Equivalent Fraction Models

Overview

In this second lesson of Fraction Foundations and Operations, students develop equivalent fractions by multiplying or dividing the numerator and denominator by the same non-zero number. Visual models and number lines support the transition from concrete representations to simplifying fractions in lowest terms.

Learning intentions

Students will:

• explain why multiplying or dividing the numerator and denominator by the same non-zero number preserves a fraction’s value
• generate equivalent fractions using visual models, number lines and calculations
• simplify fractions to their lowest terms
• communicate and check their reasoning using mathematical language

Success criteria

• I can show equivalent fractions using a diagram or number line.
• I can multiply or divide the numerator and denominator by the same non-zero number.
• I can simplify a fraction until the numerator and denominator have no common factor greater than 1.
• I can explain why my equivalent fraction has the same value.

Curriculum links

• Number — use the four operations with rational numbers, selecting efficient strategies.
• Number — recognise and represent fractions and decimals appropriately, including terminating and recurring decimals where relevant.
• Measurement — use mathematical modelling to solve practical problems involving ratios and rates.
• Mathematical proficiency — reasoning, problem-solving, fluency and communicating mathematical ideas.

Lesson structure (60 minutes)

1. 0–7 min · Hook and diagnostic. Display the opening question in the equivalent fractions introduction deck: “Are (2/3) and (8/12) the same amount? How could you prove it?” Ask the student to sketch or explain an initial response, then identify prior knowledge about numerator, denominator and common factors.

2. 7–18 min · Build the visual model. Use the visual modelling slides to show one whole divided into equal parts, comparing (1/2), (2/4), (3/6) and (4/8) with aligned fraction strips and number lines. Teacher models that multiplying both terms by the same non-zero number changes the number of parts but not the position or amount: [ \frac{1}{2}\times\frac{2}{2}=\frac{2}{4} ] The student explains what has changed and what has stayed the same.

3. 18–30 min · Guided examples. Teacher works through examples from the guided examples slides and records each step clearly: [ \frac{3}{5}=\frac{3\times2}{5\times2}=\frac{6}{10} ] and [ \frac{12}{18}=\frac{12\div6}{18\div6}=\frac{2}{3}. ] The student completes matching examples on the equivalent fractions practice worksheet, drawing a model or number line before writing the calculation. Pause after each example to check that the same operation was applied to both numerator and denominator.

4. 30–43 min · Independent practice and conference. Distribute the equivalent fractions practice worksheet. The student completes tasks in three stages: generate equivalent fractions, identify equivalent pairs using models or number lines, and simplify fractions such as (8/12), (15/25), (18/24) and (21/28). Teacher checks working after the first question, prompting: “What common factor can divide both numbers?” and “How do you know the value has not changed?”

5. 43–53 min · Reasoning challenge. Return to the reasoning challenge slides. Present statements such as “To make an equivalent fraction, add 2 to the numerator and denominator” and “(6/9) is in lowest terms.” The student decides whether each statement is always true, sometimes true or false, and justifies the decision with a counterexample, diagram or calculation. Include the open task: “Find three different fractions equivalent to (3/4), then simplify one fraction back to (3/4).”

6. 53–60 min · Review and exit check. Use the plenary and exit-check slides to revisit the key rule and ask the student to explain it aloud. On the final section of the equivalent fractions practice worksheet, the student answers: a) Complete (5/6=\square/18). b) Simplify (24/36) to lowest terms. c) Explain why dividing the numerator and denominator by the same number keeps the fraction equivalent. Teacher records whether the student can calculate, represent and explain independently.

Resources

• the equivalent fractions introduction deck
• the equivalent fractions practice worksheet
• Whiteboard and markers
• Fraction strips or rectangles prepared by the teacher
• Student ruler and pencil
• Optional calculator for checking division
• Exercise book for additional representations

Assessment

• During modelling, check whether the student understands that both numerator and denominator must be multiplied or divided by the same non-zero number.
• Review the worksheet for accurate calculations, visual representations, number-line placement and correct use of common factors.
• Use the exit check to identify whether the student can generate, simplify and justify equivalent fractions without prompting.

Differentiation

• Support with pre-drawn fraction bars and number lines, a list of common factors, and the sentence starter: “The fractions are equivalent because both the numerator and denominator were…”
• Model one operation at a time and use colour to connect the original numerator and denominator to the new pair. Read instructions aloud and allow oral explanations before written responses.
• For the individual class setting, provide frequent short conferences and adjust the worksheet sequence according to the student’s responses.
• Extend by asking the student to find all fractions equivalent to (2/3) with denominators below 30, or explain why multiplying by zero cannot produce an equivalent fraction.