Properties of number: factors, multiples, squares and cubes

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Distinguish a factor from a multiple using division
Recognise a square or cube through repeated equal factors

Vocabulary: factor · multiple · square · cube · divisible

Retrieve previous learning

What is 6 × 4?

Check: Six groups of four contain 24; the multiplication fact also gives factor pair 6 and 4 for 24.

Learn the concept

A factor divides a whole number with zero remainder. Since 6 × 4 = 24, both 6 and 4 are factors of 24; 24 is a multiple of each. The relation has a direction: 24 is not a factor of 6. When unsure, write a multiplication sentence or divide and check the remainder. Every positive whole number has 1 and itself as factors.

A square number has the form n × n, such as 6² = 36. A cube number has the form n × n × n, such as 3³ = 27. The small raised number states how many equal factors to multiply, not what number to multiply by. Both properties can overlap: 64 = 8² = 4³, so the categories are not exclusive.

For a quick diagnostic, classify 24, 27 and 36 with evidence. Twenty-four is a multiple of 6 because 24 ÷ 6 = 4; 27 is a cube because 3 × 3 × 3 = 27; 36 is a square because 6 × 6 = 36. One number can have many factors and multiples. A list of multiples continues without end, whereas a positive whole number has finitely many positive factors.

Worked model

Write 6 × 4 = 24 and identify 6 and 4 as factors of 24.
Reverse the relation: 24 is a multiple of both 6 and 4.
Compare 36 with 6 × 6 to justify that it is square.
Compare 27 with 3 × 3 × 3 to justify that it is cube.

Guided practice

Is 48 a multiple of 8, and is 49 a square?

Hint: Use division for the first and equal factors for the second.

Check the guided answer

48 ÷ 8 = 6 with no remainder, so 48 is a multiple of 8. Also 7 × 7 = 49, so 49 is square.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“If 8 is a factor of 48, then 48 is a factor of 8.”

Correction: Factor language is directional: 8 divides 48, while 48 does not divide 8 into a whole-number quotient.

Extend and reflect

Find another positive integer that is both a square and a cube.

Guidance: Any sixth power works: 2⁶ = 64 and 3⁶ = 729; show each as both a square and a cube.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Distinguish a factor from a multiple using division. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Use multiplication facts to secure factor pairs

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

List factor pairs without omission
Explain how the factor-pair list establishes divisibility

Vocabulary: factor pair · divisor · product · systematic list

Retrieve previous learning

Give two factors of 24 besides 1 and 24.

Check: For example 3 and 8, because 3 × 8 = 24.

Learn the concept

Multiplication facts provide a systematic search for factors. For 48, start at 1 × 48, then test 2 × 24, 3 × 16 and 4 × 12. Five is not a factor because 48 does not end in 0 or 5. Six × 8 completes the list; after that the same pairs appear in reverse. The positive factors are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.

A factor pair is more informative than a bare list because it shows why each member divides the target. If you see 3 × 16 = 48, you can infer 48 ÷ 3 = 16 and 48 ÷ 16 = 3. The number 48 has no repeated central factor because it is not a square. In contrast 49 has 7 × 7, so that central pair contributes only one distinct factor.

You can stop checking small divisors once the next trial divisor exceeds the square root. If a smaller factor existed after that point, its partner would already have appeared before it. Year 7 pupils may use the simpler practical rule of stopping when the smaller member of a pair would pass its partner. Explain the stopping condition rather than declaring a list complete from memory alone.

Worked model

Start a pair table for 48 with 1 × 48 and 2 × 24.
Test 3 and 4 to obtain 3 × 16 and 4 × 12.
Reject 5 because 48 is not a whole multiple of five.
Record 6 × 8, then stop when the next candidate would repeat a previous pair.

Guided practice

List factor pairs of 36 and identify the central pair.

Hint: Try small divisors in increasing order.

Check the guided answer

The pairs are 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. The last is central because 36 is square.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A square's middle pair must be written twice in the factor list.”

Correction: The pair 6 × 6 for 36 supplies one distinct factor value, six; do not double-count it.

Extend and reflect

Why do positive square numbers have an odd number of positive factors?

Guidance: Noncentral factors occur in pairs. The repeated central square root adds one unpaired distinct factor.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: List factor pairs without omission. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

List multiples and common multiples

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Generate multiples by repeated addition or multiplication
Find common multiples from two ordered lists

Vocabulary: multiple · common multiple · sequence · least common multiple

Retrieve previous learning

What are the first four positive multiples of 6?

Check: They are 6, 12, 18 and 24; each is six times a positive whole number.

Learn the concept

Multiples of a positive whole number arise by multiplying it by 1, 2, 3 and so on. The first positive multiples of 6 are 6, 12, 18, 24 and 30. Repeatedly adding six gives the same list, but multiplication makes a distant term easier: the tenth positive multiple is 6 × 10 = 60. Zero is also a multiple if zero is allowed as the multiplier; name whether a question asks for positive multiples.

A common multiple belongs to two multiple lists. List multiples of 6 and 8 side by side: 6, 12, 18, 24, 30, 36, 42, 48; and 8, 16, 24, 32, 40, 48. Their positive common multiples begin 24, 48 and 72. The smallest positive one is 24. You can check a candidate by dividing by both numbers and looking for whole-number quotients.

