056i - Converting Between Forms

Lesson 56 — Converting Between Fractions, Decimals and Percentages

Strand: Number | Descriptor: AC9M7N04 | Duration: 45 minutes

Learning Intentions

• To convert fluently between fractions, decimals and percentages.
• To select the most efficient conversion method for a given number.

Success Criteria

I can:

1. Convert a fraction to a decimal by division or by scaling the denominator.
2. Convert a decimal to a fraction using place value, then simplify.
3. Convert to and from percentages by multiplying or dividing by .
4. Recall the common conversions from memory.

Warmup

(6 minutes — recall race, mini whiteboards)

Convert to a decimal:

(Answers: ; ; ; ; ; .)

The core facts to memorise — display all lesson:

Fraction
Decimal
Percentage

Activities

Activity 1 — The Six Conversions (16 min)

The conversion map:

        ×100
Fraction ──→ Percentage
   │  ↖        │  ↗
   │   ÷100    │
   ↓           ↓
     Decimal

Rather than six unrelated rules, teach three pairs, each with its inverse.

Pair 1 — Decimal ↔ Percentage.

Pair 2 — Fraction ↔ Decimal.

Fraction to decimal: divide the numerator by the denominator.

Shortcut for friendly denominators: scale to a power of .

Decimal to fraction: use place value, then simplify.

Naming the place value is the key step: one decimal place → tenths, two → hundredths, three → thousandths.

Pair 3 — Fraction ↔ Percentage.

Fraction to percentage: multiply by .

Percentage to fraction: write over , then simplify.

We do — a mixed set, choosing the efficient route each time:

1. to a percentage.
2. to a fraction.
3. to a decimal.
4. to a decimal.
5. to a fraction.

Note on Q5: when the percentage is not a whole number, multiply top and bottom by first to clear the decimal.

Activity 2 — Independent Practice (10 min)

You do — complete the table.

Fraction	Decimal	Percentage

(Answers: ; ; ; ; ; ; .)

Discussion of the row. A percentage can exceed . It means more than one whole — useful for growth, price rises and comparisons. Ask for a real example. (A price rising to of its original value.)

Activity 3 — Inquiry: Ordering Mixed Forms (8 min)

Pairs.

Order each set from smallest to largest. You must show your method.

1. , , ,
2. , , ,
3. , , ,

Socratic scaffolding:

Prompt	Purpose
Understand: what makes this hard?	The numbers are in three different notations.
What would make comparison easy?	Converting them all to one form.
Which form is easiest to order?	Decimals — compare digit by digit from the left.
Convert set 1.	, , , .
Now order.	.
Translate back.	.
Looking back — a caution	Give the answer in the original forms, as the question presented them.

Answers: 1. ; 2. ; 3. .

Note on set 2: and are equal. Students must state this rather than inventing an order between them.

Checks for Understanding

(5 minutes — exit ticket)

1. Convert to a decimal and a percentage.
2. Convert to a fraction in lowest terms.
3. Convert to a decimal and a fraction.
4. Order from smallest: , , .
5. Reasoning. Explain why converting everything to decimals is usually the easiest way to order a mixed set.

Answers: 1. and ; 2. ; 3. and ; 4. ; 5. Decimals share a common place value structure, so they can be compared digit by digit from the left — no common denominator is needed.

Common Misconceptions

Misconception	How to pre-empt it
Moving the decimal point the wrong way when converting to a percentage.	Sanity-check: a percentage should be about times bigger than the decimal. , not .
Writing as .	Name the place value aloud: one decimal place means tenths.
Dividing the wrong way round: for .	The numerator goes inside the division — top divided by bottom.
Believing percentages cannot exceed .	The row addresses this.
Forgetting to simplify after converting.	Require lowest terms in every final fraction answer.
Treating as .	, which is larger than . Compare on a number line.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which is largest: , , or ?

Answer

As decimals: , , . The largest is .

E2 (AMC Junior style). Write as a fraction in lowest terms.

Answer

(Dividing top and bottom by .)

E3 (Challenge). A shirt costs 40115%$ of the original. What is the new price?

Answer

E4 (Challenge). Write as a decimal and as a percentage, giving the percentage to one decimal place.

Answer

, so the percentage is .

E5 (Challenge). Arrange in ascending order: , , , .

Answer

As decimals (to 4 d.p.): ; ; ; .

Order: .

Homework

1. Convert to a decimal: (a) (b) (c) (d) (e) .

2. Convert to a fraction in lowest terms: (a) (b) (c) (d) (e) .

3. Convert to a percentage: (a) (b) (c) (d) (e) .

4. Convert to a decimal and a fraction: (a) (b) (c) (d) (e) .

5. Complete the table:

   Fraction	Decimal	Percentage

6. Order from smallest to largest: (a) , , (b) , , (c) , , .

7. Reasoning. Explain the difference between and .

8. Reasoning. Explain why cannot be written as an exact percentage.

9. Challenge. A jacket is reduced by from 85$. Write the remaining fraction of the price, then find the sale price.

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) (b) (c) (d) (e) . Q3 — (a) (b) (c) (d) (e) . Q4 — (a) (b) (c) (d) (e) . Q6 — (a) (b) (c) . Q7 — , one hundredth of . Q8 — , which recurs forever, so any written percentage is an approximation. Q9 — remains; 59.50$.