Expressions and Equations

Overview

Ākonga explore algebraic expressions by writing, evaluating and simplifying expressions involving integers and rational values. They use commutative, associative and distributive properties to justify equivalent forms, then apply order of operations and inverse operations to solve and rearrange equations in practical contexts. Mathematical discussion may move between te reo Māori and English as appropriate.

Learning intentions

• WALT translate practical situations into algebraic expressions and equations.
• WALT evaluate expressions using integer and rational values and order of operations.
• WALT use algebraic properties to simplify and identify equivalent linear expressions.
• WALT solve linear equations and rearrange formulae with more than one variable.
• WALT explain and justify algebraic methods using precise mathematical language.

Success criteria

• I can identify terms, coefficients, variables and constants in an expression.
• I can evaluate an expression accurately, showing the order of operations.
• I can simplify expressions by collecting like terms and using the distributive property.
• I can solve or rearrange an equation and check that my answer is reasonable.

Curriculum links

• Pāngarau — Tau me te Taurangi: expressing, evaluating, simplifying and solving algebraic relationships.
• Pāngarau — mathematical reasoning, representation and communication through symbolic and practical contexts.
• Te Reo Rangatira — whakarongo, kōrero, pānui and tuhituhi as ākonga discuss, interpret and record mathematical thinking.
• Marau ā-Kura — use a locally meaningful context, such as a kura event, kai stall or local journey, when adapting examples.

Pangarau Vocabulary | Maths Vocabulary

English term	Te reo Māori term	Student-friendly meaning
expression	kīanga taurangi	A group of numbers, symbols and operations without an equals sign.
equation	whārite	A statement that two mathematical expressions are equal.
variable	taurangi	A letter or symbol that represents a number that can change.
coefficient	tauwehe	The number multiplied by a variable.
constant	pūmau	A number that does not change.
term	kupu taurua	A number, variable, or product separated by addition or subtraction signs.
simplify	whakangāwari	Rewrite an expression in an equivalent, simpler form.
substitute	whakakapi	Replace a variable with a known value.
solve	whakaoti	Find the value that makes an equation true.
solution	otinga	The value or values that satisfy an equation.
expand	whakawhānui	Multiply out brackets to remove them.
factorise / factor	whakatauwehe	Write an expression as a product of factors.
like terms	kupu taurite	Terms with the same variable parts and powers.
inverse operation	mahi kōaro	An operation that undoes another operation, such as addition and subtraction.
equal	ōrite	Having the same value.

Using the vocabulary: Introduce and revisit two or three terms at each key stage. Ask students to use both languages when labelling examples, explaining a method to a partner, and answering the exit assessment.

Lesson structure (90 minutes)

1. 0–8 min · Hook and prior knowledge. Teacher opens the hook and learning intentions with a practical prompt: “A kai stall charges $4 to enter and $2.50 for each item. How could we describe the total cost?” Students think-pair-share, suggest a rule such as (4+2.5n), and identify what the variable represents. Introduce the terms kīanga taurangi (expression), taurangi (variable), tauwehe (coefficient) and kēmu (constant), using the class’s preferred local language conventions.

2. 8–23 min · Explicit teaching: expressions. Teacher uses the worked examples on expressions and properties to model substitution into (3x-2), (\frac12x+4), and (2(a+3b)), emphasising brackets, negative values and order of operations. Model that (x+3x=4x), while (x+3) cannot be combined, and demonstrate commutative, associative and distributive properties. Students annotate the guided expressions and properties worksheet and explain each step to a partner.

3. 23–40 min · Collaborative simplification. Teacher displays the pair-task instructions and discussion prompts and assigns pairs a progression of expression questions on the guided expressions and properties worksheet. Students simplify expressions such as (4x+7-2x+3), (-3(2x-5)+x), and (\frac12x+\frac34x-2), recording the property or operation used. Partners must compare two expressions and decide whether they are equivalent, then justify their decision by simplifying both or testing a suitable value.

4. 40–48 min · Mid-lesson check. Teacher pauses for a mini-whiteboard check using the hinge questions: simplify (5x-2+3x+6), evaluate (2p^2-3) when (p=-2), and identify the error in a worked solution. Students show answers simultaneously, explain one correction, and receive immediate feedback. Teacher notes misconceptions about signs, powers, brackets and unlike terms for targeted support.

5. 48–68 min · Solving and rearranging equations. Teacher models on the equation-solving and rearranging examples how to solve (3x+5=20), (-2(x-4)=14), and (\frac{x}{3}+2=7), maintaining equality by applying inverse operations to both sides and checking by substitution. Then model rearranging (C=2\pi r) to make (r) the subject and (y=mx+c) to make (m) the subject. Students complete the equation section of the guided expressions and properties worksheet independently for five minutes, then compare methods in pairs. Invite students to explain why each transformation preserves equality.

6. 68–83 min · Context problem-solving. Teacher presents the practical problem and plenary prompts. Groups of three choose or receive a context, such as a phone plan, shared transport cost or fundraising target, and use an expression or equation to represent it. Students solve a related question, check their result, and prepare a two-minute explanation using a table, words and symbols. Groups share selected solutions; peers ask “He aha te take?” (“Why?”) and identify where a property or inverse operation was used.

7. 83–90 min · Exit assessment and reflection. Teacher distributes the final section of the independent exit questions. Students complete: simplify (3(2x-1)+4x); evaluate (2a-\frac12b) for (a=3,b=-4); solve (5x-7=18); and rearrange (A=\frac12bh) to make (h) the subject. Students add one sentence explaining a method they can now justify. Collect responses to plan the next sequence.

Resources

• Expressions, properties and equations slide deck
• Guided expressions, equations and exit questions worksheet
• Mini-whiteboards, pens and erasers
• Calculators for checking rational-value calculations
• Projector or interactive display
• Prepared locally relevant context cards or prompts
• Exercise books and graph paper

Assessment

• Listen for accurate use of terms and justification during pair and group discussion; question students about why unlike terms cannot be combined.
• Use mini-whiteboard responses to identify errors with negative numbers, brackets, order of operations and maintaining equality.
• Mark the exit questions for expression manipulation, substitution, equation solving and rearranging; use errors to group students for the next lesson.

Differentiation

• Support: provide a worked-example strip, colour-code like terms, allow a four-function calculator, and use sentence starters such as “These expressions are equivalent because…” and “I applied … to both sides.”
• Support learners who need it with partially completed equations, a term/operation bank and a partner rehearsal before recording; read instructions aloud and reduce the number of practice questions without reducing the reasoning demand.
• For EAL or developing te reo Māori learners, pair symbols with plain-language explanations, accept bilingual mathematical discussion, and display key terms in te reo Māori and English.
• Extend confident ākonga by asking them to create two different-looking equivalent expressions for a local context, prove equivalence algebraically, and rearrange a formula involving fractions and more than two variables.