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Course Orientation
Institutional Mandate, Vision & Mission
ILOILO STATE UNIVERSITY OF FISHERIES SCIENCE AND TECHNOLOGY
COLLEGE OF EDUCATION
To provide advanced education, higher technological, professional instruction and training in fisheries technology, arts and sciences, education, industrial technology, engineering, aquaculture, seaweed farming and other fields of study and as may be relevant to national development. It shall undertake research, extension services and production activities in support of leadership in its areas of specialization.
A leading and empowering research university in fisheries, agriculture, education, maritime studies, arts, sciences, and technology in Southeast Asia by 2030.
To produce globally competitive and empowered graduates in fisheries, agriculture, education, maritime studies, arts, sciences, and technology, and to capacitate individuals and communities towards sufficiency for nation-building.
- I - Integrity
- S - Social Justice
- D - Discipline
- Academic Excellence
To develop competent teachers imbued with positive values and rich ideals of Filipino life and culture that are responsive to a sound educational technology that could help hasten the social, economic, and ecological development of the nation, especially in Western Visayas.
- Produce quality graduates who will contribute to the advancement of their chosen field
- Conduct viable research, develop and disseminate technologies, and provide technical assistance to the community for increased production
- Offer curricular programs that are responsive to the needs of the community and industries for national development
- Promote self-employment and entrepreneurship
- Strengthen opportunities for scholarships and training among students, faculty, and staff
- Strengthen and develop linkages with other agencies or institutions to achieve the goals and objectives both in the national and international levels
- Improve facilities and structures that will efficiently and effectively carry out quality instruction, research and development, extension, and production
Course Information
Course Details
| Field | Detail |
|---|---|
| Course Title | Problem Solving, Mathematical Investigation and Modeling |
| Course Number | MAT 3113 |
| Credit | 3 units |
| Program | BSEd Mathematics |
Course Description: This course aims to enhance students' knowledge and skills in dealing with real-life and non-routine applications of mathematics. Students will have the opportunity to explore the use of problem-solving strategies or heuristics as they engage in mathematical investigations, formulate and justify conjectures, make generalizations, and communicate mathematical ideas.
- Demonstrate comprehensive understanding of problem-solving processes and strategies
- Apply various problem-solving heuristics to diverse mathematical situations
- Design and implement mathematical investigations for diverse learners
- Create and apply mathematical models to real-world situations
- Conduct and present mathematical investigations following proper methodology
- Unit 0: Course Orientation
- Unit 1: Problem Solving and Mathematics Education
- Unit 2: Problem Solving Heuristics
- Unit 3: Concepts on Mathematical Investigation and Modelling
- Unit 4: Mathematical Investigation/Modeling
Grading System
Grade Components — Lecture Courses
| Component | Percentage |
|---|---|
| Midterm/Final Examination | 40% |
| Quizzes/Activities/Worksheets/Tasks | 30% |
| Outputs/Projects | 20% |
| Oral Participation/Oral Presentation | 10% |
| Total | 100% |
Final Grade = (Midterm Grade + Tentative Final Grade) / 2
Problem Solving and Mathematics Education
1.1 Problem Solving: Definition and Process
Welcome, future math educators! This course is a departure from traditional, procedure-heavy mathematics. Here, we will not be given a problem and shown the method to solve it. Instead, we will become mathematicians — posing our own questions, exploring patterns, making conjectures, and building arguments.
- Articulate a definition of problem solving
- Differentiate between routine and non-routine problems
- Outline the steps in problem solving from personal practice vis-a-vis those outlined by Polya and others
- Explain the significance of problem-solving in mathematics education
What is Mathematical Investigation?
Routine Problem-Solving: "Solve for x: 2x + 5 = 11." (One path, one correct answer).
Mathematical Investigation: "Explore the patterns and properties of the sequence generated by 2x + 5 for different integer values of x." (Multiple paths, multiple discoveries).
Definition: A mathematical investigation is an open-ended, systematic exploration of a mathematical situation. It is a process-oriented activity where the focus is on the journey of discovery: asking "what if?" questions, testing hypotheses, and formulating generalizations.
Problem solving is the process of applying mathematical knowledge, skills, and reasoning to find solutions to unfamiliar or non-routine situations where the solution path is not immediately apparent.
- Fosters Critical Thinking — moves students beyond memorization
- Develops Perseverance — encourages learning from "productive struggle"
- Models Authentic Mathematics — shows students what mathematicians actually do
- Promotes a Growth Mindset — emphasizes process over the "right answer"
Polya's Four-Step Process
Polya's Four-Step Process is a systematic method for solving problems, developed by the renowned mathematician George Polya in his 1945 classic book, "How to Solve It." It is not a rigid formula but a flexible cycle of thinking that guides a problem-solver from confusion to solution.
The four steps are:
- Understand the Problem
- Devise a Plan
- Carry Out the Plan
- Look Back
A farmer has chickens and pigs. There are 50 heads and 140 legs. How many chickens and pigs does the farmer have?
