Order of operations and arithmetic properties

Begin with repeated changes, then explain the signs

Two changes of −3 give −3 + (−3) = −6, so (−3) × 2 = −6. Reversing the order of the factors gives the same product: 2 × (−3) = −6. Positive times negative, in either order, produces a negative result. We can model a positive whole-number count of changes directly; a negative multiplier needs an extension of this idea.

Keep multiplication consistent with splitting a sum into its parts. Since 2 + (−2) = 0, multiplying that whole sum by −3 gives zero. Multiplying the parts separately must also give zero: (−3) × 2 + (−3) × (−2) = 0. The first product is −6, so the second must be 6 to cancel it. Thus (−3) × (−2) = 6. This uses the distributive property, which we explore more fully below.

For two nonzero numbers, matching signs give a positive product and differing signs give a negative product. Find the size by multiplying the absolute values, then choose the sign. If either factor is zero, the product is zero: (−3) × 0 = 0. Zero is neither positive nor negative. Multiplying by −1 gives the opposite of the other factor.

Find a quotient by asking which product returns the starting value

For 12 ÷ (−3), ask: which number multiplied by −3 gives 12? The answer is −4, because (−4) × (−3) = 12. For (−12) ÷ (−3), the answer is 4, because 4 × (−3) = −12. Division by a nonzero number undoes multiplication by that number. Check that the quotient times the divisor gives the dividend.

A nonzero dividend and nonzero divisor with matching signs give a positive quotient; differing signs give a negative quotient. A zero dividend is different: 0 ÷ (−3) = 0, checked by 0 × (−3) = 0. Division by zero is undefined in ordinary arithmetic. For 12 ÷ 0, no number times zero gives 12. For 0 ÷ 0, every number times zero gives zero, so this test cannot choose a unique quotient. Neither case defines a quotient.

Use parentheses to show what belongs together

Suppose you have six loose stones and three groups of four stones. The total is 6 + 3 × 4. The groups contain twelve stones, and adding the six loose stones gives eighteen. If instead you have four groups, each containing six stones and another three, the total is (6 + 3) × 4, which gives thirty-six. The numbers are the same, but the arrangement is different.

Parentheses tell us to treat the enclosed expression as one quantity. In (6 + 3) × 4, add within the parentheses before multiplying. In 6 + 3 × 4, the usual convention gives multiplication priority over addition. Add parentheses when you need a different grouping or want to make the intended grouping visible. If parentheses contain other parentheses, work on the innermost group first.

Read the order in levels, including equal priorities

A positive whole number written as a small raised number can name repeated multiplication: 2³ means 2 × 2 × 2, which is 8. Here 2 is the base and 3 is the exponent; together they form a power. We only need this simple meaning here. To evaluate an expression, first handle grouped expressions using the same rules within them, then powers, then multiplication and division, and finally addition and subtraction.

Multiplication and division share one level. In an ungrouped chain at that level, work from left to right: 24 ÷ 6 × 2 = 4 × 2 = 8. Doing 6 × 2 first would silently change the expression into 24 ÷ (6 × 2). Addition and subtraction also share one level: 12 − 5 + 2 = 7 + 2 = 9. Doing 5 + 2 first would change it into 12 − (5 + 2). Read the written structure before choosing a shortcut.

Choose and explain the next operation

Use the explorer to choose a worked expression and select its next operation. Before selecting, look for parentheses and identify the highest remaining priority. In 6 + 3 × 4, multiply 3 × 4 first. In (6 + 3) × 4, add within the parentheses first. Explain why the same addition is ready in one expression but must wait in the other.

Try 24 ÷ 6 × 2 and 12 − 5 + 2. In each ungrouped chain of operations with the same priority, start on the left. Read the explanation if a choice is incorrect, then try again. A step replaces just the chosen part with its value and keeps the rest of the expression. Continue until one value remains. Use the worked choices to practise reading structure, rather than guessing an answer from the numbers.

Check which quantity a power applies to

For 2³, multiply three factors of 2; do not multiply 2 by 3. In 5 + 2³ × 3, the power gives 8 first, then 8 × 3 gives 24, then adding 5 gives 29. The raised number belongs to its base, not to every number before it.

Compare −3² and (−3)². Under the usual convention, −3² means the opposite of 3²: −(3 × 3) = −9. In (−3)², the parentheses make the whole negative number the base, so (−3) × (−3) = 9. Use parentheses to include a sign in the base. A leading minus and a power have different jobs; deciding their scope comes before calculating.

Independent parts may be evaluated in either order. In −3² + (−3)², you can work out either square first, but the addition waits for both values: −9 + 9 = 0. This does not change the grouping of a connected multiplication/division or addition/subtraction chain. It is a choice between two separate parts, each of which keeps its own structure.

Reorder or regroup when a property allows it

Adding 8 and 2 gives the same sum as adding 2 and 8. Multiplying 3 and 4 gives the same product as multiplying 4 and 3. These are the commutative properties of addition and multiplication: a + b = b + a, and a × b = b × a. The letters stand for numbers. Subtraction and division do not have these properties: 8 − 2 = 6 but 2 − 8 = −6; 8 ÷ 2 = 4 but 2 ÷ 8 is not 4.

Grouping can help too. Rewrite 7 + (3 + 8) as (7 + 3) + 8 to make ten first. Rewrite 4 × (25 × 3) as (4 × 25) × 3 to make one hundred first. The associative properties say that (a + b) + c = a + (b + c), and (a × b) × c = a × (b × c). They change grouping, not the factors or addends themselves. Subtraction and division are not associative: (12 − 5) − 2 = 5, but 12 − (5 − 2) = 9; (24 ÷ 6) ÷ 2 = 2, but 24 ÷ (6 ÷ 2) = 8.

