Expressions and equations

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

Describe a rule that works for every pattern number
Test a rule with a new case

Vocabulary: term · position · generalise · constant

Retrieve previous learning

What is the difference between adding 3 each time and multiplying a number by 3?

Check: Adding 3 gives a fixed increase; multiplying by 3 scales the chosen number.

Learn the concept

Suppose a pattern has 4, 7, 10, 13 tiles. The increase of 3 tells us how consecutive terms relate, but the rule for any position also needs a starting adjustment. At position 1, three lots of the position gives 3, so add 1. In words: triple the position, then add one.

A rule must work beyond the first few cases. Position 10 gives 3 × 10 + 1 = 31. Checking positions 1 and 2 gives 4 and 7; this supports the rule, while a visual explanation of three tiles added at each step explains why it continues.

Worked model

List position → tiles: 1 → 4, 2 → 7, 3 → 10.
The change of 3 suggests three times position.
At position 1, 3 needs one more tile, giving triple position plus 1.
Test position 4: 3 × 4 + 1 = 13.

Guided practice

A pattern has 6, 10, 14 tiles at positions 1, 2, 3. Give a word rule.

Hint: Use the change of 4, then adjust the value at position 1.

Check the guided answer

Multiply the position by 4 and add 2. Position 3 gives 4 × 3 + 2 = 14.

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 common difference is the complete rule for any position.”

Correction: It describes the step; a direct rule also needs the position and an adjustment.

Extend and reflect

Can two rules agree for the first three terms but differ later?

Guidance: Yes. Finite examples alone do not prove a unique rule; explain the construction of the pattern.

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

This lesson practises: Describe a rule that works for every pattern number. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Keep an equation balanced with a bar model

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

Represent an unknown as an equal-length bar
Solve by preserving equality

Vocabulary: equation · balance · inverse · unknown

Retrieve previous learning

What number added to 5 gives 12?

Check: 7, because 12 − 5 = 7.

Learn the concept

An equation says two expressions have the same value. Draw one bar for x + 5 and an equal bar of length 17. Removing a length of 5 from each side leaves x equal to 12. This is the inverse of adding 5.

A different equation, 3x + 2 = 20, has three equal x segments and a 2 segment. Remove 2 from both equal bars, leaving 3x = 18; divide both bars into three equal parts, so x = 6. Substitute back: 3 × 6 + 2 = 20.

Worked model

Draw x + 5 and 17 as equal bars.
Remove 5 from both sides: x = 17 − 5.
Calculate x = 12 and check 12 + 5 = 17.

Guided practice

Solve 2x + 3 = 15 using equal bars.

Hint: Remove the 3 segment, then split the remaining bar in two.

Check the guided answer

2x = 12, so x = 6. Check 2 × 6 + 3 = 15.

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

“Moving a term across the equals sign automatically changes its sign.”

Correction: State the operation performed on both sides; this preserves equality and explains the sign change.

Extend and reflect

Solve 3(x + 2) = 21 with a bar model.

Guidance: Three equal groups each measure 7, so x + 2 = 7 and x = 5.

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

This lesson practises: Represent an unknown as an equal-length bar. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Use a letter for any number in a general rule

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

Write a general rule using a letter
Distinguish one example from a general statement

Vocabulary: variable · generalised number · expression · substitute

Retrieve previous learning

What operation turns a number into three more than itself?

Check: Add 3 to the number.

Learn the concept

The phrase 'three more than a number' can be written n + 3. The letter n does not secretly mean one fixed chosen number; in a general expression it can take many values. For n = 4 the expression is 7, and for n = 10 it is 13.

A rule such as twice a number plus one is 2n + 1. Order matters: 2(n + 1) means double the sum, which is 2n + 2. Substitute n = 3 to see 7 versus 8, a counterexample to treating them as the same.

Worked model

Choose n for the input number.
Translate 'five less than double the number' to 2n − 5.
At n = 6, calculate 12 − 5 = 7; the expression remains general.

Guided practice

Write 'four times a number, then subtract two' using p and evaluate at p = 5.

Hint: The multiplication happens before the subtraction.

