Problem-solving Models and Strategies

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.

Explore our app and discover over 50 million learning materials for free.

- Applied Mathematics
- Calculus
- Decision Maths
- Discrete Mathematics
- Geometry
- Logic and Functions
- Mechanics Maths
- Probability and Statistics
- Pure Maths
- ASA Theorem
- Absolute Convergence
- Absolute Value Equations and Inequalities
- Abstract algebra
- Addition and Multiplication of series
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebra of limits
- Algebra over a field
- Algebraic Fractions
- Algebraic K-theory
- Algebraic Notation
- Algebraic Representation
- Algebraic curves
- Algebraic geometry
- Algebraic number theory
- Algebraic topology
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Associative algebra
- Average Rate of Change
- Banach algebras
- Basis
- Bijective Functions
- Bilinear forms
- Binomial Expansion
- Binomial Theorem
- Bounded Sequence
- C*-algebras
- Category theory
- Cauchy Sequence
- Cayley Hamilton Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Clifford algebras
- Cohomology theory
- Combinatorics
- Common Factors
- Common Multiples
- Commutative algebra
- Compact Set
- Completing the Square
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Congruence Equations
- Conic Sections
- Connected Set
- Construction and Loci
- Continuity and Uniform convergence
- Continuity of derivative
- Continuity of real valued functions
- Continuous Function
- Convergent Sequence
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Coupled First-order Differential Equations
- Cubic Function Graph
- Data Transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Derivative of a real function
- Deriving Equations
- Determinant Of Inverse Matrix
- Determinant of Matrix
- Determinants
- Diagonalising Matrix
- Differentiability of real valued functions
- Differential Equations
- Differential algebra
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Dimension
- Direct and Inverse proportions
- Discontinuity
- Disjoint and Overlapping Events
- Disproof By Counterexample
- Distance from a Point to a Line
- Divergent Sequence
- Divisibility Tests
- Division algebras
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Eigenvalues and Eigenvectors
- Ellipse
- Elliptic curves
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Equicontinuous families of functions
- 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
- Fermat's Little Theorem
- Field theory
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding The Area
- First Fundamental Theorem
- First-order Differential Equations
- Forms of Quadratic Functions
- Fourier analysis
- 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
- Gram-Schmidt Process
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs And Differentiation
- Graphs Of Exponents And Logarithms
- Graphs of Common Functions
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Grothendieck topologies
- Group Mathematics
- Group representations
- Growth and Decay
- Growth of Functions
- Gröbner bases
- Harmonic Motion
- Hermitian algebra
- Higher Derivatives
- Highest Common Factor
- Homogeneous System of Equations
- Homological algebra
- Homotopy theory
- Hopf algebras
- Hyperbolas
- Ideal theory
- Imaginary Unit And Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Injective linear transformation
- Instantaneous Rate of Change
- Integers
- Integrating Ex And 1x
- Integrating Polynomials
- Integrating Trigonometric Functions
- Integration
- Integration By Parts
- Integration By Substitution
- Integration Using Partial Fractions
- Integration of Hyperbolic Functions
- Interest
- Invariant Points
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Inverse of a Matrix and System of Linear equation
- Invertible linear transformation
- Iterative Methods
- Jordan algebras
- Knot theory
- L'hopitals Rule
- Lattice theory
- Law Of Cosines In Algebra
- Law Of Sines In Algebra
- Laws of Logs
- Leibnitz's Theorem
- Lie algebras
- Lie groups
- Limits of Accuracy
- Linear Algebra
- Linear Combination
- Linear Expressions
- Linear Independence
- Linear Systems
- Linear Transformation
- 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 Calculations
- Matrix Determinant
- Matrix Multiplication
- Matrix operations
- Mean value theorem
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modelling with First-order Differential Equations
- Modular Arithmetic
- Module theory
- Modulus Functions
- Modulus and Phase
- Monoidal categories
- Monotonic Function
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplicative ideal theory
- Multiplying And Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Non-associative algebra
- Normed spaces
- Notation
- Number
- Number Line
- Number Systems
- Number Theory
- Number e
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations With Matrices
- Operations with Decimals
- Operations with Polynomials
- Operator algebras
- Order of Operations
- Orthogonal groups
- Orthogonality
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Hyperbolas
- Parametric Integration
- Parametric Parabolas
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Pointwise convergence
- Poisson algebras
- Polynomial Graphs
- Polynomial rings
- 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 Determinants
- Properties of Exponents
- Properties of Riemann Integral
- Properties of dimension
- Properties of eigenvalues and eigenvectors
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic forms
- Quadratic functions
- Quadrilaterals
- Quantum groups
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Ratio and Root test
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Rearrangement
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Reduced Row Echelon Form
- Reducible Differential Equations
- Remainder and Factor Theorems
- Representation Of Complex Numbers
- Representation theory
- Rewriting Formulas and Equations
- Riemann integral for step function
- Riemann surfaces
- Riemannian geometry
- Ring theory
- Roots Of Unity
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Products
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Fundamental Theorem
- Second Order Recurrence Relation
- Second-order Differential Equations
- Sector of a Circle
- Segment of a Circle
- Sequence and series of real valued functions
- Sequence of Real Numbers
- Sequences
- Sequences and Series
- Series Maths
- Series of non negative terms
- Series of real numbers
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Similarity and diagonalisation
- Simple Interest
- Simple algebras
- 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
- Spanning Set
- Special Products
- Special Sequences
- Standard Form
- Standard Integrals
- Standard Unit
- Stone Weierstrass theorem
- Straight Line Graphs
- Subgroup
- Subsequence
- Subspace
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Summation by Parts
- Supremum and Infimum
- Surds
- Surjective functions
- Surjective linear transformation
- System of Linear Equations
- Tables and Graphs
- Tangent of a Circle
- Taylor theorem
- The Quadratic Formula and the Discriminant
- Topological groups
- Torsion theories
- Transformations
- Transformations of Graphs
- Transformations of Roots
- 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
- Uniform convergence
- Unit Circle
- Units
- Universal algebra
- Upper and Lower Bounds
- Valuation theory
- Variables in Algebra
- Vector Notation
- Vector Space
- Vector spaces
- Vectors
- Verifying Trigonometric Identities
- Volumes of Revolution
- Von Neumann algebras
- Writing Equations
- Writing Linear Equations
- Zariski topology
- Statistics
- Theoretical and Mathematical Physics