Do not confuse a common multiple with a common factor. A common factor of 6 and 8 must divide both, so it cannot exceed 6; their common multiples can be arbitrarily large. Listing is useful for small numbers, while prime factors later provide a more efficient route for large ones. The two ideas answer different questions about how quantities fit together.

Worked model

List positive multiples of 6 until at least 48.
List positive multiples of 8 until at least 48.
Circle 24 and 48 in both lists.
Divide 24 by 6 and by 8 to check it is the first positive common multiple.

Guided practice

Find the first positive common multiple of 4 and 10.

Hint: List each sequence beyond ten.

Check the guided answer

Multiples of 4 include 4, 8, 12, 16, 20; multiples of 10 include 10, 20. The first shared positive value is 20.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The first common multiple of 6 and 8 is 48 because 6 × 8 = 48.”

Correction: Their product is a common multiple, but 24 is a smaller positive one: 24 = 6 × 4 = 8 × 3.

Extend and reflect

Explain why every positive multiple of 24 is a common multiple of 6 and 8.

Guidance: Since 24 itself is divisible by both, multiplying it by any positive whole number preserves both divisibilities.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Generate multiples by repeated addition or multiplication. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Explain divisibility by two and three

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Apply the final-digit test for two
Justify the digit-sum test for three using place value

Vocabulary: divisibility · even · digit sum · remainder

Retrieve previous learning

Is 42 even?

Check: Yes, because its final digit is 2 and 42 = 2 × 21.

Learn the concept

A whole number is divisible by 2 exactly when its final digit is even: 0, 2, 4, 6 or 8. Tens, hundreds and larger place values are already divisible by 2, so only the units digit can change the remainder. Thus 246 is divisible by 2, while 153 is not. This is a test for integer divisibility, not a claim about a decimal such as 24.6.

For 3, add the decimal digits. Since 10 leaves remainder 1 on division by 3, each power of 10 does too. Consequently 153 and its digit sum 1 + 5 + 3 = 9 have the same remainder on division by 3; 153 ÷ 3 = 51. The sum of 246's digits is 12, so 246 is also divisible by 3. A sum such as 11 signals a remainder, not divisibility.

Explain a decision with the actual digits and the relevant rule. A number can pass one test and fail the other: 153 passes the 3 test but is odd. A number passing both will later pass the test for 6, but that conclusion depends on the factors 2 and 3 being combined. When large numbers are involved, tests are quicker than long division and can serve as a check afterward.

Worked model

Read 246's final digit 6 and confirm it is even.
Add its digits: 2 + 4 + 6 = 12.
Recognise 12 as a multiple of 3.
Conclude that 246 is divisible by both 2 and 3, checking 246 ÷ 3 = 82 if needed.

Guided practice

Which tests does 153 pass?

Hint: Read the units digit and add all three digits.

Check the guided answer

153 is odd, so it fails the 2 test. Its digit sum is 9, so it passes the 3 test.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A number is divisible by 3 when its final digit is 3, 6 or 9.”

Correction: The test uses the sum of every digit: 123 is divisible by 3 although its final digit is 3, while 13 is not.

Extend and reflect

Show why adding 9 to any integer leaves its divisibility-by-3 status unchanged.

Guidance: Nine is a multiple of 3, so the new and old numbers have the same remainder on division by 3.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Apply the final-digit test for two. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Combine tests for six and nine

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Apply the combined test for six
Use a digit sum to test divisibility by nine

Vocabulary: composite number · digit sum · multiple · divisibility

Retrieve previous learning

Which two small factors multiply to 6?

Check: Two and three multiply to six; checking both tests establishes divisibility by six.

Learn the concept

A whole number is divisible by 6 exactly when it is divisible by both 2 and 3. This works because 2 and 3 have no common factor above one. Check the final digit for evenness, then add the digits for divisibility by 3. The number 378 ends in 8 and has digit sum 18, so 378 ÷ 6 = 63. The number 189 has digit sum 18 but is odd, so it fails the test for 6.

A number is divisible by 9 when its digit sum is divisible by 9. Each power of ten leaves remainder one on division by nine, so replacing the place values by their digits preserves the remainder. The digit sum of 378 is 18 and 18 = 9 × 2, hence 378 ÷ 9 = 42. One can repeat the digit sum on a large number, but the reasoning still concerns the original number's remainder.

Do not assume that every multiple of 6 is a multiple of 9: 42 = 6 × 7 but 42's digit sum is 6, so 9 does not divide it. Nor does a multiple of 9 have to be even: 189 is a counterexample. Always state the separate checks rather than labelling a number merely from its size. A short inverse multiplication can verify the quotient after the divisibility decision.

Worked model

Read the last digit of 378: 8 is even, so 2 divides it.
Add its digits: 3 + 7 + 8 = 18, so 3 divides it.
Combine the two conditions to conclude that 6 divides 378.
Since 18 is also divisible by 9, conclude that 9 divides 378 and check by multiplication.

Guided practice

Is 189 divisible by 6, by 9, or both?

Hint: Do not use its digit sum alone for six.

