Inequalities and Budget Constraints

Overview

This OLNA numeracy session focuses on solving and interpreting linear inequalities in everyday contexts, including representing solutions on a coordinate plane and communicating the solution in words. It uses the OLNA Numeracy Practice Workbook to provide short, practical tasks that mirror OLNA-style thinking.

Learning intentions

• Students will solve linear inequalities in 2 variables and interpret solutions as a region on a graph.
• Students will use a test point to check whether a coordinate satisfies an inequality.
• Students will communicate what the inequality solution means in the given real-life scenario (e.g. “spending up to a budget”).
• Students will recognise the difference between inclusive and exclusive inequalities (≤ or <) and represent the boundary correctly.

Success criteria

• I can choose the correct boundary line for an inequality (<, ≤, >, ≥).
• I can shade the correct side of the boundary to show the solution region.
• I can use a test point to confirm whether a point satisfies the inequality.
• I can describe the solution in context using clear everyday language.

Curriculum links

• Algebra — solve linear inequalities and simultaneous linear equations in 2 variables; interpret solutions graphically and communicate solutions in terms of the situation.
• Algebra — graphing regions corresponding to inequalities in the Cartesian plane and verifying using a test point.
• General numeracy skills — interpreting representations and explaining reasoning clearly in context (OLNA numeracy focus).

Lesson structure (60 minutes)

1. 0–8 min · Warm-up (OLNA style quick checks). Teacher shows two inequalities on the board (one strict, one inclusive) and asks: “Would the boundary line be solid or dashed?” Students answer on mini whiteboards and justify with a one-sentence explanation.

2. 8–18 min · Worked example (graph + test point in context). Teacher uses the workbook problem: a simple budget-style inequality such as “Movie tickets cost $12 and ice-skates cost $21. Your total must be up to $150.” Students identify variables, write the inequality (for example, 12m + 21s ≤ 150), then the teacher demonstrates:

• rearranging to get the boundary line in y = mx + c form,
• drawing the boundary with the correct line type (solid for ≤, dashed for <),
• shading the correct side,
• using a test point (e.g. (0,0) or another easy point) to confirm it satisfies the inequality. Students copy the steps onto their workbook page or a dedicated “OLNA method” page.

3. 18–30 min · Guided practice (complete workbook items with prompts). Teacher allocates small steps with scaffolds:

• Step A: rewrite the inequality to graph,
• Step B: draw the boundary correctly,
• Step C: shade the solution region,
• Step D: test a point and decide if it works. Students work in pairs through 2–3 workbook questions. Teacher circulates and uses targeted prompts such as:
• “Does this inequality include equality?”
• “What does the solid/dashed line mean?”
• “Which side should be shaded and why?”
• “Pick a point that’s easy to substitute—does it satisfy?”

4. 30–45 min · Independent practice (OLNA numeracy momentum). Students complete the next workbook question independently (or one per student if questions are short). They must:

• show their working for the inequality form,
• sketch/label the boundary line,
• state the shading direction clearly,
• write a 1–2 sentence interpretation of the solution in context. Teacher provides a “must show” checklist on the board (boundary type, shading, test point, context sentence).

5. 45–55 min · Whole-class check and error correction. Teacher selects one common error (e.g. shading the wrong region, using the wrong line style for < vs ≤, or choosing a test point on the boundary line incorrectly). Students vote on the corrected approach, then the class agrees on the final method.

6. 55–60 min · Exit ticket (quick assess). Students complete 2 short items:

• Item 1: “For 2x + 3y < 24, should the boundary be solid or dashed? Give one reason.”
• Item 2: “Does the point (0,0) satisfy 2x + 3y < 24? Show substitution briefly.” Teacher collects immediately for review.

Resources

• OLNA Numeracy Practice Workbook (teacher and student copies)
• Mini whiteboards and markers (warm-up)
• Graph paper or printed coordinate plane grids
• Rulers and coloured pencils (for shading regions)
• Calculator only if workbook problem requires it (otherwise mental/substitution focus)
• Timer/visual countdown for each timed stage

Assessment

• Formative: warm-up line-style responses checked quickly via mini whiteboards.
• Formative: teacher observation during guided practice for boundary type, shading choice, and test point reasoning.
• Summative (short): exit ticket responses focusing on boundary type and point substitution.

Differentiation

• Support: provide sentence starters for interpreting solutions in context (e.g. “This inequality means you can…” / “The acceptable combinations are…”).
• Support: offer a worked “checklist” on the board during guided practice (boundary → shading → test point → answer in words).
• Extension (for fast finishers during independent practice): ask them to choose an additional test point and confirm whether it satisfies or not, then explain why their result matches the shaded region.
• EAL/SEN: allow verbal explanation alongside written work; provide extra practice substituting values into inequalities with smaller numbers before graphing.

Exit ticket questions

• 1. For 2x + 3y < 24, should the boundary line be solid or dashed? State why.
• 2. Does the point (0,0) satisfy 2x + 3y < 24? Substitute and decide yes/no.