Welcome to Multi-Step Problem Solving and Modelling!

Have you ever looked at a long word problem in math and thought, "Where do I even start?" Don't worry—most people feel that way! In this chapter, we are going to learn how to be "Mathematical Detectives." We will learn how to break down big, scary problems into small, friendly steps and how to create mathematical models to represent real-life situations.

By the end of these notes, you will have a toolkit of strategies to tackle any problem, even if you haven't seen one like it before!

1. What is a "Multi-Step" Problem?

Most basic math questions are "one-step": you see two numbers, you add them, and you're done. A multi-step problem is more like a journey. You might need to find one piece of information first (an "intermediate" answer) before you can find the final answer.

The "Step-by-Step" Strategy:

1. Read and Highlight: Underline the important numbers and what the question is actually asking for.
2. Identify the "Middle Man": Ask yourself, "What do I need to know first before I can solve this?"
3. Plan your operations: Decide if you need to use addition, subtraction, multiplication, or division at each stage.

Example: A cinema ticket costs \(£8\). A family buys \(4\) tickets and \(3\) bags of popcorn. The total bill is \(£41\). How much does one bag of popcorn cost?

Step 1: Find the total cost of the tickets. \(4 \times 8 = 32\).
Step 2: Subtract that from the total bill to see what's left for popcorn. \(41 - 32 = 9\).
Step 3: Divide that "leftover" amount by the number of bags. \(9 \div 3 = 3\).
Answer: One bag of popcorn costs \(£3\).

2. Modelling Situations

Modelling sounds fancy, but it just means "translating" a real-world story into math symbols. We use algebraic expressions or formulae to describe how things work.

Translating English to Math

When you see these words, you can turn them into math "models":

  • "The sum of" or "Total": Use addition \(+\)
  • "Difference" or "Less than": Use subtraction \(-\)
  • "Product of" or "Times": Use multiplication \(\times\) (or write them together like \(2x\))
  • "Shared" or "Per": Use division \(\div\) (or a fraction bar \(/\))

Did you know? We often use letters (variables) to represent numbers we don't know yet. For example, if a taxi charges \(£3\) plus \(£2\) for every mile (\(m\)), the mathematical model is: \(2m + 3\).

3. Using Formal Knowledge

To solve these problems, you need to use the tools you've learned in other parts of math. This might include:

  • Financial Mathematics: Calculating costs, profits, or simple interest. (For more on this, see the Financial Mathematics chapter).
  • Geometry: Using the properties of shapes to find missing lengths or areas.
  • Units: Making sure you aren't trying to add centimeters to meters! Always convert your units so they match.

Quick Tip: If a problem feels too abstract, draw a diagram! Whether it's a sketch of a garden or a bar model, seeing the problem usually makes the steps much clearer.

4. Evaluating Your Results

Once you have an answer, don't just close your book! You need to evaluate it. This means asking: "Does this answer actually make sense?"

The "Reality Check" List:

  • Estimation: If you round the numbers and do a quick mental calculation, is your answer in the right ballpark? (See Estimation and Checking Results for more tips).
  • Context: If the question asks for the number of people on a bus and you get \(22.5\), something has gone wrong! You can't have half a person.
  • Units: Did you remember to put the \(£\), \(cm\), or \(kg\) sign at the end?

5. Solving "Unfamiliar" Problems

Sometimes you will see a problem that looks totally new. Don't panic! This is what "Working Mathematically" is all about.

Try these steps when you're stuck:

  1. Try a simpler version: Change the big decimals to easy whole numbers. How would you solve it then? Use that same method for the hard numbers.
  2. Work backwards: If you know the end result, can you "undo" the steps (using inverse operations) to get back to the start?
  3. Look for patterns: Does the problem repeat? Can you describe the pattern with an expression?

Common Mistakes to Avoid

1. Doing everything at once: Many students try to put the whole problem into a calculator in one go. It’s much safer to write down the result of each step as you go.

2. Forgetting BIDMAS/BODMAS: Remember that the order of operations still applies in multi-step problems! Always do Brackets and Powers (Indices) before adding or subtracting.

3. Misreading the question: Sometimes the question asks for "the change left over," but students just calculate the total cost. Always re-read the final sentence of the question before you finish.

Key Takeaways

- Break it down: Turn one big problem into several small ones.

- Model it: Use letters and symbols to represent the situation (like \(y = mx + c\)).

- Check it: Use estimation to make sure your answer is sensible.

- Be brave: If a problem looks weird, use a diagram or a simpler example to find a way in.