Check the guided answer

The sum 1 + 8 + 9 = 18 makes it divisible by 9. It is odd, so it is not divisible by 6.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“If the digits add to 9, the number must be divisible by 6.”

Correction: A digit sum of 9 proves divisibility by 3 and 9, but the number must also be even to be divisible by 6.

Extend and reflect

Construct a three-digit multiple of 9 that is odd and explain why it fails the six test.

Guidance: For example 171 has digit sum 9 and is odd; it is divisible by 9 but not by 6.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Apply the combined test for six. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Use the final digit to test five and ten

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Distinguish the tests for five and ten
Explain the tests using base-ten place value

Vocabulary: units digit · multiple · place value · divisibility

Retrieve previous learning

What is 5 × 9?

Check: Five times nine is 45, a multiple of five ending in 5.

Learn the concept

Multiples of 10 end in zero because they are whole groups of ten. In 280, the 28 tens leave no extra units, so 280 = 10 × 28. A number such as 285 has five units after the 28 tens, so it is not divisible by 10. The rule applies to whole numbers represented in base ten and follows directly from the units column.

Multiples of 5 end in zero or five. Any whole number can be split into tens and its units digit; every group of ten already contains two fives. A final zero adds no remainder, and a final five adds exactly one more group of five. Thus 275 = 5 × 55 but is not a multiple of 10, whereas 280 passes both tests. A final digit 2 cannot give an exact quotient by five.

The implication has one direction: every multiple of 10 is a multiple of 5, since 10 = 2 × 5. The reverse is false, with 275 as a counterexample. Writing one counterexample is stronger than saying that a rule 'sometimes fails' without evidence. For a large number, inspect only the last digit, then use division or multiplication to verify if necessary.

Worked model

Read the final digit of 275 as five.
Conclude that five divides it, and verify 275 ÷ 5 = 55.
Reject divisibility by ten because its final digit is not zero.
Compare 280, which ends in zero and so passes both tests.

Guided practice

Which tests do 1,230 and 1,235 pass?

Hint: Check zero versus five in the units position.

Check the guided answer

1,230 ends in zero and passes both tests; 1,235 ends in five and passes only the five test.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Numbers ending in five are divisible by both five and ten.”

Correction: A final five passes the five test only; divisibility by ten requires a final zero.

Extend and reflect

How many positive integers from 1 to 100 are divisible by both 5 and 10?

Guidance: They are precisely the ten multiples of ten from 10 to 100, because passing the ten test already passes the five test.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Distinguish the tests for five and ten. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Test divisibility by four and eight

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Use the final two digits to test four
Use the final three digits to test eight and explain the place-value reason

Vocabulary: last two digits · last three digits · remainder · place value

Retrieve previous learning

Why is 1,200 divisible by four?

Check: Twelve hundred is 12 × 100, and each hundred contains 25 groups of four.

Learn the concept

A hundred is divisible by 4, so all complete hundreds can be ignored when checking a whole number for divisibility by 4. The remainder depends on the last two digits. The number 1,236 ends in 36, and 36 ÷ 4 = 9, so 4 divides 1,236. In contrast 1,238 ends in 38, which leaves remainder 2 on division by 4. Leading zeroes in the final two-digit block count: 2,004 ends in 04 and passes the test.

A thousand is divisible by 8, so only the final three digits matter for 8. In 1,536 the last block is 536, and 536 ÷ 8 = 67 exactly, so the whole number is divisible by 8. The last block 236 of 1,236 is not: 8 × 29 = 232, leaving 4. A number can therefore be divisible by 4 without being divisible by 8. Every multiple of 8 is a multiple of 4, because 8 = 2 × 4.

The final digit alone is insufficient for either test. Both 12 and 22 end in 2, yet only 12 is divisible by 4. A robust explanation states why the ignored part is divisible, tests the remaining block and gives a quotient or remainder. For an eight test, do not accidentally inspect only the final two digits; a block such as 112 is divisible by 8 even though 12 is not.

Worked model

Split 1,536 into one thousand and final block 536.
Recognise 1,000 = 8 × 125, so the first part is divisible by eight.
Calculate 536 ÷ 8 = 67 with no remainder.
Conclude 1,536 is divisible by eight and therefore also by four.

Guided practice

Test 1,236 for four and eight.

Hint: Use 36 for four and 236 for eight.

Check the guided answer

36 ÷ 4 = 9, so four divides 1,236. But 236 = 8 × 29 + 4, so eight does not.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The final two digits decide divisibility by eight.”

Correction: A hundred is not divisible by eight, but a thousand is; therefore the final three digits are needed.

Extend and reflect

Find a four-digit number ending in 112 that passes the eight test.

Guidance: Any thousands digit works because 1,000 is divisible by eight; for instance 7,112.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Use the final two digits to test four. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Build square and cube numbers from equal factors

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Calculate squares and cubes of positive integers
Distinguish the geometric interpretation from multiplying by two or three

Vocabulary: square · cube · equal factors · exponent

Retrieve previous learning

What are 5 × 5 and 5 × 5 × 5?

Check: They are 25 and 125 respectively; the first uses two equal factors, the second three.

