The shape of the lesson
By the end of the lesson, learners can
- One number between 1 and 10: Type a number into the lab and watch the decimal point hop until exactly one non-zero digit sits in front of it.
- Comparing and ordering: To order numbers in standard form, compare the powers first.
- Calculating in standard form: For multiplying and dividing, deal with the front numbers and the powers of ten separately, using the laws of indices.
- Standard form on a calculator: On Paper 3 and Paper 4 you may use the calculator's \(\times 10^{x}\) (or EXP) key.
- Shape: \(A \times 10^{n}\) with \(1 \le A < 10\) and \(n\) an integer.
- Large or small: numbers of 10 or more have \(n > 0\); numbers below 1 have \(n < 0\).
- Multiply/divide: front numbers together, then add/subtract the powers.
- Add/subtract: match the powers first (or use ordinary numbers).
- Tidy up: if the front number is 10 or more (or below 1), adjust it and the power.
One picture every idea returns to
A number in standard form is \(A \times 10^{n}\), where \(1 \le A < 10\) and \(n\) is an integer. The power \(n\) counts how many places the digits move: positive for large numbers, negative for small ones.
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, Big and small, one shape.. The mass of the Earth and the width of an atom are both easy to write once every number has the same shape: a number from 1 up to 10, times a power of ten. 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: One number between 1 and 10 | Type a number into the lab and watch the decimal point hop until exactly one non-zero digit sits in front of it. Project tab 3 of the lab (Powers and accuracy lab, standard form) and let a student drive it while the class predicts. Model the worked example Into standard form on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Comparing and ordering | To order numbers in standard form, compare the powers first. | Pairs |
| 17 to 23 min | Teach and check: Calculating in standard form | For multiplying and dividing, deal with the front numbers and the powers of ten separately, using the laws of indices. Model the worked example Multiply without a calculator (Extended) on the board, one line at a time. Your turn: Work out \((2 \times 10^{-3})^{2}\). Give your answer in standard form. | Pairs |
| 23 to 30 min | Teach and check: Standard form on a calculator | On Paper 3 and Paper 4 you may use the calculator's \(\times 10^{x}\) (or EXP) key. | Frayer Model |
| 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, 13 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 Standard form on a calculator, 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.
Frayer Model
Pairs fill a four-box grid for the key term: definition, characteristics, examples and non-examples. The non-examples box is where the misconceptions surface, so give it the most time.
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 |
|---|---|
| \(34 \times 10^{5}\) is a correct number but it is NOT in standard form, because \(34\) is not less than 10. | Examiners give no mark for it. Fix it to \(3.4 \times 10^{6}\). |
| With negative powers, the bigger-looking power is the smaller number: \(10^{-5}\) is a hundred times SMALLER than \(10^{-3}\). | Return to Comparing and ordering: ask a pair to state the correct version and show it on the lab before moving on. |
| You cannot add powers when adding numbers: \(2 \times 10^{5} + 3 \times 10^{4}\) is NOT \(5 \times 10^{9}\). | Index laws are for multiplying and dividing only. |
Standard form: worksheet
6 questions, 13 marks. Written in the style of the papers; no past paper question is reproduced.
Write in standard form (a) \(820\,000\,000\) (b) \(0.000\,407\).
Write as ordinary numbers (a) \(5.03 \times 10^{5}\) (b) \(1.9 \times 10^{-2}\).
Write these numbers in order of size, starting with the smallest.
\(4.5 \times 10^{-3}\) \(0.045\) \(4.5 \times 10^{-4}\) \(4.05 \times 10^{-3}\)
Without a calculator, work out \((8 \times 10^{6}) \times (7 \times 10^{-2})\). Give your answer in standard form.
Without a calculator, work out \(3.2 \times 10^{4} + 9 \times 10^{3}\). Give your answer in standard form.
The distance from the Earth to the Sun is \(1.5 \times 10^{8}\) km. Light travels at \(3 \times 10^{5}\) km per second.
Calculate the time, in minutes, that light takes to travel from the Sun to the Earth.
Answer key and marking notes
- (a) \(8.2 \times 10^{8}\)
- (b) \(4.07 \times 10^{-4}\)
- (a) \(503\,000\)
- (b) \(0.019\)
- Write \(0.045 = 4.5 \times 10^{-2}\); the smallest power is \(-4\).
- Order: \(4.5 \times 10^{-4},\ 4.05 \times 10^{-3},\ 4.5 \times 10^{-3},\ 0.045\)
- \(8 \times 7 = 56\) and \(10^{6} \times 10^{-2} = 10^{4}\), so \(56 \times 10^{4}\).
