Foundations of Mathematical Thinking · Junior Infants

Active learning ideas

Introduction to Linear Inequalities

Active learning works for introducing linear inequalities because students need to see, touch, and test relationships between numbers rather than just memorize symbols. When students manipulate physical or visual representations, they build mental models for abstract inequalities that persist long after the lesson ends.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Strand 3: Algebra - A.1.13

• 25–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Four Corners
• 35 min · Pairs

Balance Scale Challenges: Inequality Testing

Provide scales, counters, and cards with inequalities like 'left side > right side by 2'. Students add or remove objects to satisfy each inequality, record the symbol used, and solve for the variable. Discuss sign changes with negative amounts. End with graphing on personal number lines.

Differentiate between solving an equation and solving an inequality.

Facilitation Tip

During Balance Scale Challenges, circulate and ask groups to explain why adding or removing counters changes the balance in one direction but not the other.

What to look for

Give students the inequality x - 5 < 10. Ask them to: 1. Solve the inequality. 2. Graph the solution on a number line. 3. Write one number that is in the solution set and one number that is not.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Social Awareness

Activity 02

• Four Corners
• 25 min · Small Groups

Number Line Relay: Graphing Solutions

Mark a floor number line with tape. Give teams inequality cards like x - 3 ≥ 5. One student solves aloud, another hops to graph the endpoint with a circle marker. Rotate roles. Teams justify endpoints and directions.

Explain why the inequality sign sometimes reverses when solving.

Facilitation Tip

For Number Line Relay, stand near the graphing station to gently correct open versus closed circles before students move to the next station.

What to look for

Present the inequality 2x ≤ 8 and the inequality -2x ≤ 8. Ask students: 'What is the solution for each inequality? How are the solutions different? Explain why the sign changed in the second case when we solved it.'

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Social Awareness

Activity 03

• Four Corners
• 40 min · Individual

Candy Budget Game: Real-World Inequalities

Students get play money and candy prices. Set budgets like 'total ≤ €5'. They select candies, write inequalities, solve for maximum items, and graph feasible sets. Share strategies in whole-class debrief.

Construct a graph to represent the solution set of a one-step inequality.

Facilitation Tip

In Candy Budget Game, limit supplies to force students to plan quantities before calculating, reinforcing the need for inequalities over exact numbers.

What to look for

Write the scenario 'You need more than 20 points to pass.' Ask students to write an inequality for this scenario. Then, ask them to write the solution set and graph it on a number line. Check for correct inequality symbol and graph notation.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Social Awareness

Activity 04

• Four Corners
• 30 min · Small Groups

Sign Flip Stations: Negative Operations

Set up stations with problems like 2x < -6 or -4x ≥ 12. Students solve using manipulatives, predict sign direction, then verify. Rotate, compare results, and graph all solutions on group posters.

Differentiate between solving an equation and solving an inequality.

What to look for

Give students the inequality x - 5 < 10. Ask them to: 1. Solve the inequality. 2. Graph the solution on a number line. 3. Write one number that is in the solution set and one number that is not.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Social Awareness

A few notes on teaching this unit

Teach inequalities by starting with real, relatable scenarios so students see the relevance of ranges instead of single answers. Avoid rushing to the algorithm; let students test values on number lines to internalize why solutions are continuous. Use carefully sequenced activities that build from concrete to abstract, and always ask students to predict before solving to uncover misconceptions early.

Students will move from seeing an inequality as a single point to understanding it as a continuous range of solutions. They will correctly solve inequalities, graph them with proper notation, and explain when and why the inequality sign reverses. Peer discussions and hands-on activities will reveal their reasoning and correct misunderstandings in real time.

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Watch Out for These Misconceptions

• During Balance Scale Challenges, watch for students who treat inequalities like equations and expect one exact balance point.

  Have students test multiple counter values on the same side to see that the balance shifts gradually, reinforcing the idea of a range of solutions. Ask them to explain why adding 2 counters makes the scale tip in one direction but not the other.

• During Sign Flip Stations, watch for students who reverse the inequality sign in all operations, not just when multiplying or dividing by negatives.

  Use the balance scale with negative counters to show that adding a negative is like removing positives, so the scale tips predictably without sign changes. Only when multiplying by a negative do students see the scale flip entirely, requiring a sign reversal.

• During Number Line Relay, watch for students who use closed circles for strict inequalities or open circles for inclusive ones.

  Provide a sorting task with inequality cards and matching number line segments so students physically pair strict (open) or inclusive (closed) symbols with correct graph endpoints. Peer discussion helps correct misconceptions through shared reasoning.

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Methods used in this brief

• Four Corners