Learn the concept

Squaring a whole number means multiplying it by itself, not doubling it. Eight squared is 8² = 8 × 8 = 64. An eight-by-eight array has 64 unit squares, which is why the word square is apt. A square number can also be recognised from the equal factor pair: 49 = 7 × 7. Numbers between consecutive squares, such as 50 between 49 and 64, are not square numbers.

Cubing uses three equal factors: 4³ = 4 × 4 × 4 = 64. A physical cube with four unit lengths on every edge contains 64 unit cubes, but its volume units are cubic units, not square units. The same numerical result 64 is both 8² and 4³. Do not infer from this coincidence that every square is a cube or every cube a square.

The exponent gives the number of copies of the base in the multiplication. Compare 3² = 9 with 3³ = 27 and with 3 × 2 = 6. For positive numbers above one, cubing produces a larger result than squaring the same base. This size check catches some errors, though a correctly calculated answer still needs the multiplication shown. Later powers follow the same repeated-factor idea.

Worked model

Write 7² as 7 × 7 and compute 49.
Write 3³ as 3 × 3 × 3.
Multiply the first pair 3 × 3 = 9, then 9 × 3 = 27.
Compare with 3 × 3 = 9 to show why a cube needs a third factor.

Guided practice

Find 6² and 2³ and explain the notation.

Hint: Expand each exponent into equal factors.

Check the guided answer

6² = 6 × 6 = 36; 2³ = 2 × 2 × 2 = 8. The raised numbers count factors.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Five cubed is five times three, or fifteen.”

Correction: The exponent three counts copies of five: 5³ = 5 × 5 × 5 = 125.

Extend and reflect

Find a positive whole number below 1,000 that is both a square and a cube besides 1 and 64.

Guidance: A sixth power works: 3⁶ = 729 = 27² = 9³.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Calculate squares and cubes of positive integers. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Reverse squares and cubes with roots

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Find principal square roots of perfect squares
Find integer cube roots including negative cubes

Vocabulary: square root · cube root · principal root · inverse operation

Retrieve previous learning

What are 7² and 5³?

Check: 7² = 49 and 5³ = 125, giving the matching exact root facts.

Learn the concept

A square root reverses squaring. Since 7² = 49, √49 = 7 means the principal, non-negative square root. The equation x² = 49 has two real solutions, 7 and −7, because both square to 49; that is a different statement from the single symbol √49. Keeping these meanings separate prevents the common error of writing √49 = ±7.

A cube root reverses cubing. Because 5³ = 125, ∛125 = 5. A negative integer can have a negative cube root: (−5)³ = −125, hence ∛(−125) = −5. Unlike squaring, cubing preserves the sign of a negative base. Square roots of negative numbers are not real, so an ordinary Year 7 real-number calculation cannot produce √(−9).

Estimate a root with neighbouring powers before using a calculator. Since 8² = 64 and 9² = 81, √70 lies between 8 and 9. It is not an integer, so an approximate decimal needs an explicit rounding level. Similarly, 3³ = 27 and 4³ = 64 place ∛50 between 3 and 4. Such bounds give a reasonableness check independent of a keypress.

Worked model

Identify 49 as 7 × 7.
State the principal square root √49 = 7.
Identify 125 as 5 × 5 × 5 and write ∛125 = 5.
Check the inverse operations by squaring seven and cubing five.

Guided practice

Find √81 and ∛64.

Hint: Think of equal factors giving 81 and 64.

Check the guided answer

9² = 81, so √81 = 9. Also 4³ = 64, so ∛64 = 4.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The symbol √49 has two answers, 7 and −7.”

Correction: √49 denotes the principal root 7. The equation x² = 49, not the radical alone, has two real solutions.

Extend and reflect

Give whole-number bounds for √90 and ∛100.

Guidance: 9² = 81 < 90 < 100 = 10², so 9 < √90 < 10; 4³ = 64 < 100 < 125 = 5³, so 4 < ∛100 < 5.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Find principal square roots of perfect squares. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Read and write positive integer powers

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Expand a positive integer power into equal factors
Compare powers with repeated multiplication and explain parentheses

Vocabulary: base · exponent · power · repeated multiplication

Retrieve previous learning

Expand 2³.

Check: 2³ means 2 × 2 × 2 = 8, not 2 × 3.

Learn the concept

In 2⁵, the base is 2 and the exponent is 5. Expand to five equal factors: 2 × 2 × 2 × 2 × 2 = 32. The exponent is not an extra factor of five. In 3⁴, four factors of three give 81. Writing each factor first is a reliable method for small exponents and makes the meaning visible when a calculation goes wrong.

Parentheses decide which quantity is raised to a power. (−2)⁴ multiplies four negative twos and gives positive 16. The expression −2⁴, without parentheses around the negative base, means the negative of 2⁴, or −16 under the usual order of operations. Give words as well as symbols when first introducing this distinction; typography can otherwise conceal what is being multiplied.

Comparing 2⁵ and 5² shows that switching base and exponent generally changes the value: 32 versus 25. A power can be decomposed into repeated products, for example 2⁵ = 2² × 2³ = 4 × 8, because five copies can be grouped as two and three copies. This is an explanation from multiplication, not a rule to apply blindly to sums such as 2² + 2³.