Step 1: Understand the Problem
This is the most critical step. You cannot solve a problem you don't understand.
Identify the Unknowns: We need to find the number of chickens and the number of pigs.
Identify the Knowns: Total heads = 50 (total animals), Total legs = 140, Chicken = 2 legs, Pig = 4 legs.
Restate: "Two types of animals. Together 50 heads and 140 legs. How many of each?"
Constraints: The number must be whole numbers (no half animals), cannot be negative.
Step 2: Devise a Plan
Choose a strategy or "heuristic" to connect the known information to the unknowns. Consider: Guess and Check, Make a Table, Draw a Diagram, Use a Model, Write an Equation.
Choosing a Plan: Let's choose the "Make a Table" strategy.
Step 3: Carry Out the Plan
Start by guessing 25 chickens (which means 25 pigs, since there are 50 heads).
Systematic Table Approach
| Chickens | Pigs | Chicken Legs | Pig Legs | Total Legs | Check |
|---|---|---|---|---|---|
| 25 | 25 | 50 | 100 | 150 | Too Many |
| 30 | 20 | 60 | 80 | 140 | Correct! |
We found the solution! 30 chickens and 20 pigs. Heads: 30+20=50 ✓, Legs: (30×2)+(20×4)=60+80=140 ✓
Step 4: Look Back
This step is about reflection and extension. Verify the answer is correct, then consider alternative solutions.
Algebraic Approach: Let c = number of chickens, p = number of pigs.
Equation 1 (Heads): c + p = 50
Equation 2 (Legs): 2c + 4p = 140
Solve: p = 50 − c → 2c + 4(50 − c) = 140 → 2c + 200 − 4c = 140 → −2c = −60 → c = 30, p = 20
1.2 Problem Solving and Mathematics Education
What is a "Problem" in Mathematics?
Exercise: A routine task where the method of solution is immediately known. Example: Solve 2x + 5 = 15.
Problem: A task where the solution method is not immediately apparent to the solver. Example: A farmer has chickens and pigs. There are 50 heads and 140 legs. How many of each?
- Develops critical thinking and reasoning skills
- Promotes deeper conceptual understanding
- Prepares students for real-world applications
- Fosters mathematical creativity and perseverance
Problem-solving is the heart of mathematics (NCTM Standards, 1989).
Sample Classroom Activity: "The Handshake Problem" — If 10 people each shake hands with everyone else, how many handshakes occur? This encourages multiple solution strategies (drawing, tables, patterns).
1.3 Problem Solving and the Conceptual Framework of the K to 12 Mathematics Curriculum
Integration in K to 12 Curriculum
The K to 12 Mathematics Curriculum views mathematics as:
- A vehicle for developing critical thinking and problem-solving
- A tool for understanding and interacting with the world
- A language for expressing patterns and relationships
- Content Standards: Understand mathematical concepts, apply mathematics to problem-solving, communicate mathematical thinking
- Performance Standards: Formulate and solve problems, use appropriate strategies and tools, justify and explain solutions
Spiral Progression Approach
Problem-solving skills are developed across grade levels:
- Grades 1-3: Basic problem solving with whole numbers
- Grades 4-6: More complex problems with fractions, decimals, basic geometry
- Grades 7-10: Algebraic thinking and more abstract problems
- Senior High: Applied problems in specialized contexts
Sample Integration in K to 12
Grade 7 Example Competency: Solves problems involving algebraic expressions and linear equations.
Sample Problem: Maria is 5 years older than Jose. The sum of their ages is 31. Find their ages.
Teaching Approach: Use bar models to represent the problem. Develop algebraic equations: Let x = Jose's age, then x + (x+5) = 31 → 2x + 5 = 31 → 2x = 26 → x = 13 (Jose), x+5 = 18 (Maria).
1.4 Reflections on Research Studies on Implementing Problem Solving
Findings from Research
- Students in problem-centered classrooms show better conceptual understanding and retention
- Teachers need extensive professional development in problem-solving pedagogy
- Cultural factors influence implementation success
- A 2019 study showed Filipino students improved problem-solving performance when teachers used local context problems (e.g., market situations, transportation scenarios)
- Start Small: Begin with one problem-solving lesson per week. Use 'low-floor, high-ceiling' problems accessible to all students
- Focus on Process: Assess and give credit for reasoning, not just correct answers. Celebrate multiple solution strategies
- Create a Supportive Environment: Establish classroom norms that value risk-taking. Encourage collaboration and discussion
- Problem-solving is a skill that can be taught. It is not just for "smart" students
- The process is as important as the answer. Focus on how students think, not just what they produce
- Your role is to facilitate thinking. Ask good questions and create a safe environment for learners to explore
- Start where your students are. Use appropriate problems and provide necessary scaffolding
- Be a problem-solver yourself. The best way to teach problem-solving is to model it
Chapter 1 Worksheets
Required Format (40 points total)
- Step 1: Understand the Problem (10 pts) — Restate in your own words, identify known/unknown, list constraints
- Step 2: Devise a Plan (10 pts) — List at least two strategies, explain choice, outline approach
- Step 3: Carry Out the Plan (10 pts) — Show all work clearly, explain each step, present final answer
- Step 4: Look Back (10 pts) — Verify solution, suggest alternative method, propose extension
