The shape of the lesson
By the end of the lesson, learners can
- Brackets and the order of operations: A calculator follows the order of operations exactly.
- Entering times and values: Time is not decimal: there are 60 minutes in an hour.
- Reading the display: The calculator does not know the units.
- Estimate first with 1 s.f. values, so you can spot a keying error.
- Brackets round every numerator, denominator and expression under a root.
- Times: minutes ÷ 60 gives the decimal of an hour; decimal × 60 gives minutes.
- Write down the full display before rounding when asked for all the figures.
- Money to 2 decimal places; other answers to 3 s.f.
One picture every idea returns to
Put brackets round every top, every bottom and everything under a root. Round only the final answer, and translate the display: \(4.8\) in money means \(\$4.80\), and \(3.25\) hours means 3 hours 15 minutes.
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, Key it in right.. A calculator only does what you type. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Brackets and the order of operations | A calculator follows the order of operations exactly. 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 A fraction on the board, one line at a time. Your turn: Work out \(\dfrac{4.2^{3}}{1.9 + 0.75}\). Write down all the figures on your display, then give the answer to 3 significant figures. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Entering times and values | Time is not decimal: there are 60 minutes in an hour. Model the worked example Time from a calculator on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Reading the display | The calculator does not know the units. Model the worked example Do not round too early on the board, one line at a time. | Corners |
| 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, 14 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 Reading the display, 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.
Corners
Put one answer or viewpoint in each corner of the room. Students move to the corner they agree with, justify the choice to a partner there, then hear one voice from each corner before the reveal.
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 |
|---|---|
| Typing \(5.6 + 3.82 \div 2.4 \times 1.15\) divides only the 3.82. | Without brackets the calculator does the division before the addition. |
| 2 hours 40 minutes is NOT 2.4 hours. \(38.50 \times 2.4 = \$92.40\) undercharges by more than \(\$10\). | Return to Entering times and values: ask a pair to state the correct version and show it on the lab before moving on. |
| Rounding in the middle of a calculation is the commonest way to lose an accuracy mark. | Keep the full value on the calculator and round once, at the end. |
Using a calculator efficiently: worksheet
6 questions, 14 marks. Written in the style of the papers; no past paper question is reproduced.
(a) Write 1.4 hours in hours and minutes. (b) Write 4 hours 36 minutes in hours, as a decimal.
Work out \(\dfrac{8.3^{2} - 2.65}{1.7 + \sqrt{6.2}}\). Write down all the figures on your calculator display, then give your answer to 3 significant figures.
Work out \(\sqrt{7.2^{2} + 3.5^{2}}\), giving your answer to 3 significant figures.
A calculation for the cost of some fabric gives the display \(12.6\). Write down the cost in dollars.
A circle has area 60 cm2. Ali finds the radius as 4.4 cm, then works out the circumference as \(2 \times \pi \times 4.4 = 27.6\) cm.
Explain Ali's mistake and find the correct circumference to 3 significant figures.
A plumber charges a call-out fee of \(\$45\) plus \(\$38.50\) per hour. She starts a job at 09 50 and finishes at 12 30.
(a) How long does the job take?
(b) Calculate the total charge.
Answer key and marking notes
- (a) \(0.4 \times 60 = 24\): 1 hour 24 minutes
- (b) \(36 \div 60 = 0.6\): 4.6 hours
- Numerator \(= 68.89 - 2.65 = 66.24\); denominator \(= 1.7 + 2.48997... = 4.18997...\)
- Display: \(15.809\,144\,98...\)
- \(15.8\) (3 s.f.)
- \(7.2^{2} + 3.5^{2} = 51.84 + 12.25 = 64.09\)
- \(\sqrt{64.09} = 8.0056... = 8.01\) (3 s.f.). Note: it is not \(7.2 + 3.5 = 10.7\).
\(\$12.60\)
- Ali rounded the radius before using it. Keep the full value.
- \(r = \sqrt{60 \div \pi} = 4.370\,19...\) cm
- Circumference \(= 2\pi \times 4.370\,19... = 27.458... = 27.5\) cm (3 s.f.)