Worked model

Identify 3 as the base and 4 as the exponent in 3⁴.
Expand 3⁴ into 3 × 3 × 3 × 3.
Multiply 3 × 3 = 9 and the next 3 × 3 = 9.
Multiply the two groups 9 × 9 = 81 and check that four factors were used.

Guided practice

Find 4³ and say what the raised three means.

Hint: Write three equal factors of four.

Check the guided answer

4³ = 4 × 4 × 4 = 64; the exponent counts three factors of four.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The exponent tells you what number to multiply the base by once.”

Correction: It counts the copies of the base in a repeated product: 2⁵ has five factors of two.

Extend and reflect

Explain why 2² × 2³ = 2⁵ without quoting an exponent rule.

Guidance: Expand each power into two and three factors of two, then count five factors in the combined product.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Expand a positive integer power into equal factors. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Use a calculator to check powers and roots

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Enter a power or root with appropriate keys
Separate exact form from a rounded calculator display

Vocabulary: calculator · exact value · approximation · rounding · principal root

Retrieve previous learning

Between which whole numbers does √70 lie?

Check: Between 8 and 9 because 8² = 64 and 9² = 81.

Learn the concept

Use calculator keys as a check on reasoning, not as a replacement for it. For 7², enter 7 then the square key or calculate 7 × 7; the display should show 49. For a cube, a power key might require 4, power, 3 and equals; check 4³ = 64 by repeated multiplication. Key layout differs by device, so confirm the entered expression on the display before trusting the result.

For √50, a calculator displays approximately 7.0710678, but the exact value remains √50. Rounding to one decimal place gives 7.1 because the next digit is 7. This is an approximation: 7.1² = 50.41, not exactly 50. A response should say √50 ≈ 7.1 to one decimal place rather than claim that √50 = 7.1. The bounds 7² = 49 and 8² = 64 already show the answer is between 7 and 8.

An exact cube can be checked in reverse: if ∛216 shows 6, cube six to get 216. If a display gives a long decimal, record only the digits needed for the question and keep the original expression when exactness matters. Entering a negative base or a minus sign outside a power can change a result; use parentheses deliberately and compare with a mental sign check.

Worked model

Estimate √50 between seven and eight using 49 and 64.
Enter the square-root operation for 50 and read about 7.0710678.
Round to one decimal place: 7.1, because the hundredths digit is 7.
Use the approximation sign and verify 7.1² is close to, but not equal to, 50.

Guided practice

Use a calculator to estimate ∛70 to one decimal place.

Hint: First bound it between consecutive integer cubes.

Check the guided answer

4³ = 64 and 5³ = 125, so ∛70 lies between 4 and 5; calculator value is about 4.121, so 4.1 to one decimal place.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A rounded calculator display is always equal to the exact root.”

Correction: A rounded decimal is approximate: √30 ≈ 5.5, while 5.5² = 30.25 rather than 30.

Extend and reflect

Why is √64 exact but √65 normally reported approximately?

Guidance: Eight squared is exactly 64. Sixty-five lies between adjacent square numbers, so its positive root is not a whole number.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Enter a power or root with appropriate keys. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

List factors by paired division

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

List all positive factors of a larger integer
Stop a factor search without missing pairs

Vocabulary: factor · factor pair · divisor · square root

Retrieve previous learning

What factor pair includes 8 for 72?

Check: 8 × 9 = 72, so 8 and 9 are paired factors.

Learn the concept

To list the positive factors of 72, build factor pairs in ascending order of the smaller member: 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12 and 8 × 9. Five and seven do not divide 72 exactly. Once the smaller tested number would exceed the larger paired value, every later pair reverses an earlier one. This gives a completeness argument rather than a list recalled from memory.

Write the distinct factors in numerical order: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Each listed value can be checked through its pair. The factors of a fixed positive integer form a finite set, unlike its multiples. A prime number has only the end pair 1 × itself; a composite number has at least one additional pair. The word factor refers to exact whole-number division here.

A square can have an unpaired central factor, as 64 has 8 × 8. If a list repeats eight, it is a list of pair positions, not distinct factor values. For a non-square such as 72, each pair supplies two different values. This pattern helps audit a list: an omitted paired value signals a missed division fact. A calculator can confirm uncertain quotients, but it does not decide which trials were needed.

Worked model

Test divisors from one upwards and write factor pairs of 72.
Record 1 × 72, 2 × 36, 3 × 24 and 4 × 18.
Reject five and seven, then record 6 × 12 and 8 × 9.
Stop before pairs repeat and sort all twelve distinct factors.

Guided practice

Find all positive factors of 30.

Hint: Build pairs, then sort the values.

Check the guided answer

The pairs are 1 × 30, 2 × 15, 3 × 10 and 5 × 6; factors are 1, 2, 3, 5, 6, 10, 15 and 30.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Once 8 is a factor of 72, 9 can be omitted because it belongs to the same pair.”

Correction: A pair gives two distinct factors when its values differ: both 8 and 9 divide 72 exactly.

Extend and reflect

Find a positive number below 50 with exactly five distinct factors.

Guidance: Sixteen has factors 1, 2, 4, 8 and 16. Explain how the central 4 × 4 pair gives the odd count.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: List all positive factors of a larger integer. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Check whether a whole number is prime

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Define prime and composite numbers
Test possible divisors systematically rather than relying on appearance

Vocabulary: prime · composite · divisor · factor pair

Retrieve previous learning

What positive factors does 7 have?