Chapter 1 Quiz
Chapter 1 Quiz
1. According to Polya's problem-solving process, what is the first step?
2. Which of the following best describes a non-routine problem?
3. In the chickens and pigs problem, what strategy involved organizing guesses systematically?
4. What is the main difference between an exercise and a problem in mathematics?
5. Which step in Polya's process involves verifying the solution and considering alternatives?
6. According to research, what is a major barrier to implementing problem-solving approaches?
7. What does the K to 12 Mathematics Curriculum view as the primary role of mathematics?
8. In the 'Look Back' step, what should students do?
9. Which problem-solving strategy involves starting from the desired result?
10. What is a 'low-floor, high-ceiling' problem?
11. According to research, students in problem-centered classrooms typically show:
12. Which of the following is an example of an exercise?
13. What is the primary role of the teacher in a problem-solving classroom?
14. In the spiral progression approach of K to 12, when are algebraic thinking problems introduced?
15. Which strategy was used in the algebraic approach to the chickens and pigs problem?
16. What cultural factor was mentioned as affecting problem-solving implementation in the Philippines?
17. Which of the following promotes a growth mindset in mathematics?
18. What does 'productive struggle' refer to in problem-solving?
19. In the handshake problem with 10 people, what mathematical concept is primarily developed?
20. Which research finding supports the use of local context problems?
21. What is the key question in the 'Understand the Problem' step?
22. Which characteristic is emphasized in mathematical investigation but not in routine problem-solving?
23. What should teachers do to address implementation challenges according to research?
24. In the age problem (Maria and Jose), what strategy helps visualize the relationship?
25. What is the main benefit of the 'Look Back' step?
26. According to K to 12 performance standards, what should students be able to do?
27. Which problem-solving strategy involves testing specific cases to find patterns?
28. What does research say about traditional vs. problem-based teaching?
29. What is essential for creating a supportive problem-solving environment?
30. Why is it important for teachers to be problem-solvers themselves?
Problem Solving Heuristics
Introduction to Heuristics
- Identify methods appropriate for solving a problem
- Demonstrate how strategies can be applied to problems in real life
- Solve problems by applying appropriate problem-solving techniques
- Solve problems correctly in at least two ways
- Appreciate the relevance of problem-solving in real life through active participation
What are Heuristics?
Heuristics are general strategies or "rules of thumb" for solving problems. They are not guaranteed algorithms but rather flexible approaches that often lead to solutions. Heuristics are like tools in a toolbox — you select the right tool for the specific job.
- Different problems require different approaches
- Students have diverse thinking styles
- Real-world problems rarely have single solution paths
- Develops mathematical flexibility and creativity
To be effective investigators, we need a toolkit of strategies:
- Looking for Patterns: Numerical, geometric, visual
- Organizing Data: Using tables, charts, lists, diagrams
- Creating Representations: Drawing pictures, graphs, manipulatives
- Working Backwards: Starting from a desired result
- Considering Special Cases: Testing with simple or extreme examples
- Generalizing: Moving from specific observations to a general rule
- Conjecturing: Stating an "I think that..." proposition clearly
2.1 Algebraic Approach
Using algebraic symbols and equations to represent and solve problems.
When to use: Problems with unknown quantities, situations with clear relationships between variables, when precise numerical answers are needed.
A number plus twice that number is 45. Find the number.
Let x = the number
Equation: x + 2x = 45
3x = 45
x = 15
Find three consecutive integers whose sum is 87.
Let x = first integer, x+1 = second, x+2 = third
Equation: x + (x+1) + (x+2) = 87
3x + 3 = 87 → 3x = 84 → x = 28
Answer: 28, 29, 30
2.2 Model Approach
Using visual representations like bar models, diagrams, or physical manipulatives.
When to use: Problems with part-whole relationships, when students struggle with abstract thinking, ratio and proportion problems.
John has 3 times as many marbles as Peter. Together they have 48 marbles. How many does each have?
Bar Model: John = 3 units, Peter = 1 unit
Total: 4 units = 48 marbles
1 unit = 12 marbles (Peter)
3 units = 36 marbles (John)
2.3 Make a Table
Organizing data systematically in a table format to identify patterns.
When to use: Problems with multiple variables, looking for patterns, when trial and error is needed.
How many different outfits can you make with 3 shirts and 2 pants?
Outfit Combinations
| Shirt | Pant 1 | Pant 2 |
|---|---|---|
| Red | Outfit 1 | Outfit 2 |
| Blue | Outfit 3 | Outfit 4 |
| Green | Outfit 5 | Outfit 6 |
Answer: 6 different outfits
2.4 Listing
Systematically listing all possibilities to ensure completeness.
