The shape of the lesson
By the end of the lesson, learners can
- Naming the numbers: In the lab, pick two numbers and watch their multiples light up on the 1 to 100 grid; then switch to factors and step through the sieve until only the primes are left.
- Prime factors: Open the prime factors tab, choose a number and press Next step: the lab divides by the smallest prime that works, again and again, until 1 is left.
- HCF and LCM: The highest common factor (HCF) is the biggest number that divides into both; the lowest common multiple (LCM) is the smallest number both divide into.
- Rational, irrational and reciprocals: A rational number can be written as a fraction \(\dfrac{a}{b}\) of two integers (\(b \ne 0\)).
- Prime: exactly two factors. 1 is not prime; 2 is the only even prime.
- Prime factors: divide by the smallest prime again and again; write the answer with indices.
- HCF: primes in both numbers, at the lowest power.
- LCM: every prime that appears, at the highest power.
- Rational: can be written as \(\dfrac{a}{b}\); simplify roots before deciding.
- Reciprocal: \(1 \div\) the number; make mixed numbers improper first.
One picture every idea returns to
Every whole number greater than 1 can be written as a product of primes in exactly one way. Write both numbers as prime products: the HCF takes the primes they share (lowest powers), the LCM takes every prime that appears (highest powers).
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, Every number has a family.. Natural numbers, integers, primes, squares, cubes, factors, multiples, rational and irrational numbers and reciprocals: the vocabulary every other topic in IGCSE Mathematics is written in. 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: Naming the numbers | In the lab, pick two numbers and watch their multiples light up on the 1 to 100 grid; then switch to factors and step through the sieve until only the primes are left. Project tab 1 of the lab (Number lab, types of number) and let a student drive it while the class predicts. Model the worked example Sorting a list on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Prime factors | Open the prime factors tab, choose a number and press Next step: the lab divides by the smallest prime that works, again and again, until 1 is left. Project tab 2 of the lab (Number lab, prime factors, HCF and LCM) and let a student drive it while the class predicts. Model the worked example Product of prime factors on the board, one line at a time. | Pairs |
| 17 to 23 min | Teach and check: HCF and LCM | The highest common factor (HCF) is the biggest number that divides into both; the lowest common multiple (LCM) is the smallest number both divide into. Model the worked example HCF and LCM from prime factors on the board, one line at a time. Your turn: Find the HCF and the LCM of 48 and 180. | Pairs |
| 23 to 30 min | Teach and check: Rational, irrational and reciprocals | A rational number can be written as a fraction \(\dfrac{a}{b}\) of two integers (\(b \ne 0\)). Model the worked example Reciprocals on the board, one line at a time. | 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, 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 Rational, irrational and reciprocals, 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 |
|---|---|
| 1 is not a prime number, and 2 is the only even prime. | A list of primes starting \(1, 3, 5, 7, 9\) loses marks twice: 1 and 9 are not prime, and 2 is missing. |
| \(72 = 8 \times 9\) is true, but it is not a product of PRIMES: 8 and 9 both split further. | Keep going until every factor is prime, then use index notation: \(2^{3} \times 3^{2}\). |
| Do not mix the rules. | The HCF can never be bigger than the smaller number, and the LCM can never be smaller than the bigger one. If your HCF of 84 and 90 is 1260, you have swapped them. |
| \(\sqrt{9}\) looks irrational because it has a root sign, but \(\sqrt{9} = 3\), which is rational. | Always simplify first. And the reciprocal of \(1\dfrac{2}{3}\) is not \(1\dfrac{3}{2}\): convert to an improper fraction before turning it over. |
Types of number: primes, factors, multiples, rational and irrational: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
(a) Write 3 040 006 in words.
(b) Write twelve million five hundred thousand in figures.
\(1 \quad 2 \quad 9 \quad 15 \quad 23 \quad 27 \quad 49 \quad 64\)
From this list, write down (a) the prime numbers (b) the numbers that are both square numbers and cube numbers.
Write 252 as a product of its prime factors.
\(120 = 2^{3} \times 3 \times 5\). Using your answer to the previous question, find (a) the HCF (b) the LCM of 252 and 120.
(a) From \(\ \sqrt{0.25},\ \sqrt{18},\ 3.14,\ 2\pi\ \) write down the irrational numbers.
(b) Find the reciprocal of \(2\dfrac{3}{4}\).
Ama has 84 red beads and 120 blue beads. She shares all of them into identical bags, so every bag has the same number of red beads and the same number of blue beads, with none left over.
Find the largest number of bags she can make, and how many beads of each colour go in each bag.
Answer key and marking notes
- (a) three million, forty thousand and six
- (b) 12 500 000
- (a) 2 and 23
- (b) \(1 = 1^{2} = 1^{3}\) and \(64 = 8^{2} = 4^{3}\), so 1 and 64.