Check: Only 1 and 7 divide seven exactly, so seven is prime.

Learn the concept

A prime number is a positive whole number greater than one with exactly two positive factors: one and itself. The number one has only one positive factor, so it is neither prime nor composite. Composite numbers have more than two. Two is prime and is the only even prime; any larger even number is divisible by two as well as one and itself. Definitions matter more than a memorised list.

To test 29, try small possible divisors. It is odd, so not divisible by 2; its digit sum 11 rejects 3; its final digit 9 rejects 5. A factor larger than √29 would have a partner smaller than √29, so no further trial is needed. Thus 29 is prime. In contrast 39 has digit sum 12 and equals 3 × 13, so it is composite. Forty-nine is 7 × 7 and is composite despite being odd and not divisible by 3 or 5.

A counterexample can disprove a claim such as 'all odd numbers are prime': 9 = 3 × 3. The reverse claim 'every prime is odd' is also false because two is prime. Make the factor search visible: a conclusion is more trustworthy when the learner can show why no missing pair exists. A prime check for larger integers may be supported by a calculator, but the logical criterion stays the same.

Worked model

State that a prime must be greater than one and have exactly two factors.
For 29, reject 2 by oddness and 3 by digit sum eleven.
Reject 5 because 29 does not end in zero or five.
Since no possible smaller factor remains before √29, conclude 29 is prime.

Guided practice

Are 1, 2 and 21 prime?

Hint: Count distinct factors; do not classify one by habit.

Check the guided answer

One is neither prime nor composite. Two has exactly factors 1 and 2, so prime. Twenty-one = 3 × 7, so composite.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Every odd number is prime.”

Correction: Nine, fifteen and forty-nine are odd composite numbers; an odd number can have odd factors besides one and itself.

Extend and reflect

Explain why checking possible divisors only up to the square root is sufficient.

Guidance: If n = ab and both a and b exceeded √n, their product would exceed n. Thus at least one factor of a composite n is no larger than √n.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Define prime and composite numbers. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Write an integer as prime factors

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Build a complete prime-factor tree
Write the final product with powers where useful

Vocabulary: prime factor · factor tree · product notation · unique factorisation

Retrieve previous learning

Is 7 prime?

Check: Yes. Its only positive factors are one and seven, so a factor tree stops there.

Learn the concept

A factor tree repeatedly replaces a composite number by two factors until every leaf is prime. For 84, split 84 = 12 × 7; seven is prime, while 12 = 3 × 4 and 4 = 2 × 2. The prime leaves give 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7. Multiplying 4 × 3 × 7 returns 84, which checks that no factor has been lost.

A different first split gives the same prime factors: 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7. The order of the prime factors may differ but the multiset does not; this is the unique factorisation property for integers greater than one. A final leaf such as four is incomplete because four can still be split into two primes. The number one should not be included as a prime leaf.

Power notation records repeats compactly. In 72 = 2 × 2 × 2 × 3 × 3, write 2³ × 3². This is not the same as 2 × 3 × 2 or 2³ + 3². Use multiplication symbols between different prime powers and verify by calculating 8 × 9 = 72. Prime factorisation later makes HCF and LCM calculations more systematic.

Worked model

Split 84 into 12 × 7 and mark 7 prime.
Split 12 into 3 × 4, then 4 into 2 × 2.
Write all prime leaves as 2 × 2 × 3 × 7 = 2² × 3 × 7.
Multiply 4 × 3 × 7 = 84 to verify completeness.

Guided practice

Prime-factorise 60.

Hint: Start with 6 × 10 or 2 × 30; split until all leaves are prime.

Check the guided answer

60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A factor tree ending in 4 and 7 is complete because both are factors.”

Correction: Only prime numbers can be final leaves; four must split into 2 × 2.

Extend and reflect

Give two different first splits of 120 and show that both yield the same prime factors.

Guidance: For example 12 × 10 and 3 × 40 both lead to 2³ × 3 × 5; write both trees or chains.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Build a complete prime-factor tree. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Evaluate prime-factor expressions

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Evaluate a product of prime powers accurately
Use a size or inverse factorisation check

Vocabulary: prime power · product · exponent · factorisation

Retrieve previous learning

What does 2³ × 3² represent?

Check: It is 8 × 9 = 72, a product of prime powers rather than a sum.

Learn the concept

A prime-factor expression names an integer without writing it in ordinary decimal form. Evaluate each power first, then multiply. For 2³ × 3² × 5, calculate 2³ = 8 and 3² = 9; then 8 × 9 × 5 = 72 × 5 = 360. This order avoids misreading 2³ as six or adding the prime powers together. The multiplication sign between factors matters.

Grouping can simplify mental arithmetic: in 2² × 5² × 7, 2² × 5² = 4 × 25 = 100, so the number is 700. This also shows that a factorisation can reveal divisibility: a product containing a factor 2 and a factor 5 is divisible by ten. A factor of 3 does not appear in 700's expression, so 700 is not divisible by three. Check against a digit-sum test as another representation.