- (a) 09 50 to 12 30 is 2 hours 40 minutes.
- (b) 2 h 40 min \(= 2\dfrac{2}{3}\) h \(= 2.666...\) h (not 2.4 h).
- Labour: \(38.50 \times 2.666... = 102.666...\)
- Total: \(45 + 102.666... = \$147.67\)
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.
Work out \(\dfrac{7.3 + 4.85}{1.6 \times 2.5}\). Give your answer correct to 3 significant figures.
- A3.04
- B14.9
- C8.51
- D19.0
Work out \(\sqrt{9.6^{2} + 4.3^{2}}\). Give your answer correct to 3 significant figures.
- A10.5
- B13.9
- C28.1
- D111
A walker covers 17 km at an average speed of 5 km/h. How long does the walk take?
- A3 hours 24 minutes
- B3 hours 40 minutes
- C3 hours 4 minutes
- D85 hours
An electrician charges \(\$42.60\) per hour. How much does she charge for 1 hour 50 minutes of work?
- A\(\$78.10\)
- B\(\$63.90\)
- C\(\$76.68\)
- D\(\$77.96\)
A circle has area 80 cm2. Find its circumference, correct to 3 significant figures.
- A31.7 cm
- B31.4 cm
- C15.9 cm
- D56.2 cm
The time period of a pendulum is \(T = 2\pi\sqrt{\dfrac{L}{g}}\). Find \(T\) when \(L = 0.85\) and \(g = 9.81\), correct to 3 significant figures.
- A1.85
- B0.591
- C0.738
- D0.544
Quiz key
Top \(= 12.15\), bottom \(= 4\), so the answer is \(12.15 \div 4 = 3.0375 = 3.04\) (3 s.f.). The tempting 14.9 comes from typing \(7.3 + 4.85 \div 1.6 \times 2.5\) with no brackets, so only 4.85 is divided. 8.51 brackets the bottom but not the top, and 19.0 brackets the top but then multiplies by 2.5 instead of dividing.
\(9.6^{2} + 4.3^{2} = 92.16 + 18.49 = 110.65\), and \(\sqrt{110.65} = 10.519\ldots = 10.5\) (3 s.f.). The tempting 13.9 just adds 9.6 and 4.3, as if the root and the squares cancelled. 28.1 puts only \(9.6^{2}\) under the root, and 111 forgets the square root.
Time \(= 17 \div 5 = 3.4\) hours, and \(0.4 \times 60 = 24\) minutes, so 3 hours 24 minutes. The tempting 3 hours 40 minutes reads the display 3.4 as if the decimal were minutes. 3 hours 4 minutes is the same misreading, and 85 hours multiplies instead of dividing.
1 hour 50 minutes \(= 1\dfrac{50}{60} = \dfrac{11}{6}\) hours, so the charge is \(42.60 \times \dfrac{11}{6} = \$78.10\). The tempting \(\$63.90\) uses 1.5 hours, typing the time as 1.50. \(\$76.68\) uses 1.8 hours, and \(\$77.96\) rounds the time to 1.83 hours too early.
\(r = \sqrt{\dfrac{80}{\pi}} = 5.0462\ldots\) cm. Keeping the full value, \(C = 2\pi \times 5.0462\ldots = 31.706\ldots = 31.7\) cm. The tempting 31.4 cm rounds the radius to 5.0 first, which loses the accuracy. 15.9 cm uses \(\pi r\), and 56.2 cm uses \(r = \sqrt{80}\), forgetting to divide by \(\pi\).
\(\dfrac{0.85}{9.81} = 0.08664\ldots\), \(\sqrt{0.08664\ldots} = 0.29435\ldots\), and \(2\pi \times 0.29435\ldots = 1.8495\ldots = 1.85\) (3 s.f.). The tempting 0.591 closes the root bracket after \(L\), giving \(\dfrac{2\pi\sqrt{0.85}}{9.81}\). 0.738 puts \(2\pi\) inside the root, and 0.544 leaves out the root altogether.
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.