Check the guided answer

4p − 2; at p = 5 the value is 4 × 5 − 2 = 18.

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 letter always stands for one unknown answer.”

Correction: A letter can represent any permitted input in a general rule; an equation may instead constrain it to particular values.

Extend and reflect

Invent two expressions equal at n = 2 but unequal at n = 3.

Guidance: For example n + 2 and 2n are both 4 at n = 2, but are 5 and 6 at n = 3.

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

This lesson practises: Write a general rule using a letter. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Write multiplication, powers and division clearly

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

Read 3a, a² and a/b correctly
Use brackets where grouping changes meaning

Vocabulary: coefficient · power · quotient · brackets

Retrieve previous learning

What is 4 × 4 × 4?

Check: 64.

Learn the concept

Algebra removes unnecessary multiplication signs: 3a means 3 × a, while ab means a × b. The notation a² means a × a, not 2a. If a = 4, then a² = 16 and 2a = 8.

The expression a/b means a divided by b, with b nonzero. Brackets show a grouped operation: 2(a + b) doubles both terms, while 2a + b doubles only a. Write 1a simply as a, and 0a as 0.

Worked model

Translate 'three times x plus its square' to 3x + x².
At x = 2, calculate 3 × 2 + 2² = 6 + 4.
The value is 10, not (3 × 2 + 2)².

Guided practice

Write 5 × y × y without multiplication signs and evaluate at y = 3.

Hint: The two y factors make a square.

Check the guided answer

5y²; at y = 3 this is 5 × 9 = 45.

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 on a letter means multiply the letter by two.”

Correction: A square is multiplication by itself: a² = a × a.

Extend and reflect

Why is a/b undefined when b = 0?

Guidance: No number times zero can recover a nonzero numerator; division by zero is not defined.

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

This lesson practises: Read 3a, a² and a/b correctly. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Name the parts of an algebraic statement

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 terms, coefficients and factors
Distinguish an expression, equation and identity

Vocabulary: term · coefficient · factor · expression · equation · identity

Retrieve previous learning

How many terms are in 2x + 7?

Check: Two: 2x and 7.

Learn the concept

In 5x² − 3x + 8, the terms are 5x², −3x and 8. The coefficient of x² is 5, and the coefficient of x is −3. In 5x² the factors include 5, x and x. A factor multiplies; a term is separated by addition or subtraction at the top level.

An expression such as 2x + 5 has no equals sign. An equation such as 2x + 5 = 11 asks which x makes it true. An identity such as 2(x + 3) = 2x + 6 is true for every x because of the distributive law. An equation can sometimes have more than one solution, so do not define an equation as having exactly one.

Worked model

Write 4y − 7 as terms 4y and −7.
Identify the coefficient of y as 4.
Identify 4 and y as factors of 4y.
2(4 + 3) = 2 × 4 + 2 × 3 illustrates an identity.

Guided practice

In 6a² + 2a − 9, give the coefficient of a², coefficient of a and constant.

Hint: Keep the sign with its term.

Check the guided answer

They are 6, 2 and −9 respectively.

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 statement with an equals sign is an identity.”

Correction: An identity holds for all permitted values; an equation may hold only for certain values.

Extend and reflect

Explain why x + x = 2x is an identity.

Guidance: Both sides have the same value for every x, since adding two copies of x is doubling x.

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

This lesson practises: Identify terms, coefficients and 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Distinguish a specific unknown from a variable input

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

Interpret a letter in an equation
Interpret a letter in a changing rule

Vocabulary: unknown · variable · input · solution

Retrieve previous learning

Evaluate 2n + 1 when n = 5.

Check: 11.

Learn the concept

In x + 7 = 15, x is an unknown constrained by the equation: x must be 8. In y = 2x + 1, x may vary and y changes with it. A table for x = 0, 1, 2 gives y = 1, 3, 5; this rule describes many input-output pairs.

The same symbol can play different roles in different contexts. State what the domain permits: in a ticket-counting model, x might be a whole number at least zero; in an abstract linear graph, negative or fractional x may be permitted.