When to use: Combinatorial problems, when order or arrangement matters, probability problems.
List all 2-digit numbers you can make with digits 2, 4, 6 without repetition.
Solution: 24, 26, 42, 46, 62, 64
2.5 Looking for Patterns
Identifying and extending numerical or geometric patterns.
When to use: Sequence problems, problems with repeated operations, when a rule or formula can be generalized.
What is the 10th term in the sequence: 2, 5, 8, 11, ...?
Pattern Recognition: Terms increase by 3 each time
nth term = 3n − 1
10th term = 3(10) − 1 = 29
2.6 Guess and Check
Making educated guesses and refining based on results.
When to use: When other methods seem too complex, problems with limited possible answers, to build intuition before formalizing.
I'm thinking of two numbers. Their product is 48, and their difference is 2. What are the numbers?
Guessing Process: Try 8 and 6: product 48 ✓, difference 2 ✓
Answer: 8 and 6
2.7 Working Backwards
Starting from the result and reversing the operations.
When to use: Problems with a known end point, when operations are reversible, age problems.
I did some shopping. I spent half of my money, then ₱100 more. I have ₱200 left. How much did I start with?
Before last spending: ₱200 + ₱100 = ₱300
This was half of original: Original = ₱300 × 2 = ₱600
2.8 Geometric Approach
Using geometric principles, diagrams, or spatial reasoning.
When to use: Problems involving shapes, space, or measurement, optimization problems, when visualization helps understanding.
What's the area of a walkway 2m wide around a rectangular pool 20m × 15m?
Outer rectangle: 24m × 19m (adding 2m each side)
Area = (24×19) − (20×15) = 456 − 300 = 156 m²
2.9 Using Softwares/Applications
Leveraging technology to solve complex problems.
When to use: Large datasets, complex calculations, visualization needs, real-world modeling.
- GeoGebra — for geometric investigations
- Desmos — for graphing and equations
- Spreadsheets — for data analysis
- Python/R — for statistical modeling
Implementing Heuristics in the Classroom
- Understand the problem: What is being asked?
- Analyze the information: What data is given?
- Consider constraints: Are there limitations?
- Choose strategy: Which heuristic fits best?
- Evaluate: Does this approach make sense?
Teaching Multiple Approaches: The Handshake Problem
If 10 people each shake hands with everyone else, how many handshakes occur?
Approach 1 - Systematic Counting: 9 + 8 + 7 + ... + 1 = 45
Approach 2 - Combinatorial Reasoning: C(10,2) = 10!/(2!8!) = 45
Approach 3 - Geometric Model: Represent as complete graph with 10 vertices
- Algebraic: The sum of three consecutive even numbers is 78. Find the numbers. (24, 26, 28)
- Model: A recipe calls for 2 cups flour to 1 cup sugar. For 6 cups flour, how much sugar? (3 cups)
- Pattern: What is the sum of the first 50 odd numbers? (2500 = 50²)
- Working Backwards: I doubled my money, spent ₱100, and have ₱300 left. How much did I start with? (₱200)
Chapter 2 Worksheet: Case Studies
Chapter 2 Quiz
Chapter 2 Quiz
1. What are heuristics in problem-solving?
2. Which approach is MOST appropriate for finding all possible combinations of 4 ice cream flavors chosen 2 at a time?
3. In the sequence 2, 5, 8, 11, ..., what pattern is being used?
4. A student has ₱500. She spends half on books, then ₱100 on supplies, and has ₱150 left. Which heuristic is BEST?
5. Which problem is BEST solved using the model approach with bar models?
6. For 'Find three consecutive integers whose sum is 87,' letting x, x+1, x+2 represent the numbers demonstrates which heuristic?
