The four-step order of operations, the left-to-right rule for operations of equal priority, grouping symbols including the fraction bar, whether a calculator can be trusted to apply the order itself, and evaluating a multi-operation expression that models a real situation.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 1 — Connections to Algebra
Order of Operations
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 15-22 — the lesson these objectives are drawn from
Warm-up
Two people can follow the same expression honestly and disagree. That is the problem this lesson solves.
Discussion prompt
Work out 3 plus 4 times 5 in your head. Then work it out again pretending you must go strictly left to right. Do the two answers agree, and if not, which one should win?
Hint: There are only two possible answers here, and both of them are defensible until someone writes a rule down.
Answer:
\[ 3 + 4 \cdot 5 = 3 + 20 = 23 \qquad \text{versus} \qquad (3 + 4) \cdot 5 = 35 \]
Twenty-three is the agreed answer, because multiplication is carried out before addition. Nothing about arithmetic forces that choice — it is a convention, adopted so that every expression written anywhere has exactly one value. Once you see it as an agreement rather than a law, the four steps become much easier to remember.
Concept
An expression containing more than one operation would have several possible values if everybody evaluated it in their own order. The order of operations is the agreement that fixes a single value, and it is followed by every mathematician, every textbook and most calculators.
order of operations — The agreed sequence for evaluating an expression: grouping symbols first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right.
Grouping symbols exist so that you can override the default order deliberately when you need a different one.
Figure (svg): Three versions of the same digits with different groupings, giving three different answers
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 15-15
Section
Section 1
Concept
There are four steps and they are always attempted in the same sequence. A step with nothing to do is simply skipped, but no step is ever taken out of turn.
Notice that there are four steps for six operations, because two pairs of operations share a step.
Figure (svg): Four stacked steps showing the order of operations: grouping symbols, then powers, then multiplication and division left to right, then addition and subtraction left to right
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 15-15 — the Order of Operations key-concept box
Picture it
Each rung is narrower than the one above it, because each step has fewer candidates left to act on.
Figure (svg): Four stacked steps showing the order of operations: grouping symbols, then powers, then multiplication and division left to right, then addition and subtraction left to right
The two shared rungs are the interesting ones. Multiplication and division sit together, and addition and subtraction sit together, which is why the left-to-right rule is needed at all.
Worked example
This is Example 1 from the textbook, at x equal to 4. Three operations, and the order settles all of them.
\[ \text{Evaluate } 3x^2 + 1 \text{ when } x = 4. \]
Substitute 4 for x
Why: Substitution happens before the order of operations is applied, because the order applies to numbers.
\[ 3 \cdot 4 ^{2} + 1 \]
Evaluate the power
Why: Step two. There are no grouping symbols, so step one passes and the exponent goes first.
\[ 3 \cdot 16 + 1 \]
Multiply
Why: Step three. Three times sixteen is forty-eight.
\[ 48 + 1 \]
Add
Why: Step four, and the last thing that happens.
\[ 49 \]
Figure (svg): The solution to Worked example evaluate three x squared plus one shown as a ladder of expressions, one row per algebraic move
\[ 3x^2 + 1 = 3 \cdot 4^2 + 1 = 3 \cdot 16 + 1 = 49 \]
Verify: check what would have happened in the wrong order
Why: Multiplying before squaring would give twelve squared, which is 144, and adding one gives 145 — nowhere near 49. The gap between the two is large enough that a quick estimate catches the error, which is a good reason to estimate before computing.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 15-15
Ranking
Four steps, one correct sequence. Two of them are the ones people swap.
Put in order
Why: Grouping symbols come first because they are an explicit instruction to override the default order. Powers come next, which is the step most often skipped past. Multiplication and division share the third step, and addition and subtraction share the fourth. The pairing in those last two steps is why a left-to-right rule is needed at all.
Worked example
Guided Practice 1 to 4. Same routine four times, with the operations shuffled.
\[ \text{Evaluate at } x = 2: \quad 2x^2 + 5, \quad 8 - x^2, \quad 6 + 3x^3, \quad 20 - 4x^2. \]
Substitute 2 everywhere, then evaluate every power first
Why: Two squared is four and two cubed is eight, and those values are fixed before any multiplying starts.
