Welcome to "Considering Wider Problems"!
You’ve already learned how to organize and process information to solve a puzzle or a math-based problem. But in the real world, problems don’t stay still! Sometimes the situation changes, or the "model" we use to solve the problem needs a little bit of extra work to make it accurate. In this chapter, we will learn how to look at the "big picture" and adapt our solutions when things get a bit more complicated.
4.1 Identify the Impact of a Change to a Problem
Imagine you have planned the perfect route to get to school in 20 minutes. You’ve solved the problem! But suddenly, you hear on the news that the main bridge is closed for repairs. Your solution is no longer perfect. This is what we call considering the implications of a change to the scenario.
What is a "Scenario"?
A scenario is simply the setting or the set of rules in which a problem exists. If you change a rule or a piece of the setting, the answer to your problem might change too.
How to Handle Changes
When a change happens, you need to follow these steps:
1. Identify the Change: What exactly is different? (e.g., A road is closed).
2. Find the Impact: How does this change affect your variables? (e.g., Travel time increases by 15 minutes).
3. Adjust the Solution: Does your original answer still work? If not, what is the new answer? (e.g., I must leave 15 minutes earlier to arrive on time).
Real-World Example:
A theater manager has scheduled three plays to start at 18:00, 19:00, and 20:00. However, the city announces a massive parade that will block the streets until 18:30. The manager must consider the implications: if the first play starts at 18:00, nobody will be able to get there! The manager must adjust the solution by shifting all the start times later.
Quick Review Box:
When the scenario changes, don't panic! Just ask yourself: "How does this new piece of news change the numbers I already calculated?"
4.2 Develop a Model
Don't worry if the word "model" sounds like something from a science lab. In Problem Solving, a model is just a set of rules or a formula we use to represent a real-life situation.
Identifying Features to Include
Sometimes, a model is too simple. It might leave out important details. Your job is to identify features of the situation that need to be added to make the model better.
Analogy:
Think of a model like a drawing of a house. If your drawing only shows the doors and windows but forgets the roof, it’s not a very good "model" of a house! To make it better, you have to identify that the "roof" is a feature that needs to be included.
Adjusting a Model
Once you find a new feature, you have to adjust the model to incorporate it. This usually involves adding a new calculation or a new rule.
Example: Taxi Fares
Imagine a taxi company has a simple model for prices: \( \text{Price} = \$2 \text{ per kilometer} \).
However, you notice that even for a very short trip, the driver charges a minimum of $5. The current model doesn't show this! To adjust the model, you would add a "base fee" feature:
\( \text{Price} = \$5 + (\$2 \times \text{kilometers traveled}) \)
Common Mistakes to Avoid
1. Over-complicating: Only add features that are relevant to the problem. If the color of the taxi doesn't change the price, don't include it in the model!
2. Ignoring Constraints: Always check if your new model still fits the original data you were given.
Key Takeaway
Developing a model is a two-step process: first, look for what is missing (Identify features); second, figure out how to write that into your rules (Adjust the model).
Chapter Summary
In this section of Problem Solving, we learned that:
• Problems are rarely static; a change in the scenario requires an adjustment to the solution.
• A model is a representation of reality using rules or math.
• To improve a model, we must identify missing features and incorporate them into our calculations.
Did you know?
Video game designers use models all the time! They create "physics models" to decide how a character jumps. If the jump feels too floaty, they adjust the model by changing the gravity variable until it feels just right.
Memory Aid: The "Tailor" Trick
Think of a problem-solver as a tailor. If the "customer" (the scenario) grows taller, the "suit" (the solution) must be adjusted to fit. If the suit is missing a pocket (a feature), the tailor must sew one on (develop the model)!