The Equality Investigation

Overview

In lesson 1 of Algebra Detectives: Solve, Simplify, Share, students investigate equality as a balance rather than “the answer comes next”. They interpret, sort and justify number sentences, then use inverse operations to find missing values involving addition, subtraction, multiplication and division.

Learning intentions

• WALT recognise equality as a balanced relationship.
• WALT explain what number sentences mean.
• WALT use reasoning to decide whether statements are true or false.
• WALT find missing values using inverse operations.
• WALT communicate mathematical thinking respectfully with a partner and an audience.

Success criteria

• I can identify equal and unequal statements.
• I can justify why a number sentence is true or false.
• I can complete missing values using +, −, × and ÷.
• I can explain my reasoning using mathematical vocabulary.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: connected challenge activities that complement classroom mathematics learning.
• Mathematics and Statistics — Additional resources for advanced learners: reasoning, explanation and non-routine problem solving.
• Te Mātaiaho capabilities and values: communicating mathematical ideas, reasoning, collaboration, Manaaki and Atamai.

Lesson structure (45 minutes)

1. 0–5 min · Hook: mystery sort. Display the opening challenge in the equality investigation deck: “Which of these statements belong together?” Give pairs six mystery number sentences, including examples such as 7 + 5 = 12, 12 = 7 + 5, 9 × 2 = 9 + 9 and 15 − 6 = 8; students silently predict true/false, then share one reason with a partner.

2. 5–11 min · Build the meaning. Model 8 + 4 = 12 with counters on both sides of the Algebra Balance Mat, then deliberately change one side to create an unequal statement. Use the equality investigation deck to introduce equal, unequal, true, false, expression, value, and inverse operation; students explain what the equals sign means in their own words.

3. 11–23 min · Paired research stations. Set up three stations and give each pair the equality investigation worksheet to record findings. At Station A, students use counters and the balance mat to test missing values; at Station B, they use a number line or mental strategies to investigate addition and subtraction; at Station C, they use related multiplication and division facts to complete and check statements. Students must write a true statement, a false statement and a reason at each station, using the vocabulary cards provided.

4. 23–30 min · Reasoning conference. Pause the stations and display two contrasting examples in the equality investigation deck, such as 18 ÷ 3 = 2 × 3 and 24 − 7 = 10 + 7. Ask, “How could we prove this without simply calculating both sides?” Students compare methods with their partner, then two pairs share different strategies; clarify that both sides must have the same value and that either side may come first.

5. 30–40 min · Maths court. Assign each pair one statement from the equality investigation worksheet as its “case”. One student acts as lawyer and one as mathematician, presenting a verdict of true or false with evidence from a model, calculation, inverse operation or number line. The class acts as the court, listening for convincing evidence and asking one respectful question, showing Manaaki and Atamai.

6. 40–45 min · Plenary and exit check. Return to the opening sort in the equality investigation deck and invite students to revise one answer or explanation. Students complete the final three questions on the equality investigation worksheet: □ + 9 = 16, 4 × □ = 28, and “Explain whether 20 − 6 = 7 + 7 is true or false”; collect responses as the exit check.

Resources

• the equality investigation deck
• the equality investigation worksheet
• the Algebra Balance Mat
• Counters or connecting cubes
• Mini-whiteboards and pens
• Number lines
• Equality vocabulary cards
• Station instruction cards
• Pencils and highlighters

Assessment

• Listen for whether students interpret = as “has the same value as”, rather than “the answer is”.
• At stations, check that students test missing values and explain their decisions using a model, calculation or inverse operation.
• Use the exit check to identify students needing further work with equality, inverse operations or explaining mathematical reasoning.

Differentiation

• Support learners with teacher-guided examples, counters, the balance mat, number lines and vocabulary cards with visuals. Begin with whole-number statements and provide sentence starters: “This is true because…” and “The two sides are equal/unequal because…”.
• Pair students deliberately so that partner talk is supportive and both learners have a defined role. Read statements aloud and provide enlarged, uncluttered recording spaces for students who need visual or processing support.
• For EAL learners, rehearse same value, both sides, true, false, equal, unequal orally before writing; accept labelled diagrams, gestures and verbal explanations alongside equations.
• Offer challenge cards containing decimals and negative integers, such as −3 + 8 = 5 and 2.5 + □ = 6, requiring students to justify a strategy rather than guess.

Extension

• Students create six number sentences containing exactly three true and three false statements, using at least two different operations. They swap with another pair, who classify each sentence and provide evidence.
• Students write one deliberately misleading statement where the equals sign appears at the beginning or in the middle, then explain why its structure is still mathematically valid.