General Problem Solving: Learn It 1

  • Extract relevant information from word problems and interpret mathematical notation in real-world contexts
  • Apply problem-solving strategies such as breaking down complex problems, using trial and error, pattern recognition, and logical reasoning
  • Utilize technology like graphing calculators, spreadsheets, and mathematical software to enhance problem-solving abilities
  • Evaluate the reasonableness of a claim and rewrite quantitative statements to improve clarity
  • Interpret data from graphs, charts, and tables to solve mathematical problems

In previous math courses, you’ve no doubt run into the infamous “word problems.” Unfortunately, these problems rarely resemble the type of problems we actually encounter in everyday life. In math books, you usually are told exactly which formula or procedure to use, and are given exactly the information you need to answer the question. In real life, problem solving requires identifying an appropriate formula or procedure, and determining what information you will need (and won’t need) to answer the question.

Strategies for Reading and Understanding Math Problems

Reading and understanding math problems is an essential skill for successful problem-solving. Let’s explore effective strategies to help you navigate and comprehend math problems with ease. By applying these strategies, you will develop the ability to extract key information, identify problem objectives, and confidently solve mathematical problems.

    1. Read Carefully: Begin by reading the problem statement attentively. Understand the context, identify the problem’s objective, and note any important details or constraints. Pay close attention to numerical values, units of measurement, and keywords that indicate mathematical operations.
    2. Identify the Question: Determine what the problem is asking you to find or solve. Look for phrases like “Find,” “Calculate,” or “Determine.” Understanding the question will guide your approach and help you focus on the relevant information.
    3. Highlight Key Information: Identify and highlight the essential details and quantities provided in the problem. This includes numerical values, units, and any other relevant data. Use labels or variables to represent unknowns or quantities that need to be calculated. Highlighting the key information allows you to focus on what is important information and ignore unimportant distracting information.
    4. Visualize the Problem: Create mental or written representations of the problem to aid your understanding. Draw diagrams, charts, or graphs if appropriate. Visualizing the problem can help you see the relationships between quantities and identify potential solution paths.
    5. Break It Down: Break the problem into smaller parts or steps. Analyze each part individually to understand its purpose and how it contributes to the overall solution. Identify the mathematical operations or concepts required for each step.
    6. Use Problem-Solving Strategies: Familiarize yourself with problem-solving strategies, such as working backward, making a table or chart, using logical reasoning, or applying relevant formulas or equations. Choose the strategy that best suits the problem at hand. We will explore problem-solving strategies more later in this section.
    7. Solve Step-by-Step: Once you have a plan in mind, solve the problem step-by-step. Show your work and perform the necessary calculations, ensuring accuracy and attention to detail.
    8. Check Your Answer: After finding a solution, verify if it makes sense in the given context. Re-read the problem, check your calculations, and assess whether the answer aligns with the question’s requirements. If possible, use estimation or alternative methods to confirm the reasonableness of your answer.
    9. Reflect on the Solution: Take a moment to reflect on the problem-solving process. Evaluate the effectiveness of your strategy and identify any insights or learnings that can be applied to similar problems in the future.

By employing these strategies, you can improve your ability to read and understand math problems, identify relevant information, and approach problem-solving with clarity and confidence. Regular practice with a variety of problems will enhance your skills and deepen your mathematical understanding.

Classifying the Types of Problems

In mathematics, different types of problems require distinct approaches and techniques to arrive at a solution. Are we trying to solve for an unknown variable? Maybe we need to simplify a complex expression. Or perhaps, we’re asked to calculate a certain quantity. Sometimes, we might even need to graph a function or navigate a multi-step problem with multiple operations and parts to consider. Let’s look at various problem classifications to help you identify the nature of a problem and apply the appropriate problem-solving strategies.

  1. Solve: In these problems, we’re typically trying to find the value of an unknown. For example, “solve for [latex]x[/latex] in the equation [latex]2x + 3 = 7[/latex]“. To solve it, we’ll need to isolate [latex]x[/latex] on one side of the equation.
  2. Simplify: In a simplification problem, we might be given a complicated expression, like [latex](3x^2)^2[/latex]. Here, our job is to simplify it to a more manageable form, in this case, [latex]9x^4[/latex].
  3. Calculate: Calculation problems ask us to compute a specific numerical value. “Calculate the area of a circle with radius [latex]3[/latex]” is an example. Here, we need to use the formula for the area of a circle ([latex]πr^2[/latex]) to get our answer.
  4. Graph: Graphing problems usually involve plotting a function or equation on a coordinate plane. For example, “Graph the function [latex]y = 2x - 1[/latex]“. We would find several values of [latex]y[/latex] for different [latex]x[/latex]-values and plot those points on the graph.
  5. Multi-step: These problems are a mix of the above types and involve multiple steps. They often require careful planning and understanding of the order of operations. For instance, “Solve for [latex]x[/latex] in the equation [latex]2x + 3 = 7[/latex], then calculate the value of [latex]y[/latex] in the equation [latex]y = 3x - 2[/latex]“.

It’s important to remember that these types aren’t mutually exclusive. A problem can have elements of more than one type. For example, a multi-step problem may require us to solve an equation, simplify an expression, and then calculate a final value. Remember, the key to conquering any math problem is understanding what it’s asking you to do. Once you’ve got that down, you’re halfway there!