You will learn
- State the order.
- Evaluate expressions.
- Compare results with and without parentheses.
- Spot errors.
Version 0.2 · expanded edition · built from cited public sources Content is based on the official frameworks and sources listed under Sources and standards. No official endorsement is claimed. Written in the style of common primary school assessments. No exam result is promised.
Student lesson
- Brackets first.
- Multiply and divide next, left to right.
- Add and subtract last, left to right.
- Brackets can change the answer.
Words to know
Sequence of steps.
Brackets.
Before add/subtract.
Find the value.
1. Discover: 8 + 3 × 5
Is 8 + 3 × 5 equal to 55 or 23?
2. Understand: order of operations
Parentheses, then × and ÷, then + and − (left to right).
- 8 + 3 × 5 = 8 + 15 = 23.
- (8 + 3) × 5 = 11 × 5 = 55.
- 36 ÷ 6 + 4 = 6 + 4 = 10.
- 10 − 2 × 3 + 1 = 10 − 6 + 1 = 5.
Idea 1: Without brackets
9 + 6 ÷ 3 = 9 + 2 = 11. 40 − 5 × 6 = 40 − 30 = 10.
Idea 2: With brackets
(9 + 6) ÷ 3 = 15 ÷ 3 = 5. 4 × (7 − 2) + 1 = 20 + 1 = 21.
Idea 3: Left to right
24 ÷ 4 × 2 = 6 × 2 = 12.
Why left to right? Multiplication and division are equal in rank.
3. Worked examples
Example 1 (I show you): Evaluate
9 + 6 ÷ 3
- Divide first.
Answer: 11
Example 2 (We do it together): Evaluate
4 × (7 − 2) + 1
- Brackets, ×, then +.
Answer: 21
Example 3 (You try): Evaluate
24 ÷ 4 × 2
Show the steps and answer
- Left to right.
Answer: 12
Did you know?
Different orders give different answers, so we agree on one.
Why this matters
Every calculator and computer follows these rules.
4. Explore: evaluate
You need paper.
- Write 8 + 3 × 5.
- Multiply first.
- Add.
- Compare with (8 + 3) × 5.
Think about it: what did you notice?
23 and 55.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- First I do ___.
- Next I do ___.
- Last I do ___.
- The answer is ___.
Guided practice
Question 1. 8 + 3 × 5?
23.
Question 2. (8 + 3) × 5?
55.
Question 3. 10 − 2 × 3 + 1?
5.
Independent practice
Level 1: Start
- 9 + 6 ÷ 3
- (9 + 6) ÷ 3
- 40 − 5 × 6
- 4 × (7 − 2) + 1
- 24 ÷ 4 × 2
Level 2: Grow
- 6 + 3 × 2
- (6 + 3) × 2
- 20 − 10 ÷ 5
Level 3: Stretch
- 2 × (3 + 4) + 5 × 2
- 100 ÷ (5 + 5) − 4
- Ben says 8 + 3 × 5 = 55. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
A pupil says 8 + 3 × 5 = 55. What is the error?
Show a sample answer
Added first; it is 23.
Another one. Ana buys 2 books at 6 each and a pen for 3. Write and evaluate the cost.
Show a sample answer
2 × 6 + 3 = 15
7. Common mistakes
Fix: Multiply before adding.
Fix: Multiply and divide first.
Fix: Brackets come first.
8. Lesson summary and revision card
- I can state the order.
- I can evaluate expressions.
- I can compare results with and without parentheses.
- I can spot errors.
9. Check and master
Answer all 14 questions. You will see feedback after each one.
Your result appears here after you answer every question.
10. Key takeaways
- Parentheses.
- × ÷.
- + −.
11. Next lesson
Next lesson: Unknown quantities. Keep going with the next idea in this course.
Exam practice
These questions are written in the style of common school and international primary assessments. Try them on your own, without looking back. Then check your answers in the marking guide. The marks are shown in brackets.
- Find 9 + 6 ÷ 3. [2 marks]
- Find (9 + 6) ÷ 3. [2 marks]
- Find 40 − 5 × 6. [2 marks]
- Find 4 × (7 − 2) + 1. [3 marks]
- Ali says 24 ÷ 4 × 2 = 3. Is he right? Explain. [3 marks]
Total: 12 marks.
Marking guide and model answers
- 6 ÷ 3 = 2 [1]; 11 [1]
- 9 + 6 = 15 [1]; 5 [1]
- 5 × 6 = 30 [1]; 10 [1]
- 7 − 2 = 5 [1]; 4 × 5 = 20 [1]; 21 [1]
- No [1]. Left to right: 6 × 2 [1]. The answer is 12 [1].
Score guide: 11 to 12 = secure. 8 to 10 = nearly there, revise the parts you missed. Under 8 = work through the lesson again with a grown-up.
