The Power of Worked Examples
What Makes a Great Worked Example?
A worked example is more than just showing the correct answer. It is a guided tour of the problem-solving process. Effective worked examples share several key characteristics that make them powerful learning tools.
First, they start by clearly stating the problem. This seems simple, but it's crucial. The learner needs to know exactly what question is being answered. Next, they outline the goal or what a successful solution will achieve. The core of the example is the step-by-step solution. Each step should be small, logical, and explained in plain language. Crucially, the reasoning behind each step is included—the "why" is just as important as the "what." Finally, a great worked example ends with a conclusion that reviews the answer and often points out how this method can be applied to other, similar problems.
The Step-by-Step Blueprint
While the content changes from subject to subject, the structure of a strong worked example remains consistent. Think of it as a recipe for solving problems.
| Step | Description | Purpose |
|---|---|---|
| 1. Problem Statement | Clearly present the problem to be solved. | Sets the context and defines the challenge. |
| 2. Identify Knowns & Unknowns | List what information is given and what you need to find. | Organizes the information and clarifies the goal. |
| 3. Choose a Strategy/Formula | Select the appropriate method, law, or formula to use. | Connects the problem to relevant knowledge. |
| 4. Execute the Solution | Work through the calculations or logical steps. | Demonstrates the application of the strategy. |
| 5. Review the Answer | Check if the answer makes sense and is reasonable. | Promotes critical thinking and verification. |
Worked Example in Mathematics: Solving for X
Let's see the blueprint in action with a classic algebra problem.
Problem Statement: Solve for $x$: $3(x - 5) + 7 = 16$
Step 1: Identify the Goal. We need to find the value of the variable $x$ that makes the equation true. The equation has parentheses and addition, so we will need to simplify and isolate $x$.
Step 2: Apply the Distributive Property. We start by simplifying the left side. The distributive property tells us that $a(b + c) = ab + ac$. Here, we have $3(x - 5)$.
$3(x - 5) + 7 = 16$
$(3 \times x) - (3 \times 5) + 7 = 16$
$3x - 15 + 7 = 16$
Step 3: Combine Like Terms. Now we combine the constant terms on the left side: $-15 + 7$.
$3x - 8 = 16$
Step 4: Isolate the Variable Term. To get $3x$ by itself, we add $8$ to both sides of the equation. We do this to cancel out the $-8$ on the left.
$3x - 8 + 8 = 16 + 8$
$3x = 24$
Step 5: Solve for the Variable. Now, $3x$ means $3$ multiplied by $x$. To get $x$ alone, we do the opposite operation, which is division. We divide both sides by $3$.
$\frac{3x}{3} = \frac{24}{3}$
$x = 8$
Step 6: Check Your Answer. It's always a good idea to verify. Substitute $x = 8$ back into the original equation:
$3(8 - 5) + 7 = 3(3) + 7 = 9 + 7 = 16$.
This matches the original equation, so our answer is correct.
Conclusion: The solution is $x = 8$. The key steps were simplifying using the distributive property, combining like terms, and using inverse operations to isolate the variable.
Worked Example in Science: Calculating Density
Worked examples are just as vital in science. Let's tackle a common physics/chemistry problem.
Problem Statement: A small rectangular block of metal has a mass of 450 g. Its length is 5 cm, width is 3 cm, and height is 2 cm. What is the density of the metal?
Step 1: Identify Knowns and Unknowns.
Known: Mass ($m$) = 450 g
Known: Length ($l$) = 5 cm, Width ($w$) = 3 cm, Height ($h$) = 2 cm
Unknown: Density ($D$)
Step 2: Recall the Relevant Formula. Density is defined as mass per unit volume. The formula is:
$D = \frac{m}{V}$
Where $D$ is density, $m$ is mass, and $V$ is volume.
Step 3: Calculate the Volume. We have a rectangular block, so its volume is given by:
$V = l \times w \times h$
Substituting the known values:
$V = 5 \, \text{cm} \times 3 \, \text{cm} \times 2 \, \text{cm}$
$V = 30 \, \text{cm}^3$
Step 4: Apply the Density Formula. Now we plug the mass and volume into the density formula.
$D = \frac{450 \, \text{g}}{30 \, \text{cm}^3}$
Step 5: Perform the Calculation and State the Units.
$D = 15 \, \text{g/cm}^3$
We divide the numbers (450 / 30 = 15) and keep the units (g/cm^3). Density is always expressed with units.
Step 6: Interpret the Result. The density of the metal block is 15 g/cm³. This is a relatively high density; for comparison, water has a density of 1 g/cm³, so this metal is much denser than water.
How to Learn Effectively from Worked Examples
Simply reading a worked example is not enough. To truly benefit, you need to engage with it actively.
1. Cover and Solve: Read the problem statement, then cover the solution. Try to solve the problem on your own. After you've attempted it, uncover the worked example and compare your steps to the ones provided. This highlights where your understanding is strong and where it needs improvement.
2. Self-Explain: As you read each step of the example, ask yourself "Why was this step taken?" Explain the reasoning behind it in your own words. This deepens your understanding of the underlying principles.
3. Look for Patterns: Worked examples teach you how to recognize types of problems. After studying an example, try to find another problem that looks different on the surface but requires the same solution method. This builds your ability to transfer knowledge.
4. Create Your Own: Once you feel comfortable with a concept, try creating a worked example for a friend or for your future self. Teaching a concept is one of the best ways to master it.
Common Mistakes and Important Questions
Q: I can follow the worked example, but I can't solve a new problem on my own. Why?
This is a very common experience. It often means you are memorizing the steps for that specific problem instead of understanding the "why" behind each step. To fix this, focus on the self-explanation technique. For each step, pause and ask, "Why is this the logical next thing to do?" Also, make sure you are practicing with a variety of problems that use the same core concept but in different contexts.
Q: Should I look at the answer first or try to solve it myself first?
A mix of both strategies is best. If a concept is completely new to you, it's okay to study the worked example first to understand the process. However, once you have a basic understanding, you will learn more deeply by trying the problem yourself first, even if you struggle. The struggle is part of the learning process! When you then look at the worked example, your brain is primed to see the solution path you missed.
Q: What is the biggest mistake people make when creating their own worked examples?
The most common mistake is skipping steps or failing to explain the reasoning. The creator, who already understands the problem, might see two or three steps as one logical leap. But for a learner, that leap can be a huge chasm. A good worked example breaks down the solution into the smallest, most logical steps possible and explicitly states why each step is necessary.
Worked examples are a bridge between not knowing and understanding. They demystify the problem-solving process by making expert thinking visible. By studying them actively—self-explaining, comparing, and pattern-finding—you can build a robust mental toolkit for tackling challenges in math, science, and beyond. Remember, the ultimate goal is not to replicate a single solution, but to internalize the process so you can confidently navigate unfamiliar problems on your own. The next time you face a difficult problem, seek out a worked example; it's like having a personal tutor guiding you one step at a time.
Footnote
[1] Distributive Property: A fundamental property in algebra that states that multiplying a number by a sum is the same as doing each multiplication separately. It is written as $a(b + c) = ab + ac$. This property is essential for simplifying expressions and solving equations.
[2] Density: A physical property of matter defined as mass per unit volume. The standard formula is $D = \frac{m}{V}$. It is an intensive property, meaning it is the same for any sample size of a given substance under the same conditions.