- \(252 \div 2 = 126,\ 126 \div 2 = 63,\ 63 \div 3 = 21,\ 21 \div 3 = 7,\ 7 \div 7 = 1\)
- \(252 = 2^{2} \times 3^{2} \times 7\)
- (a) Shared primes at the lowest power: \(2^{2} \times 3 = 12\)
- (b) All primes at the highest power: \(2^{3} \times 3^{2} \times 5 \times 7 = 2520\)
- (a) \(\sqrt{0.25} = 0.5 = \dfrac{1}{2}\) and \(3.14 = \dfrac{314}{100}\) are rational, so the irrational numbers are \(\sqrt{18}\) and \(2\pi\).
- (b) \(2\dfrac{3}{4} = \dfrac{11}{4}\), so the reciprocal is \(\dfrac{4}{11}\).
- The number of bags must divide both 84 and 120, and be as large as possible: the HCF.
- \(84 = 2^{2} \times 3 \times 7\), \(120 = 2^{3} \times 3 \times 5\), so HCF \(= 2^{2} \times 3 = 12\) bags.
- Each bag: \(84 \div 12 = 7\) red and \(120 \div 12 = 10\) blue beads.
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.
Which list contains only prime numbers?
- A2, 3, 13, 29
- B1, 3, 5, 7
- C3, 9, 11, 17
- D2, 11, 21, 23
Write 180 as a product of its prime factors.
- A\(2^{2} \times 3^{2} \times 5\)
- B\(4 \times 9 \times 5\)
- C\(2^{3} \times 3 \times 5\)
- D\(2^{2} \times 3^{3} \times 5\)
One bus leaves a station every 12 minutes and another every 18 minutes. Both leave together at 09:00. At what time do they next leave together?
- A09:36
- B09:06
- C09:30
- D12:36
Which of these numbers is irrational?
- A\(\sqrt{20}\)
- B\(\sqrt{25}\)
- C\(0.\dot{3}\)
- D\(\dfrac{22}{7}\)
\(A = 2^{3} \times 3^{2} \times 5\) and \(B = 2^{2} \times 3^{4} \times 7\). Find the lowest common multiple of \(A\) and \(B\), in index form.
- A\(2^{3} \times 3^{4} \times 5 \times 7\)
- B\(2^{2} \times 3^{2}\)
- C\(2^{5} \times 3^{6} \times 5 \times 7\)
- D\(2^{3} \times 3^{4}\)
\(72 = 2^{3} \times 3^{2}\). What is the smallest whole number that 72 must be multiplied by to give a square number?
- A2
- B3
- C8
- D6
Quiz key
2, 3, 13 and 29 each have exactly two factors, so all four are prime. The list 1, 3, 5, 7 is the tempting one, but 1 has only one factor, so it is not prime. 9 = 3 × 3 and 21 = 3 × 7, so the other two lists each contain a number that is not prime.
\(180 = 2 \times 90 = 2 \times 2 \times 45 = 2 \times 2 \times 3 \times 15 = 2 \times 2 \times 3 \times 3 \times 5\), so \(180 = 2^{2} \times 3^{2} \times 5\). \(4 \times 9 \times 5\) does equal 180, but 4 and 9 are not prime, so it is not a product of PRIME factors. \(2^{3} \times 3 \times 5 = 120\) and \(2^{2} \times 3^{3} \times 5 = 540\).
The buses leave together again after the lowest common multiple of 12 and 18. \(12 = 2^{2} \times 3\) and \(18 = 2 \times 3^{2}\), so the LCM is \(2^{2} \times 3^{2} = 36\) minutes, giving 09:36. 09:06 uses the HCF (6), which is a factor, not a common multiple. 09:30 adds 12 + 18, and 12:36 uses \(12 \times 18 = 216\) minutes, a common multiple but not the lowest.
20 is not a square number, so \(\sqrt{20} = 4.472\ldots\) never ends or repeats: it is irrational. \(\sqrt{25}\) has a root sign but equals 5, which is rational. \(0.\dot{3} = \dfrac{1}{3}\) is rational because it recurs, and \(\dfrac{22}{7}\) is a fraction of integers (it is only an approximation to \(\pi\)).
For the LCM take every prime that appears, each to its HIGHEST power: \(2^{3}\), \(3^{4}\), \(5\) and \(7\), so LCM \(= 2^{3} \times 3^{4} \times 5 \times 7 = 22\,680\). \(2^{2} \times 3^{2}\) is the HCF (shared primes, lowest powers). \(2^{5} \times 3^{6} \times 5 \times 7\) is \(A \times B\), a common multiple but not the lowest. \(2^{3} \times 3^{4}\) leaves out the 5 and the 7, so it is not a multiple of either number.
A square number has every prime to an even power. \(3^{2}\) is already even, but \(2^{3}\) is odd, so one more 2 is needed: \(72 \times 2 = 144 = 2^{4} \times 3^{2} = 12^{2}\). Multiplying by 8 also gives a square (\(576 = 24^{2}\)), but 8 is not the SMALLEST multiplier. \(72 \times 3 = 216\) and \(72 \times 6 = 432\) are not square numbers.
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.