Mapping Rational Numbers

Overview

Students connect fractions, decimals and percentages as different representations of the same quantity. They use visual models, meaningful Aotearoa contexts and number lines, while explicitly identifying the whole.

Learning intentions

• WALT represent fractions, decimals and percentages in meaningful ways.
• WALT connect visual, symbolic and contextual representations.
• WALT recognise rational numbers as quantities that can be represented in different forms.
• WALT explain the whole and locate values on a number line.

Success criteria

• I can represent a quantity as a fraction, decimal and percentage where appropriate.
• I can explain what the whole represents.
• I can use a diagram to prove that two representations are equivalent.
• I can place a value between 0 and 1 on a number line.

Curriculum links

• Represent and interpret rational numbers using fractions, decimals and percentages.
• Use visual, symbolic and contextual representations to explain mathematical ideas.
• Locate, compare and order numbers on number lines.
• Communicate mathematical thinking, justify conclusions and use appropriate mathematical language.

Lesson structure (60 minutes)

1. 0–5 minutes – Connect to prior knowledge

Display the opening question in the introduction and hook slides: “Which is more: 1/2, 0.5 or 50%?” Students think, pair-share and justify their ideas. Briefly revisit unit fractions, place value to thousandths, common fractions and familiar percentages.

2. 5–15 minutes – Teacher modelling

Use a 100-square to shade half. Name the whole clearly: “The whole is one complete square, or 100 equal parts.” Model the equivalence 1/2 = 50/100 = 0.5 = 50%, linking the shaded diagram, symbols and spoken explanation. Show the same quantity on a number line from 0 to 1. Use the modelling slides to reveal each representation gradually.

3. 15–25 minutes – Guided representation

Organise students into mixed-ability pairs. Give each pair fraction strips, hundred grids, decimal grids and number lines. Work through 1/4, 3/4, 0.2 and 75%, asking students to build, draw, write and locate each value. Pause after each example to ask: “What is the whole?”, “How does the diagram prove equivalence?” and “Where would this number sit between 0 and 1?” Provide the fraction number-line vocabulary cards for groups needing vocabulary support.

4. 25–42 minutes – Representation gallery

Place students in groups of four or five. Distribute the rational-number representation gallery worksheet. Each group chooses two accessible contexts, such as rainfall recorded as a percentage of a target, completed waka journeys, or planting in the school garden. For each context, groups create a display showing a quantity as a fraction, decimal and percentage where appropriate, plus a diagram and number-line position. Use the relevant task instructions in the gallery activity slides.

5. 42–50 minutes – Gallery walk and mathematical discussion

Groups view other displays, leaving one question or positive comment about the whole, equivalence or number-line position. Select two examples to discuss. Address common errors, such as treating the numerator as the whole, confusing 0.2 with 0.02, or assuming that a larger denominator always means a larger quantity.

6. 50–56 minutes – Consolidate and extend

Ask students to represent a value greater than one, such as 1 1/2, using a mixed number, decimal and percentage where meaningful. Discuss why percentages can exceed 100% and how the whole must still be identified. Students who need consolidation instead complete a supported example using a pre-partitioned grid and fraction wall.

7. 56–60 minutes – Exit ticket

Students complete the final prompt on the rational-number representation gallery worksheet: “Represent 3/5 in two ways and explain what it means.” Invite two students to share different representations. Collect responses and display the closing reflection prompt in the plenary and exit-ticket slides.

Resources

• the introduction and hook slides / the modelling slides / the gallery activity slides / the plenary and exit-ticket slides
• the rational-number representation gallery worksheet
• Fraction strips and fraction walls
• Hundred grids and decimal grids
• Number lines from 0 to 1
• Pencils, coloured pencils and glue
• Prepared local rainfall, sports or school-garden data
• Large paper or whiteboards for gallery displays
• Pre-partitioned grids for support

Assessment

• Observe whether students identify the whole, partition models accurately and connect equivalent representations.
• Listen for explanations using numerator, denominator, decimal, percentage and number line appropriately.
• Use the gallery worksheet and exit response to identify misconceptions and group needs for Lesson 2.

Differentiation

• Support: provide pre-partitioned grids, fraction walls, partially completed number lines, reduced choices and oral rehearsal before writing.
• EAL learners: explicitly teach and display bilingual vocabulary where possible; pair terms with diagrams, gestures and sentence frames such as “The whole is ___ because ___.”
• SEN: use larger manipulatives, clear colour coding, short instructions and mixed-ability pairs; allow students to explain orally or by pointing and annotating.
• Extension: represent values greater than one and explain the relationship between improper fractions, mixed numbers, decimals and percentages. Challenge students to create two different contexts for the same rational number.