Percentages as Parts

Overview

In this tenth lesson of “Mapping Data and Change”, students connect percentages to fractions, decimals and hundredths. They use visual models and efficient strategies to find common percentages of quantities, explaining how the representations show the same amount.

Learning intentions

• WALT connect percentages, fractions and decimals using hundredths.
• WALT find common percentages of quantities.
• WALT use visual models to explain percentage reasoning.
• WALT communicate mathematical thinking clearly and check whether an answer is reasonable.

Success criteria

• I can explain that a percentage is an amount out of 100.
• I can match common percentages with equivalent fractions and decimals.
• I can find 10%, 25%, 50% and 75% of a quantity.
• I can use a diagram, equation or words to justify my answer.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: connecting challenge tasks with relevant mathematical ideas and prior learning.
• Mathematics and Statistics — Additional resources for advanced learners: reasoning, explanation and multiple representations.
• Mathematical and statistical practices: represent, reason, calculate, communicate and evaluate.

Lesson structure (45 minutes)

1. 0–5 min · Hook and notice. Display the percentage hook slide showing a 100-square grid with 50 squares shaded and ask, “Which is the larger amount: 50%, one-half or 0.5?” Students make an individual choice, justify it to a partner, and share what they notice. Establish that the symbols look different but may describe the same amount.

2. 5–12 min · Build the connection. Use the representation slides to model a hundred-square: 1% is one square, 10% is ten squares, 25% is 25 squares and 50% is 50 squares. Link each to its fraction and decimal: 10% = 10/100 = 0.10, 25% = 25/100 = 0.25 and 50% = 50/100 = 0.50. Students record the connections and explain them using the sentence frame: “___% is equivalent to ___ because ___.”

3. 12–20 min · Model efficient strategies. Demonstrate finding 50%, 25%, 75% and 10% of quantities using familiar fraction relationships: 50% is one-half, 25% is one-quarter, 75% is three-quarters, and 10% is one-tenth. For 80, model 50% = 40, 25% = 20 and 75% = 60; for 230, model 10% = 23. Emphasise that the strategy must match the percentage and that students should estimate before calculating. Students solve two examples on mini-whiteboards and explain one using a visual or equation.

4. 20–32 min · Guided investigation. Distribute the percentages as parts worksheet and provide pairs with the fraction circles cut-out pack. Students complete the matching and quantity tasks, using fraction circles or a quick hundred-square sketch to show 25%, 50% and 75%. They compare methods with a partner. Pause at 26 minutes to check the common error of treating 25% as one-quarter of 100 only, rather than one-quarter of the given quantity.

5. 32–40 min · Reasoning challenge and discussion. Display the reasoning challenge slides: “A student says 25% of 60 is 25. Do you agree? Prove it in two different ways.” Students solve independently, then rehearse an explanation with a partner using a diagram, fraction and equation. Invite several solutions, selecting different representations. Ask, “Which method is most efficient here, and when might another method be useful?”

6. 40–45 min · Plenary and exit check. Return to the plenary slides and revisit the opening comparison. Students complete the final worksheet question: “Find 75% of 40 and show how you know.” They write one sentence explaining the connection between 75%, 3/4 and 0.75, then hand in the worksheet as they leave.

Resources

• the percentage lesson slide deck
• the percentages as parts worksheet
• the fraction circles cut-out pack
• Mini-whiteboards, pens and erasers
• Pencils and coloured pencils
• Hundred-square model displayed or drawn on the board
• 25 copies of the worksheet
• Fraction circles cut and prepared before the lesson

Assessment

• Listen during the hook and partner explanations for whether students recognise equivalent representations, rather than simply naming memorised facts.
• Use mini-whiteboard responses to identify misconceptions about the whole quantity, especially confusing 25% of a quantity with 25.
• Check worksheet models, equations and explanations. Use the final question to decide who is ready for less familiar percentages and who needs further work with hundredths and benchmark fractions.

Differentiation

• Support students with a visible hundred-square, fraction circles, a percentage–fraction–decimal reference table and the sentence frame: “___% of ___ is ___ because ___.”
• Work with a small group during guided practice, beginning with quantities divisible by 2, 4 or 10. Allow students to draw or physically build the fraction before calculating.
• For students needing language support, pre-teach “percent”, “of”, “equivalent”, “whole” and “quantity”; pair them with a supportive peer and accept oral explanations alongside written work.
• Extend confident students by asking them to find 20%, 5% or 15% of a quantity using 10% and 5%, then create a convincing error for a partner to solve and explain.