Equivalent Number Forms

Overview

Students connect fractions, decimals and percentages as equivalent representations of the same quantity. They build on fraction equivalence, place value and multiplication by 10 and 100, using visual matching, explicit modelling and differentiated problem-solving.

Learning intentions

Students will:

• recognise equivalent fractions, decimals and percentages;
• simplify fractions before converting;
• scale fractions to a denominator of 100 where appropriate;
• use place value and multiplication by 100 to convert decimals to percentages.

Success criteria

• I can explain why (10/20 = 1/2 = 0.5 = 50%).
• I can simplify a fraction and convert it into a decimal and percentage.
• I can convert fractions with denominators of 25 by scaling to 100.
• I can compare representations and justify that they are equivalent.

Curriculum links

• Determine percentages of quantities and identify equivalent benchmark fractions, decimals and percentages.
• Compare and order decimals to three decimal places using place value.
• Recognise fractions as division and use equivalence to represent fractional quantities.

Lesson structure (60 minutes)

1. 0–7 min · Hook and diagnostic. Display a whole divided into 100 equal squares using the opening visual and diagnostic question and ask, “Which is greater: 0.6, 60%, or 3/5?” Students make a quick individual choice, justify it to a partner and share strategies. Note common methods and misconceptions without confirming every answer immediately.

2. 7–20 min · Explicit modelling. Use the conversion model slides to model that equivalent forms name the same amount. Start with (10/20): simplify by dividing numerator and denominator by 10 to get (1/2); interpret (1 ÷ 2 = 0.5); then multiply (0.5 × 100 = 50%). Model the connected chain: [ 10/20 = 1/2 = 0.5 = 50% ] Emphasise that simplifying does not change the value. Model (30/50 = 3/5 = 0.6 = 60%), explaining that (3 ÷ 5 = 0.6). Link decimal place value to percentage: multiplying by 100 moves the digits two places left in the written decimal notation, so (0.6 × 100 = 60).

3. 20–30 min · Denominator 25 and guided practice. Model (7/25) by scaling both parts by 4: [ 7/25 = 28/100 = 0.28 = 28%. ] Repeat with (18/25 = 72/100 = 0.72 = 72%). Explain that a denominator of 25 can become 100 by multiplying by 4. Students complete, with teacher prompting, (9/25), (12/25) and (21/25) in their books. Check answers using mini-whiteboards and ask, “What operation did you apply to the denominator, and why must you apply it to the numerator too?”

4. 30–43 min · Matching and discussion. In pairs, students use the fraction-decimal-percentage matching cards to place equivalent representations together, recording one explanation for each match. Pairs must explain one match using simplifying, one using scaling to 100 and one using division. Circulate and ask: “How do you know these values are equal?” and “Could you use a different strategy?” Select two contrasting explanations for a brief class discussion.

5. 43–55 min · Independent differentiated practice. Distribute the differentiated equivalent-representations worksheet. Students work independently, then check selected answers with a partner.

• Support: Complete conversion chains for (1/2), (1/4), (3/5), (7/25) and (18/25), using a place-value chart, fraction wall or multiplication prompts.
• Core: Simplify and convert (15/30), (24/40), (13/25), (19/25) and (0.45). Order (0.6), (58%), (3/5) and (0.625), explaining the comparison.
• Challenge: A student says (8/25 = 0.8 = 80%). Explain the error and correct it. Then create two different fractions equal to (64%), showing all representations.

6. 55–60 min · Plenary and exit ticket. Revisit the opening question with the plenary and reflection slides. Students complete the final questions on the exit-ticket section: (a) Convert (14/25) into a decimal and percentage; (b) explain why (12/20) and (0.6) are equivalent; (c) identify the error in “(0.37 = 3.7%)”. Collect responses to plan the next lesson.

Resources

• the equivalent-representations slide deck
• the differentiated equivalent-representations worksheet
• the fraction-decimal-percentage matching cards
• Mini-whiteboards, markers and erasers
• Student exercise books and pencils
• Fraction strips or fraction walls
• 100-square or percentage grid
• Place-value charts
• Projector or interactive display

Assessment

• Use the opening comparison, mini-whiteboard responses and guided examples to identify whether students understand equivalence, simplifying and scaling.
• During pair work, listen for explanations that preserve the value by applying the same operation to numerator and denominator.
• Use the exit ticket to assess conversion, comparison and misconception correction. Group students for follow-up based on whether errors relate to place value, fraction equivalence or percentage notation.

Differentiation

• Provide support students with a completed conversion example, a 100-square, fraction strips, a place-value chart and sentence starters such as “I know these are equivalent because…”.
• Allow students who need adjustments to use calculators only after estimating, reduce the number of worksheet items, and respond orally or by annotating a conversion chain.
• Pre-teach and display the terms equivalent, simplify, scale, decimal and percentage, with gestures and visual examples for EAL/D learners.
• Extend confident students through the error-analysis and creation tasks, requiring more than one strategy and a written justification.

Common misconceptions to address are treating (0.6) as (6%) rather than (60%), moving the decimal point inconsistently, multiplying only the denominator when scaling, and believing a larger denominator always means a larger fraction. Reinforce that percentages are “out of 100” and that equivalent forms represent the same point on a number line.