Worked model

For x + 7 = 15, subtract 7 from both sides to get x = 8.
For y = 2x + 1, choose x = 8 to obtain y = 17.
Changing x to 9 changes y to 19, so x varies in the rule.

Guided practice

Is t a specific unknown or a variable in t/2 = 6 and in p = t/2?

Hint: Ask whether one value is required or many inputs are allowed.

Check the guided answer

In t/2 = 6 it is constrained to t = 12. In p = t/2 it can vary with p, subject to the context.

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 variable has no restrictions because it can change.”

Correction: A variable can vary only within the values allowed by its definition or context.

Extend and reflect

Can an equation have a variable on both sides and still specify one unknown?

Guidance: Yes: 2x + 3 = x + 8 has x = 5, found by preserving equality.

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

This lesson practises: Interpret a letter in an equation. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Generalise a relationship and test its scope

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

Write a statement that holds for all permitted inputs
Use a counterexample to reject an incorrect claim

Vocabulary: generalisation · identity · counterexample · consecutive

Retrieve previous learning

What is the next whole number after n?

Check: n + 1.

Learn the concept

Three consecutive whole numbers can be written n, n + 1 and n + 2. Their sum is 3n + 3 = 3(n + 1), so it is three times the middle number for every whole number n. The algebra explains the pattern, rather than checking only a few examples.

A counterexample disproves a universal claim. 'The square of every number is greater than the number' fails at 0, since 0² = 0, and at 1. Some statements need a stated domain; 'the sum of two odd whole numbers is even' works because (2a + 1) + (2b + 1) = 2(a + b + 1).

Worked model

Represent consecutive numbers as n, n + 1, n + 2.
Add to obtain 3n + 3.
Factor to 3(n + 1), which is three times the middle number.

Guided practice

Show that two consecutive whole numbers always sum to an odd number.

Hint: Write the numbers as n and n + 1.

Check the guided answer

Their sum is 2n + 1, one more than an even number, so it is odd.

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

“Several confirming examples prove a statement for every number.”

Correction: Examples can suggest a rule; an algebraic argument proves it for the stated domain, while one counterexample disproves it.

Extend and reflect

Does the consecutive-square result hold for negative integers?

Guidance: Yes. The expression 2n + 1 is odd for any integer n; check the meaning of consecutive squares in context.

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

This lesson practises: Write a statement that holds for all permitted inputs. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Substitute values using correct operation order

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 an expression for a specified value
Use brackets when substituting negatives

Vocabulary: substitution · evaluate · order of operations · negative

Retrieve previous learning

What is (−3)²?

Check: 9, because (−3) × (−3) = 9.

Learn the concept

To evaluate 2x² − 3x at x = −2, write 2(−2)² − 3(−2). Square the entire negative number first: 2 × 4 + 6 = 14. Substitution with brackets avoids confusing −2², which by convention is −(2²).

A table can show how an expression changes. For y = x² + 1, inputs x = −2, 0, 2 give y = 5, 1, 5. The equal outputs at opposite inputs come from squaring; they do not imply all inputs produce the same result.

Worked model

Write 2x² − 3x with x = −2 as 2(−2)² − 3(−2).
Calculate the power first: (−2)² = 4.
Then 8 − (−6) = 14.

Guided practice

Evaluate a² + 2a for a = −3.

Hint: Use brackets for both copies of the negative input.

Check the guided answer

(−3)² + 2(−3) = 9 − 6 = 3.

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

“When x = −2, x² is −4.”

Correction: The square of the substituted number is (−2)² = 4; keep brackets around negative inputs.

Extend and reflect

Find all real x for which x² + 1 = 10.

Guidance: x² = 9, so x = 3 or x = −3.

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

This lesson practises: Evaluate an expression for a specified value. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Recognise like terms by their variable parts

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

Group terms with the same variable powers
Explain why unlike terms remain separate

Vocabulary: like terms · coefficient · variable part · power

Retrieve previous learning

How many apples are 3 apples plus 4 apples?

Check: 7 apples.

Learn the concept

The terms 3x and 4x are like terms because both count x-units; their sum is 7x. The terms 3x and 4x² count different units and cannot be combined into 7x or 7x². Similarly 2ab and 5ba are like terms because ab = ba.