7. When should 'Guess and Check' typically be used?
8. What is the main advantage of teaching multiple problem-solving strategies?
9. In the handshake problem with 10 people, which approach uses C(10,2)?
10. Which technology tool is MOST appropriate for investigating geometric transformations?
11. For 'How many rectangles of different sizes can be found in a 3x3 grid?', which heuristic is MOST appropriate?
12. What is the FIRST step in the decision framework for selecting a strategy?
13. Which real-world situation is BEST solved using 'Make a Table'?
14. In the marbles problem (John has 3x as many as Peter, total 48), what does one 'unit' represent?
15. Which characteristic makes heuristics different from algorithms?
16. For 'What is the sum of the first 100 odd numbers?', which heuristic helps discover the pattern?
17. Which problem is LEAST appropriate for the algebraic approach?
18. What does the 'Look Back' step primarily involve?
19. In the walkway problem (2m around a 20x15m pool), what is the outer rectangle's length?
20. Which software is BEST for creating dynamic graphs of equations?
21. What is the main purpose of 'Considering Special Cases'?
22. For 'If 5 workers complete a job in 12 days, how long for 8 workers?', which heuristic is MOST appropriate?
23. Which problem-solving approach is most similar to the scientific method?
24. What is the key benefit of using the 'Model Approach' with visual representations?
25. In the outfits problem (3 shirts, 2 pants), what mathematical operation is fundamentally used?
26. Which heuristic is MOST useful for solving a mystery or puzzle?
27. What does 'systematic' mean in the Listing heuristic?
28. Which real-world application BEST uses the Geometric Approach?
29. Why is it important for students to solve problems in multiple ways?
30. In the decision framework, what should you do AFTER choosing a strategy?
Concepts on Mathematical Investigation and Modelling
Learning Objectives
- Pose problems based on a given situation
- Understand the stages or parts of a Mathematical Investigation
3.1 Introduction to Mathematical Investigation and Modelling
A mathematical investigation begins with a situation that must be understood or a set of data that must be organized and explained in mathematical terms. It is essential to run the risk of proposing conjectures. Testing these conjectures and collecting more data may support them or lead to new conjectures.
A mathematical investigation allows students to examine situations using various techniques and, in the process of their exploration, develop skills that can be applied to other problems.
Mathematical Investigation refers to a systematic exploration of mathematical situations to discover patterns, relationships, and properties.
Mathematical Modeling refers to the process of using mathematics to represent, analyze, and solve real-world problems.
Mathematical Investigation vs. Mathematical Modelling
| Aspect | Mathematical Investigation | Mathematical Modelling |
|---|---|---|
| Focus | Focuses on pure mathematics | Connects math to the real world |
| Purpose | Explores patterns and relationships | Solves practical problems |
| Nature | Often open-ended | Usually has a specific application |
| Orientation | Process-oriented | Solution-oriented |
Example: Investigate the relationship between the number of sides of a regular polygon and the number of diagonals.
- Experience methods of planning, organizing, analyzing, and evaluating data
- Freely choose what aspects of the problem situation they would like to pursue and what strategies they would use
- Apply appropriate mathematics or discover a mathematical relationship
- Develop questions, approaches, and results that are, at least for them, original products
- Use the same general methods used by research mathematicians — cycles of data gathering, visualization, abstraction, conjecturing, and proof
- Communicate mathematically by describing their thinking, writing definitions and conjectures, using symbols, justifying conclusions, and reading mathematics
- Formulate their own questions from a given situation — giving teachers a clear indication of their level of knowledge and understanding
Possible Approaches for the Polygon Diagonal Investigation:
- Draw polygons and count diagonals
- Look for patterns in the numbers
- Develop a general formula
- Test and verify the formula
3.2 Stages in Conducting Mathematical Investigation
The Seven-Stage Investigation Process
First decide who will be doing the investigation — individual, group, or class. What will be investigated?
Choosing an Investigation Topic:
- Personal interest and curiosity
- Mathematical significance
- Accessibility of resources
- Potential for discovery
Use mind mapping, brainstorming, lateral thinking, 5 Es (Engage, Explore, Explain, Elaborate, Evaluate). Begin a log or journal. Generate ideas for the topic and related mathematical content. Select a topic and outline the investigation content. Develop a timeline.
Examples: What happens when you add consecutive odd numbers? Investigate patterns in Pascal's Triangle. Explore the Fibonacci sequence in nature.
Start with the simplest possible cases and build up systematically to more complex cases. Note every observation and record everything.
- Systematic listing/drawing
- Organizing relationships in tables or graphs
- Look for similarities, differences, or connections between cases
Example (Polygon Diagonals):
Triangle: 0 diagonals
Quadrilateral: 2 diagonals
Pentagon: 5 diagonals
Hexagon: 9 diagonals
Heptagon: 14 diagonals
Exploring systematically should permit some patterns or relationships to begin to emerge. Make general statements about patterns or relationships observed in the cases considered.
Forming Tentative Generalizations: Based on observed patterns, clear and testable statements, mathematically precise.
Example Conjecture: For an n-sided polygon, the number of diagonals is n(n−3)/2
Check the consistency of conjectures using existing cases. If the data for one or more instances do not agree with the conjecture, then it is false (a counterexample) and must be either rejected or revised.
Testing the Diagonal Formula:
Octagon: 8(8−3)/2 = 20 diagonals ✓
Nonagon: 9(9−3)/2 = 27 diagonals ✓
Explain why the conjectures will work for a few or all cases. Prove the conjectures (by mathematical induction, direct/indirect proof, visual proof). See the connection among conjectures. Get a deeper understanding of the investigation as you work longer on it. Gestation may occur.
Justifying the Diagonal Formula: Each vertex connects to n−3 others (excluding itself and neighbors). n vertices × (n−3) connections. Divide by 2 to avoid double-counting: n(n−3)/2.
Present findings through clear written reports, visual representations, oral presentations, and proper mathematical notation.
Pose new questions: What if conditions change? Can this be generalized further? Related investigations?
Extensions for Diagonal Investigation: "What about 3D polyhedra?" "Investigate intersecting diagonals" "Explore relationships between diagonals and triangles formed"
3.3 Mathematical Problem Posing
The art of creating new mathematical problems or reformulating existing ones.