Do the multiplications
Why: Two times four is eight; three times eight is twenty-four; four times four is sixteen. The second expression has no multiplication.
\[ 8, -, 24, 16 \]
Do the additions and subtractions last
Why: Eight plus five, eight minus four, six plus twenty-four, twenty minus sixteen.
\[ 13, 4, 30, 4 \]
Collect the four answers
Why: Every one followed the same four steps, and two of them landed on the same value by coincidence.
\[ 13, 4, 30, 4 \]
Figure (svg): The solution to Worked example four expressions at x equal to 2 shown as a ladder of expressions, one row per algebraic move
\[ 13, \quad 4, \quad 30, \quad 4 \]
Verify: re-check the two that agree
Why: Eight minus two squared is eight minus four, which is four. Twenty minus four times two squared is twenty minus sixteen, which is also four. Two different routes to the same number is a coincidence here, not an error — and confirming it separately is what tells you which.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 15-15
Trap
\[ 3x^2 \text{ at } x = 4 \]
Multiply the 3 by the 4 first, then square the result
Why: The 3 and the x are written next to each other, so the multiplication looks like it comes first.
\[ (3 \cdot 4)^2 = 12^2 = 144 \quad \text{(wrong)} \]
This is the same error as putting brackets in that were never written. Nothing in the expression grouped the 3 with the x.
\[ 3x^2 \text{ at } x = 4 \]
Evaluate the power before the multiplication, because powers are step two and multiplication is step three
Why: An exponent with no brackets attaches only to the symbol immediately in front of it, exactly as in Lesson 1.2.
\[ 3 \cdot 4^2 = 3 \cdot 16 = 48 \]
Forty-eight against one hundred and forty-four. Being adjacent on the page does not make two symbols a group — only a bracket does that.
Sorting
For each expression, name the operation that has to be carried out first.
Sort into buckets
Sort each expression by which step acts first.
Notice how often the first three steps simply pass. The order of operations is a priority list, not a set of tasks that all have to be performed.
Elimination
All four are attempts at the same expression, evaluated at x equal to 3.
Eliminate the wrong options
Which correctly evaluates 2x squared plus 4 at x equal to 3?
Survives elimination: A
Why: Three squared is nine, twice nine is eighteen, and eighteen plus four is twenty-two. The three wrong answers each move one operation out of turn, and each lands somewhere quite different — which is a compact demonstration that the order is doing real work rather than tidying up.
Prediction
Sometimes adding brackets makes no difference at all. Decide before you compute.
Predict first
Does 4 plus 5 times 2 have the same value as 4 plus the quantity 5 times 2?
Correct: Yes, both are 14.
\[ 4 + 5 \cdot 2 = 4 + 10 = 14 \qquad 4 + (5 \cdot 2) = 14 \]
\[ (4 + 5) \cdot 2 = 9 \cdot 2 = 18 \quad \text{— this is the one that differs} \]
Why: The default order already multiplies before adding, so bracketing the multiplication merely writes down what was going to happen anyway. Brackets only change a value when they force an operation to happen out of its normal turn — putting them round the 4 plus 5 instead would give 18, and that is the version that changes something.
Section
Section 2
Concept
Multiplication and division share a step, and addition and subtraction share a step. When two operations of the same priority meet, you carry them out in order from left to right, exactly as you read.
left-to-right rule — When operations have the same priority, perform them in order from left to right.
\[ 24 - 8 - 6 = (24 - 8) - 6 \qquad 15 \cdot 2 \div 6 = (15 \cdot 2) \div 6 \]
Multiplication does not beat division, and addition does not beat subtraction. Nothing but position separates them.
Figure (svg): Two chains showing that subtraction and division are worked from left to right, with 24 minus 8 minus 6 giving 10 and 15 times 2 divided by 6 giving 5
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 16-16 — the Left-to-Right Rule box and Example 2
Picture it
In both of these, working from the right would give a different and wrong answer.
Figure (svg): Two chains showing that subtraction and division are worked from left to right, with 24 minus 8 minus 6 giving 10 and 15 times 2 divided by 6 giving 5
Twenty-four minus eight minus six is ten, not twenty-two. Fifteen times two divided by six is five, not forty-five. Reading order is the whole rule.
Worked example
Example 2 from the textbook, all three parts. The third one mixes the tiers.
\[ \text{Evaluate } \; 24 - 8 - 6, \quad 15 \cdot 2 \div 6, \quad 16 + 4 \cdot 2 - 3. \]
In the first, subtract from left to right
Why: Both operations are subtractions, so they share a tier and position decides. Twenty-four minus eight is sixteen, and sixteen minus six is ten.
\[ 24 - 8 - 6 = 10 \]
In the second, multiply and divide from left to right
Why: Multiplication and division share a tier. Fifteen times two is thirty, and thirty divided by six is five.
\[ 15 \cdot 2 \div 6 = 5 \]
In the third, do the multiplication first because it outranks the other two
Why: Four times two is eight. Only now are the addition and subtraction in play.
\[ 16 + 8 - 3 \]
Then add and subtract from left to right
Why: Sixteen plus eight is twenty-four, and twenty-four minus three is twenty-one.
\[ 24 - 3 = 21 \]
Figure (svg): The solution to Worked example three left-to-right chains shown as a ladder of expressions, one row per algebraic move
\[ 24 - 8 - 6 = 10, \quad 15 \cdot 2 \div 6 = 5, \quad 16 + 4 \cdot 2 - 3 = 21 \]
Verify: re-run the third one with brackets written in
Why: Writing it as sixteen plus the quantity four times two, all minus three, gives the same twenty-one. Making the invisible brackets visible is the fastest way to confirm that the tiers were respected.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 16-16
Discrimination
For each expression, decide whether working right to left instead of left to right would change the answer.
