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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.

## Problem-solving strategies and model descriptions

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.

## Types of problem-solving models and strategies

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

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.

### Understand the problem

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").

### Devise a plan

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.

### Carry out the plan

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.

### Look back

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.

## IDEAL problem-solving model

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.

## Examples of problem-solving models and strategies

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 and models - Key takeaways

- 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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##### Frequently Asked Questions about Problem-solving Models and Strategies

What are problem-solving models?

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

What are types of problem-solving?

The most basic types of problem-solving are Polya's four-step problem-solving model and the IDEAL problem-solving model.

What are the strategies to problem-solve efficiently?

The strategies to solve a problem efficiently are to understand it, determine the correct method, solve it and verify and learn from it.

What are the lists of problem-solving models in algebra?

In algebra, any problem can be solved using Polya's four-step problem-solving model and IDEAL problem-solving model.

What are the 5 problem-solving strategies?

The 5 problem-solving strategies are 1. Identify The Problem, 2. Define An Outcome, 3. Explore Possible Strategies, 4. Anticipate Outcomes & Act, 5. Look And Learn.

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