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmeldenHave 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.

What are the steps to solve a problem efficiently?

1. Understand the problem

2.Plan

3.Solve

4.Check

Name the two problem-solving models.

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

State two problem-solving strategies when devising a plan.

Use the guess and check method.

Apply the specific formula for the problem.

Step 01: What do you know?

- Mrs. Grave gives 1 penny on Day 1, 2 pennies on Day 2, and 4 pennies on Day 3.
- Each day the amount of money will double.
- Paul does his tasks for 5 days.

Step 02: What do you want to know?

You curious to figure out how much money will Paul have in total after 5 days of doing his tasks. We want to solve the problem by formulating a simpler one.

31

Step 01: What does David know?

- The number of players starting the tournament:8
- Only winners can advance to the next round

Step 02: What does David want to know?

David wants to compare the number of players in the second round to the number that starts the tournament. To solve the problem, David can use a diagram.

4/8

Step 01: What do you know?

- Slices of tomatoes and cucumber were used.
- The total number of slices used is 60.
- The ratio of cucumbers to tomatoes is 4:6
- The ratio simplifies to 2:3

Step 02: What do you want to know?

We need to know the number of both cucumber and tomato slices. We want to solve the problem by doing a table.

24 cucumber slices and 36 tomato slices is one solution

Already have an account? Log in

Open in App
More about Problem-solving Models and Strategies

The first learning app that truly has everything you need to ace your exams in one place

- Flashcards & Quizzes
- AI Study Assistant
- Study Planner
- Mock-Exams
- Smart Note-Taking

Sign up to highlight and take notes. It’s 100% free.

Save explanations to your personalised space and access them anytime, anywhere!

Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.

Already have an account? Log in

Already have an account? Log in

The first learning app that truly has everything you need to ace your exams in one place

- Flashcards & Quizzes
- AI Study Assistant
- Study Planner
- Mock-Exams
- Smart Note-Taking

Sign up with Email

Already have an account? Log in