Te Ao Polynomial

Overview

In this first and only lesson of the unit, ākonga develop a shared language for polynomials and connect symbolic manipulation to concrete algebra-tile models. They progress from monomials and exponent laws to polynomial operations, perimeter, and multiplication, explaining strategies in te reo Māori and English.

Learning intentions

• WALT identify monomials, binomials, trinomials, coefficients, constants, terms and like terms.
• WALT apply exponent laws to powers, multiplication and division of monomials.
• WALT model, add and subtract polynomials using algebra tiles.
• WALT expand a monomial by a polynomial and multiply binomials, including special cases.
• WALT use polynomial addition to find a perimeter and communicate mathematical thinking clearly.

Vocabulary / Kupu Pangarau

• Polynomial — taupū-maha: an expression made from one or more terms.
• Term — wāhanga: a part of an expression separated by + or −.
• Coefficient — whakarea: the number multiplying a variable.
• Constant — pūmau: a number without a variable.
• Variable — taurangi: a letter or symbol representing a number.
• Exponent/index — taupū: the small number showing repeated multiplication.
• Degree — tohu: the highest exponent in a polynomial.
• Expression — kīanga: numbers, variables and operations without an equals sign.
• Like terms — ngā wāhanga ōrite: terms with the same variables and exponents.
• Simplify — whakangāwari: rewrite in an equivalent, simpler form.
• Expand — whakawhānui: multiply out brackets.
• Factorise — whakawehe tauwehe: write an expression as a product of factors.
• Factor — tauwehe: a quantity multiplied by another quantity.
• Equation — whārite: a statement that two expressions are equal.
• Evaluate — aromatawai: find the value by substituting and calculating.

Success criteria

• I can classify a polynomial and explain its parts using appropriate mathematical language.
• I can simplify monomial expressions using exponent laws.
• I can combine like terms and use tiles to justify polynomial addition or subtraction.
• I can expand products accurately and explain how I checked my answer.

Curriculum links

• Pāngarau — Tau me te Taurangi: algebraic language, expressions, exponent laws and operations with polynomials.
• Pāngarau — Ine me te Āhuahanga: representing and finding perimeter using algebraic expressions.
• Te Reo Rangatira — Whakarongo, Kōrero and Tuhituhi: listening, discussing and recording mathematical reasoning in te reo Māori.
• Marau ā-Kura: connect examples and mathematical communication to the kura’s local language conventions and contexts where appropriate.

Lesson structure (87 minutes)

1. 0–10 min · Kupu and concept warm-up. Open with the hook and vocabulary slides: display (3x^2-5x+7) and ask, “He aha ngā wāhanga o tēnei kīanga?” Teacher introduces or revisits kīanga taurangi (algebraic expression), tauwehe (coefficient), wāhanga (term), tau pūmau (constant), monomial, binomial, trinomial, and wāhanga ōrite (like terms). Students sort examples into categories, annotate the displayed expression, and rehearse a sentence such as “Ko ___ te tauwehe o ___.” Check pronunciation and accept te reo Māori, English, or both while building precise bilingual usage.

2. 10–30 min · Monomial powers and operations. Use the exponent-law worked examples and distribute the polynomial foundations practice sheet. Teacher models (x^a x^b=x^{a+b}), (x^a\div x^b=x^{a-b}), ((x^a)^b=x^{ab}), including numerical coefficients and the condition that the divisor is non-zero. Students complete increasingly difficult examples: (3x^2\cdot4x^3), (12a^7\div3a^2), and ((2m^3)^2), then explain one answer to a partner. Pause for mini-whiteboard checks, asking, “He aha te ture taupū i whakamahia e koe?” Address the common error of adding exponents when multiplying unlike bases.

3. 30–50 min · Model with algebra tiles. In groups of three, provide algebra tiles representing (x^2), (x), and constants, with different orientations or colours for positive and negative values. Use the algebra-tile modelling instructions. Teacher models (x^2+3x+2), then demonstrates that subtraction means removing tiles and that zero pairs can be added when needed. Students build, record and compare (2x^2+3x-1), ((x^2+2x+3)+(x^2-x-5)), and ((3x^2+x+2)-(x^2+2x-4)). Each group appoints a kaiwhakamārama (explainer), kaiwhakamātau (checker), and kaiwhakahaere rauemi (resource manager), rotating roles once.

4. 50–70 min · Add, subtract and find perimeter. Display the perimeter diagram and discussion prompts. Teacher models collecting like terms and applies the method to a rectangle whose side lengths are (2x+3) and (x+5), finding (P=2(2x+3)+2(x+5)=6x+16). Students complete the related perimeter problems on the polynomial foundations practice sheet, first representing the sides, then adding all boundary expressions and simplifying. Partners must justify why unlike terms cannot be combined. Invite explanations using: “Tuatahi…”, “Nō te mea…”, “Nō reira…”. Confer with groups and check that students do not multiply side lengths when the question asks for perimeter.

5. 70–82 min · Expand and multiply. Use the multiplication and special-cases slides. Teacher models the distributive property (3x(2x-5)=6x^2-15x), then a grid or area model for ((x+4)(x+2)=x^2+6x+8). Students practise products on the worksheet, including ((2x-3)(x+5)), ((x+6)^2), and ((x+7)(x-7)). Explicitly identify the special cases ((a+b)^2=a^2+2ab+b^2) and ((a+b)(a-b)=a^2-b^2). Students use a second method, such as distribution or an area model, to check one answer.

6. 82–87 min · Exit assessment and kōrero. Return to the plenary slide. Students complete the final five-minute section of the polynomial foundations practice sheet independently: classify (4x^2-3x+1), simplify (5a^2\cdot2a^3), expand (2x(x-4)), and state one strategy in te reo Māori or English. Students self-rate confidence from 1–5 and hand in the response as they leave.

Resources

• the complete polynomial foundations slide deck
• the polynomial foundations practice sheet
• Algebra tiles, including positive and negative tiles
• Mini-whiteboards and pens
• Projector or interactive display
• Group role cards or board display
• Exercise books and pencils
• Bilingual mathematical language bank prepared by the teacher

Assessment

• Listen for accurate vocabulary and reasoning during the warm-up, partner explanations and group modelling; record students needing support with terms or exponent laws.
• Use mini-whiteboard responses and the perimeter task to identify errors with like terms, signs, distribution and perimeter.
• Mark the exit assessment for classification, exponent laws, expansion and mathematical communication; use results to plan follow-up or individual conferencing.

Differentiation

• Provide a bilingual word bank, colour-coded worked examples, partially completed algebra-tile diagrams and sentence starters such as “Ka taea te whakakotahi ēnei wāhanga nā te mea…”.
• Pair ākonga strategically and allow students to explain orally before recording symbolic steps. Use concrete tiles and enlarged notation for students with processing, visual or fine-motor needs.
• Reduce the number of terms or provide a step-by-step exponent-law reference for students who need support, while retaining the same mathematical purpose.
• Challenge confident ākonga to create a polynomial perimeter problem, solve it in two ways, or prove a special case using an area model; they must justify equivalence rather than only provide an answer.