If you are given a factorisation and asked whether a stated integer is possible, multiply back and compare. For instance 2⁴ × 3 = 16 × 3 = 48, not 24. The unique factorisation property also says that an integer cannot have two genuinely different collections of prime factors; a mismatch indicates an arithmetic or copying error. Keep exponents attached to their own bases.

Worked model

Evaluate 2³ as eight and 3² as nine.
Write the remaining product 8 × 9 × 5.
Multiply 8 × 9 = 72, then 72 × 5 = 360.
Check that 360 is divisible by two, three and five, as its factors predict.

Guided practice

Calculate 2² × 3 × 7.

Hint: Evaluate the square before multiplying.

Check the guided answer

2² = 4, so 4 × 3 × 7 = 84.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“To evaluate 2³ × 3², add eight and nine.”

Correction: The prime powers are joined by multiplication: 2³ × 3² = 8 × 9 = 72.

Extend and reflect

Which integer is 2³ × 3 × 5², and what is its largest power-of-ten factor?

Guidance: The value is 8 × 3 × 25 = 600. It contains two 2s and two 5s, so 100 is its largest power-of-ten factor.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Evaluate a product of prime powers accurately. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Find the highest common factor

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Find HCF by listing factors or comparing prime powers
Interpret HCF as the largest equal grouping size

Vocabulary: common factor · highest common factor · prime power · divisor

Retrieve previous learning

Prime-factorise 36.

Check: 36 = 2² × 3², since 4 × 9 = 36.

Learn the concept

The highest common factor of two positive integers is the largest positive whole number that divides both. For 36 and 60, list common factors such as 1, 2, 3, 4, 6 and 12; none above 12 divides both. The HCF is 12. It can be the size of the largest identical groups when both collections must be split without leftovers. HCF is a factor, so it cannot exceed either original positive number.

Prime factors give a systematic method. Write 36 = 2² × 3² and 60 = 2² × 3 × 5. Take only primes present in both, each with the smaller exponent: 2² × 3 = 12. Three squared appears in 36 but only one three appears in 60, so only one can be shared. Including five would fail because 36 has no factor five. Verify 36 ÷ 12 = 3 and 60 ÷ 12 = 5.

The word 'highest' does not mean choose the greatest prime factor seen in either expression. A shared product of several primes may be larger. For three numbers, retain prime powers common to all three, again using the smallest exponent. If one number divides the other, the smaller number is the HCF, such as HCF(8, 24) = 8. Context must still explain what each group represents.

Worked model

Prime-factorise 36 = 2² × 3².
Prime-factorise 60 = 2² × 3 × 5.
Choose common powers 2² and 3, using the smaller exponent in each case.
Multiply 4 × 3 = 12 and verify both original numbers divide by 12 exactly.

Guided practice

Find HCF(24, 40).

Hint: Use 24 = 2³ × 3 and 40 = 2³ × 5.

Check the guided answer

Only 2³ is common, so the HCF is 8; 24 ÷ 8 = 3 and 40 ÷ 8 = 5.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“HCF(36, 60) is 5 because five is a large prime in 60.”

Correction: Five is not a factor of 36. The highest common factor is 12, from shared 2² × 3.

Extend and reflect

Find HCF(36, 60, 84) by prime powers.

Guidance: 36 = 2² × 3², 60 = 2² × 3 × 5, 84 = 2² × 3 × 7. Shared powers are 2² × 3 = 12.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Find HCF by listing factors or comparing prime powers. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Find the lowest common multiple

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Find LCM from multiples or prime powers
Check that the answer is divisible by both originals

Vocabulary: common multiple · lowest common multiple · prime power · divisibility

Retrieve previous learning

What are the first three positive multiples of 12?

Check: They are 12, 24 and 36.

Learn the concept

The lowest common multiple is the smallest positive integer divisible by each original number. Multiples of 12 begin 12, 24, 36, 48; multiples of 18 begin 18, 36, 54. Their first shared positive value is 36, so LCM(12, 18) = 36. The answer must be at least as large as either original positive number. Zero is mathematically a common multiple, but 'lowest common multiple' in school problems means the smallest positive one.

Using prime factors, 12 = 2² × 3 and 18 = 2 × 3². To hold both numbers as factors, take every prime seen with the larger exponent: 2² × 3² = 4 × 9 = 36. Taking the smaller exponents instead would find the HCF, not the LCM. The product 12 × 18 = 216 is a common multiple but need not be the least; shared factors cause it to be unnecessarily large.

Verify a proposed LCM in two parts. First divide it by each original number: 36 ÷ 12 = 3 and 36 ÷ 18 = 2. Then explain why no smaller positive shared multiple exists, for example by the prime-power method or a short ordered list. A timing context may use LCM for the first time cycles coincide after the starting moment, but actual schedules also need the same starting time.

Worked model

Factorise 12 = 2² × 3 and 18 = 2 × 3².
Select each prime at its largest exponent: 2² and 3².
Multiply 4 × 9 to obtain 36.
Verify 36 ÷ 12 = 3 and 36 ÷ 18 = 2, then note smaller prime powers cannot contain both factors.

Guided practice

Find LCM(8, 12).

Hint: Compare 8 = 2³ and 12 = 2² × 3.

Check the guided answer

Use 2³ × 3 = 24; both eight and twelve divide 24 exactly.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The LCM of two numbers is always their product.”