Like terms have the same letters raised to the same powers, even when written in a different order. A constant has no variable part, so 6 and −2 are like terms. Identify the complete signed term before collecting.

Worked model

In 3x + 2y − 5x + 4y, group 3x and −5x.
Group 2y and 4y separately.
The two variable parts differ, so do not combine x with y.

Guided practice

Which terms in 4a² + 3a − 7a² + 2 are like 4a²?

Hint: Compare both the letter and its exponent.

Check the guided answer

Only −7a² is like 4a²; 3a has a different power and 2 is a constant.

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

“Any terms containing the same letter are like terms.”

Correction: The entire variable part, including every exponent, must match.

Extend and reflect

Are 3a²b and −ba² like terms?

Guidance: Yes: multiplication is commutative, so both have variable part a²b.

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

This lesson practises: Group terms with the same variable 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Collect like terms without changing an expression's value

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

Simplify sums of signed terms
Verify equivalence by substitution

Vocabulary: simplify · equivalent · collect · signed term

Retrieve previous learning

What is 5x − 2x?

Check: 3x.

Learn the concept

Collecting like terms changes the form, not the value. For 5a − 2b − 8a + 6b, combine the a terms as −3a and the b terms as 4b, giving −3a + 4b. The minus sign belongs to each term after it.

A substitution check can catch a sign error. At a = 2 and b = 3, the original is 10 − 6 − 16 + 18 = 6; the simplified form is −6 + 12 = 6. One check does not prove equivalence for every input, but the collection law does.

Worked model

Rewrite 7p − 4 + 2p + 9 as (7p + 2p) + (−4 + 9).
Add the coefficients: 9p + 5.
At p = 1, both forms give 14.

Guided practice

Simplify 8x − 3y − 11x + 5y.

Hint: Group x terms and y terms with their signs.

Check the guided answer

(8 − 11)x + (−3 + 5)y = −3x + 2y.

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

“Subtracting a term removes its negative sign when terms are reordered.”

Correction: Move each signed term intact; −4 and −9 sum to −13.

Extend and reflect

Show 2x + 3y + 5x − y = 7x + 2y for every x and y.

Guidance: Use commutativity and associativity to group like terms; no particular substitution is needed.

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

This lesson practises: Simplify sums of signed terms. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Distribute a constant across every term

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 constant times a bracket
Preserve signs when the constant is negative

Vocabulary: distribute · expand · bracket · equivalence

Retrieve previous learning

What is 3 × (4 + 2)?

Check: 18.

Learn the concept

Three copies of (x + 4) contain three x-units and twelve ones: 3(x + 4) = 3x + 12. Multiplication applies to every term in the bracket. Check at x = 2: both sides give 18.

For −2(3x − 5), multiply −2 by 3x and by −5 separately to get −6x + 10. A useful area model labels the signed lengths algebraically, but do not treat a negative geometric area as a physical area; use the arithmetic law for signed numbers.

Worked model

In 4(2a − 3), multiply 4 by 2a to get 8a.
Multiply 4 by −3 to get −12.
Write 8a − 12; at a = 2, both forms equal 4.

Guided practice

Expand −3(y + 4).

Hint: Multiply −3 by both y and 4.

Check the guided answer

−3y − 12.

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 multiplier applies only to the first term in a bracket.”

Correction: The distributive law multiplies each term inside the bracket.

Extend and reflect

Factorise 12x − 20 using the inverse of expansion.

Guidance: Both terms contain 4, so 12x − 20 = 4(3x − 5).

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

This lesson practises: Expand a constant times a bracket. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Expand a bracket multiplied by an algebraic term

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

Multiply coefficients and variable powers
Check the degree of each resulting term

Vocabulary: monomial · coefficient · power · distribute

Retrieve previous learning

What is x × x and what is 2 × 3?

Check: x² and 6.

Learn the concept

To expand 2x(3x + 4), multiply 2x by each term: 2x × 3x = 6x² and 2x × 4 = 8x. Thus the result is 6x² + 8x. The power rises because x × x = x², not because the coefficients are added.

