The shape of the lesson
By the end of the lesson, learners can
- Rounding to a place value: In the lab, press Next number, choose decimal places, significant figures or a nearest power of ten, and decide which way it rounds before the lab shows you.
- Significant figures: The first significant figure is the first non-zero digit, counting from the left.
- Estimating a calculation: Round every number to 1 significant figure, then work it out in your head.
- A sensible degree of accuracy: Answers in context must make sense.
- Deciding digit: 5 or more rounds up; 4 or less does not.
- Significant figures start at the first non-zero digit.
- Place holders: fill with zeros in whole numbers; keep trailing zeros in decimals (8.00).
- Estimate: every number to 1 s.f., show the rounded numbers, then calculate.
- Context: round up or down so the answer makes sense; money to 2 d.p.
One picture every idea returns to
Find the last digit you are keeping, then look at the next digit only: 5 or more rounds up, 4 or less leaves it. Keep the size of the number with zeros (38 652 is 38 700 to 3 s.f., not 387). To estimate, round every number to 1 significant figure first.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Roughly right, quickly.. Round to decimal places, significant figures or the nearest 10, 100 or 1000; estimate a calculation by rounding every number to 1 significant figure; and give answers to a sensible accuracy for the situation. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 11 min | Teach and check: Rounding to a place value | In the lab, press Next number, choose decimal places, significant figures or a nearest power of ten, and decide which way it rounds before the lab shows you. Project tab 4 of the lab (Powers and accuracy lab, rounding and estimating) and let a student drive it while the class predicts. Model the worked example Nearest 10, 100, 1000 on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Significant figures | The first significant figure is the first non-zero digit, counting from the left. Model the worked example Three cases on the board, one line at a time. | Pairs |
| 17 to 23 min | Teach and check: Estimating a calculation | Round every number to 1 significant figure, then work it out in your head. Model the worked example Rounding to 1 s.f. on the board, one line at a time. | Pairs |
| 23 to 30 min | Teach and check: A sensible degree of accuracy | Answers in context must make sense. Model the worked example Up or down? on the board, one line at a time. Your turn: A 2-litre bottle of juice is poured into cups that each hold 0.3 litres. How many cups can be filled completely? | Connect-Extend-Challenge |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 16 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in A sensible degree of accuracy, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Connect-Extend-Challenge
After the new idea, each student writes three lines: how it connects to what they knew, how it extends their thinking, and what still challenges them. Pairs compare, then the challenges become the plenary.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| 7.9963 to 2 d.p. is 8.00, not 8. | The zeros show the accuracy asked for; writing 8 gives only 0 decimal places. |
| 38 652 to 3 s.f. is 38 700, not 387. | Rounding must not change the size of the number. |
| Dividing by 0.5 gives a bigger answer, not a smaller one: \(400 \div 0.5 = 800\), not 200. | Return to Estimating a calculation: ask a pair to state the correct version and show it on the lab before moving on. |
| Do not round half-way through a calculation. | Keep the full calculator value and round only the final answer, or your 3 s.f. answer may be wrong. |
Rounding and estimation: worksheet
6 questions, 16 marks. Written in the style of the papers; no past paper question is reproduced.
Write (a) 5764 correct to the nearest hundred (b) 0.073 586 correct to 3 significant figures (c) 19.996 correct to 2 decimal places.
Write (a) 0.000 806 1 correct to 2 significant figures (b) 2 049 600 correct to 3 significant figures.
By rounding each number to 1 significant figure, estimate the value of \(\dfrac{19.6 \times 3.08}{0.51}\). Show your working.
(a) Estimate \(\sqrt{98.6} \times 4.92\), showing your working.
(b) Use your calculator to find the exact value, correct to 3 significant figures.
(c) Is your estimate an overestimate or an underestimate? Explain why.
Three friends share a restaurant bill of $86.35 equally.
(a) Work out each share, correct to the nearest cent. (b) Explain why each friend should actually pay $28.79.