- \(56 = 5.6 \times 10^{1}\), so the answer is \(5.6 \times 10^{5}\).
- \(9 \times 10^{3} = 0.9 \times 10^{4}\).
- \(3.2 + 0.9 = 4.1\), so \(4.1 \times 10^{4}\).
- Time = distance ÷ speed = \(\dfrac{1.5 \times 10^{8}}{3 \times 10^{5}} = 0.5 \times 10^{3} = 500\) seconds.
- \(500 \div 60 = 8.33\) minutes (3 s.f.).
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 0.000 47 in standard form.
- A\(4.7 \times 10^{-4}\)
- B\(4.7 \times 10^{4}\)
- C\(47 \times 10^{-5}\)
- D\(4.7 \times 10^{-3}\)
Write \(3.06 \times 10^{5}\) as an ordinary number.
- A306 000
- B30 600 000
- C3 060 000
- D30 600
Which of these numbers is written in standard form?
- A\(9.9 \times 10^{-2}\)
- B\(0.99 \times 10^{-1}\)
- C\(99 \times 10^{-3}\)
- D\(10 \times 10^{-3}\)
Work out \(\left(6.4 \times 10^{7}\right) \div \left(1.6 \times 10^{-3}\right)\). Give your answer in standard form.
- A\(4 \times 10^{10}\)
- B\(4 \times 10^{4}\)
- C\(4 \times 10^{-10}\)
- D\(4 \times 10^{-21}\)
Without a calculator, work out \(\left(5 \times 10^{4}\right) \times \left(7 \times 10^{6}\right)\). Give your answer in standard form.
- A\(3.5 \times 10^{11}\)
- B\(35 \times 10^{10}\)
- C\(3.5 \times 10^{10}\)
- D\(3.5 \times 10^{24}\)
Without a calculator, work out \(4.2 \times 10^{5} + 3 \times 10^{4}\). Give your answer in standard form.
- A\(4.5 \times 10^{5}\)
- B\(7.2 \times 10^{9}\)
- C\(7.2 \times 10^{5}\)
- D\(4.5 \times 10^{4}\)
Quiz key
The first non-zero digit is 4, so \(A = 4.7\). The point moves 4 places to the right (0.000 47 to 4.7), and the number is small, so the power is \(-4\): \(4.7 \times 10^{-4}\). \(4.7 \times 10^{4} = 47\,000\) has the wrong sign. \(47 \times 10^{-5}\) has the right value but 47 is not between 1 and 10, so it is not standard form. \(4.7 \times 10^{-3} = 0.0047\) miscounts the places.
Move the decimal point 5 places to the right: \(3.06 \to 306\,000\). The tempting 30 600 000 writes 3.06 and then adds five zeros, which moves the point 7 places. 3 060 000 moves it 6 places and 30 600 moves it 4.
In \(9.9 \times 10^{-2}\), \(A = 9.9\) satisfies \(1 \le A < 10\), so it is standard form. \(0.99\) is less than 1 and \(99\) is not less than 10. \(10 \times 10^{-3}\) is tempting because 10 looks like a boundary value, but \(A\) must be strictly less than 10 (it should be \(1 \times 10^{-2}\)).
\(6.4 \div 1.6 = 4\) and \(10^{7} \div 10^{-3} = 10^{7 - (-3)} = 10^{10}\), so the answer is \(4 \times 10^{10}\). The tempting \(4 \times 10^{4}\) works out \(7 - 3\), losing the minus sign of the \(-3\). \(4 \times 10^{-10}\) subtracts the wrong way round, and \(4 \times 10^{-21}\) multiplies the indices.
\(5 \times 7 = 35\) and \(10^{4} \times 10^{6} = 10^{10}\), giving \(35 \times 10^{10}\). That is not standard form because 35 is not less than 10, so write \(35 = 3.5 \times 10^{1}\): the answer is \(3.5 \times 10^{11}\). \(35 \times 10^{10}\) has the right value but is not in standard form. \(3.5 \times 10^{10}\) changes 35 to 3.5 without increasing the power, and \(3.5 \times 10^{24}\) multiplies the indices.
\(4.2 \times 10^{5} = 420\,000\) and \(3 \times 10^{4} = 30\,000\). Their sum is \(450\,000 = 4.5 \times 10^{5}\). \(7.2 \times 10^{9}\) adds the front numbers and the powers, but the index laws are only for multiplying and dividing. \(7.2 \times 10^{5}\) adds the front numbers as if both powers were \(10^{5}\), and \(4.5 \times 10^{4} = 45\,000\) uses the wrong power.
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.