A negative term changes signs. For −3a(2a − 5), the products are −6a² and +15a, giving −6a² + 15a. At a = 2, both original and expansion equal −3 × 2 × (4 − 5) = 6.

Worked model

Write 4y(y − 3) as 4y × y − 4y × 3.
Calculate 4y² − 12y.
At y = 2, original 8(−1) and expanded 16 − 24 both equal −8.

Guided practice

Expand 5m(2m + 1).

Hint: Multiply 5m by each term; m × m = m².

Check the guided answer

10m² + 5m.

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

“Multiplying x by x leaves x.”

Correction: x × x = x²; count the repeated variable factors.

Extend and reflect

Expand −a²(2a − 3).

Guidance: Multiply each term: −2a³ + 3a².

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

This lesson practises: Multiply coefficients and variable 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Expand separate products and then collect

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 each product accurately
Collect only like terms afterwards

Vocabulary: expand · collect · equivalent · signed product

Retrieve previous learning

Expand 2(x + 3).

Check: 2x + 6.

Learn the concept

In 2(x + 3) + 3(x − 1), expand each bracket before collecting: 2x + 6 + 3x − 3 = 5x + 3. The plus between products means both expanded expressions contribute to the sum.

Subtraction needs extra care. For 4(y + 2) − 2(y − 5), first write 4y + 8 − (2y − 10). The entire second product is subtracted, so 4y + 8 − 2y + 10 = 2y + 18. Substituting y = 1 gives 20 in both forms.

Worked model

Expand 3(2a − 1) + 2(a + 4) to 6a − 3 + 2a + 8.
Collect a terms and constants separately.
Obtain 8a + 5.

Guided practice

Simplify 2(p + 5) + 4(p − 2).

Hint: Expand both products before grouping.

Check the guided answer

2p + 10 + 4p − 8 = 6p + 2.

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 minus sign between products changes only the first term of the second bracket.”

Correction: Subtract the entire product, or distribute a negative multiplier to every term.

Extend and reflect

Simplify −2(3x − 4) − 3(2x + 1).

Guidance: The expansion is −6x + 8 − 6x − 3 = −12x + 5.

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

This lesson practises: Expand each product 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Choose an efficient order for equivalent 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

Simplify a shared structure before expanding
Justify when an efficient rearrangement is valid

Vocabulary: factor · equivalent · distributive law · efficiency

Retrieve previous learning

Simplify 3x + 2x.

Check: 5x.

Learn the concept

In 3(2x + 5) + 2(2x + 5), both products share the same bracket. Treat that bracket as one quantity B: 3B + 2B = 5B. The compact result 5(2x + 5) can then expand to 10x + 25 if needed. This is the distributive law in reverse.

Efficiency does not license changing grouping. For 3(x + 2) + 2(x − 2), the brackets differ; calling both B would be wrong. Expand to 3x + 6 + 2x − 4 = 5x + 2, or use 5x plus the separately signed constants.

Worked model

Identify repeated bracket (a − 4) in 7(a − 4) − 2(a − 4).
Collect the bracket-units: (7 − 2)(a − 4).
Obtain 5(a − 4), equal to 5a − 20.

Guided practice

Simplify 6(p + 1) + 4(p + 1) without expanding first.

Hint: Count copies of the same bracket.

Check the guided answer

10(p + 1), or 10p + 10 expanded.

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

“Any two brackets containing x can be combined as identical units.”

Correction: They must be exactly the same expression before their coefficients can be collected directly.

Extend and reflect

Simplify a(x + 2) + b(x + 2).

Guidance: Factor the common bracket to (a + b)(x + 2), valid for all a, b and x.

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

This lesson practises: Simplify a shared structure before expanding. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Find a common numerical and algebraic 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 the HCF of coefficients and variable powers
Check a proposed factor by multiplication

Vocabulary: highest common factor · coefficient · power · factor

Retrieve previous learning

What is the highest common factor of 12 and 18?

Check: 6.