A room has 47.8 m2 of wall to paint, and it needs two coats. One litre of paint covers 11.5 m2. Paint is sold in 2.5-litre tins.
(a) By rounding each number to 1 significant figure, estimate the number of litres needed.
(b) Calculate the number of tins that must be bought.
Answer key and marking notes
- (a) 5800
- (b) 0.0736
- (c) 20.00
- (a) 0.000 81
- (b) 2 050 000
- \(\dfrac{20 \times 3}{0.5}\)
- \(= \dfrac{60}{0.5} = 120\)
- (a) \(\sqrt{100} \times 5 = 50\)
- (b) \(48.854\dots = 48.9\)
- (c) Overestimate: both 98.6 and 4.92 were rounded UP.
- (a) \(86.35 \div 3 = 28.783\dots = \$28.78\)
- (b) \(3 \times 28.78 = 86.34\), which is 1 cent short of the bill, so each must pay $28.79 to cover it.
- (a) \(\dfrac{2 \times 50}{10} = 10\) litres
- (b) Litres: \(2 \times 47.8 \div 11.5 = 8.313\dots\)
- Tins: \(8.313\dots \div 2.5 = 3.33\dots\)
- 3 tins would not be enough, so round up: 4 tins.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Write 0.040 78 correct to 2 significant figures.
- A0.041
- B0.04
- C0.040
- D0.0408
Write 49 650 correct to 3 significant figures.
- A49 700
- B497
- C49 600
- D50 000
Write 6.9973 correct to 2 decimal places.
- A7.00
- B7
- C6.99
- D7.0
By rounding each number to 1 significant figure, estimate the value of \(\dfrac{61.3 \times 4.82}{0.487}\).
- A600
- B150
- C300
- D6000
By rounding each number to 1 significant figure, estimate the value of \(\dfrac{\sqrt{38.6 + 61.9}}{0.213}\).
- A50
- B500
- C2
- D5
Use a calculator to work out \(\dfrac{12.7^{2}}{3.9 + 5.84}\), giving your answer correct to 3 significant figures.
- A16.6
- B16.5
- C47.2
- D16.56
Quiz key
The first significant figure is the 4, the second is the 0 after it. The next digit is 7, so round up: 0.041. The tempting 0.04 counts the zero before the 4 as significant (it has only 1 s.f.). 0.040 cuts the number off without rounding up, and 0.0408 is 3 s.f.
The first three figures are 4, 9, 6. The next digit is 5, so round up: 49 700. 497 has the right digits but changes the size of the number: rounding must keep it close to 49 650. 49 600 rounds down when the digit is 5, and 50 000 is correct to only 1 or 2 s.f.
Keep 2 decimal places: 6.99|73. The next digit is 7, so round up: \(6.99 + 0.01 = 7.00\). The tempting 7 has the right value but shows 0 decimal places; the zeros show the accuracy asked for. 7.0 shows only 1 decimal place, and 6.99 rounds down when it should round up.
\(\dfrac{60 \times 5}{0.5} = \dfrac{300}{0.5} = 600\) (the exact value is 607 to 3 s.f.). The tempting 150 multiplies by 0.5 instead of dividing, and 300 forgets the division altogether. 6000 makes a place value error when dividing by 0.5.
\(\dfrac{\sqrt{40 + 60}}{0.2} = \dfrac{\sqrt{100}}{0.2} = \dfrac{10}{0.2} = 50\) (the exact value is 47.1 to 3 s.f.). The tempting 500 forgets the square root: \(100 \div 0.2\). 2 multiplies by 0.2 instead of dividing, and 5 divides by 2 instead of 0.2.
\(12.7^{2} = 161.29\) and \(3.9 + 5.84 = 9.74\), so \(161.29 \div 9.74 = 16.5595\ldots = 16.6\) (3 s.f.). 16.5 truncates instead of rounding: the next digit is 5, so round up. 47.2 types it without brackets, working out \(12.7^{2} \div 3.9 + 5.84\). 16.56 is correct to 4 s.f., not 3.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.