Order of Operations
Evaluating expressions using PEMDAS — includes custom operation symbol problems (e.g., a◆b = a − 3b)
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Imagine making a giant, gooey pizza. You wouldn't put the cheese on the pan, bake it, and THEN add the raw dough on top, right? 🍕 That would be a messy disaster! Just like making a pizza, math has a special recipe called the Order of Operations. If you do the math steps in the wrong order, you get a totally weird answer!
To keep us from making math disasters on the ISEE, we use a secret code: PEMDAS. 🕵️♂️ You might know it as 'Please Excuse My Dear Aunt Sally.' It tells us exactly which part of a math problem to tackle first. First, we do Parentheses (the crust). Then Exponents (the sauce). Next comes Multiplication and Division (the cheese and pepperoni). Finally, we finish with Addition and Subtraction (baking it to perfection).
On the ISEE, you'll see these questions in both the Quantitative Reasoning and Mathematics Achievement sections. The test-makers love to try and trick you by putting addition at the very beginning of a problem, hoping you'll just read left to right like a book. But you're too smart for that! 🧠 Always follow your PEMDAS recipe, write down your steps, and you'll serve up the right answer every single time. Let's practice being math master chefs! 👨🍳
Practice Questions
3 practice questions for ISEE Upper Level
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- Evaluate both columns using the order of operations (PEMDAS).
Column A:
First, evaluate the exponent: .
Next, perform the multiplication: .
Finally, perform the subtraction: .
The value of Column A is 0.
Column B:
First, evaluate the expression inside the parentheses: .
Next, evaluate the exponent: .
Finally, multiply the two results: .
The value of Column B is 36.
Since 36 is greater than 0, the quantity in Column B is greater.
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- When evaluating an expression with a fraction bar, treat the numerator and the denominator as if they are enclosed in parentheses. Evaluate each separately using the order of operations.
Numerator:
- 1. Multiply first:
- 2. Subtract:
Denominator:
- 1. Evaluate the exponent first:
- 2. Subtract:
Finally, divide the evaluated numerator by the evaluated denominator: .
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- Follow the order of operations (PEMDAS). First, evaluate the expression inside the parentheses: .
Inside the parentheses, division must be performed before subtraction:
- 1. Divide:
- 2. Subtract:
Now substitute 15 back into the original expression in place of the parentheses:
.
Tips & Strategies
- Multiplication and Division are twins! They are on the exact same level. If you have a problem like , you go left-to-right. Don't multiply first just because 'M' comes before 'D' in PEMDAS. , and .
- On Quantitative Comparison questions, test-makers love to see if you'll assume is the same as . As we saw in question 2, they are NOT the same! Always do the math to prove it.
- Write down every single step, one line at a time underneath the original problem. Trying to hold all the numbers in your head is a quick way to accidentally drop a number!
Common Mistakes
- Watch out for reading math problems like a book! Just because starts with addition doesn't mean you add first. You MUST multiply first to get .
- Don't forget that a fraction bar is a sneaky grouping symbol. If you see , you have to figure out the entire top () and the entire bottom () before you divide them!
Frequently Asked Questions
What if I see a weird symbol like a star or a smiley face in a math problem?
Don't panic! The ISEE sometimes invents 'custom operations' like . Just plug the numbers into the formula they give you and follow standard PEMDAS rules to solve it.
What do I do if there are parentheses inside of parentheses?
Work from the inside out! Find the deepest, most buried set of parentheses, solve the math inside them first, and then work your way outward.
Is there a penalty for guessing if I run out of time on a long PEMDAS problem?
Nope! On the ISEE, there is NO penalty for wrong answers. If you are totally stuck or running out of time, pick your favorite letter and bubble it in. Never leave a question blank!