You will learn
- Estimate first.
- Check with the inverse.
- Spot unreasonable answers.
- Correct 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
- Estimate before you calculate.
- Check with the inverse operation.
- Spot answers that are far from the estimate.
- Fix errors and re-check.
Words to know
A close guess.
Makes sense.
Opposite operation.
A mistake.
1. Discover: does it make sense?
You calculate 48 × 19 and get 1,792. Is that sensible?
2. Understand: checking
- Estimate: 50 × 20 = 1,000. The true answer is near 1,000 (48 × 19 = 912), so 1,792 is too big.
- Check 912 ÷ 19 = 48 by division.
- 3.2 + 4.9 ≈ 3 + 5 = 8; the exact sum is 8.1.
- A subtraction can be checked by adding back.
Idea 1: Estimate
48 × 19 ≈ 50 × 20 = 1,000. So 1,792 is too large; the true answer is 912.
Idea 2: Decimals
3.2 + 4.9 ≈ 3 + 5 = 8.
Idea 3: Inverse
912 ÷ 19 = 48; check 19 × 48 = 912. 50 − 18 = 32; check 18 + 32 = 50.
Why estimate first? It catches big mistakes quickly.
3. Worked examples
Example 1 (I show you): Estimate
48 × 19
- 50 × 20.
Answer: 1,000
Example 2 (We do it together): Inverse
50 − 18 = 32
- 18 + 32.
Answer: 50, correct
Example 3 (You try): Judge
1,792 for 48 × 19
Show the steps and answer
- Compare with 1,000.
Answer: too large
Did you know?
The inverse of subtraction is addition; the inverse of division is multiplication.
Why this matters
Checking prevents silly mistakes and builds trust in answers.
4. Explore: check
You need paper.
- Round the numbers.
- Estimate.
- Compare.
- Use the inverse.
Think about it: what did you notice?
912.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- My estimate is ___.
- My answer is close to / far from it.
- The inverse check is ___.
- The answer is / is not reasonable.
Guided practice
Question 1. 50 × 20?
1,000.
Question 2. Is 1,792 sensible?
No.
Question 3. 3.2 + 4.9 ≈ ?
8.
Independent practice
Level 1: Start
- Estimate 48 × 19
- Is 1,792 sensible?
- Estimate 3.2 + 4.9
- 912 ÷ 19
- Check 50 − 18 = 32 by adding
Level 2: Grow
- Estimate 31 × 29
- Estimate 6.8 + 3.1
- Check 72 ÷ 8 = 9 by multiplying
Level 3: Stretch
- Is 400 sensible for 19.8 + 20.2?
- Check 85 − 37 = 48 by adding
- Ben says check 50 − 18 = 32 by 50 + 18. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
You get 1,792 for 48 × 19. Reasonable?
Show a sample answer
No, estimate is about 1,000.
Another one. A pupil says 7.2 ÷ 0.9 = 0.8. Use an estimate to find the right answer.
Show a sample answer
About 7 ÷ 1, so the answer is 8
7. Common mistakes
Fix: Always estimate.
Fix: Compare with an estimate.
Fix: Use the inverse.
8. Lesson summary and revision card
- I can estimate first.
- I can check with the inverse.
- I can spot unreasonable answers.
- I can correct 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
- Estimate.
- Use the inverse.
- Fix errors.
11. Next lesson
Next lesson: Applying mathematics to real situations. 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.
- Estimate 48 × 19. [2 marks]
- Is 1,792 a sensible answer for 48 × 19? Explain. [2 marks]
- Estimate 3.2 + 4.9. [2 marks]
- Check that 912 ÷ 19 = 48. [2 marks]
- Ali says to check 50 − 18 = 32 by working out 50 + 18. Is he right? Explain. [3 marks]
Total: 11 marks.
Marking guide and model answers
- Round to 50 × 20 [1]; 1,000 [1]
- The estimate is about 1,000 [1]; 1,792 is too large [1]
- 3 + 5 [1]; 8 [1]
- 19 × 48 [1]; 912, so it is correct [1]
- No [1]. Use the inverse: 18 + 32 [1]. It equals 50, so the answer is correct [1].
Score guide: 10 to 11 = secure. 7 to 9 = nearly there, revise the parts you missed. Under 7 = work through the lesson again with a grown-up.
Printable worksheet
- Estimate 48 × 19
- 1,792 sensible? (yes/no)
- Estimate 3.2 + 4.9
- 912 ÷ 19
- 18 + 32
- Estimate 48 × 19
- Is 1,792 sensible for 48 × 19?
- Estimate 3.2 + 4.9
- 912 ÷ 19
- Check 50 − 18 = 32: 18 + 32
Homework
Check three of your homework answers by estimating.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| Estimate 48 × 19 | 1,000 | 2 |
| 1,792 sensible? (yes/no) | no | 2 |
| Estimate 3.2 + 4.9 | 8 | 2 |
| 912 ÷ 19 | 48 | 2 |
| 18 + 32 | 50 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 1,000; 2 no; 3 8; 4 48; 5 50.
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 1,000; 7 too large; 8 8; 9 48; 10 50.
Independent practice answers. Level 1: 1 1,000; 2 too large; 3 8; 4 48; 5 18 + 32 = 50, so it is correct. Level 2: 1 900; 2 10; 3 8 × 9 = 72. Level 3: 1 too large; 2 37 + 48 = 85; 3 Use the inverse: 18 + 32 = 50..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Estimate first | 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 Check with the inverse | 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 Spot unreasonable answers | 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 Correct 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: Trusting the calculator blindly. Always estimate.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Rounding.
- 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 learns to check answers for sense. You do not need to be good at mathematics to help.
- Estimate shopping totals.
- Ask: does the answer make sense?
- Check with the opposite operation.
- 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: estimate first; check with the inverse; spot unreasonable answers; correct errors.
Prerequisite knowledge. Lesson 5.
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. Is it close to my estimate? What does the inverse show?
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. Trusting the calculator blindly. Always estimate.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 11-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. Rounding. 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): Standards for Mathematical Practice. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core MP1 and 5.NBT.B; Cambridge Primary Mathematics Stage 6 (estimation and checking). These sources informed the lesson.