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Problem-solving Models and Strategies

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits at Infinity and Asymptotes
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Power
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Spearman's Rank Correlation Coefficient
- Standard Deviation
- Standard Error
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

Have you ever been confronted with a challenging problem and had no idea how to even begin working on it? For instance, let's say you have two upcoming exams on the same day, and you are unsure how to prepare for them. Or, let's say you are solving a complex math problem, but you are stuck and don't know how to proceed. In these moments, **problem-solving strategies and models** can help us tackle difficult problems by guiding us with well-known approaches or plans to follow.

In this article, we explore problem-solving strategies and models that can be applied to solve problems. Then, we practice applying these models in some example exercises.

Oftentimes in mathematics, there is more than one way to solve a problem. Using problem-solving strategies can help you approach problems in a structured and logical manner to improve your efficiency.

**Problem-solving strategies** are models based on previous experience that provide a recommended approach for solving problems or analyzing potential solutions.

Problem-solving strategies involve steps like understanding, planning, and organizing, for example. While problem-solving strategies cannot guarantee an easier solution to a problem, they do provide techniques and tools that act as a guide for success.

Many models and strategies are developed based on the nature of the problem at hand. In this article, we discuss two well-known models that are designed to address various types of problems, including:

**Polya's four-step problem-solving model****IDEAL problem-solving model**

Let's look at these two models in detail.

A mathematician named George Polya developed a model called the Polya four-step problem-solving model to approach and solve various kinds of problems. This method has the following steps:

**Understand the problem****Devise a plan****Carry out the plan****Look back**

John Bransford and Barry Stein also proposed a five-step model named IDEAL to resolve a problem with a sound and methodical approach. The IDEAL model is based on the following steps:

**Identify The Problem****Define An Outcome****Explore Possible Strategies****Anticipate Outcomes & Act****Look And Learn**

Using either of these two models to help you identify and approach problems methodically can help make it easier to solve them.

Polya's four-step problem-solving model can be used to solve day-to-day problems as well as mathematical and other academic problems. As seen briefly, the steps of this problem-solving model include: understanding the problem, creating and carrying out a plan, and looking back. Let's look at these steps in more detail to understand how they are used.

This is a critical initial step. Simply put, if you don't fully understand the problem, you won't be able to identify a solution. You can understand a problem better by reviewing all of the inputs and available information, including its conditions and circumstances. Reading and understanding the problem helps you to organize the information as well as assign the relevant variables.

The following techniques can be applied during this problem-solving step:

Read the problem out loud to process it better.

List or summarize the important information to find out what is given and what is still missing.

Sketch a detailed diagram as a visual aid, depending on the problem.

Visualize a scenario about the problem to put it into context.

Use keyword analysis to identify the necessary operations (i.e., pay attention to important words and phrases such as "how many," "times," or "total").

Now that you have taken the time to properly understand the problem, you can devise a plan on how to proceed further to solve it. During this second step, you identify what strategy to follow to arrive at a solution. When considering a strategy to use, it's important to consider exactly what it is that you want to know.

Some problem-solving strategies include:

Identify the pattern from the given information and use it.

Use the guess-and-check method.

Work backward by using potential answers.

Apply a specific formula for the problem.

Eliminate the possibilities that don't work out.

Solve a simpler version of the problem first.

Form an equation and solve it.

During this third step, you solve the problem by applying your chosen strategy. For example, if you planned to solve the problem by drawing a graph, then during this step, you draw the graph using the information gathered in the previous steps. Here, you test your problem-solving skills and find if the solution works or not.

Below are some points to keep in mind when solving the problem:

Be systematic in your approach when implementing a strategy.

Check the work and see whether the solution works in all relevant cases.

Be flexible and change the strategy if necessary.

Keep solving and don't give up.

At this fourth step, you check your solution. This can be done by solving the problem in another way or simply by confirming that your solution makes sense. This step helps you decide if any improvements are needed for your solution. You may choose to check after solving an individual problem or after solving an entire set. Checking the problem carefully also helps you to reflect on the process and improve your methods for future problem solving.

The IDEAL problem-solving model was developed by Bransford and Stein as a guide for understanding and solving problems. This method is used in both education and industry. The IDEAL problem-solving model consists of five steps: identifying the problem, describing the outcome, exploring the possible strategies, anticipating the outcome, and looking back to learn. Let us explore these steps in detail by considering them one by one.

** Identify the problem** - In this first step, you identify and understand the problem. To do this, you evaluate which information is provided and available, and you identify the unknown variables and missing information.

** Describe the outcome** - In this second step, you define the result you are seeking. This matters because a problem might have multiple potential results, so you need to clarify which outcomes in particular you are aiming for. Defining an outcome clarifies the path that must be taken to solving the problem.

** Explore possible strategies** - Now that you have considered the desired outcome, you are ready to brainstorm and explore different strategies and techniques to solve your particular problem.