Multiply each part when splitting a sum

Four bags each hold ten blue stones and two white stones. Count all stones with 4 × (10 + 2), or count the blue and white stones separately with 4 × 10 + 4 × 2. Both give 48. The factor four multiplies each part, not just the first. This is the distributive property: a × (b + c) = a × b + a × c. It also works over subtraction: a × (b − c) = a × b − a × c.

Keep signs attached to the parts you multiply. For (−2) × (5 − 3), calculating inside the parentheses gives (−2) × 2 = −4. Distributing gives (−2) × 5 − (−2) × 3 = −10 − (−6) = −4. A minus before a group means taking its opposite, which is multiplying the whole group by −1: −(5 − 3) = −5 + 3 = −2. In −(5 + 3), both addends change to their opposites, giving −5 − 3 = −8.

Keep every equality true

An equals sign states that the quantities on both sides have the same value. Write 6 + 3 × 4 = 6 + 12 = 18. Every expression in that chain names eighteen. If you want to record only the multiplication, write it separately: 3 × 4 = 12. Do not write 6 + 3 × 4 = 12 and then append + 6: the first equality would already be false.

Rewrite the whole remaining expression after each step, retaining the numbers and signs you have not yet used. A shortcut is valid when a property justifies the new expression. For example, 19 × 4 = (20 − 1) × 4 = 20 × 4 − 1 × 4 = 76. You changed the form while preserving the value; you did not change the question.

Check the value, the structure and the meaning

For six loose stones and three groups of four, eighteen is sensible: the three groups already contain twelve, and six more give eighteen. Thirty-six would describe a different grouping. A rough size check can catch a mistake, but it may not distinguish every error. Return to the original words or diagram and ask whether your expression represented them correctly.

Check a product by dividing by a nonzero factor, and a quotient by multiplying by its nonzero divisor. For (−3) × 4 = −12, (−12) ÷ 4 = −3 checks the product. For (−12) ÷ (−3) = 4, 4 × (−3) = −12 checks the quotient. You can also evaluate a distributed expression by the original grouping. These checks complement one another: a calculator can check arithmetic, but entering the wrong grouping still answers the wrong question.

Worked examples

Try it yourself

1. Evaluate 7 + 2 × 5 and (7 + 2) × 5. Which operation comes first in each?

   Show the answer and reasoning

   17 and 45 respectively. Multiply first in the first expression; add inside the parentheses first in the second.

   7 + 2 × 5 = 7 + 10 = 17. In the grouped expression, (7 + 2) × 5 = 9 × 5 = 45. The parentheses make seven plus two a single quantity to multiply.

2. Find 24 ÷ 6 × 2 and 12 − 5 + 2. Why should you not do 6 × 2 or 5 + 2 first?

   Show the answer and reasoning

   8 and 9. Each pair of operations shares a priority, so work left to right.

   24 ÷ 6 × 2 = 4 × 2 = 8; 12 − 5 + 2 = 7 + 2 = 9. Doing the suggested right-hand parts first adds an unwritten grouping and changes the values.

3. Find 2³, −3² and (−3)². Explain which number is the base in each power.

   Show the answer and reasoning

   8, −9 and 9 respectively. The bases are 2, 3 and −3.

   2³ = 2 × 2 × 2 = 8. In −3², square 3 and then take its opposite. In (−3)², multiply two factors of −3; their product is positive. Parentheses include the sign in the base.

4. Is (6 + 4) + 9 equal to 6 + (4 + 9)? Is (10 − 4) − 2 equal to 10 − (4 − 2)? Explain.

   Show the answer and reasoning

   The sums are both 19. The differences are 4 and 8, so they are unequal.

   Addition is associative, so regrouping the same addends preserves the sum. Subtraction is not associative: (10 − 4) − 2 = 6 − 2 = 4, whereas 10 − (4 − 2) = 10 − 2 = 8.

5. Find (−2) × (7 − 4) by distribution, then find −(7 − 4). Check both by calculating inside the parentheses.

   Show the answer and reasoning

   −6 and −3 respectively.

   (−2) × (7 − 4) = −14 − (−8) = −14 + 8 = −6, matching (−2) × 3. Also −(7 − 4) = −7 + 4 = −3, matching the opposite of 3. Distribute the factor to both parts.

6. Evaluate (−18) ÷ (−3) and 0 ÷ (−3). Are 18 ÷ 0 and 0 ÷ 0 defined? Explain using multiplication.

   Show the answer and reasoning

   6 and 0 respectively. Both divisions by zero are undefined.

   6 × (−3) = −18 and 0 × (−3) = 0. No number times zero gives 18. Every number times zero gives zero, so 0 ÷ 0 does not identify one quotient. A valid inverse division check needs a nonzero divisor.

What to carry forward

• Signed products follow consistent addition and distribution; signed quotients can be checked by multiplying by a nonzero divisor.
• Read grouping first, then powers, then multiplication and division, then addition and subtraction. Work left to right within each ungrouped chain at a shared level.
• A power applies to its base. −3² and (−3)² have different bases and different values.
• Addition and multiplication permit reordering and regrouping. Subtraction and division do not generally permit the same changes.
• Distribution multiplies every part of a sum or difference, including its sign. Keep each equality true and check the original meaning.