Equations on Grids

Overview

This five-day sequence develops Year 6 students’ understanding of unknowns, equality and simple algebraic equations, then connects equations to ordered pairs and plotting on a coordinate grid. Students use visual models, tables, reasoning and discussion to explain patterns and solve increasingly challenging problems.

Learning intentions

Day 1 — Understanding equality and unknowns

• WALT recognise that an equation is a statement of equality.
• WALT represent an unknown using a symbol or letter.
• WALT solve simple one-step equations using inverse operations.

Day 2 — Solving and checking equations

• WALT solve one-step equations involving addition, subtraction, multiplication and division.
• WALT use substitution to check whether a solution is correct.
• WALT explain the steps used to solve an equation.

Day 3 — Coordinates and plotting

• WALT identify the horizontal and vertical axes on a grid.
• WALT read and write ordered pairs in the correct order.
• WALT plot points accurately in the first quadrant.

Day 4 — Connecting equations and coordinates

• WALT use a rule to generate ordered pairs.
• WALT plot pairs and describe the pattern they make.
• WALT connect a simple equation with its table and graph.

Day 5 — Applying and explaining

• WALT solve a problem involving an equation and a coordinate grid.
• WALT communicate mathematical thinking using diagrams, tables and words.
• WALT check whether a graph and equation match.

Success criteria

Students can say:

• I can explain that both sides of an equation must have the same value.
• I can solve an unknown and check it by substituting my answer.
• I can label axes and plot an ordered pair correctly.
• I can make a table of values from a simple rule.
• I can explain the relationship between a rule, a table and plotted points.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: challenge is connected to relevant textbook content and prior learning.
• Mathematics and Statistics — Mathsteasers / Additional resources for advanced learners: extension through non-routine reasoning, justification and multiple representations.
• Te Mātaiaho practice: students reason, represent, communicate and make connections between mathematical ideas.

Lesson structure (5 × 45 minutes)

Day 1 — Equality and unknowns

1. 0–5 min · Hook. Display the puzzle (□ + 7 = 15) using the opening puzzle slide and ask, “What must the box contain, and how do you know?” Students solve mentally and explain their reasoning to a partner.
2. 5–15 min · Model. Use the equality and balance slides to demonstrate equations as balanced statements and model (x+7=15), (x-4=9), (3x=18) and (x÷5=4). Students identify the inverse operation.
3. 15–30 min · Guided practice. Distribute the equations and balance worksheet. Students solve visual and symbolic equations independently, then compare methods in pairs.
4. 30–40 min · Challenge discussion. Present the reasoning challenge slide: “Can two different-looking equations have the same answer?” Students find examples and justify them.
5. 40–45 min · Exit check. Students complete: (x+9=17), and write one sentence explaining how they checked the answer.

Day 2 — Solving and checking

1. 0–7 min · Retrieval. Revisit answers through the retrieval and misconception slides. Students correct a deliberately incorrect solution and identify the error.
2. 7–17 min · Explicit teaching. Model solving with inverse operations, recording one equal step at a time. Demonstrate checking by substitution, for example (x-6=11), (x=17), then (17-6=11).
3. 17–32 min · Practice. Students complete the next section of the solving and checking questions. They must show a step and a check for every equation.
4. 32–40 min · Partner reasoning. Display the always-sometimes-never prompts. Pairs discuss statements such as “You can check an answer without solving the equation first.”
5. 40–45 min · Exit check. Students solve (4x=28), check it, and circle the part they found most useful: model, inverse operation or substitution.

Day 3 — Coordinates and plotting

1. 0–5 min · Hook. Show a treasure-map grid on the grid mystery hook and ask, “How could we describe the exact location of the treasure?”
2. 5–15 min · Teach. Model the origin, horizontal axis, vertical axis and ordered pairs. Emphasise “across first, then up” and plot ((2,5)), ((6,1)) and ((0,4)).
3. 15–30 min · Guided activity. Students use the coordinate-grid section of the axes and plotting worksheet to label axes, read coordinates and plot points. Partners check each point aloud.
4. 30–40 min · Spot the error. Use the plotting errors slides to compare reversed coordinates and misplaced points. Students explain how to correct each example.
5. 40–45 min · Exit check. Students plot ((4,3)) and write the coordinates of a second point chosen by the teacher.

Day 4 — Rules, tables and graphs

1. 0–7 min · Warm-up. Use the input-output warm-up to complete a table for (y=x+2). Students describe what changes as (x) increases.
2. 7–17 min · Model. Demonstrate choosing values for (x), calculating (y), writing ordered pairs and plotting them. Keep all values in the first quadrant.
3. 17–32 min · Investigation. Students complete rule-and-table tasks on the rules, tables and grids worksheet, then plot each set of ordered pairs.
4. 32–40 min · Discuss patterns. Display the pattern discussion slide. Students describe direction, spacing and how the equation creates the pattern.
5. 40–45 min · Exit check. For (y=2x+1), students give two ordered pairs and explain how they found them.

Day 5 — Apply and explain

1. 0–5 min · Review. Use the five-day review slides for a quick retrieval quiz covering equations, axes, ordered pairs and rules.
2. 5–12 min · Introduce task. Present the final investigation prompt: students must use a rule, table and coordinate grid to solve a mystery pattern.
3. 12–32 min · Independent application. Students complete the final section of the integrated equations and graph task. They solve, plot, identify a pattern and explain how they know their answer is reasonable.
4. 32–40 min · Share and critique. Pairs compare solutions using the prompts “I notice…” and “I wonder…”. Students revise one explanation for clarity.
5. 40–45 min · Reflection. Students complete a short reflection: one new idea, one strategy and one question they still have.

Resources

• the five-session teaching and discussion slide deck
• the five-session practice and application worksheet
• Whiteboard and markers
• Pencils, rulers and erasers
• Enlarged coordinate grids
• Individual mini-whiteboards
• Document camera or interactive display

Assessment

• Listen for accurate use of equality, inverse operation, variable, axis and ordered pair during partner explanations.
• Check worksheets for one-step solutions, substitution checks, correctly labelled axes and accurately plotted points.
• Use daily exit checks and Day 5 explanations to identify students needing reteaching or further challenge.

Differentiation

• Support students with partially completed equations, colour-coded axes, enlarged grids, a “across, then up” prompt and worked examples kept visible. Allow oral explanations or use of a mini-whiteboard before writing.
• Provide dyslexia-friendly copies: clear sans-serif font, large spacing, uncluttered pages, short numbered instructions, reduced copying and optional text-to-speech or adult read-aloud.
• Pair students strategically and rehearse vocabulary orally before independent work. Accept equations, diagrams, tables or spoken explanations as evidence of thinking.
• Advanced learners create a rule that produces a chosen pattern, compare two rules, find a missing coordinate or explain whether two different equations could generate the same plotted points.