Fractions, decimals and percentages

Classroom resources

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Students will be able to

• I can convert between fractions, decimals and percentages.
• I can explain why 0.5 = 50%, not 5%.
• I can compare amounts written in different forms.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need Equivalent fractions (the lesson before) and place value with decimals (Year 6).

The one idea: percent means out of 100, so hundredths connect all three. The common mistake is reading the digits after the point as the percentage (0.5 = 5%); it is planted in Max's working and named in the success criteria.

Decimal answers are marked by form (3/8 is told to write a decimal); a percentage can be typed with or without the % sign.

Running the lesson

1. Predict: take a show of hands. Students who chose 34% are reading digits.
2. Run: ask how many squares one quarter is, then three.
3. Investigate: model each line of the worked example with the grid beside it.
4. Modify: before the 1/3 task, ask students to shade 1/3 of the grid exactly.
5. Make: ask why odd numbers of eighths cannot be shaded exactly.

When students get stuck

Answers to the checks

1. three quarters: Which is the same amount as $\tfrac{3}{4}$?
   75%. 3/4 = 75/100 (× 25 on the top and the bottom), and out of 100 is a percentage: 75%. As a decimal, 0.75.
2. shade the grid: The bar is $\tfrac{3}{4}$. Shade the same amount of the hundred grid. Was your prediction right?
   
3. Three ways: Write $\tfrac{3}{8}$ as a decimal.
   0.375
   3 ÷ 8 = 0.375. As a percentage: 37.5%.
4. Find the mistake: Max writes $0.5$ as a percentage. Click the line where his working goes wrong.
   line 4
   Line 4: 0.5 is 5 tenths, which is 50 hundredths. Out of 100: 50%. Multiply a decimal by 100 for the percentage.
5. Percentage to fraction: Write $16\%$ as a fraction in its simplest form.
   4/25
   16% = 16/100 = 4/25 (÷ 4).
6. Fifths: The bar is $\tfrac{2}{5}$. Shade the same amount of the grid.
   
   2/5 = 40/100 = 40%.
7. Decimal to percentage: Write $0.6$ as a percentage (a number of percent).
   60
   0.6 × 100 = 60%.
8. Percentage to decimal: Write $35\%$ as a decimal.
   0.35
   35 ÷ 100 = 0.35.
9. Fraction to percentage: Write $\tfrac{7}{20}$ as a percentage.
   35
   7/20 = 35/100 = 35%.
10. A decimal that never ends: Edge case: what is $\tfrac{1}{3}$ as a decimal?
    0.333…, with the 3s going on for ever
    1 ÷ 3 = 0.333…: a recurring decimal. Still a rational number (a fraction); the grid shows about 33.3 squares.
11. Decimal to fraction: Write $0.45$ as a fraction in its simplest form.
    9/20
    0.45 = 45/100 = 9/20.
12. Fraction to percentage: Write $\tfrac{3}{5}$ as a percentage.
    60
    3/5 = 60/100 = 60%.
13. Order them: Which is smallest: $0.6$, $\tfrac{2}{3}$ or $65\%$?
    0.6
    Write each as a decimal: 0.6, 0.666… and 0.65. The smallest is 0.6.
14. Review: simplest form: Review: write $\tfrac{9}{12}$ in its simplest form.
    3/4
    ÷ 3: 3/4.
15. eighths that are whole percentages: Choose a number of eighths ($a$) that is a whole-number percentage, then shade the same amount on the grid.
    
    2/8 = 25%, 4/8 = 50%, 6/8 = 75% (and 8/8 = 100%). Odd numbers of eighths end in .5%.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Fifths

Open task: any program that does what the step asks.

one change at a time › Decimal to percentage

Open task: any program that does what the step asks.

one change at a time › Percentage to decimal

Open task: any program that does what the step asks.

one change at a time › Fraction to percentage

Open task: any program that does what the step asks.

one change at a time › A decimal that never ends

Open task: any program that does what the step asks.

eighths that are whole percentages

Open task: any program that does what the step asks.

Stretch and support

Stretch: Which fractions with a bottom number up to 12 give decimals that stop, and which recur? What do the stopping ones have in common? (In simplest form, bottoms made only of 2s and 5s.)

Support: Use the hundred grid for every conversion; for eighths, allow half squares.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Fractions, decimals and percentages".

The idea: Converting between fraction, decimal and percentage: I can Students convert fluently between fraction, decimal and percentage representations of rational numbers.

What understanding looks like:
- Correctly converts a given rational number between fraction, decimal and percentage forms.
- Identifies when representations are equivalent without performing full conversion.
- Does not confuse 0.5 with 5% (confusing decimal and percentage forms).

Mistakes students often make here:
- A student writes 0.5 as 5% (Max's mistake).
- A student writes 3/4 as 3.4 or 0.34.
- A student cannot simplify a percentage as a fraction.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Equivalence and simplification of fractions), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Fractions, decimals and percentages".

Exit ticket

Write 3/5 as a decimal and as a percentage, and 0.08 as a percentage.

0.6 and 60%; 0.08 = 8%.

What each step is for

• three quarters: Committing to a percentage for 3/4 shows whether they treat the digits as the percentage (34%) or think in hundredths.
• shade the grid: Shading the hundred grid to match 3/4 turns the fraction into hundredths, which is what a percentage and a two-place decimal both are.
• converting: Each conversion goes through a fraction over 100 or a division: fraction to decimal by dividing, decimal to percentage by × 100, percentage to fraction over 100 then simplified. Max's mistake reads the digits after the point as the percentage.
• one change at a time: One change at a time: fifths on the grid, each direction of conversion, then the edge case of a decimal that never ends.
• all three: Conversions in every direction, mixed, with digit-reading planted again; ordering mixed forms; the review keeps simplifying fresh.
• eighths that are whole percentages: Searching eighths for whole-number percentages shows which fractions fit hundredths exactly, and that the rest (3/8 = 37.5%) need a decimal.

Where it leads

Next: Fractions and decimals on the line (node 3).

Where it leads: percentage of an amount, percentage change and discounts (Year 8), and probability.