Sort into buckets
Sort each expression by whether direction changes its value.
Worked example
Two expressions that look almost identical. One is order-sensitive and one is not.
\[ \text{Evaluate } \; 100 \div 5 \cdot 2 \; \text{ and } \; 100 \div 5 \div 2. \]
In the first, divide before multiplying because it stands further left
Why: Both operations share a tier, so position alone decides. One hundred divided by five is twenty.
\[ 20 \cdot 2 \]
Finish the first
Why: Twenty times two is forty.
\[ 40 \]
In the second, divide twice from left to right
Why: One hundred divided by five is twenty, then twenty divided by two is ten.
\[ 20 \div 2 \]
Finish the second
Why: The answer is ten, a quarter of the first answer.
\[ 10 \]
Figure (svg): The solution to Worked example where left-to-right actually matters shown as a ladder of expressions, one row per algebraic move
\[ 100 \div 5 \cdot 2 = 40 \qquad 100 \div 5 \div 2 = 10 \]
Verify: try the wrong order on the first one deliberately
Why: Multiplying first would give one hundred divided by ten, which is ten — the same answer as the second expression, and wrong for the first. That collision is exactly why the rule has to be stated rather than guessed.
Error analysis
The student was asked to evaluate two chains. Both answers are wrong, and both errors have the same cause.
Annotate
On: \( 24 - 8 - 6 = 24 - 2 = 22 \qquad 15 \cdot 2 \div 6 = 15 \cdot \tfrac{1}{3} = 5 \)
The second line is worth dwelling on. Checking answers alone would have passed it, and checking the method is what catches it.
Fill the middle
Rewriting a chain with brackets is the fastest way to make the left-to-right rule visible.
Fill in the blanks
24 - 8 - 6 \;=\; ( 24 - 8 ) - 6 \;=\; 16 - 6 \;=\; 10
Why: The bracket makes explicit what the left-to-right rule was already saying: the leftmost subtraction happens first. Twenty-four minus eight is sixteen, and sixteen minus six is ten. Writing the brackets in is not required, but it converts a rule you have to remember into a picture you can read.
Two truths and a lie
Look at the direction each chain has to be worked in.
Eliminate the wrong options
Which of these does NOT equal 10?
Survives elimination: D
Why: The fourth chain has three subtractions rather than two, and worked from left to right it gives 14, then 10, then 8. The first three all land on ten by different routes, which makes the odd one out genuinely worth checking rather than guessable from its shape.
Socratic
Addition and multiplication do not need a direction rule. Subtraction and division do.
Discussion prompt
Explain why the left-to-right rule never changes the answer for a chain of additions, but always matters for a chain of subtractions. Use a specific pair of examples in your answer.
Hint: Think about whether swapping two of the numbers changes anything.
Answer:
Addition can be done in any order without changing the total: two plus three plus four is nine however you group or reorder it. Subtraction cannot. Twenty minus six minus four is ten, but doing the right-hand subtraction first gives twenty minus two, which is eighteen — the four has switched from being taken away to being given back.
The same split runs through the other tier: multiplication is order-free and division is not. So the left-to-right rule is really a rule about the two operations that are order-sensitive, stated in a way that covers all four at once. Chapter 2 will give those order-free operations their proper names.
Section
Section 3
Concept
Parentheses and brackets are the obvious grouping symbols. A fraction bar is a third one: it groups everything above it and everything below it, then divides, even though no brackets are written.
\[ \frac{1 + 2}{4 - 1} = (1 + 2) \div (4 - 1) = 3 \div 3 = 1 \]
Rewriting a fraction with brackets before evaluating it removes every ambiguity at once.
Figure (svg): A fraction bar shown acting as a grouping symbol, with the numerator and denominator each bracketed before the division happens
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 16-16 — the note that a fraction bar can act as a grouping symbol
Picture it
The bar does two jobs at once: it divides, and it groups both of the things it is dividing.
Figure (svg): A fraction bar shown acting as a grouping symbol, with the numerator and denominator each bracketed before the division happens
This is why a numerator with three terms in it does not need brackets on the page — the bar has already supplied them. Written on one line, they become compulsory.
