Equivalent Number Forms

Overview

Students connect fractions, decimals and percentages as equivalent representations of the same positive rational number. They build on prior understanding of place value, equivalent fractions and simple percentages, then apply efficient conversion strategies to practical examples.

Learning intentions

Students will:

• convert between common fractions, decimals and percentages;
• explain why equivalent representations have the same value;
• use efficient mental and written strategies to solve conversion problems;
• choose a suitable representation and check whether an answer is reasonable.

Success criteria

• I can convert a fraction to a decimal and percentage.
• I can convert a decimal or percentage to an equivalent fraction.
• I can explain my strategy using mathematical language.
• I can check that all three representations have the same value.

Curriculum links

• Number — finding equivalent representations of rational numbers, including fractions, decimals and percentages.
• Number — using efficient mental and written strategies with positive rational numbers.
• Number — selecting suitable representations to solve numerical problems.
• Number — using estimation and reasonableness checks.

Lesson structure (30 minutes)

1. 0–4 min · Hook and retrieval. Teacher opens with the hook and retrieval slides and displays: “Which is greater: 0.4, 40% or (\frac{4}{10})?” Ask the student to justify the answer without calculating on a calculator. The student records an answer, explains their thinking aloud and recalls that a percentage means “out of 100”.

2. 4–10 min · Direct teaching. Teacher uses the conversion strategy slides to model a conversion triangle:

• fraction → decimal: divide numerator by denominator;
• decimal → percentage: multiply by 100;
• percentage → decimal: divide by 100;
• fraction → percentage: create an equivalent fraction with denominator 100 where efficient, or convert through a decimal.

Model (\frac{3}{4}=0.75=75%), emphasising that (\frac{3}{4}) can be multiplied by (\frac{25}{25}) to make (\frac{75}{100}). The student annotates the worked example and answers short teacher questions such as, “Why does the value stay the same?”

3. 10–16 min · Guided practice. Teacher opens the guided conversion examples and completes the first two questions collaboratively: (\frac{1}{2}), 0.25 and 60%. Prompt the student to select the most efficient route and explain each step. The student completes the next three examples, using a number line or place-value reasoning if helpful. Check after each item and address misconceptions, particularly confusing 0.5 with 5% or moving the decimal in the wrong direction.

4. 16–23 min · Independent application. Teacher directs the student to the conversion and comparison questions. The student completes a short set involving common fractions, terminating decimals and percentages, including:

• (\frac{2}{5}), 0.2, 35%, (\frac{7}{8}), 0.125 and 12.5%;
• a comparison question: “A student says 0.6 is greater than 55%. Is the claim correct?”
• a practical question: “A class has completed 75% of a project. Write this as a fraction in simplest form and a decimal.”

Teacher observes strategy choice, asks the student to show working and provides immediate feedback without completing the calculation for them.

5. 23–27 min · Reasonableness check and discussion. Teacher displays the error analysis and discussion slides with the statement: “(\frac{3}{5}=0.35=35%).” The student identifies and corrects the error, explaining that (\frac{3}{5}=0.6=60%). Teacher asks the student to estimate whether each answer should be less than, equal to or greater than one, reinforcing that a percentage greater than 100% represents a value greater than one.

6. 27–30 min · Exit assessment. Teacher uses the plenary and exit prompt and asks the student to answer on the bottom of the worksheet:

• Convert (\frac{7}{10}) into a decimal and percentage.
• Convert 45% into a decimal and a fraction in simplest form.
• Explain one method used today.

The student reads one answer aloud and self-rates confidence from 1–5. Teacher collects the worksheet to determine the next lesson’s starting point.

Resources

• the equivalent representations teaching deck
• the fractions, decimals and percentages worksheet
• Mini whiteboard or exercise book
• Pencil and ruler
• Calculator for checking only, if available
• Place-value chart or number line
• Teacher answer guide

Assessment

• During modelling and guided practice, question the student about why multiplying or dividing by 100 changes a decimal representation but not the underlying value.
• During independent work, check accuracy, simplification of fractions, strategy selection and use of mathematical explanations.
• Use the exit assessment to identify whether the student can convert in both directions and recognise an unreasonable or incorrect equivalence.

Differentiation

• Support with a conversion triangle, a place-value chart, a 0–1 number line and sentence starters such as “I converted this by…” and “These are equivalent because…”.
• Begin with benchmark values such as (\frac{1}{2}), (\frac{1}{4}), (\frac{3}{4}), 0.1, 0.5 and 75% before moving to less familiar values.
• For EAL learners, explicitly teach and display “numerator”, “denominator”, “equivalent”, “percentage” and “simplest form”; allow oral explanation before written explanation.
• For additional challenge, ask the student to create three different representations of 0.375 and explain why converting through a decimal may be more efficient than finding a denominator of 100.