Change the conditions of existing problems.
Original Problem: Find the area of a rectangle with length 8 and width 5.
What-if-not Questions: What if it's not a rectangle but a parallelogram? What if we fix the perimeter instead of dimensions? What if we want maximum area with fixed perimeter?
Make problems broader or more specific.
Original: Solve x² − 5x + 6 = 0
Generalize: For what values of k does x² − 5x + k = 0 have real roots?
Specialize: Find a quadratic with roots 2 and 3.
Start from solution and work backward.
Original: Prove the Pythagorean theorem.
Reverse: Given a² + b² = c², what can you say about the triangle?
Express the same concept differently.
Algebraic → Geometric: Show geometrically why (a+b)² = a² + 2ab + b²
Numerical → Graphical: Graph the relationship between roots and coefficients.
Mathematical Modelling Process — The Modeling Cycle
Step 1: Understand the Real-World Problem
Step 2: Formulate the Mathematical Model
Step 3: Solve the Mathematical Problem
Step 4: Interpret the Results
Step 5: Validate the Model
Real Problem: Predict the population of a city in 10 years.
Mathematical Model: Assume exponential growth: P(t) = P₀e^(rt). P₀ = current population, r = growth rate, t = time in years.
Solution and Interpretation: Calculate P(10) using the formula. Consider factors affecting accuracy. Suggest model improvements.
Investigation Projects
Your Turn: Choose ONE of the following to investigate over the next two weeks. You will submit a written report and give a short presentation.
Scenario: A school has 1000 lockers, all closed. 1000 students walk by. The first student opens every locker. The second student closes every second locker. The third student changes the state of every third locker. This continues until the 1000th student.
Investigate: Which lockers are open at the end? Why? What is the pattern? Can you generalize for 'n' lockers and 'n' students?
Scenario: A palindromic number reads the same forwards and backwards (e.g., 121, 1331, 404).
Investigate: Take any number. Reverse its digits and add it to the original. Repeat the process with the sum. Does this always eventually produce a palindrome? (e.g., 57 → 57+75=132 → 132+231=363, a palindrome). Are there numbers that don't seem to work? (These are called Lychrel numbers).
Scenario: You have 24 one-meter fencing segments to make a rectangular pen.
Investigate: What different rectangles can you make? How does the area change as the dimensions change? What shape gives the maximum area? Can you generalize for 'p' segments of fencing?
Chapter 3 Worksheets
Chapter 3 Quiz
Chapter 3 Quiz
1. What is the main focus of mathematical investigation?
2. Which stage involves forming tentative generalizations based on observed patterns?
3. The problem posing strategy that involves changing conditions of existing problems is called:
4. In the polygon diagonals investigation, what was the formula for an n-sided polygon?
5. Which is a key characteristic of mathematical modelling?
6. During the 'Testing' stage, what should students primarily do?
7. Changing 'Solve x²-5x+6=0' to 'For what k does x²-5x+k=0 have real roots?' demonstrates which strategy?
8. What is the final step in the mathematical modelling cycle?
9. In the exploration stage of polygon diagonals, how many diagonals does a hexagon have?
10. Which stage involves presenting findings clearly through reports or presentations?
11. Starting from a solution and working backward is called:
12. What does the 'Extending' stage involve?
13. Which is mathematical modelling rather than pure investigation?
14. The justification for the diagonal formula n(n-3)/2 involved:
15. Which strategy changes an algebraic problem to a geometric one?
16. What is the primary purpose of the 'Starting Point' stage?
17. In P(t) = P₀e^(rt), what does r represent?
18. Which stage involves developing logical arguments and proofs?
19. 'What about 3D polyhedra?' in the diagonal investigation represents which stage?
20. Which characteristic best describes mathematical investigation?
21. When testing the diagonal formula for a nonagon (9 sides), what result confirms the conjecture?
22. 'What-if-not' applied to rectangle area might ask:
23. Which is an example of mathematical investigation?
24. In the modelling cycle, which step comes after 'Formulate the Model'?
25. What is the key difference between investigation and modelling?
26. Which stage involves gathering empirical evidence and looking for patterns?
27. The reverse thinking strategy might transform 'Prove the Pythagorean theorem' into:
28. What makes a good mathematical investigation topic?
29. In the population growth model, what does P₀ represent?
30. The main purpose of mathematical problem posing is to:
Mathematical Investigation/Modeling
Learning Objectives
- Participate actively and collaboratively in the initial conduct of a mathematical investigation
- Pose problems related to the situation identified by the group
- Participate actively and collaboratively in formulating conjectures
- Formulate conjectures gathered from gathered data
- Participate actively and collaboratively in verifying conjectures
- Verify conjectures using an appropriate verification method
- Participate actively and collaboratively in justifying conjectures
- Prove conjectures using an appropriate method of proving
- Participate actively and collaboratively in formulating the summary and possible extensions of their MI paper
- Participate actively in the MI Congress
- Diverse Perspectives — different group members see different aspects
- Cognitive Synergy — group thinking that surpasses individual capabilities
- Social Learning — learning mathematical communication and argumentation skills
- Real-World Preparation — mirrors how mathematicians actually work
4.1 Getting Started
Choosing Investigation Topics:
- Personal interest and curiosity
- Availability of resources
- Mathematical significance
- Real-world relevance
Example Starter: Investigate the Fibonacci sequence in nature.
