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“Mentally solve: 3 × (x + 4) when x = 5.

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“Mentally solve: 3 × (x + 4) when x = 5.

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Post your strategy or explanation in one of the boxes below.”

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Operations

"Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.

But, when you see something like ...

... what part should you calculate first?

Start at the left and go to the right?
Or go from right to left?
Warning: Calculate them in the wrong order, and you can get a wrong answer !
So, long ago people agreed to follow rules when doing calculations, and they are:

Order of Operations

Do things in Parentheses First

Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract

Multiply or Divide before you Add or Subtract

Otherwise just go left to right

How Do I Remember It All ... ? PEMDAS !

Divide and Multiply rank equally (and go left to right).

Add and Subtract rank equally (and go left to right)

So do it this way:

After you have done "P" and "E", just go from left to right doing any "M" or "D" as you find them.

Then go from left to right doing any "A" or "S" as you find them.

Pudgy Elves May Demand A Snack
Popcorn Every Monday Donuts Always Sunday
Please Eat Mom's Delicious Apple Strudels
People Everywhere Made Decisions About Sums

You may prefer GEMS ( Grouping, Exponents, Multiply or Divide, Add or Subtract).
In the UK they say BODMAS (Brackets, Orders, Divide, Multiply, Add, Subtract).
In Canada they say BEDMAS (Brackets, Exponents, Divide, Multiply, Add, Subtract).
It all means the same thing! It doesn't matter how you remember it, just so long as you get it right.

Examples

Example: How do you work out 3 + 6 × 2 ?

Multiplication before A ddition:

Example: How do you work out (3 + 6) × 2 ?

P arentheses first:

Example: How do you work out 12 / 6 × 3 / 2 ?

Multiplication and D ivision rank equally, so just go left to right:

A practical example:

Example: Sam threw a ball straight up at 20 meters per second, how far did it go in 2 seconds?

Sam uses this special formula that includes the effects of gravity:

height = velocity × time − (1/2) × 9.8 × time ²

Sam puts in the velocity of 20 meters per second and time of 2 seconds:

height = 20 × 2 − (1/2) × 9.8 × 2 ²

Now for the calculations!
The ball reaches 20.4 meters after 2 seconds
Exponents of Exponents ...

What about this example?

Exponents are special: they go top-down (do the exponent at the top first). So we calculate this way:
And finally, what about the example from the beginning?
297, 1566, 1572, 298, 1568, 381, 1570, 3854, 3855, 383

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"Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation. But, when you see something like ... ... what part should you calculate first? Start at the left and go to the right? Or go from right to left?

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Algebra Basics: The Distributive Property - Math Antics

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Error 1: Incorrect Substitution

Expression:
Evaluate 3 × (x + 4) when x = 5

Student Work:
3 × (x + 4) → 3 × (x) + 4 → 3 × 5 + 4 = 19

Why It's Wrong:
The student split the expression incorrectly, applying multiplication only to x, not to the whole parenthetical expression.

Discussion Prompt:

“Why can’t we break the parentheses apart like this?”

“What operation should be done first?”

🔸 Error 2: Ignoring Parentheses

Student Work:
3 × (x + 4) → 3 × x + 4 → 3 × 5 + 4 = 19

Why It's Wrong:
The student ignored the parentheses, treating it as 3x + 4 instead of distributing 3 to both x and 4.

Discussion Prompt:

“How do parentheses change the meaning of the expression?”

“Would 3x + 4 be the same as 3(x + 4)? Why or why not?”

🔸 Error 3: Adding Before Substituting

Student Work:
3 × (x + 4) → 3 × (x) + (4) → 3 × x = 15, then 15 + 4 = 19

Why It's Wrong:
The student mixed substitution and evaluation in the wrong order.

Discussion Prompt:

“What should we do first: substitute or evaluate?”

“Does x + 4 mean you wait to add until you know x?”

🔸 Error 4: Misapplying the Order of Operations

Student Work:
3 × (x + 4) = 3 × x + 4 = 3 × 5 + 4 = 19
(Doesn’t follow PEMDAS strictly)

Why It's Wrong:
This shows a lack of understanding of the precedence of parentheses.

Discussion Prompt:

“What does PEMDAS say about expressions inside parentheses?”

“Can you evaluate the expression in a different order?”

🔸 Error 5: Incorrect Arithmetic

Student Work:
3 × (x + 4) → 3 × (5 + 4) = 3 × 8 = 24

Why It's Wrong:
Simple arithmetic mistake (5 + 4 = 9, not 8).

Discussion Prompt:

“Let’s recheck our addition. What’s 5 + 4?”

“Even if your process was right, what went wrong here?”