Worked example
Example 3 from the textbook. Numerator and denominator are each finished before the division happens.
\[ \text{Evaluate } \; \frac{8 + 7 \cdot 4}{7^2 - 1}. \]
Treat the numerator as its own bracketed expression
Why: Multiplication before addition inside it: seven times four is twenty-eight, and eight plus twenty-eight is thirty-six.
\[ \text{numerator } = 36 \]
Treat the denominator as its own bracketed expression
Why: The power first: seven squared is forty-nine, and forty-nine minus one is forty-eight.
\[ \text{denominator } = 48 \]
Now do the division the bar was asking for
Why: Thirty-six over forty-eight, which is a fraction that can be reduced.
\[ \frac{36}{48} \]
Simplify the fraction
Why: Twelve divides both, giving three over four.
\[ \frac{3}{4} \]
Figure (svg): The solution to Worked example an expression with a fraction bar shown as a ladder of expressions, one row per algebraic move
\[ \frac{8 + 7 \cdot 4}{7^2 - 1} = \frac{36}{48} = \frac{3}{4} \]
Verify: rewrite it on one line with brackets and re-evaluate
Why: Written as the quantity eight plus seven times four, all divided by the quantity seven squared minus one, the same two numbers appear and the same quotient results. If the one-line version needs brackets that the fraction did not, the bar was doing exactly the grouping job claimed for it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 16-16
Translation
Each fraction on the left has to be rewritten on one line. Pair it with the correct version.
Match the pairs
Why: Every multi-term numerator and every multi-term denominator needs its own bracket once the bar is gone. The first two show that it matters which side the sum is on: one plus two over three is one, while one over two plus three is one fifth. The bar keeps those apart visually, and only brackets can keep them apart on one line.
Worked example
Guided Practice 7. When one grouping sits inside another, the innermost is always finished first.
\[ \text{Evaluate } \; \frac{x^2 + 1}{2} + x - 5 \; \text{ when } x = 1. \]
Substitute 1 for x throughout
Why: Every occurrence of the letter is replaced, including the one inside the fraction.
\[ \frac{1 ^{2} + 1}{2} + 1 - 5 \]
Finish the numerator
Why: The power first, then the addition: one squared is one, and one plus one is two.
\[ \frac{2}{2} + 1 - 5 \]
Do the division the bar asks for
Why: Two over two is one.
\[ 1 + 1 - 5 \]
Add and subtract from left to right
Why: One plus one is two, and two minus five is negative three.
\[ -3 \]
Figure (svg): The solution to Worked example nested grouping symbols shown as a ladder of expressions, one row per algebraic move
\[ \frac{1^2 + 1}{2} + 1 - 5 = 1 + 1 - 5 = -3 \]
Verify: check the sign of the answer against the sizes
Why: The two positive contributions total two, and five is being taken away, so the answer must land three below zero. A negative answer here is correct rather than a warning sign — Chapter 2 is entirely about that side of the number line.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 16-16
Trap
\[ \frac{1 + 2}{4 - 1} \;\rightarrow\; 1 + 2 \div 4 - 1 \]
Rewrite the fraction on one line by replacing the bar with a division sign
Why: The bar looks like a single operation, so a single sign seems like a fair swap.
\[ 1 + 2 \div 4 - 1 = 1 + 0.5 - 1 = 0.5 \quad \text{(wrong)} \]
The correct value is one. The rewrite silently discarded two sets of brackets that the bar was providing free of charge.
\[ \frac{1 + 2}{4 - 1} \;\rightarrow\; (1 + 2) \div (4 - 1) \]
Replace the bar with a division sign and a pair of brackets on each side
Why: The bar was doing three jobs — group above, group below, divide — so the one-line version needs three symbols, not one.
\[ (1 + 2) \div (4 - 1) = 3 \div 3 = 1 \]
This is the single most common error when typing mathematics into a calculator or a spreadsheet, where everything has to go on one line.
Elimination
The fraction has 4 plus 8 on top and 2 underneath.
Eliminate the wrong options
Which one-line expression has the same value as that fraction?
Survives elimination: A
Why: The numerator is 4 plus 8, which is 12, and twelve divided by two is six. Any one-line version that does not bracket the whole numerator divides only part of it, and each of the three wrong options divides a different part — landing on 8, 8 and 10 respectively rather than 6.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Symbol | What it groups | Worked first? |
|---|---|---|
| parentheses | everything inside them | Yes, innermost first |
| fraction bar | everything above it and everything below it | Yes, both parts before dividing |
| no symbol at all | nothing — the default order applies | No |
The third row is the one worth remembering. Symbols written next to each other are not grouped merely by being adjacent; only an actual grouping symbol groups them.
Edge cases
Find an example where dropping the bar's brackets does the most damage.
Discussion prompt
Invent a fraction whose one-line version, written without any brackets, gives an answer more than ten times too large. Show both values, and say what feature of your fraction made the gap so big.