Printable worksheet
- 9 + 6 ÷ 3
- (9 + 6) ÷ 3
- 40 − 5 × 6
- 4 × (7 − 2) + 1
- 24 ÷ 4 × 2
- 8 + 3 × 5 =
- (8 + 3) × 5 =
- 36 ÷ 6 + 4 =
- 10 − 2 × 3 + 1 =
- 24 ÷ 4 × 2 =
Homework
Write five expressions needing the order of operations.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| 9 + 6 ÷ 3 | 11 | 2 |
| (9 + 6) ÷ 3 | 5 | 2 |
| 40 − 5 × 6 | 10 | 2 |
| 4 × (7 − 2) + 1 | 21 | 2 |
| 24 ÷ 4 × 2 | 12 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 11; 2 5; 3 10; 4 21; 5 12.
Quiz answers: 1 A, 2 B, 3 C, 4 A, 5 B, 6 A, 7 B, 8 C, 9 A, 10 B, 11 C, 12 A, Challenge 1 C, Challenge 2 C.
Extra worksheet answers (items 6 onward): 6 23; 7 55; 8 10; 9 5; 10 12.
Independent practice answers. Level 1: 1 11; 2 5; 3 10; 4 21; 5 12. Level 2: 1 12; 2 18; 3 18. Level 3: 1 24; 2 6; 3 He added first. Multiply first: 8 + 15 = 23..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 State the order | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O2 Evaluate expressions | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O3 Compare results with and without parentheses | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O4 Spot errors | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
Courseversity badge thresholds (overall quiz score): Gold = 90–100%; Silver = 80–89.9%; Bronze = 70–79.9%. A badge is never a full record of progress.
If a learner struggles
- Find the misconception: Left to right only. Multiply before adding.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Practise two operations.
- Guided practice: work through one example together and let the learner try the next one.
- Reassess: use the quiz again or the worksheet.
- Move on when the learner shows Silver-level evidence independently.
For parents and guardians
In this lesson your child practises the order of operations. You do not need to be good at mathematics to help.
- Make up calculations for each other.
- Ask: what comes first?
- Compare with a calculator.
- Ask “how do you know?” after each answer, and listen.
- Praise effort, not only right answers. Mistakes are how we learn.
- Ten to fifteen minutes a day is better than one long session.
Teacher package
1–5. Objectives, prerequisites, materials, preparation, duration
Lesson objectives. Learners will: state the order; evaluate expressions; compare results with and without parentheses; spot errors.
Prerequisite knowledge. Lesson 3.
Materials. Counters (stones, seeds or bottle tops), paper, pencils, board or chalkboard. Low-resource options: use stones, seeds, sticks, bottle tops or drawings in the sand or on the board instead of printed materials.
Preparation. Gather the counters and draw the example pictures on the board before the lesson.
Suggested duration. 45–60 minutes. It can be split into two shorter sessions.
6–9. Teaching procedure, explanation notes, guided and independent activities
Introduction (5 min). Start with the Discover situation and let learners suggest ideas before you teach.
Explanation (10 min). Teach the Understand section with real objects first, then pictures, then symbols and words.
Guided activity (10 min). You need paper. Learners follow the Explore steps in pairs.
Independent activity (10 min). Learners complete the independent practice and the worksheet.
Assessment and reflection (5 min). Ask each learner to say one thing they learned and how they know it.
Key questions. Brackets? Multiply or divide? Add or subtract?
Teacher explanation notes. Use the concrete, pictorial, abstract order: real objects first, then pictures, then symbols and words. Return to real objects whenever a learner needs support.
10–12. Differentiation, formative assessment, common misconceptions
Support. Use real objects and work with a partner; say each step aloud together. Extension. Ask learners to make up their own example and explain it to a partner.
Formative assessment. Listen to learner explanations, ask “how do you know?”, and check the independent work.
Misconceptions. Left to right only. Multiply before adding.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 12-mark exam practice scheme, the 10-mark scheme and all answers are in the answer sections above. Re-check every answer and the marking scheme before publication.
16–19. Remediation, enrichment, homework, teacher reflection
Remediation. Practise two operations. Enrichment. Invite learners to find or create three more examples at home or in the classroom and explain them. Homework. See above.
Teacher reflection. After teaching, record: which concepts learners understood; which misconceptions occurred; which activities worked; which learners need more practice; and what to improve next time.
Sources and standards
This lesson is based on the sources below, and no official endorsement is claimed. Reference starting points named in the Courseversity standard: Common Core State Standards for Mathematics, Cambridge Primary and the IB Primary Years Programme. Where a standard code is named, it points to the official framework.
Official standards for this topic area. Common Core State Standards for Mathematics (official site): Grade 5 » Operations & Algebraic Thinking. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core 5.OA.A.1; Cambridge Primary Mathematics Stage 5 and 6 (order of operations). These sources informed the lesson.