1. Teaching through the Mathematical Processes
Session 2: Problem Solving with the Mathematical Processes in Mind
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2. Find Someone Who . . .
Find someone in the group who satisfies a criteria on the card.
Each square must have a different name.
First BINGO - diagonals
Second BINGO – full card
This activity is designed to help build a community of learners by getting to know each other.
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3. Mathematical Processes
Take the participants on a tour through the TIPS Mathematical Processes package.
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4. Mathematical Processes
Teaching through the Mathematical Processes addresses differentiated instruction by attending to the needs of different learners and learning groups. These Mathematical Processes are addressed in more depth through the grades and support lifelong learning.
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5. Mathematical Processes
For each of the Mathematical Processes, there is a page that identifies <click and read > Role of Students, Instructional Strategies, Sample Questions and Sample Feedback.
There are a number of examples under each category. The suggestions are to assist you in focusing on a particular process with your students.
Rather than trying to focus on all the Mathematical Processes at the same time, focus on a single process for a period of time; perhaps one where you have identified a gap. Involving your colleagues in this same process allows for a collaborative focus.
A Mathematical Process page can serve as a guide for yourself and your students. The Student Success Differentiated Instruction package has the same information on tent cards.
A suggested approach:
Inform students which process will be the focus for a group of lessons.
Share the Student Role suggestions so they know what you want them to do and what you’ll be looking for.
As this unfolds, you may wish to incorporate portions of the Sample Questions and Feedback suggestions.
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6. Exploring Mathematical Processes
Individually, explore the Mathematical Processes package with particular attention to a “different” process from what you studied earlier.
Give participants time to examine a different process from Session 1, e.g., 5 minutes.
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7. Big Idea is Problem Solving
Problem solving forms the basis
of effective mathematics programs and should be the mainstay of mathematical instruction.
The Ontario Curriculum Grades 1 – 8, Mathematics, Revised 2005
Problem Solving is fundamental to learning mathematics. Most if not all important mathematics concepts and procedures can be taught through problem solving.
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8. Problem Solving with the Mathematical Processes in Mind
With your partner(s) select one of the given problems to solve.
Ask questions using the Mathematical Process package prompts.
Note when a Mathematical Process is being used.
<<Some solutions can be found as hidden slides at the end of this presentation. Further solutions are posted on the MP page of the CAMPPP gains-camppp.wikispaces.com>>
MYRNA: IS THE ABOVE STATEMENT GOING TO BE TRUE OVER TIME?
You will be solving problems using the Mathematical Processes as a lens.
Pay attention to the questions both you and your partner(s) use as well as your thinking: “Am I reflecting back on this? Am I connecting it with some other concept or idea in the real world?”
Explain the task:
Choose on of the two problems that follow and ask the question prompts from the Mathematical Processes package.
You have already taken a detailed look at your Expert Process - depending on how you decide to approach the problem, you could branch out into the other Mathematical Processes you were examining,.
Make note of any Mathematical Process used in your solution strategy.
<Some solutions can be found as hidden slides at the end of this presentation.>
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9. Problem Solving with the Mathematical Processes in Mind
Deck Problem
DECK = І
You have been hired to build a deck attached the second floor of a cottage using 48 prefabricated 1m x 1m sections. Determine the dimensions of at least 2 decks that can be built in the configuration shown.
Will different decks require the same amount of railing? Explain.
COTTAGE
Read the Deck Problem aloud.
Pose the question: How will the shape and area of the deck impact the perimeter, or partial perimeter of railing?
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10. Problem Solving with the Mathematical Processes in Mind
Trapezoid Problem
Three employees are hired to tar a rectangular parking lot of dimensions 20 m by 30 m. The first employee tars one piece and leaves the remaining shape, shown below, for the other 2 employees to tar equal shares.
Show how they can share the job. Justify your answer.
Read the Trapezoid Problem aloud and ask if there are any questions before starting.
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11. Problem Solving with the Mathematical Processes in Mind
Revisit the problem.
Solve the problem in two more different ways:
- ask questions using the Mathematical Process package prompts
- note when a Mathematical Process is being used.
Read instructions on the slide.
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12. Graphical Representation
Deck Problem: Multiple Strategies
Graphical Representation
Numerical Representation
Short Edge
Long Edge 1 2 3 4 6 8
24.5 13
9.5 7
Concrete Representation
This slide illustrates multiple solutions to the deck problem. You might use it as part of your group discussion once the participants complete the activity.
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Algebraic Representation
2xy – x2 = 48
Cottage

13. Deck Problem: Tiles Even Number of Tiles Remaining Perfect Square
Cottage
48 – 12 = 47
48 – 22 = 44
48 – 32 = 37
48 – 42 = 32
48 – 52 = 23
48 – 62 = 12
<<This slide illustrates multiple solutions to the deck problem. You might use it as part of your group discussion once the participants complete the activity.>>

14. Problem Solving Across the Grades
A1= 120 m2
This series of slides illustrates solutions to the trapezoid problem. You might use it as part of your group discussion once the participants complete the activity.
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A = 180 m2
A = 240 m2
A2 = 60 m2

15. Problem Solving Across the Grades
A = 180 m2
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A1= 120 m2
A = 240 m2
A = 180 m2
A2= 60 m2
x = 12 m
x = 6 m

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17. H <<Click to next slide>> 12 cm

18. x + y = 30 A1 = A2 . . . y = x + 6 <<Click to next slide>>

19. Problem Solving Across the Grades
y = x + 6
( 12, 18)
x = and y = 18
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y = 30 - x

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21. Problem Solving Across the Grades3 3
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15 m
15 m

22. Problem Solving Across the Grades
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18 cm

23. Problem Solving Across the Grades
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24. Problem Solving Across the Grades
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25. Discuss
How did solving this problem in more than one way encourage and promote the use of different Mathematical Processes?
Poll the group.
Raise your hand if you think:
You used the Mathematics Processes Reasoning and Proving fairly well by doing the problem that you did.
You did some Reflecting.
You had to choose to use some Tools and/or Computational Strategies.
Did some Representing.
Did some Connecting.
In the classroom, when students are exposed to rich problems, problem solving naturally includes Mathematical Processes.
You have to be specific and explicit and may have to ask leading questions to encourage using a Mathematical Process. By asking students to problem solve in another way you can elicit additional Mathematical Processes.
Honour all approaches as valid. When students solve problems with each other and use mathematical language they develop confidence and comfort with using mathematical terms.
We want every student to feel like what they do is important. Whether the answer is right or wrong, we all learn from it.
MYRNA: DO YOU WANT TO LEAVE THE ABOVE BOLDED STATEMENTS IN – the sense is in the previous statement?
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26. Home Activity Reflection Journal:
Write about the interconnectivity of the Mathematical Processes and problem solving.
Investigate other ways to solve the problem you were given.
Remind the participants of the date and location of the next session.
Assign the Home Activity.