Hint: A denominator with a subtraction in it that nearly cancels is a good place to look.
Answer:
\[ \frac{20}{10 - 9} = \frac{20}{1} = 20 \qquad \text{but} \qquad 20 \div 10 - 9 = 2 - 9 = -7 \]
Here the unbracketed version is not merely wrong but the wrong sign. The damage is largest when the denominator is a difference that comes out small: the true division is by a small number and therefore large, while the unbracketed version divides by the first term only and then subtracts.
This is exactly the failure mode that produces wildly wrong answers in spreadsheets, where everything is typed on one line and no bar is available to do the grouping for you.
Section
Section 4
Concept
A scientific or graphing calculator normally applies the order of operations for you. A simple four-function calculator often performs each operation the moment you press the next key, which is strictly left to right. You need to know which one you are holding.
There is a one-line test that settles it, and it takes about five seconds.
Figure (svg): Two calculator displays for the same keystrokes, one showing 6 because it obeys the order of operations and one showing 1 because it does not
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 17-17 — Example 4, Use a Calculator
Picture it
Neither machine is broken. They are following two different conventions.
Figure (svg): Two calculator displays for the same keystrokes, one showing 6 because it obeys the order of operations and one showing 1 because it does not
The machine on the right is not wrong about arithmetic — it is doing exactly what it was asked, one operation at a time. It is simply not applying the agreement, so you have to apply it for it.
Worked example
Example 4 from the textbook. Work out both possible displays by hand before you press anything.
\[ \text{Enter } 10 - 6 \div 2 - 1. \text{ Does the display show } 6 \text{ or } 1? \]
Work it out using the order of operations
Why: Division is step three and comes before the two subtractions: six divided by two is three.
\[ 10 - 3 - 1 \]
Finish the subtractions left to right
Why: Ten minus three is seven, and seven minus one is six.
\[ 10 - 3 - 1 = 6 \]
Now work it out strictly left to right instead
Why: Ten minus six is four, four divided by two is two, and two minus one is one.
\[ ((10 - 6) \div 2) - 1 \]
Compare the two results
Why: Six against one. The two conventions give different displays, which is exactly what makes this a usable test.
\[ 6\text{ or } 1 \]
Figure (svg): The solution to Worked example the calculator test shown as a ladder of expressions, one row per algebraic move
\[ 10 - 6 \div 2 - 1 = 6 \qquad ((10 - 6) \div 2) - 1 = 1 \]
Verify: check the left-to-right version step by step
Why: Ten minus six is four; four divided by two is two; two minus one is one. And with the order applied, six divided by two is three, ten minus three is seven, seven minus one is six. Two different numbers from one set of keystrokes, which is the point of the test.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 17-17
Prediction
A four-function calculator that performs each operation as the next key is pressed.
Predict first
You enter 2 plus 3 times 4 on such a calculator. What does it display?
Correct: 20.
\[ \text{left to right: } (2 + 3) \cdot 4 = 20 \]
\[ \text{order of operations: } 2 + (3 \cdot 4) = 14 \]
Why: The machine adds two and three the moment the times key is pressed, giving five, and then multiplies by four to give twenty. A calculator that applies the order of operations would multiply first and display fourteen. The gap between the two displays is exactly the value of knowing which machine you are holding.
Worked example
If your calculator does not apply the order, you have to supply it — either with brackets or by reordering.
\[ \text{Enter } 3 + 4 \cdot 5 \text{ on a calculator that works strictly left to right.} \]
Work out what the correct answer is by hand
Why: Multiplication before addition: four times five is twenty, and three plus twenty is twenty-three.
\[ \text{correct answer } 23 \]
Notice what the naive keystrokes would give
Why: Three plus four is seven, and seven times five is thirty-five. Wrong by twelve.
\[ \text{naive result } 35 \]
Reorder so that the higher-priority operation is entered first
Why: Enter four times five, then add three. Same expression, keystrokes rearranged.
\[ 4 \cdot 5 + 3 \]
Or use brackets if the calculator has them
Why: Bracket keys let you write the expression as it appears on the page.
\[ 3 + (4 \cdot 5) \]
Figure (svg): The solution to Worked example entering an expression on a left-to-right machine shown as a ladder of expressions, one row per algebraic move
\[ 3 + 4 \cdot 5 = 23 \quad \text{entered as} \quad 4 \cdot 5 + 3 \]
Verify: check that the reordering was legal
Why: Addition may be carried out in any order, so moving the three from the front to the back does not change the value. That permission is what makes the reordering safe here — it would not be safe if the leading term were being subtracted.
Trap
\[ 10 - 6 \div 2 - 1 \]
Type the expression exactly as it appears and copy down whatever the display says
Why: The calculator is a machine, so it is assumed to be right by definition.
