part Lesson as a vehicle to Teach Through Problem-solving: (Activate Thinking, Working on it, Reflect and Connect) A good instructional task captures students’ interests and imagination and also satisfies the following criteria. The Task(s) Are aligned with the Cambridge Objective(s). Provides a learning situation related to key concept or big ideas. Or problem is meaningful relevant and interesting to students. Cognitively demanding (solution is not immediately obvious) and there may be more than one solution) Or problem promotes the use of one or more problem solving strategies (multiple entry or exit points) Differentiated Requires decision making above and beyond the choosing of a mathematical operation. May encourage collaboration in seeking solutions. Resources, materials, manipulatives prepared in advanced. Assessment Variety of assessment tools to access students throughout the lesson Questioning Questions are prepared in advance to encourage mathematical thinking and communication of mathematical reasoning.

What types of questions will I ask students to promote thinking and to assess? Teachers promote the sharing of ideas by asking the following kinds of questions: What did you do to find out…? How could you show that…? Can you explain why…? How do you know that…? How do you know that your idea is correct? Can someone explain a different strategy?

Formative Assessment Example Always, sometimes or Never True

Teacher Reflection & Homework Have I including all of the necessary requirements of my lesson? What will I need to bring or prepare for the next session?