Learn the concept

The terms 12x² and 18x each contain a factor 6x: 12x² = 6x × 2x and 18x = 6x × 3. The numerical HCF is 6, while the shared algebraic part is one x, not x² because 18x has only one x factor.

For 8a²b and 12ab³, the numerical HCF is 4 and the shared letters are a and b, so the HCF is 4ab. Dividing gives 2a and 3b², which multiply back to the original terms.

Worked model

Find HCF(15, 25) = 5.
For 15p³ and 25p², the smaller p power is p².
The algebraic HCF is 5p²; check factors 3p and 5.

Guided practice

Find the HCF of 18m²n and 24mn².

Hint: Use the smaller exponent for each shared letter.

Check the guided answer

6mn, since the numerical HCF is 6 and each term has at least one m and one n.

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 HCF uses the largest power seen in either term.”

Correction: A common factor must divide every term; choose the lowest shared power.

Extend and reflect

Find the HCF of 24a³b², 36a²b and 60ab³.

Guidance: The coefficient HCF is 12 and minimum powers are a¹b¹, so 12ab.

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

This lesson practises: Find the HCF of coefficients and variable 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Factorise a sum by reversing distribution

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

Take out a common factor
Check factorisation by re-expanding

Vocabulary: factorise · common factor · expand · equivalent

Retrieve previous learning

Expand 3(x + 4).

Check: 3x + 12.

Learn the concept

Factorising rewrites a sum as a product. For 12x + 18, the highest common numerical factor is 6, giving 6(2x + 3). Re-expand to check: 6 × 2x + 6 × 3 = 12x + 18.

With algebraic factors, 15a² − 10a contains 5a in both terms. Divide each term by 5a to obtain 5a(3a − 2). Retain the negative sign in the bracket and re-expand. For a = 0, both forms are zero; the algebraic law establishes equivalence for all a.

Worked model

Find the HCF of 14y² and 21y as 7y.
Divide 14y² by 7y to get 2y and 21y by 7y to get 3.
Write 14y² + 21y = 7y(2y + 3), then expand to check.

Guided practice

Factorise 8p − 12 using the highest common factor.

Hint: Both coefficients divide by 4.

Check the guided answer

4(2p − 3); expanding gives 8p − 12.

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 common factor is removed only from the first term.”

Correction: Divide every term by the extracted factor; re-expansion checks the result.

Extend and reflect

Can 6x² + 9x be factorised with 3x even if x = 0?

Guidance: Yes, 3x(2x + 3) is an identity; avoid deriving it by dividing both sides by x at x = 0.

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

This lesson practises: Take out a common factor. 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 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Model and solve a two-plan cost problem

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

Translate a context into expressions
Solve and interpret an equality condition

Vocabulary: model · fixed fee · rate · equation · break-even

Retrieve previous learning

Simplify 4n + 6n.

Check: 10n.

Learn the concept

Two fictional club plans charge A = £12 + £4 per visit and B = £30 + £2 per visit. For n visits, costs are A(n) = 12 + 4n and B(n) = 30 + 2n. Equal cost requires 12 + 4n = 30 + 2n, so 2n = 18 and n = 9. Substitute: both cost £48.

The algebraic solution must fit the situation: visits are nonnegative whole numbers. For fewer than nine visits, A is cheaper (at n = 5: £32 versus £40); for more than nine, B is cheaper (at n = 12: £60 versus £54). The model assumes exactly these fees and no other charges.

Worked model

Let n be the number of visits and state n is a nonnegative integer.
Write costs 12 + 4n and 30 + 2n.
Solve 12 + 4n = 30 + 2n to find n = 9.
Check both at n = 9: £48.

Guided practice

Plan C costs £5 plus £3 per visit and D costs £17 plus £1 per visit. When do they cost the same?

Hint: Write 5 + 3n = 17 + n.

Check the guided answer

2n = 12, so n = 6; both cost £23.

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 plan with the lower fixed fee is always cheaper.”

Correction: The variable rate also matters; solve or compare costs at the relevant quantity.

Extend and reflect

What changes if visits can be cancelled but the fixed fee remains?

Guidance: The formulas still apply to completed visits, but refund or cancellation terms must be added before claiming a real-life cost.