Critical Thinking and Problem Solving Strategies
MAT 1275 and MAT 1275 CO
Introduction
The critical thinking strategies and sample problems discussed here are based on the book chapter:
Rojas, E., and Benakli, N. (2020). Mathematical Literacy and Critical Thinking, In Teaching College-Level Disciplinary Literacy (pp. 197-226). Palgrave Macmillan, Cham. https://link.springer.com/book/10.1007/978-3-030-39804-0
Sample Questions for MAT1275
Solving Quadratic Equations by Factoring
- Sample question 1: Solve the equation by factoring:
\[3x^2-6=7x\]
- Sample Question 2: Solve the equation by factoring:
\[6-3x^2=-7x\]
Context:
- Equation
- Quadratic
Observations:
- Solve
- Factoring
- \(x^2\) and the \(x\) terms are on different sides of the equal sign
- The coefficient of \(x^2\) is not 1
- There is no common factor
- There are three terms
- There is only one quadratic term
Questions:
- Is it helpful that there is a common factor on the left side?
- What is a variable?
- What is a constant?
- What is a term?
- What is a factor?
- What is a coefficient?
- What is an equation?
- Does it matter that the variable is \(x\) or \(t\) or something else?
- What happens if I factor the left side?
- Can I divide the left side by 3? Would I need to also divide the right side by 3?
- What does “solving” mean?
- What does “solve by factoring” mean?
- What does this equation mean?
- How many solutions can I expect?
- Do I need to collect the terms on the left side? Or the right side?
- What is the form of the answer?
- Do I need to check the answer?
- Why might I need to solve a similar equation?
- What is the first step?
- What is the form of the last step?
- Are there other ways, besides factoring, to solve this equation?
- What are other ways, in general, for solving a quadratic equation?
- Why might I want to write this in ’standard form”?
- What is ‘standard form’?
- If there are other ways of solving this, why might solving by factoring be a more appropriate or better method?
- What is the main idea behind solving by factoring?
- What can I conclude about \(p\) and \(q\) if I know \(pq=0\)?
- How do I find the solutions after I factor?
- Can I always factor a quadratic?
Strategies:
- Decide which terms to add to each side of the equation.
- The first step is to write in standard form (add or subtract).
- Guess the factoring (check your guess) \((3x \pm ?)(x \pm ?)\).
- Let’s use the AC method/grouping (demonstrate what I can do if I can’t guess).
- If \(x\) satisfies the original equation then give the linear equations it must satisfy.
Concepts:
- Factoring
- Solving quadratic and linear equations
- The zero product property
- Checking a solution
- Evaluating a polynomial
- Creating equivalent equations
- GCF
- Grouping and factoring common factors
- Answers, solutions
Conclusion:
- If \(x\) satisfies the original equation then \(x\) satisfies \(x=a\) or \(x=b\).
- Check my answer from the original source to guard against error.
- Conclude using a complete sentence.
Solving Radical Equations
Solve the radical equation:
\[\sqrt{x - 5} - 2 = x - 13\]
Context:
- What does this mean?
- What does it mean to solve?
- What am I looking for here? What does the answer look like? Can I guess?
Observations:
- What kind of ‘things’ are in the equation?
- Where does the variable appear?
Questions:
- What experience do I have in solving equations similar to this?
- How is this different from/same as those similar equations?
- What is a square root?
- What are the properties of the square root?
- What are the challenges I anticipate in solving the equation?
Strategies:
- Can I make it look like one of those similar (simpler) problems?
- Check a few numbers to get a feel for the equation.
Concepts:
- How can I see that \(\sqrt{x-5}\) is not equal to \(\sqrt{x} - \sqrt{5}\)?
- Give an example showing that \(a^2+b^2\) is not equal to \((a+b)^2\).
- How do I “undo” a square root?
- What do I get when I square 3, and when I square \(-3\)? What do I make of that?
Conclusion:
- Do I have to check my answer?
- Does my answer make sense?
- Write a complete sentence to conclude.