The shape of the lesson
By the end of the lesson, learners can
- Half a unit either way: In the lab, set the rounded value and the accuracy, then drag the true value along the line.
- Bounds in context: Read the accuracy from the wording: 'to the nearest 5 kg', 'to the nearest tenth of a second', 'to 1 decimal place'.
- Bounds of calculations: Extended.
- Unit: read it from the accuracy (nearest 5 kg, 1 d.p., 2 s.f.).
- Bounds: value \(\pm\) half a unit.
- Interval: lower \(\le x <\) upper.
- Add, multiply (Extended): max with max, min with min.
- Subtract, divide (Extended): max with the OTHER value's min, and vice versa.
One picture every idea returns to
The bounds are half a rounding unit below and above the given value: for 17 m to the nearest metre, \(16.5 \le L < 17.5\). (Extended) For a maximum, combine the values that make the answer biggest: when dividing, that is the upper bound over the LOWER bound.
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, Half a unit either way.. A length given as 17 m to the nearest metre could really be anything from 16.5 m up to (but not including) 17.5 m. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Half a unit either way | In the lab, set the rounded value and the accuracy, then drag the true value along the line. Project tab 5 of the lab (Powers and accuracy lab, limits of accuracy) and let a student drive it while the class predicts. Model the worked example Nearest metre on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Bounds in context | Read the accuracy from the wording: 'to the nearest 5 kg', 'to the nearest tenth of a second', 'to 1 decimal place'. Model the worked example Nearest 5 on the board, one line at a time. Your turn: A sprinter's time is 9.8 s, correct to the nearest tenth of a second. Write the error interval for the time \(t\). | Pairs |
| 21 to 30 min | Teach and check: Bounds of calculations | Extended. Model the worked example Perimeter and area (Extended) on the board, one line at a time. Your turn: (Extended) Two rods are 15 cm and 6 cm long, each to the nearest centimetre. Find the largest possible difference in their lengths. | Jot Thoughts |
| 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, 15 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 Bounds of calculations, 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.
Jot Thoughts
Teams write ideas on slips, one idea per slip, as many as they can in two minutes, saying each aloud as they place it in the middle. Then they sort the slips, which is where the structure of the topic appears.
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 |
|---|---|
| The upper bound of 17 m is 17.5 m, not 17.4 or 17.49. | Cambridge writes the upper bound as 17.5, even though 17.5 itself is excluded from the interval. |
| 60 kg to the nearest 5 kg is NOT \(59.5 \le m < 60.5\). | The half unit is half of 5, not half of 1. |
| For the lower bound of a speed, do NOT use both lower bounds: \(99.5 \div 12.35 = 8.06\) is bigger than the true minimum. | Dividing by the larger time gives the smaller answer. |
Limits of accuracy: upper and lower bounds: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
The length of a pencil is 23 cm, correct to the nearest centimetre. Write down the lower and upper bounds of the length.
The mass \(m\) kg of a melon is 0.8 kg, correct to 1 decimal place. Write down the error interval for \(m\).
The population of a town is 45 000, correct to the nearest thousand. Write down the lower and upper bounds.
A sprinter's time \(t\) seconds is 9.8 s, correct to the nearest 0.1 s.
(a) Write down the error interval for \(t\). (b) Could the actual time have been 9.85 s? Explain.
(Extended) A rectangle measures 12 cm by 7 cm, each correct to the nearest centimetre.
Calculate (a) the upper bound of the area (b) the lower bound of the perimeter.
(Extended) A car travels 250 km, correct to the nearest 10 km, in 3.2 hours, correct to the nearest 0.1 hour.
(a) Calculate the upper bound and the lower bound of the average speed. Give your answers correct to 3 significant figures.
(b) The driver says her average speed was definitely below 80 km/h. Is she right? Explain.
Answer key and marking notes
- Lower bound 22.5 cm
- Upper bound 23.5 cm
- Half of 0.1 is 0.05
- \(0.75 \le m < 0.85\)
- Lower bound 44 500
- Upper bound 45 500
- (a) \(9.75 \le t < 9.85\)
- (b) No: 9.85 s would round to 9.9 s, so it is outside the interval.
