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

2. 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.

3. 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 –
Content Specialist Teacher for Mathematics

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

5. 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

6. 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

7. 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

8. Exploring Fraction Operations –Adding Fractions
Where and Why is it Wrong? A student adds these two fractions: = 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

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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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 =1
5 − 3 =1 =3
6 𝑛 + 5 𝑛 = 1 2
1 𝑛 − 5 𝑛 =2
𝑛 7 − 5 𝑛 = 1 2

17. 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, =
2 1 2 −1 1 4 = =
p.125 Act 8.16 JVdW 6-8
May need to discuss how imperial divisions work
Give ruler

18. 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

19. 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

20. 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 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 cup of milk in each row. How much milk did Wilma use?
p.130 Act from JVdW 6-8

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

22. Exploring Fraction Operations –Multiplying Fractions
How much?
How much is of 2 3
How would you work this problem without using standard algorithm?
How much is 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

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

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

25. 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

26. 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

27. 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

28. 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

29. 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

30. Effective Teaching Process for Fraction Operations
p 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 Effective Teaching Process
Read p.140 Teaching Considerations for Teaching Fractions

31. 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

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

33. 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?

34. …The Three-Part Lesson

35. 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.

37. 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.

38. 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?

39. Formative Assessment Example
Always, sometimes or Never True

40. 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?