** Anticipate outcomes and act** - From the previous step, you already have explored different strategies and techniques. During this step, you review and evaluate them in order to choose the best one to act on. Your selection should consider the benefits and drawbacks of the strategy and whether it can ultimately lead to the desired outcome. After making your selection, you act on it and apply the technique to the given problem.

** Look and learn** - The final step to solving problems with this method is to consider whether the applied technique worked and if the needed results were obtained. Also, an additional step is learning from the current problem and its methods to make problem solving more efficient in the future.

Here are some solved examples of the problem-solving models and strategies discussed above.

Find the number when two times the sum of \(3\) and that number is thrice that number plus \(4\). Solve this problem with **Polya's four-step problem-solving model**.

**Solution:** We will follow the steps of **Polya's four-step problem-solving model** as mentioned above to find the number.

__Step 1__: Understand the problem.

By reading and understanding the question, we denote the unknown number as \(x\).

__Step 2__: Devise a plan.

We see that two times \(x\) is added to \(3\) to make it equal to thrice the \(x\) plus \(4\). So, we can determine that forming an equation to solve the mathematical problem is a reasonable plan. Therefore, we form an equation by going step by step:

First we add \(x\) with \(3\) and multiply it with \(2\).

\begin{equation}\tag{1}\Rightarrow 2(x+3)\end{equation}

Then, we form the second part of the equation for thrice the \(x\) plus \(4\).

\begin{equation}\tag{2}\Rightarrow 3x+4\end{equation}

Hence, equating both sides \((1)\) and \((2)\) we get,

\[2(x+3)=3x+4\]

__Step 3__: Carry out the plan.

Now, we algebraically solve the equation above.

\begin{align}2(x+3) &=3x+4 \\2x+6 &= 3x+4 \\3x-2x &= 6-4 \\x &=2\end{align}

__Step 4__: Look back.

By inputting the value of 2 in our equation, we see that two times \(2+3\) is \(10\) and three times \(2\) plus \(4\) is also 10. Hence, the left side and right side are equal. So, our solution is satisfied.

Hence, the number is \(2\).

A string is \(48 cm\) long. It is cut into two pieces such that one piece is three times that of the other piece. What is the length of each piece?

**Solution****:** Let us work on this problem using the IDEAL problem-solving method.

__Step 1__: Identify the problem.

We are given a length of a string, and we know that it is cut into two parts, whereby one part is three times longer than the other. As the length of the longer piece of string is dependent on the shorter string, we assume only one variable, say \(x\).

__Step 2__: Describe the outcome.

From the problem, we understand that we need to find the length of each piece of string. And we need the results such that the total length of both the pieces should be \(48 cm\).

__Step 3__: Explore possible strategies.

There are multiple ways to solve this problem. One way to solve it is by using the trial-and-error method. Also, as one length is dependent on another, the other way is to form an equation to solve for the unknown variable algebraically.

__Step 4__: Anticipate outcomes and act.

From the above step, we have two methods by which we can solve the given problem. Let's find out which method is more efficient and solve the problem by applying it.

**Method 1**

For the trial-and-error method, we need to assume value(s) one at a time for the variable and then solve for it individually until we get the total of 48.

That is, suppose we consider \(x=1\).

Then, by the condition, the second piece is three times the first piece.

\[\Rightarrow 3x=3(1)=3\]

Then the length of both pieces should be:

\[\Rightarrow 1+3=4\neq 48\]

Hence, our assumption is wrong. So, we need to consider another value. For this method, we continue this process until we find the total of \(48\). We can see that proceeding this way is time-consuming. So, let us apply the other method instead.

**Method 2**

In this method, we form an equation and solve it to obtain the unknown variable's value. We know that one piece is three times the other piece. Therefore, let the length of one piece be \(x\). Then the length of the other piece is \(3x\).

Now, as the string is \(48 cm\) long, it should be considered as a sum of both of its pieces.

\begin{align}&\Rightarrow x+3x=48 \\&\Rightarrow 4x=48 \\&\Rightarrow x=\frac{48}{4} \\&\Rightarrow x=12 \\\end{align}

So, the length of one piece is \(12cm\). The length of the other piece is \(3x=3(12)=36cm\).

Step 5: Look and learn

Let's take a look to see if our answers are correct. The unknown variable value we obtained is \(12\). Using it to find the other piece we get a value of \(36\). Now, adding both of them, we get:

\[\Rightarrow 12+36=48\].

Here, we got the correct total length. Hence, our calculations and applied method are right.

- Problem-solving strategies are models developed based on previous experience that provide a recommended approach for analyzing potential solutions for problems.
- Two common models include Polya's Four-Step Problem-Solving Model and the IDEAL problem-solving model.
- Polya's Four-Step Problem-Solving Model has the following steps: 1) Understand the problem, 2) Devise a plan, 3) Carry out the plan, and 4) Looking back.
- The IDEAL model is based on the following steps: 1) Identify The Problem, 2) Define An Outcome, 3) Explore Possible Strategies, 4) Anticipate Outcomes and Act, 5) Look And Learn.

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