9.1 Use a Problem Solving Strategy
3 min read•june 25, 2024
Problem-solving strategies are essential tools for tackling math challenges. These methods help break down complex problems into manageable steps, making them easier to solve. By following a systematic approach, you can confidently tackle a wide range of mathematical puzzles.
From understanding the problem to reviewing your solution, each step plays a crucial role. These strategies also teach you to translate words into numbers, identify key information, and use . Mastering these techniques will boost your problem-solving skills across various math topics.
Problem Solving Strategy
Systematic approach for word problems
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- Read problem carefully identifies given information (values, quantities, conditions)
- Determine unknown value or quantity to be found ()
- Identify constraints or conditions mentioned (limitations, restrictions)
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- Break down problem into smaller, manageable steps (simplify, organize, )
- Identify appropriate mathematical concepts or needed (, formulas)
- Consider different approaches selects most suitable one (, )
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- Carry out chosen approach step by step (systematic, organized)
- Perform necessary calculations or manipulations (, algebraic)
- Simplify expressions or solve equations as needed (reduce, isolate )
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- Verify solution makes sense in context of problem (reasonable, logical)
- Check for computational errors or inconsistencies (accuracy, precision)
- Ensure solution satisfies all given conditions or constraints (completeness, validity)
Step-by-step problem-solving strategy
- Identify the type of problem
- Recognize patterns or similarities with previously encountered problems (analogies, connections, )
- Classify problem based on characteristics (, ratio, )
- Extract relevant information
- Identify given values, quantities, or measurements (data, inputs)
- Note special conditions, constraints, or relationships mentioned (assumptions, dependencies)
- Represent the problem mathematically
- Assign variables to unknown quantities (symbols, letters)
- Express relationships between quantities using mathematical symbols or equations (formulas, expressions)
- Solve the mathematical representation
- Apply appropriate formulas, algorithms, or problem-solving techniques (, )
- Perform calculations or manipulations to find solution (simplify, evaluate)
- Interpret the solution
- Relate mathematical solution back to original problem context (meaning, significance)
- Express answer in appropriate units or format (labels, precision)
- Verify solution satisfies given conditions and makes logical sense (reasonableness, consistency)
Verbal to numerical translation
- Identify key words and phrases
- Recognize words that indicate mathematical operations (, , , )
- Identify words that suggest relationships or comparisons (more than, less than, )
- Assign variables to unknown quantities
- Use letters or symbols to represent unknown values mentioned (x, y, z)
- Choose meaningful variable names that relate to problem context (age, distance, price)
- Translate verbal descriptions into mathematical expressions
- Convert words or phrases into mathematical symbols or operations (plus, minus, times)
- Combine variables and constants using appropriate mathematical operators (, )
- Simplify resulting expression if necessary (combine like terms, distribute)
- Construct equations
- Use translated expressions to form equations that represent relationships between quantities (equality, inequality)
- Ensure equations accurately reflect conditions or constraints mentioned (balance, equivalence)
- Solve the equations
- Apply appropriate algebraic techniques to solve for unknown variables (isolation, substitution)
- Isolate desired variable by performing inverse operations on both sides of equation (addition, multiplication)
- Simplify solution and express it in required form (decimal, fraction)
Additional Problem-Solving Techniques
- Use logical reasoning to analyze problem structure and relationships
- Apply techniques to check reasonableness of solutions
- Utilize methods to represent problem scenarios graphically
Key Terms to Review (28)
Algebraic: Algebraic refers to the use of algebra, a branch of mathematics that employs symbols, variables, and equations to represent and analyze quantitative relationships. Algebraic methods and concepts are fundamental in problem-solving strategies, as they allow for the systematic manipulation and generalization of mathematical expressions.
Arithmetic: Arithmetic is the branch of mathematics that deals with the fundamental operations of addition, subtraction, multiplication, and division. It is the basic language of mathematics and is essential for problem-solving and quantitative reasoning in various areas of life and study.
Devise a Plan: Devise a plan refers to the process of creating a strategic approach or method to solve a problem or achieve a goal. It involves carefully considering the steps, resources, and strategies needed to effectively address a specific challenge or task.
Difference: The difference between two numbers is the amount by which one number exceeds the other. It is the result of subtracting one number from another and represents the magnitude of the separation between the two values.
Elimination: Elimination refers to the process of removing or getting rid of something, especially waste products or unwanted materials from the body or a system. It is a crucial component of problem-solving strategies, as it involves identifying and removing unnecessary or irrelevant information to arrive at the desired solution.
