Welcome back! In our last lesson, we focused on the different ways to represent parts of a whole by converting between fractions, decimals, and percentages. Continuing our journey through the essential number skills you'll need for Year 9, this lesson tackles another crucial foundation of mathematics: the rules that govern calculations.
Imagine you and a friend are asked to solve 5 + 10 ÷ 5. You might add first to get 15 ÷ 5 = 3, while your friend might divide first to get 5 + 2 = 7. Who is right? To avoid this confusion and ensure everyone gets the same correct answer, mathematicians follow an agreed-upon set of rules. Today, you'll master these rules, known as the order of operations, and learn how to apply them to calculations involving both positive and negative numbers (integers).
The Rules of the Road: BIDMAS
When you write an essay, you follow grammar and punctuation rules to make your meaning clear. In mathematics, the order of operations serves the same purpose. A common way to remember this order in New Zealand is the acronym BIDMAS.
- Brackets
- Indices (or Orders, like squares and cubes)
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
It's very important to note that Division and Multiplication have equal priority, as do Addition and Subtraction. When you have a calculation with both, you simply work from left to right.
The following video from Cognito provides a clear introduction to why we need an order of operations and how BIDMAS works.
How to use BODMAS (Order of Operations)
Watch this video to understand the logic behind having a specific order for calculations.
Pay close attention to the explanation of the BIDMAS acronym from the beginning until the first worked example is complete. Note how each step of BIDMAS is applied in sequence to simplify the expression.
Adding Negative Numbers to the Mix
Now that we have the order, let's add another layer of complexity: negative numbers. An integer is any whole number, which can be positive, negative, or zero (...-3, -2, -1, 0, 1, 2, 3...). Applying BIDMAS correctly is even more important when negative integers are involved, as the signs can easily lead to mistakes.
Before we combine BIDMAS with negative numbers, let's quickly review the rules for adding, subtracting, multiplying, and dividing them. The following resource from BBC Bitesize is a great reference.
Order of operations and negative numbers - GCSE Maths Revision - BBC Bitesize
This article provides a solid review of the rules for working with negative numbers, which we'll need for the rest of this lesson.
First, read the section on adding and subtracting negative numbers. Start from the heading "Adding and subtracting with negative numbers". The text explains the key rules, such as "subtracting a negative is the same as adding a positive." Next, scroll down to the section "How to multiply and divide negative numbers". Focus on the rules for determining the sign of the answer: If the signs are the same (e.g., negative × negative), the result is positive. If the signs are different (e.g., positive × negative), the result is negative.
Putting It All Together
You now have the two key ingredients: the order of operations (BIDMAS) and the rules for integers. The final step is to combine them to solve more complex problems. This requires careful, step-by-step work, paying close attention to brackets and signs.
The next video works through two excellent examples that show how to apply these rules together. The second example is particularly important as it highlights a very common point of confusion: the difference between an expression like and .
Order of Operations with Integers | Positives and Negatives | PEMDAS | Math with Mr. J
Watch how Mr. J applies the order of operations to expressions involving negative integers.
First, watch the first example. Notice how he deals with the brackets first, then works from left to right for the division and multiplication. Then, watch the second, more complex example. Pay very close attention to the explanation of exponents with negative numbers around this timestamp. Understanding this distinction is crucial for getting the right answer.
A Note on Calculators
When you use a scientific calculator, it is programmed to follow the order of operations automatically. This is a great tool, but you still need to know how to enter the expression correctly.
Here are two key tips:
- Use the correct negative button: Most scientific calculators have a subtraction button (
-) and a separate negative/sign-change button, which might look like(-)or+/-. Use the negative button to enter a negative number, like(-3). - Use brackets: When dealing with powers of negative numbers, use brackets just as you saw in the video. If you type
-3^2into your calculator, it will likely calculate first and then make it negative, giving you -9. To calculate , you must enter(-3)^2to get the correct answer of 9.
Key Takeaways
Excellent work navigating some tricky calculations! Being methodical and careful is the key to success with the order of operations.
- BIDMAS provides the standard order for performing calculations: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
- Operations with equal priority (D/M and A/S) are performed from left to right.
- When multiplying or dividing integers, two numbers with the same sign give a positive result, while two numbers with different signs give a negative result.
- Be extra careful with indices and negative numbers. Remember that is different from .
In our next lesson, we'll continue to strengthen your calculation skills by focusing on how to add, subtract, multiply, and divide fractions and decimals.
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