4.2 Close vs Open-ended Problems
Close-ended: Single correct answer.
Example: "Solve 2x + 5 = 15"
Open-ended: Multiple approaches and solutions.
Example: "Find as many ways as possible to make 50 using operations and digits 2, 3, 4"
4.3 Problem Identification
Formulating Clear Research Questions:
- Specific and focused
- Mathematically meaningful
- Investigable with available resources
Example: Instead of "Study triangles," use "Investigate the relationship between triangle side lengths and area."
4.4 Organizing Data
Systematic Data Collection: Tables and charts, systematic experimentation, careful measurement and recording.
Example: Investigating Rectangle Areas with Fixed Perimeter (P = 20)
| Length | Width | Perimeter | Area |
|---|---|---|---|
| 1 | 9 | 20 | 9 |
| 2 | 8 | 20 | 16 |
| 3 | 7 | 20 | 21 |
| 4 | 6 | 20 | 24 |
| 5 | 5 | 20 | 25 |
4.5 Formulating Conjectures
Developing Testable Hypotheses: Based on observed patterns, clear and specific statements, mathematically precise.
Example: For rectangles with fixed perimeter, the area is maximized when the rectangle is a square.
4.6 Verifying Conjectures
Testing with Multiple Cases: Try various examples, look for counterexamples, consider edge cases.
Example: Test the rectangle conjecture with perimeters 12, 16, 20, etc.
4.7 Justifying Conjecture
Developing Mathematical Proofs: Logical reasoning, algebraic justification, geometric arguments.
Rectangle Justification: Let perimeter P = 2(l + w), so w = P/2 − l. Area A = l × w = l(P/2 − l) = (P/2)l − l². This is a quadratic, maximum at l = P/4, so w = P/4 → square.
4.8 Writing the Summary and Possible Extensions
Structuring the Investigation Report:
- Introduction and problem statement
- Methodology and data collection
- Findings and conjectures
- Justification and proofs
- Conclusions and reflections
- Extensions and new questions
Example Extensions: "What about 3D shapes? What if we fix surface area instead of perimeter?"
4.9 Finalizing the Paper
- Title & Introduction: State your investigation question and its origins
- The Process: Detail your exploration. What strategies did you use? Include your tables, charts, and failed attempts — this is key!
- Findings & Conjectures: Clearly state the patterns you found and the conjectures you made
- Justification & Proof: Provide your reasoning for why your conjectures are true. This can be logical argument, visual proof, or algebraic proof
- Conclusions & Reflections: Summarize your main discovery. What was most challenging? What new questions arose?
- Extensions: Suggest one or two ways this investigation could be taken further
4.10 Mathematical Investigation/Modeling Congress
- Audience Analysis: Tailoring presentation for mathematical peers
- Visual Aid Design: Creating clear, engaging presentation materials
- Time Management: Practicing within time constraints (10-12 minutes)
- Q&A Preparation: Anticipating and preparing for questions
- Hook the audience with your initial question
- Briefly show your process of discovery
- Clearly state your main conjecture and your explanation for it
- Conclude with your most interesting reflection or extension
- Use visual aids (poster, slides, whiteboard) effectively
Situations for Mathematical Investigations
The following are 25 suggested investigation topics for students to explore:
- 1. Happy Numbers: Square each digit and add the squares repeatedly. If sequence reaches 1, the number is 'happy'; otherwise 'sad.' Investigate.
- 2. Lines: Lines are drawn on a plane. Investigate.
- 3. Ants: Ants emerge from their hole at the top of a wire grid and always walk downwards. Investigate the paths they may take.
- 4. Consecutive Sums: Some numbers can be expressed as the sum of consecutive positive integers. 9=2+3+4, 11=5+6, 18=3+4+5+6. Investigate.
- 5. Returns: A robot walks one pace and turns 90° right, two paces and turns 90° right, three paces and turns 90° right, repeating. Investigate.
- 6. Birthdays: On what day of the week will your 100th birthday fall? Investigate this and other days of interest.
- 7. Corners: A figure has two different types of corners. Investigate.
- 8. Triangles: Draw triangles with sides of integral length and longest side of length 5 units. Investigate.
- 9. Rectangles: How many rectangles are there in a diagram? Investigate.
- 10. Piles: Start with two unequal piles of coins. Shift enough coins from the larger pile to double the smaller. Continue. Investigate.
- 11. Sums of Squares: Some numbers can be written as sums of squares. Investigate.
