Represent and Interpret

Overview

Lesson 1 of 5 in Equivalent Proportional Representations. Students connect fractions, decimals, percentages, diagrams and words to describe the same proportional relationship in familiar contexts.

Learning intentions

• WALT identify the whole and the part in a proportional relationship.
• WALT represent the same quantity as a fraction, decimal, percentage, diagram and words.
• WALT interpret what each representation means in context.
• WALT recognise that equivalent representations have the same value.

Success criteria

• I can identify the whole before finding the part.
• I can represent a quantity in at least three ways.
• I can explain what each representation means in context.
• I can recognise that equivalent representations have the same value.

Curriculum links

• Number: represent, compare and interpret rational numbers using fractions, decimals and percentages.
• Number: recognise and use equivalent forms to describe proportional relationships.
• Mathematical practices: use representations, explain reasoning, and communicate with precise mathematical vocabulary.
• Key competencies: thinking; using language, symbols and texts; participating and contributing.

Lesson structure (60 minutes)

1. 0–10 minutes – Launch: What is the whole? Open with the launch and context slides. Display a 100-square and ask: “If 25 squares are shaded, what could this tell us?” Connect responses to familiar contexts such as 25 out of 100 students participating in a sport, or one quarter of a food portion. Establish that the whole is the total amount being considered. Introduce: whole, part, fraction, numerator, denominator, decimal, percentage, equivalent and proportion.

2. 10–25 minutes – Teacher modelling Model 1/4 using a shaded 10-by-10 grid, fraction notation, decimal notation, percentage and a sentence: “One quarter of the class chose football.” Think aloud: the denominator tells the number of equal parts; the numerator tells the parts selected. Show that 1/4 = 0.25 = 25% because each represents the same value. Repeat briefly with 1/2 and 3/4. Use the modelling slides and check understanding with mini-whiteboards.

3. 25–30 minutes – Hinge question and grouping Ask students to independently show the match for 0.25: 1/4, 25%, 1/25 or 2/8. Students hold up answers, then explain their choice to a partner. Address misconceptions before assigning students to differentiated stations. Ask: “What is the whole in this example?” and “How do you know the representations are equivalent?”

4. 30–55 minutes – Differentiated representation stations Use the station instruction slides to explain expectations. Students work in groups of 3–4 and rotate or complete the task matched to their readiness.

• Support: Use fraction strips, shaded grids and counters to build 1/2, 1/4, 3/4 and 1/10. Complete the partially filled representation table on the representation table worksheet. Adult support prompts: “What is the whole?” and “How many equal parts?”
• Core: Match fraction, decimal, percentage, diagram and context cards. Then create a small representation poster for one match, including a sentence explaining the whole and part. Use the ratio table and double number line mat where helpful to connect equivalent quantities.
• Extension: Represent 7/8 in at least four ways, explain why the forms are equivalent, and place the value on a number line. Students should justify their thinking rather than only calculate.

Circulate with an observation checklist, noting accurate use of vocabulary, identification of the whole, and explanations of equivalence.

5. 55–60 minutes – Reflection and exit ticket Return to the reflection and exit-ticket slide. Students complete the final section of the representation table worksheet: represent 3/5 in three forms and explain one representation in context. Invite two students to share different forms. Preview the next lesson: comparing and ordering equivalent proportional representations.

Resources

• the introduction, modelling, station and reflection slide deck
• the representation table worksheet
• 10-by-10 grids and coloured pencils
• Fraction strips and counters
• Teacher-prepared fraction, decimal, percentage, diagram and context cards
• Number lines
• Mini-whiteboards and pens
• Calculators for checking, not replacing, reasoning
• Teacher observation checklist
• Culturally responsive contexts: class participation, local sports statistics, recipes and food portions

Assessment

• Use questioning throughout: “What is the whole?”, “What does the denominator tell us?” and “How do you know these are equivalent?”
• Use the hinge question to identify students needing further support before stations.
• Record observations of representation accuracy and explanations; collect the exit ticket to plan the next lesson.

Differentiation

• Provide fraction strips, shaded grids, counters, colour coding and a partially completed table for students needing concrete or visual support.
• Pair students strategically and provide sentence frames: “The whole is…”, “The part is…”, and “These are equivalent because…”
• Allow EAL and students with literacy needs to explain orally, draw, label, or use a bilingual glossary before writing.
• Extend confident students with 7/8, additional number-line placement, and a justification of equivalence using multiplication or division.