Equal To: The term 'equal to' refers to a mathematical relationship where two quantities or expressions have the same value. It signifies that the two sides of an equation or comparison are identical in magnitude or worth.
Equations: Equations are mathematical statements that express the equality between two expressions. They are used to represent relationships between variables and constants, and to solve for unknown quantities by finding the values that satisfy the equality.
Estimation: Estimation is the process of approximating or making an informed guess about a value or quantity without performing precise calculations. It involves using available information and reasonable assumptions to arrive at a reasonable estimate or range of values.
Execute the Plan: Executing the plan is the critical step in the problem-solving process where the individual takes action to implement the strategy or solution they have developed. This involves putting the plan into practice and following through on the steps necessary to reach the desired outcome.
Exponents: Exponents are a mathematical notation used to represent repeated multiplication of a number by itself. They indicate the number of times a base number is multiplied by itself, providing a concise way to express large or small quantities.
Formulas: Formulas are mathematical expressions that represent relationships between variables, constants, and operations. They provide a concise and standardized way to express and calculate specific quantities or values in a wide range of applications, including problem-solving strategies.
Geometry: Geometry is the branch of mathematics that deals with the study of shapes, sizes, and spatial relationships. It explores the properties and measurements of various geometric objects, such as points, lines, angles, surfaces, and solids.
Graphical: Graphical refers to the visual representation of information, data, or concepts using graphs, charts, diagrams, or other visual elements. It involves the use of visual aids to convey complex information in a clear and concise manner, often to aid in problem-solving, decision-making, or understanding a particular topic or process.
Logical Reasoning: Logical reasoning is the process of using rational, systematic, and analytical thinking to draw conclusions or make decisions. It involves the ability to identify patterns, make inferences, and evaluate the validity and soundness of arguments.
Parentheses: Parentheses are punctuation marks used to enclose additional information or expressions within a sentence. They serve to separate certain parts of a mathematical expression or an equation, providing clarity and indicating the order of operations to be performed.
Pattern Recognition: Pattern recognition is the ability to identify and understand recurring sequences, structures, or relationships within data or information. It is a fundamental cognitive process that allows individuals to make sense of the world around them by detecting and interpreting patterns in various contexts.
Percentage: A percentage is a way of expressing a part of a whole as a fraction of 100. It is a unit used to represent a proportional relationship or rate of change between two values.
Problem Decomposition: Problem decomposition is the process of breaking down a complex problem into smaller, more manageable sub-problems that can be solved independently. It is a fundamental strategy in problem-solving that allows for a systematic and organized approach to addressing complex challenges.
Product: The product is the result of multiplying two or more numbers or quantities together. It represents the combined or cumulative effect of the factors involved in the multiplication operation.
Quotient: The quotient is the result of dividing one number by another. It represents the number of times the divisor goes into the dividend, and is the answer to a division problem.
Review and Check the Solution: Reviewing and checking the solution is a crucial step in the problem-solving process. It involves carefully examining the steps taken to arrive at the solution, ensuring accuracy, and verifying that the solution meets the original requirements or constraints of the problem. This step helps to identify any errors or alternative approaches that could lead to a better solution.
Substitution: Substitution is the process of replacing one value or expression with another in an equation or formula. It is a fundamental technique used to solve equations, simplify expressions, and manipulate formulas to isolate a specific variable.
Sum: The sum is the result of adding two or more numbers or quantities together. It represents the total or combined value of the addends. The sum is a fundamental concept in mathematics that is essential for understanding addition and its applications in various mathematical topics.
Target Variable: The target variable, also known as the dependent variable, is the key outcome or response that a researcher aims to predict or explain in a study. It is the variable of primary interest and is influenced by the independent variables or predictors in the analysis.
Understand the Problem: Understanding the problem is a critical step in the problem-solving process. It involves carefully analyzing the given information, identifying the key elements, and determining what the problem is asking to be solved. This foundational step ensures that the problem-solver has a clear and accurate comprehension of the task at hand before attempting to find a solution.
Variables: Variables are symbolic representations of values or quantities that can change or vary within the context of a problem or mathematical expression. They serve as placeholders for unknown or changing information, allowing for the manipulation and analysis of data.
Visualization: Visualization is the ability to form mental images or pictures in one's mind. It is a cognitive process that involves the creation, manipulation, and interpretation of visual representations to aid in understanding and problem-solving.
Word Problems: Word problems are mathematical questions presented in the form of a written description of a real-world scenario. They require the solver to extract the relevant information, identify the appropriate mathematical operations, and apply them to find the solution.