- 12. Primes: Look at a table of prime numbers. How are primes spread through the counting numbers? Investigate.
- 13. Co-primes: Two positive integers are co-prime if their GCF is 1. The four positive integers less than 10 co-prime with 10 are 1,3,7,9. Investigate.
- 14. Tiles: Two small squares are printed on a square grid. Can the remainder be covered with 2×1 tiles? Investigate.
- 15. Straws: Squares are made from colored straws. Investigate.
- 16. Coins: Put coins all heads up on a table. Turning over three at a time, try to get all tails up. Investigate.
- 17. Strips: Make a strip of squares. Color some whole squares black, leaving others white. Investigate.
- 18. Repeats: Unit fractions have a numerator of 1. Investigate those whose decimal equivalents recur.
- 19. Joins: Each of three points can be joined to the other two without any joins crossing. Investigate for more than three points.
- 20. Chords: Mark points on a circle and join them by chords. Investigate.
- 21. Stairs: Consider stairs made up of n layers. Investigate.
- 22. Layers of Equilateral Triangles: Draw equilateral triangles with x layers. Investigate.
- 23. Handshakes: There are n people at a party and each shakes hands with others. Investigate.
- 24. Diagonals: Diagonals of regular polygons are connected. Investigate.
- 25. Billiard Balls: Make a billiard ball triangle of varying layers and varying ball radii. Investigate.
Chapter 4 Outputs
Activity 1: Final Mathematical Investigation Paper (25 points)
| Criteria | Excellent (full pts) | Proficient | Developing | Beginning |
|---|---|---|---|---|
| Introduction (3 pts) — Investigation question and motivation | Clear, compelling question with motivation | Clear question stated | Question is vague | No clear question |
| Methodology (4 pts) — Collaborative process, data collection, challenges | Detailed description, systematic methods | Adequate methods described | Methods somewhat described | Methods unclear |
| Findings and Analysis (8 pts) — Patterns, conjectures, comprehensive analysis | Insightful patterns, clear conjectures, thorough analysis | Good patterns and conjectures | Simple observations, weak analysis | No clear findings |
| Justification and Proof (6 pts) — Mathematical reasoning, verification approaches | Rigorous proof with multiple approaches | Adequate justification | Partial justification | No valid justification |
| Conclusion and Extensions (4 pts) — Summary, reflection, new questions | Deep reflection, insightful extensions | Good summary and extension | Superficial reflection | No meaningful reflection |
Activity 2: MI Congress Presentation (20 points)
| Criteria | Excellent (full pts) | Proficient | Developing | Beginning |
|---|---|---|---|---|
| Content and Mathematical Depth (8 pts) | Exceptional mathematical depth, comprehensive | Good mathematical content | Basic content covered | Minimal mathematical depth |
| Collaboration Demonstration (4 pts) | Equal participation, smooth transitions | Good coordination among members | Unequal participation | One person dominates |
| Presentation Skills (4 pts) | Engaging delivery, confident, well-paced | Clear presentation | Somewhat disorganized | Difficult to follow |
| Visual Aids and Organization (4 pts) | Professional, enhances understanding | Clear and relevant | Basic visuals | Poor quality, distracting |
Chapter 4 Quiz
Chapter 4 Quiz
1. What is the main benefit of collaboration in mathematical investigation?
2. When choosing an investigation topic, what should be the PRIMARY consideration?
3. Which is an example of an open-ended problem?
4. What characterizes a well-formulated research question?
5. In the rectangle area investigation with fixed perimeter, what pattern was discovered?
6. Which method was used to justify the rectangle area conjecture?
7. What is the purpose of the 'Extensions' section in an investigation report?
8. When verifying conjectures, what is MOST important?
9. What does 'cognitive synergy' in collaboration refer to?
10. Which is the BEST research question for investigation?
11. What is the key difference between close-ended and open-ended problems?
12. When organizing data systematically, what is the MAIN benefit of using tables?
13. What makes a conjecture mathematically precise?
14. In the MI Congress, what should the opening of a presentation do?
15. What is the purpose of including 'failed attempts' in the investigation report?
16. Why test the rectangle conjecture with different perimeters?
17. What is the MAIN goal of 'Conclusions and Reflections'?
18. In collaborative conjecture formulation, what should members do FIRST?
19. What does 'audience analysis' involve for the MI Congress?
20. Why consider 'edge cases' when verifying conjectures?
21. What is the primary value of 'real-world preparation' in collaborative investigation?
22. When finalizing the paper, what should be checked for proper mathematical notation?
23. What is the BEST approach for Q&A preparation for the MI Congress?
24. In rectangle area optimization, what concept explains why area is maximized as a square?
25. What does 'social learning' in collaborative investigation refer to?
26. After brainstorming questions, what should the group do?
27. What is the key element of active participation in collaborative work?
28. Why is visual aid design important for MI Congress presentations?
29. What should the Methodology section include?
30. What is the ultimate goal of the Mathematical Investigation Congress?