On a simple four-function calculator that produces 1, and the answer is copied out with complete confidence. The machine did what it was asked; the assumption about what it was asked was wrong.
\[ 10 - 6 \div 2 - 1 = 6 \]
Run the five-second test once, then trust the machine accordingly
Why: Testing the tool is cheaper than checking every answer it produces.
If the test gives the order-of-operations answer, you may type expressions as written. If it does not, use brackets or reorder the keystrokes yourself. Either way you now know which of the two you are doing.
Matching
The same four expressions entered on both kinds of calculator.
Match the pairs
Why: Each expression gives two different displays depending on the machine. The order-of-operations answers come from doing the multiplication or division first; the left-to-right answers come from performing each operation as it is entered. Neither machine is faulty — they are following different conventions, and only one of them matches the convention your teacher is using.
Constraint
Your calculator works strictly left to right and has no bracket keys at all.
Discussion prompt
How would you enter the fraction with 4 plus 8 on top and 2 underneath, so that the display shows the correct value of 6? Give the keystrokes in order, and explain why your reordering is legal.
Hint: You can always finish the numerator before you start dividing.
Answer:
Enter four, plus, eight, equals — the display now shows 12, the finished numerator. Then press divide, two, equals, and the display shows 6.
The reordering is legal because pressing equals forces the machine to complete everything entered so far, which is exactly what a bracket does. On a left-to-right calculator the equals key is the bracket key, and using it deliberately is how you supply grouping the machine will not supply for you.
Estimation
An estimate is a check on the calculator, not just on your arithmetic.
Predict first
You need 48 divided by the quantity 4 plus 2. Roughly what should the answer be?
Correct: About 8.
\[ \frac{48}{4 + 2} = \frac{48}{6} = 8 \qquad \text{but} \qquad 48 \div 4 + 2 = 14 \]
Why: The denominator is six, and forty-eight divided by six is exactly eight. If a calculator displays fourteen, it divided by four and then added two — the classic missing-bracket result — and the estimate catches that instantly. Estimating before computing turns the calculator from an authority into something you can check.
Section
Section 5
Concept
A word problem with several quantities in it becomes a single expression, and the order of operations is what keeps its parts in the right sequence. Writing one expression rather than several separate sums is what makes the reasoning checkable.
The order of operations does the bookkeeping so you do not have to hold intermediate results in your head.
Figure (svg): A bar model comparing your score of 8 field goals and 2 free throws with your friend's 4 field goals, showing a 10 point lead
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 17-17 — Example 5, Evaluate a Real-Life Expression
Picture it
Field goals are worth two points and free throws one point.
Figure (svg): A bar model comparing your score of 8 field goals and 2 free throws with your friend's 4 field goals, showing a 10 point lead
Your eighteen against their eight is a ten point lead, and every one of those four numbers came from a different step of the same expression.
Worked example
Example 5 from the textbook. You make 8 field goals and 2 free throws; your friend makes half as many field goals as you and no free throws.
\[ \text{Evaluate } \; 8 \cdot 2 + 2 \cdot 1 - \tfrac{8}{2} \cdot 2. \]
Read what each piece counts before evaluating anything
Why: Eight field goals at two points, two free throws at one point, and the friend's four field goals at two points.
Do the multiplications and the division, left to right
Why: Eight times two is sixteen; two times one is two; eight over two is four, and four times two is eight.
\[ 16 + 2 - 8 \]
Add and subtract from left to right
Why: Sixteen plus two is eighteen, and eighteen minus eight is ten.
\[ 18 - 8 \]
State the answer in the language of the question
Why: The number ten is a lead in points, not a score.
\[ 10\text{ points ahead} \]
Figure (svg): The solution to Worked example how many points ahead are you shown as a ladder of expressions, one row per algebraic move
\[ 8 \cdot 2 + 2 \cdot 1 - \tfrac{8}{2} \cdot 2 = 16 + 2 - 8 = 10 \]
Verify: compute both scores separately and subtract
Why: Your score is sixteen plus two, which is eighteen. Your friend's is four field goals at two points, which is eight. Eighteen minus eight is ten, matching the single-expression answer — and the fact that the two routes agree is what confirms the expression was assembled correctly.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 17-17
Translation
Four descriptions, four expressions. Two of them need a bracket the words never mention.
Match the pairs
Why: The second needs no bracket because the order of operations already multiplies each count by its point value before adding. The third and fourth do need brackets, because a count has to be completed before it is converted into points. The test is always the same: ask which quantity must be finished before the next operation is meaningful.
