Equivalent Forms

Overview

Lesson 2 of 10 in Rational Number Knowledge Practices. Students connect familiar fractions, decimals and percentages, using visual models and calculations to explain when values are equivalent and when differently sized wholes make comparisons misleading.

Learning intentions

• WALT use multiplication, division and visual models to generate equivalent forms.
• WALT convert familiar fractions to decimals and percentages.
• WALT distinguish equivalent values from different-sized wholes.
• WALT explain why multiplying or dividing the numerator and denominator by the same number preserves value.

Success criteria

• I can make equivalent fractions using a scale factor.
• I can convert familiar fractions to decimals and percentages.
• I can use a model or calculation to justify equivalence.
• I can explain what changed and what stayed the same.

Curriculum links

• Number: recognise, represent and connect fractions, decimals and percentages.
• Number: use multiplication and division to find equivalent forms and simplify fractions.
• Mathematical practices: represent ideas in more than one way, reason, justify and communicate.
• Students develop confidence, strategic thinking and perseverance when solving and explaining problems.

Lesson structure (60 minutes)

1. 0–5 minutes – Hook and retrieval

Open with the opening comparison and retrieval prompt. Show two differently sized wholes, each with one-half shaded, and ask: “Do these show the same amount? What information do we need?” Briefly revisit fraction representations from Lesson 1. Invite students to use the words equivalent, whole, numerator and denominator.

2. 5–15 minutes – Explicit teaching and modelling

Use the worked-example slides to model:

2/5 = 4/10 = 40/100 = 0.4 = 40%

Build the example with a fraction wall and hundred grid. Emphasise that multiplying both numerator and denominator by the same number changes the names of the parts, but not the value of the fraction. Ask: “What changed and what stayed the same?” Model simplifying 4/10 back to 2/5 by dividing both parts by 2.

3. 15–25 minutes – Guided equivalence chains

In pairs, students complete the first section of the equivalent forms practice sheet. Begin with denominators 2, 4, 5 and 10, such as:

1/2 = __/10 = 0.__ = __% 3/4 = __/100 = __% = 0.__

Students may use fraction walls, hundred grids and place-value charts. Pause to discuss why 3/6 = 1/2, and check that students can describe the scale factor rather than only give an answer.

4. 25–40 minutes – Card sort and partner matching

Organise students into groups of three or four. Give each group teacher-prepared cards showing related fractions, decimals and percentages, then open the card-sort instructions and discussion prompts. Students match equivalent forms and arrange each set into a chain. They must choose one match to justify with a model or calculation.

Circulate and ask: “How do you know these are equal?” “What is the scale factor?” “Could these be different-sized wholes?” Use the fractions vocabulary cards as a language reference for students who need support.

5. 40–50 minutes – Independent application

Students complete the remaining questions on the equivalent forms practice sheet independently, then compare one answer with a partner. Include examples requiring conversion and explanation, including 7/20 = 35%. Students can use a calculator only to check an answer after showing their method.

6. 50–56 minutes – Share and address misconceptions

Use the reasoning and misconception slides to display selected responses. Discuss the difference between an equivalent value and the same-looking shaded amount in differently sized wholes. Reinforce that multiplying only the numerator, or only the denominator, changes the value.

7. 56–60 minutes – Exit ticket and reflection

Finish with the exit prompt on the plenary and exit-ticket slide. Students complete:

3/4 = __/100 = __% = __

They add one short explanation of how they know the forms are equivalent, then hand it in as they leave.

Resources

• the Equivalent Forms teaching deck
• the equivalent forms practice sheet
• Fraction walls
• Hundred grids
• Place-value charts
• Teacher-prepared fraction, decimal and percentage cards
• Mini-whiteboards or scrap paper
• Pencils and highlighters
• Calculators for checking, not replacing, reasoning
• the fractions vocabulary cards

Assessment

• Listen during modelling and guided work for accurate use of scale factors and the explanation that the whole remains unchanged.
• Check card-sort justifications and worksheet responses for connected fraction, decimal and percentage representations.
• Use the exit ticket to identify students needing further work with tenths, hundredths, percentage conversion or equivalent fractions.

Differentiation

• Support: begin with denominators 2, 4, 5 and 10; provide fraction walls, hundred grids, place-value charts and partially completed equivalence chains.
• Support language development: preteach equivalent, simplify, scale factor, tenths, hundredths and percent using the fractions vocabulary cards. Provide sentence stems: “___ is equivalent to ___ because…”
• SEN/EAL: pair students strategically, model one step at a time, allow oral explanations, enlarged models and calculator checks after working is shown.
• Extension: ask students to generate several equivalent forms and prove why 7/20 = 35%; challenge them to create a new matching set for another group.