- (a) \(12.5 \times 7.5 = 93.75\) cm2
- (b) \(2 \times (11.5 + 6.5) = 36\) cm
- Distance bounds 245 and 255 km; time bounds 3.15 and 3.25 h.
- (a) Upper bound: \(255 \div 3.15 = 80.95\dots = 81.0\) km/h
- Lower bound: \(245 \div 3.25 = 75.38\dots = 75.4\) km/h
- (b) No: the upper bound is above 80 km/h, so the speed could have been over 80 km/h.
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.
A pencil is 23 cm long, correct to the nearest centimetre. Which error interval shows its length \(L\) cm?
- A\(22.5 \le L < 23.5\)
- B\(22 \le L < 24\)
- C\(22.5 \le L < 23.4\)
- D\(22.95 \le L < 23.05\)
A bag of flour has a mass of 4.7 kg, correct to 1 decimal place. What is the upper bound of its mass?
- A4.75 kg
- B4.74 kg
- C4.8 kg
- D5.2 kg
The number of people at a concert is 300, correct to the nearest 50. What is the lower bound?
- A275
- B250
- C299.5
- D295
A number \(x\) is 5800, correct to 2 significant figures. Which is the error interval for \(x\)?
- A\(5750 \le x < 5850\)
- B\(5795 \le x < 5805\)
- C\(5799.5 \le x < 5800.5\)
- D\(5700 \le x < 5900\)
A rectangle measures 9 cm by 6 cm, each correct to the nearest centimetre. Find the lower bound of its area.
- A46.75 cm2
- B54 cm2
- C61.75 cm2
- D53.3 cm2
A train travels 180 km, correct to the nearest 10 km, in 2.5 hours, correct to the nearest 0.1 hour. Find the upper bound of its average speed.
- A75.5 km/h
- B72.5 km/h
- C71.4 km/h
- D72 km/h
Quiz key
The rounding unit is 1 cm, so half of it is 0.5 cm. Lower bound \(23 - 0.5 = 22.5\), upper bound \(23 + 0.5 = 23.5\): \(22.5 \le L < 23.5\). \(22 \le L < 24\) uses a whole unit each side. \(L < 23.4\) is tempting, but 23.45 also rounds to 23, so the upper bound is written as 23.5. \(22.95 \le L < 23.05\) uses half of 0.1 instead of half of 1.
Half of 0.1 is 0.05, so the upper bound is \(4.7 + 0.05 = 4.75\) kg. 4.74 is tempting because 4.75 would round up to 4.8, but Cambridge writes the upper bound as 4.75. 4.8 adds a whole unit of 0.1, and 5.2 adds 0.5, half of 1 instead of half of 0.1.
Half of 50 is 25, so the lower bound is \(300 - 25 = 275\). The tempting 299.5 uses half of 1, as if the number were given to the nearest whole number. 250 subtracts a whole unit of 50, and 295 subtracts 5 instead of 25.
The second significant figure (8) is in the hundreds column, so 5800 is correct to the nearest 100. Half of 100 is 50, giving \(5750 \le x < 5850\). \(5795 \le x < 5805\) treats it as the nearest 10, and \(5799.5 \le x < 5800.5\) as the nearest 1. \(5700 \le x < 5900\) uses a whole unit each side instead of half.
Lower bounds: length 8.5 cm, width 5.5 cm. Lower bound of area \(= 8.5 \times 5.5 = 46.75\) cm2. 54 is the area from the rounded values, and 61.75 \(= 9.5 \times 6.5\) is the UPPER bound. 53.3 comes from \(8.95 \times 5.95\), using half of 0.1 instead of half of 1.
Distance: \(175 \le d < 185\). Time: \(2.45 \le t < 2.55\). To make a division as large as possible, use the upper bound on top and the LOWER bound underneath: \(185 \div 2.45 = 75.51\ldots = 75.5\) km/h (3 s.f.). The tempting 72.5 uses both upper bounds (\(185 \div 2.55\)), but dividing by a bigger time gives a smaller speed. 71.4 is \(175 \div 2.45\), and 72 uses the rounded values.
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.