Comparing Relative Quantities

Overview

Lesson 7 of 10 in Rational Number Knowledge Practices. Students compare quantities fairly by identifying the reference whole, distinguishing absolute differences from relative comparisons, and using fractions, percentages and ratios to evaluate claims.

Learning intentions

• WALT compare relative quantities, not only absolute amounts.
• WALT distinguish a difference from a relative comparison.
• WALT use fractions, percentages and ratios to interpret claims.
• WALT explain conclusions using mathematical evidence.

Success criteria

• I can identify the reference whole and denominator.
• I can compare quantities fairly, even when totals differ.
• I can calculate and compare fractions or percentages.
• I can support a conclusion with evidence and clear reasoning.

Curriculum links

• Number: use fractions, decimals, percentages and ratios to represent and compare quantities.
• Number: reason about relationships between parts and wholes, including equivalent representations.
• Statistics: interpret data displays, question claims and communicate conclusions based on evidence.
• Mathematical practices: represent, calculate, justify, communicate and evaluate the reasonableness of solutions.

Lesson structure (60 minutes)

1. 0–5 minutes – Hook and retrieval

Open with the comparison hook. Display: “Which shows greater success: 12 successes out of 20, or 15 successes out of 30?” Students first vote silently, then justify their choice to a partner. Quickly revisit ordering fractions and finding percentages of quantities.

2. 5–15 minutes – Explicit teaching

Use the worked comparison slides to model the two claims. Explain that the absolute difference is 3 successes, but the relative comparison is different: 12 out of 20 is 60%, while 15 out of 30 is 50%. Introduce and clarify: relative, absolute, reference whole, difference, ratio, percentage point, claim and evidence.

Emphasise the question: “Compared with what?” Highlight that the denominator represents the reference whole. Ask: “Is the denominator the same?” and “Does ‘more’ mean a larger number or a larger proportion?”

3. 15–25 minutes – Guided claim analysis

Display three short claims from the guided claims slides, such as “More students chose cycling”, “Team A was more successful”, and “Participation increased.” In pairs, students identify the numerator, denominator and reference whole, then decide whether each claim describes an absolute amount or a relative amount.

Discuss why “more” can mean a greater count or a greater proportion. Model rewriting one unclear claim so that it is precise, for example: “Cycling was chosen by 12 students, but walking was chosen by 50% of the class.”

4. 25–43 minutes – Team investigation

Place students in groups of four and distribute the relative quantities investigation worksheet. Each group chooses one adapted environmental, population or sports dataset, such as participation in a conservation activity across two years or results from two sports teams.

Students calculate relevant fractions, percentages or ratios and answer: What is the reference whole? Are the denominators the same? Is the claim about an absolute amount or a relative amount? Groups write an evidence-based conclusion using at least one calculation. Assign roles: reader, calculator, checker and reporter.

Circulate and ask: “Compared with what?” “What does your denominator represent?” “Can you compare these percentages fairly?” “What evidence supports your conclusion?”

5. 43–52 minutes – Share and critique

Invite three groups to present their conclusion using the investigation discussion slides. The class checks whether each group identified the reference whole and used an appropriate comparison. Encourage students to challenge unclear wording respectfully and distinguish a larger number from a larger proportion.

6. 52–57 minutes – Consolidation

Return to the opening example. Students independently explain why 12 out of 20 and 15 out of 30 do not show the same relative success. Invite a few responses, reinforcing that 60% is greater than 50%, even though 15 is the larger absolute number.

7. 57–60 minutes – Exit ticket

Finish with the exit ticket section: “Which is the greater proportion: 8/10 or 12/20? Explain why.” Students must state the reference whole and show a calculation, such as converting to percentages. Collect responses to identify students needing further support.

Resources

• the complete comparison slide deck, used for the hook, modelling, guided claims, investigation instructions and plenary.
• the relative quantities investigation worksheet, including datasets, calculation tables, sentence frames and the exit ticket.
• Adapted NZ environmental, population and sports datasets.
• Graphs showing totals and proportions.
• Calculators, one per pair or group.
• Mini-whiteboards or scrap paper.
• Pens, highlighters and rulers.
• Group role cards or a visible list of roles.

Assessment

• Observe pair and group discussion, listening for accurate identification of the denominator and reference whole.
• Check investigation calculations and conclusions for fair comparisons, correct percentages or ratios, and supporting evidence.
• Use the exit ticket to assess whether students can compare 8/10 and 12/20 and explain their reasoning.

Differentiation

• Support learners with simplified datasets, smaller numbers, icons, partially completed calculations and explicit highlighting of each denominator.
• Provide sentence frames: “The reference whole is…”, “Although ___ is larger, ___ is the greater proportion because…”, and “My evidence is…”
• Allow students to use calculators and give an oral explanation or recorded presentation instead of writing a full conclusion.
• Extend confident learners by asking them to critique a misleading graph, explain how its scale affects interpretation, or construct a counterexample where a larger absolute amount represents a smaller proportion.