Worked example
Guided Practice 8. The phrase three times as many plus one has to become symbols in the right order.
\[ \text{Your friend makes } 4 \text{ field goals. You make three times as many plus one. Find your score.} \]
Translate the phrase into a count of field goals
Why: Three times as many as four is twelve, and one more makes thirteen. The plus one is one extra field goal, not one extra point.
\[ 3 \cdot 4 + 1 = 13\text{ field goals} \]
Convert field goals into points
Why: Each field goal is worth two points, so multiply the count by two.
\[ 13 \cdot 2 \]
Evaluate
Why: Thirteen times two is twenty-six.
\[ 26 \]
Check the unit of the answer
Why: The question asked for points, and the last multiplication converted goals into points.
\[ 26\text{ points} \]
Figure (svg): The solution to Worked example three times as many, plus one shown as a ladder of expressions, one row per algebraic move
\[ (3 \cdot 4 + 1) \cdot 2 = 13 \cdot 2 = 26 \text{ points} \]
Verify: test the alternative reading and rule it out
Why: Reading the plus one as one extra point instead would give twelve goals worth twenty-four points, plus one, which is twenty-five. The wording says three times as many field goals plus one, so the extra one is a field goal — and noticing that the two readings differ is more valuable than either answer.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 17-17
Trap
\[ 3 \cdot 4 + 1 \cdot 2 = 12 + 2 = 14 \]
Write the phrase down symbol by symbol as it is spoken, then evaluate
Why: Three times four plus one times two reads naturally in that order, so it gets written in that order.
Fourteen points is not the answer to any question that was asked. The expression says something different from the sentence it came from.
\[ (3 \cdot 4 + 1) \cdot 2 = 13 \cdot 2 = 26 \]
Decide what is being counted, finish that count, and only then convert
Why: The field goals have to be totalled before the conversion to points, so the total needs a bracket round it.
Word problems almost always need at least one bracket that the English never mentioned. Ask what quantity has to be complete before the next operation makes sense, and put a bracket round it.
Missing information
A question can be perfectly well written and still be unanswerable.
Discussion prompt
You make 8 field goals and 2 free throws. Your friend makes half as many field goals as you. How many points ahead are you? Say exactly what one piece of information is missing, and give the two different answers the question has depending on how that gap is filled.
Hint: Look at what the original textbook problem said about the friend that this version does not.
Answer:
The number of free throws your friend made is missing. The original problem said no free throws, which fixes the answer; without that, the friend's score is unknown.
\[ \text{friend with no free throws: } 18 - 8 = 10 \qquad \text{friend with 2 free throws: } 18 - 10 = 8 \]
A word problem is not complete until every quantity in the expression has a value. Noticing the gap is a real skill, and it is the same skill as noticing which letter in a formula has no number against it.
Elimination
A cinema charges 9 dollars per adult ticket and 6 dollars per child ticket. You buy 2 adult and 3 child tickets.
Eliminate the wrong options
Which expression gives the total cost in dollars?
Survives elimination: A
Why: Each count is multiplied by its own price and the two amounts are then added, which the default order of operations does without needing any brackets at all: eighteen plus eighteen is thirty-six dollars. The three wrong options each combine a count with a price at the wrong moment, and each one fails a units check before the arithmetic is reached.
Socratic
You could always work a word problem out in separate pieces instead of building one expression.
Discussion prompt
Give one advantage of writing a word problem as a single expression rather than as a series of separate calculations, and one situation where separate calculations would genuinely be the better choice.
Hint: Think about what happens when one of the numbers in the problem changes.
Answer:
The single expression records the whole method, not just the answers. If a number changes — the friend scores six field goals instead of four — you substitute and re-evaluate rather than redoing the reasoning. That is the beginning of the idea of a formula, and it is why Lesson 1.6 builds a whole problem-solving plan around it.
Separate calculations are better when the intermediate quantities are themselves interesting, or when they will be needed again later. Working out your score and your friend's score separately gives you two numbers you might actually want, rather than one lead you might not.
Comparison
Fill the blanks from memory before you scroll back. The third column is the one that decides ties.
Comparison matrix
| Step | Operations | Tie-breaker within the step |
|---|---|---|
| 1 | grouping symbols | innermost pair first |
| 2 | powers | no ties possible here |
| 3 | multiplication and division | left to right |
| 4 | addition and subtraction | left to right |
Steps three and four each hold two operations of equal rank, which is the entire reason the left-to-right rule has to exist. Steps one and two never need it.
Pattern
Whether the question is an arithmetic chain, a fraction, a calculator entry or a word problem, the same five moves cover it.
Step two is the one most often skipped, and it is the step that prevents the largest errors — a lost denominator bracket can change an answer by a factor of ten or flip its sign.
OpenStax Elementary Algebra 2e, §1.2 Use the Language of Algebra §1.2
Check
The basic order. Solve it before you click.
Check your understanding
Evaluate the expression 5 plus 2 times 3 squared.
