Ordering Rational Numbers

Overview

Lesson 3 of 10 in Rational Number Knowledge Practices. Students compare and order fractions, decimals and percentages by converting representations, locating values on number lines and using benchmarks such as 0, 1/2 and 1.

Learning intentions

• WALT compare fractions, decimals and percentages.
• WALT select useful benchmarks to order rational numbers.
• WALT justify comparisons using symbols, words and number lines.
• WALT recognise when converting is useful and when it is not necessary.

Success criteria

• I can order numbers shown in mixed representations.
• I can use 0, 1/2 and 1 as benchmarks.
• I can use <, > and = correctly.
• I can explain how I know one rational number is greater or less than another.

Curriculum links

• Number: recognise, represent, compare and order rational numbers in familiar contexts.
• Number: connect equivalent fractions, decimals and percentages and use number lines to reason about magnitude.
• Mathematical practices: choose efficient strategies, communicate mathematical thinking and justify conclusions.
• Key competencies: thinking; using language, symbols and texts; participating and contributing through collaborative reasoning.

Lesson structure (60 minutes)

1. 0–7 minutes – Hook and retrieval

Open with the hook and retrieval slides. Display 0.6, 55%, 5/8 and 0.65 and ask students to silently predict the order on mini-whiteboards. Invite two or three different strategies, without confirming the answer immediately.

Briefly revisit equivalent forms and number-line placement: 1/2 = 0.5, 1/4 = 0.25 and 3/4 = 0.75. Ask: “Which value is closest to 1? Which is furthest from 1?”

2. 7–17 minutes – Explicit teaching and modelling

Use the worked example slides to model ordering 0.6, 55%, 5/8 and 0.65. Show that 55% = 0.55, 5/8 = 0.625, and then order the values as 55%, 0.6, 5/8, 0.65.

Emphasise two reliable strategies: convert all values to a common representation, or locate them using benchmarks and a number line. Discuss: “Is converting always necessary?” and “How do you know 0.7 is greater than 65%?”

3. 17–29 minutes – Guided floor number line

Place a large open number line across the room from 0 to 1. Give pairs number cards showing fractions, decimals and percentages. Students place each card and explain their position to another pair.

Use the guided discussion slides for prompts: “Which benchmark helps?”, “What equivalent form could you use?”, and “Could this value be between 1/2 and 3/4?” Encourage physical movement and oral explanation before recording symbols.

4. 29–44 minutes – Collaborative context practice

In teams of four, students use the mixed-representation ordering worksheet. They order sets based on Aotearoa contexts, such as percentages of native planting completed, distances travelled or temperatures. Each team must record the ordered list, at least one comparison symbol and a written justification.

Assign roles: reader, card or number organiser, checker and explainer. Circulate and ask students to identify their benchmark or explain why conversion is, or is not, needed.

5. 44–53 minutes – Independent application

Students complete the final section of the mixed-representation ordering worksheet independently. Include examples requiring comparison of a fraction with a decimal and a percentage, plus one number-line representation.

Pause for a quick check using mini-whiteboards: “Which is greater: 0.7 or 65%? Show or explain your thinking.” Address misconceptions, particularly that a larger denominator always means a larger fraction.

6. 53–60 minutes – Exit ticket and reflection

Finish with the plenary slides and ask students to complete the exit ticket on the mixed-representation ordering worksheet: order 3/5, 58%, 0.62 and justify one comparison.

Students share with a partner using the stem, “I know ___ is greater/less than ___ because…”. Collect responses to identify who can compare accurately and who needs further work with equivalence or benchmarks.

Resources

• the ordering and comparing slide deck
• the mixed-representation ordering worksheet
• Open floor number line labelled 0 and 1
• Prepared number cards showing fractions, decimals and percentages
• Mini-whiteboards, pens and erasers
• Blu-tack or tape for placing cards
• Aotearoa distance, temperature or environmental data
• Exercise books or pencils

Assessment

• Listen for accurate use of benchmarks, equivalence and comparison vocabulary during floor-line discussion.
• Check team and independent worksheet responses for correct ordering, symbols and written justification.
• Use the exit ticket to group students for the next lesson: secure, developing or needing targeted support.

Differentiation

• Support: use fewer cards; begin with values close to 0, 1/2 and 1; provide benchmark strips and a number line from 0 to 1.
• Provide sentence frames: “___ is greater than ___ because…”, “I changed ___ into ,” and “ is between ___ and ___.”
• Allow students to physically place cards, draw diagrams or explain orally before writing. Pair students strategically and pre-teach compare, order, greater than, less than, equal to, benchmark and magnitude.
• Extension: create two different values between 0.62 and 0.63, represent them in different forms and order them. Ask students to prove that their values lie in that interval.