Introduction to Algebra — Te Tātai Tauira

Mathematics Phase 3 — Algebra. Every statement below is quoted verbatim from the live Te Mātaiaho curriculum and anchors specific lessons.

"Investigating the patterns of triangular numbers, square numbers, and cube numbers, extending the patterns, creating tables of values, and plotting the values on the coordinate plane."

"- algebraic notation - expanded form - formulae - like terms - linear equation - linear patterns."

"Forming and solving one- and two-step linear equations with integer solutions (e.g. t + 7 = 12, 5s + 3 = 18) - Checking the truth of and completing number sentences involving all four operations and including the use of inequalities (e.g. 0.8 × 12 ≤ 8 × 0.5 + 8, true or false"

How the statements map to the arc:

• Patterns of triangular, square, and cube numbers (Lessons 1–3): Investigating shape patterns (kōwhaiwhai/tukutuku), extending them, and building tables of values and rules.
• Algebraic notation, formulae, and linear patterns (Lessons 2–3): Using variables and algebraic notation to represent unknowns, and writing rules and formulae for the patterns students find.
• Forming and solving one- and two-step linear equations (Lessons 4–5 & Summative): Balancing equations with inverse operations, solving two-step equations in context, and designing a tukutuku panel with explicit algebraic rules.

📑 Full verified curriculum alignment for this unit → — Phase 3 · Years 7–8.

Throughout this 5-lesson unit, students develop algebraic thinking and pattern recognition by connecting cultural practices with symbolic notation:

• Lesson 1: Pattern Detectives — Identify, describe, and extend sequential shape and number patterns, recording values in structured tables.
• Lesson 2: The Mystery of 'x' — Use letters (variables) to represent unknown values in mathematical expressions and write simple algebraic statements from word problems.
• Lesson 3: Building with Algebra — Use algebraic rules to generate and analyse geometric patterns, linking to kōwhaiwhai and tukutuku panels.
• Lesson 4: The Balancing Act — Model equations as a physical balance scale and solve one-step linear equations using inverse operations.
• Lesson 5: The Two-Step Shuffle — Solve two-step linear equations and apply algebraic modelling to solve real-world problems in Aotearoa.

Students demonstrate progress and mastery through the following lesson-specific indicators:

• Lesson 1: Pattern Detectives
  ✓ I can find the rule for a pattern and use it to predict the next three terms in a sequence.
• Lesson 2: The Mystery of 'x'
  ✓ I can write an algebraic expression to represent a word problem containing an unknown value.
• Lesson 3: Building with Algebra
  ✓ I can create a table showing the relationship between the pattern term and the number of elements, and write its algebraic rule (e.g. y = 2x + 1).
• Lesson 4: The Balancing Act
  ✓ I can explain why an equation remains balanced when I perform the same operation on both sides.
• Lesson 5: The Two-Step Shuffle
  ✓ I can solve a two-step equation (e.g., 2x + 3 = 11) using inverse operations and show each step of my working.

Scaffold support: Provide concrete algebra tiles, balance scales, and pre-structured function machines. Use visual grids and colour-coded variables to highlight equations.

ELL / ESOL: Pre-teach key mathematical vocabulary (variable, equation, inverse, term). Use visual pattern cards and sentence starters for algebraic translations (e.g., "x is double the value of y").

Accelerated learners: Challenge students to find general rules for non-linear sequences or prove algebraic relationships using structural models.

Inclusion: Support multiple formats for solving equations (algebra tiles, balancing scales, or symbolic steps) and allow flexible recording formats.

Three pathways of genuine depth — each extends real unit substance rather than adding busy work, and each ends in a tangible deliverable.

• Entry-level — Pattern Builder. Create a physical repeating pattern using mosaic tiles or coloured blocks, document its table of values up to the 10th term, and write its linear rule.
• Developing — Visualising Coordinates. Plot the coordinate pairs from Lesson 1 and Lesson 3 linear rules on a cartesian plane, and write a paragraph explaining how the steepness of the line connects to the multiplier in the algebraic rule.
• Mastery — Tukutuku Algebra Proof. Take the summative Tukutuku panel design project, calculate the total number of stitches required for a 10x10 panel grid using algebraic formulas, and prove its accuracy using a smaller 3x3 test grid.

This unit moves students from observation to symbolic modelling to structural problem solving. It begins by grounding algebraic rules in visible sequences, discovering repeating motifs in kōwhaiwhai and number patterns (Lesson 1). Students then transition to algebraic symbols, learning to represent unknowns as variables (Lesson 2). Lesson 3 bridges the two, using linear algebra to model geometric sequences (like the number of slats needed to build a tukutuku frame).

The final part of the unit shifts to solving equations: understanding equations as balanced scales (Lesson 4), progressing to solving two-step equations in context (Lesson 5), and culminating in a summative design project where students build their own tukutuku panel based on algebraic rules.

Where it leads: This unit establishes essential foundations for secondary algebra (NCEA Level 1). It prepares students for plotting graphs, manipulating expressions, and solving simultaneous equations in senior mathematics.