Decimal Fractions

Overview

In this first lesson of the unit, students build a conceptual understanding of decimal fractions by connecting tenths and hundredths to whole numbers, common fractions, place value and number lines. The lesson uses concrete materials, visual models and mathematical discussion to establish language and representations for later decimal operations.

Learning intentions

• WALT represent tenths and hundredths using materials, diagrams and place-value charts.
• WALT connect decimal notation with fraction notation.
• WALT locate and explain decimals on a number line.
• WALT communicate mathematical thinking using precise vocabulary.

Success criteria

• I can explain what the tenths and hundredths digits represent.
• I can write a decimal as a fraction with a denominator of 10 or 100.
• I can represent a decimal in more than one way.
• I can locate and justify the position of a decimal on a number line.

Curriculum links

• Te Mātaiaho Mathematics and Statistics — Mathsteasers: challenging questions that deepen mathematical understanding.
• Te Mātaiaho Mathematics and Statistics — Mathsteasers / Alignment: connecting challenge tasks with relevant mathematical content.
• Te Mātaiaho Mathematics and Statistics — Mathsteasers / Additional resources for advanced learners.
• Mathematical practices: representing, explaining, connecting ideas, and justifying thinking.

Lesson structure (60 minutes)

1. 0–7 min · Hook and prior knowledge. Display a 10 by 10 hundredths grid and ask, “Which is greater: 0.4 or 0.35? How do you know?” using the opening comparison slide. Students make an individual estimate, then discuss with a partner, explaining what each digit might mean.

2. 7–18 min · Teacher modelling. Model one whole as a base-ten flat, one tenth as a rod or strip, and one hundredth as a small unit; connect these to a place-value chart and the notation 0.4, 4/10, 0.40 and 40/100. Use the tenths and hundredths modelling slides and invite students to help complete the chart. Students record the representations and rehearse the language: “The 4 in 0. forty represents four tenths” and “The 5 in 0.35 represents five hundredths.”

3. 18–30 min · Guided concrete investigation. In pairs, students use base-ten materials and the standard form place value slider to build teacher-called decimals such as 0.2, 0.07, 0.30, 0.46 and 0.80. Students write each decimal as a fraction, draw or describe the model, and explain why zero may be needed as a placeholder. Pause after 0.30 and 0.3 to discuss equivalent decimal fractions.

4. 30–43 min · Independent application. Distribute the decimal fractions representation worksheet. Students complete tasks involving place-value charts, base-ten or grid representations, fraction notation and matching decimals to positions on number lines. Circulate and ask, “What is the whole?” “What does this digit represent?” and “How can you prove the location is correct?” Students should show more than one representation for selected questions.

5. 43–53 min · Number-line reasoning and share-back. Display an open number line from 0 to 1 using the number-line discussion slides. Students place 0.1, 0.25, 0.5, 0.75 and 0.9, first estimating and then justifying the intervals used. Invite pairs to compare strategies, highlighting that 0.25 is 25/100 and lies halfway between 0.2 and 0.3.

6. 53–60 min · Plenary and exit check. Show the final prompts on the reflection and exit slides. Students respond on the worksheet or in their books: “Represent 0.36 as a fraction, describe a model for it, and mark it on a 0–1 number line.” They add one sentence explaining the meaning of the 3 and the 6, then share one new connection they made.

Resources

• the complete decimal fractions slide deck
• the decimal fractions representation worksheet
• Base-ten materials or place-value blocks
• Place-value charts and mini-whiteboards
• Pencils, rulers and coloured pens
• the standard form place value slider

Assessment

• Listen during partner talk for correct use of tenths, hundredths, numerator, denominator, decimal point and equivalent.
• Check models and worksheet responses for accurate links between concrete materials, place-value charts, fractions and decimals.
• Use the exit response to identify whether students can represent, locate and explain a decimal; group students for the next lesson based on these needs.

Differentiation

• Provide concrete materials, enlarged place-value charts, colour coding and teacher-guided modelling for students who need additional support.
• Offer vocabulary banks and sentence starters such as “The ___ digit represents ___ hundredths” and “I know this is equivalent because…”.
• Reduce the initial task to tenths and familiar hundredths, then gradually introduce numbers containing zero placeholders.
• For EAL learners and students requiring language or processing support, pair oral explanation with diagrams, gestures and repeated worked examples; allow thinking time and verbal rather than written rehearsal before recording.
• Extension learners investigate thousandths using the place-value slider and explain how equivalent decimal fractions are formed, for example why 0.4 = 0.40 = 0.400. They create a decimal that can be represented in three equivalent forms and justify its position on a number line.