At a glance. One lesson of about 60 minutes. Students evaluate expressions with brackets, powers, multiplication, division, addition and subtraction, and explain why the order is needed.
Calculator: no calculator. Nothing needs one. Students are learning the convention itself, and different calculators handle it differently.
What students will learn
Know
- The order: brackets, powers, multiplication and division, addition and subtraction.
- Operations on the same level are done from left to right.
- Brackets are used to change the order.
Understand
- The order is an agreement, so that an expression has one value.
- Multiplication and division share a level, as do addition and subtraction.
- Brackets state which part is done first.
Do
- Evaluate expressions with up to four kinds of operation.
- Place brackets to make a given value.
- Spot and explain left-to-right errors.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: evaluating expressions in the correct order (questions 1 to 8).
- Criterion B. Investigating patterns: the stepping figure. Ask students to find a rule for which operation the figure chooses at each step.
- Criterion C. Communicating: explaining why a worked answer is wrong and writing each step on its own line (questions 6 and 9).
- Criterion D. Applying mathematics in real-life contexts: writing a cost as one expression (questions 5 and 6).
The one thing to do with the figure
Ask for the value of the first expression before pressing anything. Take a vote between 50 and 25. Then press Next step and ask students to name the rule that picked each operation.
On the 48 ÷ 6 × 2 expression, ask what the figure does, and why it does not do the multiplication first. The pairing of division with multiplication is the thing to name aloud.
On the counters figure, ask students to predict (a + b) × c before moving a slider, then to say which panel has more groups and why the totals differ.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook, the order of operations and the stepping figure with predictions. |
| 10 min | Key idea and common mistake; students step through three more expressions. |
| 10 min | Worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 9; 2² × 3²; 20. |
|---|---|
| Worked example | 120. |
| Faded example | 40. |
| 1. Multiplication before addition | B, 17. A adds first (left to right); C drops the 5. |
| 2. Multiply before subtract | 4. 10 − 6. |
| 3. Brackets first | 24. 8 × 3. |
| 4. Divide and multiply | 18. 6 × 3. |
| 5. Power in the middle | 22. 4 + 18. |
| 6. Placing brackets | A. B and C leave the multiplication first, giving 14. |
| 7. Several operations | 12. 5, 25, 50, 5, 12. |
| 8. Longer expression | 82. 100 − 20 + 2. |
| 9. Multiply and divide share a level | B. A multiplies first (24 ÷ 8 = 3); C stops after the division; D thinks multiplication outranks division. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1; 24 in question 2 | Works strictly left to right. | Underline the multiplication first and ask why. |
| 4 in question 3 | Ignores the brackets. | Ask what job the brackets do. |
| 2 in question 4 | Multiplied before dividing. | Say they share a level and work left to right. |
| 98 or 26 in question 5 | Added before applying the power, or before multiplying. | Write each step on a new line and name the rule. |
| B or C in question 6 | Added brackets that change nothing. | Compute each option. |
| 382 or 2 in question 8 | Subtracted first, or squared a product. | Rewrite with the power evaluated first. |
| A or D in question 9 | Multiplied first, or believes multiplication outranks division. | Ask which sign is met first going left to right in 24 ÷ 4 × 2, then compare with 24 ÷ (4 × 2). |
Questions worth asking
- Why do we need an order at all? So that an expression has one value for everybody.
- Does multiplication always come before division? No: they share a level and are done left to right.
- Do brackets ever change nothing? Yes, when what is inside would be done first anyway, as in 2 + (3 × 4).
- What if there are brackets inside brackets? Start with the innermost.
Differentiation
A way in for anyone stuck
- Give a card with the four levels and an example for each.
- Rewrite each expression on a new line after every step.
- Colour the operation being done in each line.
Stretch
- Question 7, then: use the numbers 2, 3 and 4 once each, with + and × and any brackets. What is the largest value you can make, and the smallest? (24, from 2 × 3 × 4, and 9, from 2 + 3 + 4.)
- Why is the order a convention and not a fact? Where might a different convention be used? (Programming languages and some calculators.)
Students learning in English as an additional language
“Order” here means sequence, not an instruction. “Evaluate” means work out the value. Point to the operation as you say each name.
| Word | Plain meaning | Example |
|---|---|---|
| expression | numbers and operations written together, with no equals sign | 5 + 3 × 4 |
| evaluate | work out the value | evaluate 5 + 3 × 4 to get 17 |
| operation | something done to numbers: add, subtract, multiply, divide | × is an operation |
| brackets | symbols that show which part to do first | (5 + 3) × 4 |
| power | repeated multiplication of a number by itself | 3² = 9 |
| left to right | the order for operations on the same level | 10 − 4 − 3 = 3 |
Sentence starters for explaining aloud or in writing:
- I do the … first because …
- … and … are on the same level, so I work …
- The brackets mean I …
Two things not to say
“Just use a calculator.” Calculators differ, and the point of this lesson is to read an expression the same way as everyone else.
“Multiplication before division.” They share a level, which is why the phrases BEDMAS and BODMAS can mislead.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- OrderWork out 2 + 4 × 5.
22. - Earlier workWhat is 2⁵?
32.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ShoppingKofi buys 3 notebooks at 6 dirhams each and 2 pens at 4 dirhams each. Write one expression for the total and evaluate it.
3 × 6 + 2 × 4 = 26 dirhams. - SharingTwo groups of 24 and 18 students share 6 buses equally. Write one expression for how many per bus and evaluate it.
(24 + 18) ÷ 6 = 7.
3Challenge
Reasoning, generalising and working backwards.
- BracketsPut one pair of brackets in 2 + 3 × 4 + 5 so that it equals 25.
(2 + 3) × 4 + 5 = 25. - Four foursUse four 4s, with operations and no brackets, to make 17.
4 × 4 + 4 ÷ 4 = 17.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Write each expression on the board and let students call out which operation goes next.