Answer: A
Why: Powers come before multiplication, so three squared is nine first. Then two times nine is eighteen, and five plus eighteen is twenty-three. Each of the three steps happened in its own turn, with nothing taken out of sequence.
Check
The left-to-right rule. Work along the chain in order.
Check your understanding
Evaluate the expression 36 divided by 6 times 3.
Answer: A
Why: Multiplication and division share a step, so position decides: thirty-six divided by six is six, and six times three is eighteen. Neither operation outranks the other, which is exactly what the left-to-right rule is for.
Check
A fraction bar. Rewrite it with brackets before you choose.
Check your understanding
Evaluate the fraction whose numerator is 3 plus 9 and whose denominator is 5 minus 3.
Answer: A
Why: The bar groups both parts, so the numerator is twelve and the denominator is two, and twelve divided by two is six. Rewriting it as the quantity three plus nine, all divided by the quantity five minus three, makes both groupings explicit before any arithmetic happens.
Real world
You are entering a formula into a spreadsheet to work out an average. The three scores are in cells you have already added together, and the total is 240 over 3 people.
Discussion prompt
You type the total divided by three, then add a five point bonus, all on one line. Write out the two different expressions someone could mean by that description, evaluate both, and say which pair of brackets settles the ambiguity.
Hint: Does the bonus apply to the average, or is it part of the total before dividing?
Answer:
\[ 240 \div 3 + 5 = 80 + 5 = 85 \]
\[ (240 + 5) \div 3 = 245 \div 3 \approx 81.7 \]
The first adds the bonus to the average; the second adds it to the total before averaging. Both are reasonable things to want, and the English description does not distinguish them — only brackets do.
This is the single most common source of wrong numbers in real spreadsheets. A fraction on paper cannot be ambiguous because the bar groups for you; on one line, every grouping has to be typed.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
In the expression 8 divided by 4 times 2, does the multiplication happen before the division?
Correct: No, they share a step and the leftmost one goes first.
\[ 8 \div 4 \cdot 2 = 2 \cdot 2 = 4 \]
\[ 8 \div (4 \cdot 2) = 8 \div 8 = 1 \quad \text{— only correct if the brackets are written} \]
Why: Eight divided by four is two, and two times two is four. Doing the multiplication first would give eight divided by eight, which is one — a different answer, so it certainly makes a difference. The belief that multiplication outranks division is extremely common and comes from mnemonics that list the two operations in a fixed order without saying they are equal in rank.
Explain it
They are confident with each operation on its own and have never had to combine three of them in one line.
Discussion prompt
In no more than four sentences, explain why the order of operations exists at all — not what it says, but why anybody needed to agree on it. Then give them one expression whose value depends on the agreement, and say what the two possible answers are.
Hint: Start from what would happen if two people evaluated the same expression without an agreement.
Answer:
A usable answer: without an agreement, the same written expression would mean different things to different people, so nothing written down could be relied on. The order of operations is a convention, like driving on an agreed side of the road — the particular choice matters less than everyone making the same one.
A good example is three plus four times five. Without the agreement it could be twenty-three or thirty-five, and with the agreement it is twenty-three. Brackets then exist so that a writer who genuinely wants thirty-five can say so unambiguously.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: The power step is fixed by writing the four steps out from memory once a day for a week. Direction is fixed by inserting the invisible brackets on the page before evaluating. Fraction bars are fixed by making the rewrite automatic, every single time, even when the numerator has one term. Word problems are fixed by asking which quantity has to be complete before the next operation is meaningful, and bracketing that. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
Down the left of a page, write the four steps of the order of operations as a numbered ladder, and mark the two steps that hold a pair of equal-ranked operations. To the right of each step, write one short expression whose value depends on that step being done in its turn, and evaluate it. Across the bottom, write one fraction with more than one term above and below the bar, rewrite it on a single line with all its brackets shown, and evaluate it both ways to confirm they agree. Finally, in the margin, write the five-second calculator test and both of its possible displays.
Your two evaluations of the fraction must agree. If they do not, the one-line version is missing a bracket — almost always around the denominator.
Recap
Five things, and the second one is the one that quietly costs the most marks when it is missed.
| If the question says | Your first move is |
|---|---|
| Evaluate this expression | Substitute, then look for grouping symbols |
| Use the order of operations | Write the four steps down the margin |
| Simplify the fraction | Bracket the numerator and the denominator |
| Enter this in your calculator | Run the five-second test first |
| How many points ahead | Build one expression, then check its brackets |
Lesson 1.4 introduces equations and inequalities, where an expression is set equal to something and you have to decide whether a given number makes the statement true. Evaluating correctly is the whole of that decision, so this lesson is the tool the next one uses.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.3 Order of Operations §1.3, pp. 15-22 — everything on these slides traces back here
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