Correction: Their product is a common multiple, but LCM(12, 18) = 36 rather than 216 because they share factors.

Extend and reflect

Find LCM(6, 8, 9) and explain the prime-power selection.

Guidance: Use 2³ from eight and 3² from nine, giving 72; six also divides 72.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Find LCM from multiples or prime powers. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Choose HCF or LCM from a context

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Identify a greatest equal-group problem as HCF
Identify a first repeated coincidence problem as LCM

Vocabulary: highest common factor · lowest common multiple · equal groups · cycle

Retrieve previous learning

What are HCF(24, 36) and LCM(24, 36)?

Check: They are 12 and 72 respectively: 12 divides both, while 72 is the least positive multiple of both.

Learn the concept

The arithmetic words alone do not select HCF or LCM; read what a context asks. If 24 red and 36 blue tokens must be split into the greatest possible number of identical packs with none left, that number is HCF(24, 36) = 12. Each pack gets 2 red and 3 blue. The proposed pack count has to divide both collections, so it cannot exceed the smaller total of 24.

If two lights flash every 24 and 36 seconds and flash together at time zero, their next shared flash is after LCM(24, 36) = 72 seconds. Seventy-two is a multiple of both intervals: it is the third 24-second flash and second 36-second flash after the start. The answer is a time, not a pack count. If the lights start at different moments, the simple LCM answer does not by itself settle when they coincide.

Prime factorisations 24 = 2³ × 3 and 36 = 2² × 3² make both values visible. HCF takes shared minimum exponents, 2² × 3 = 12; LCM takes maximum exponents, 2³ × 3² = 72. Check the units and the relationship: for these two numbers, 12 × 72 = 24 × 36. That identity can check arithmetic, but the context still decides which result to use.

Worked model

Ask whether the answer must divide both quantities or be divisible by both intervals.
For identical packs, factorise 24 and 36 and take shared minimum powers.
Calculate HCF = 2² × 3 = 12 packs.
For first repeated coincidence, take maximum powers to obtain LCM = 2³ × 3² = 72 seconds.

Guided practice

Which method finds the first time 8-minute and 12-minute bells ring together after a shared start?

Hint: The answer is a multiple of both intervals.

Check the guided answer

LCM: the first shared positive multiple is 24 minutes.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Both grouping and repeat-time problems use the same formula because both mention two numbers.”

Correction: Grouping asks for a divisor of both totals; repeat time asks for a multiple of both intervals.

Extend and reflect

Construct a story involving 12 and 18 whose answer uses HCF, then change it to one using LCM.

Guidance: For HCF ask for greatest identical packs; for LCM ask when two cycles starting together next coincide. Give units.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Identify a greatest equal-group problem as HCF. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Solve multi-step problems with number structure

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Choose between factors, multiples, squares and cubes in a context
Check a multi-step solution using inverse operations and units

Vocabulary: model · constraint · factor · multiple · square · cube

Retrieve previous learning

What method finds the greatest number of equal packs from 72 and 90 items?

Check: Use their highest common factor: it is the greatest positive pack count dividing both totals.

Learn the concept

A multi-step problem can require more than a memorised divisibility rule. Suppose 72 red and 90 blue counters are put into the greatest number of identical packs with no leftover. Prime-factorise 72 = 2³ × 3² and 90 = 2 × 3² × 5. The HCF is 2 × 3² = 18, so make 18 packs, each with 4 red and 5 blue. Check 18 × 4 = 72 and 18 × 5 = 90. The answer 18 is a count of packs, not counters per pack.

A different story may ask for repeated events. If one signal repeats every 12 minutes and another every 18 minutes after a shared starting time, take LCM(12, 18) = 36 minutes. If a square arrangement is required instead, search for an equal factor pair: 49 tiles form a 7-by-7 square, while 50 tiles cannot make a filled whole-number square without adding or removing tiles. The story's required structure, rather than the presence of two numbers, picks the method.

Check a proposed answer against all constraints. A grouping answer must divide each stock exactly; a shared-cycle time must be a multiple of every interval; a square root is a side length and must be positive in a physical arrangement. State whether fractional objects, leftovers or different start times are allowed. If not specified, write a reasonable assumption and explain how changing it might change the result.

Worked model

Read the equal-pack condition and select HCF rather than LCM.
Factorise 72 = 2³ × 3² and 90 = 2 × 3² × 5.
Take shared prime powers 2 × 3² = 18 and calculate 18 packs.
Divide stocks to get 4 red and 5 blue per pack and multiply back to verify both totals.

Guided practice

Can 81 tiles make a filled square, and how long is each side in tiles?

Hint: Find an equal whole-number factor pair.

Check the guided answer

Yes. Since 9 × 9 = 81, a filled square has nine tiles along each side.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A multi-step problem is solved once a number has been calculated, even if it has no unit.”

Correction: The result must answer the story: eighteen packs, thirty-six minutes or a seven-tile side are distinct meanings.

Extend and reflect

A square display has 144 positions. Could equal teams of 8 and 9 fill it separately with no leftovers?

Guidance: The side is 12 because 12² = 144. Both 8 and 9 divide 144, so either team size can split all positions into equal groups.