055i - Equivalent Forms of Rational Numbers

Lesson 55 — Fractions, Decimals and Percentages as Equivalent Forms

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

Learning Intentions

• To understand that fractions, decimals and percentages are three ways of writing the same quantity.
• To find equivalent representations of rational numbers.

Success Criteria

I can:

1. Explain what a rational number is.
2. Generate equivalent fractions by multiplying or dividing numerator and denominator.
3. Simplify a fraction to its lowest terms.
4. Recognise the same quantity written as a fraction, decimal or percentage.

Warmup

(6 minutes — same or different? pairs)

Decide whether each pair represents the same quantity. Justify.

1. and
2. and
3. and
4. and
5. and

Answers: 1. Same; 2. Same; 3. Different — , while ; 4. Not exactly — recurring, so is only an approximation; 5. Same.

Discussion of Q4. This is worth several minutes. cannot be written exactly as a terminating decimal. We write or to show the digits continue forever.

Activities

Activity 1 — Explicit Instruction: what a Rational Number is (10 min)

Definition. A rational number is any number that can be written as a fraction , where and are integers and .

Everything below is rational:

Number	As a fraction	Why it qualifies
		every whole number is a fraction over
		terminating decimals convert directly
		per cent means “out of “
		negatives count too
		mixed numbers convert to improper fractions
		recurring decimals are rational as well

The key idea: fraction, decimal and percentage are three notations, not three different kinds of number. Choosing between them is like choosing between “half past three” and “3:30”.

When each form is most useful:

• Fractions — exact values, especially thirds and sevenths; comparing parts of a whole.
• Decimals — measurement, money, calculator work, ordering.
• Percentages — comparisons, statistics, discounts, interest.

Activity 2 — Equivalent Fractions and Simplifying (14 min)

The principle. Multiplying or dividing both numerator and denominator by the same non-zero number gives an equivalent fraction, because you are multiplying by a disguised .

I do — generating equivalents. Write four fractions equivalent to .

I do — simplifying. Simplify .

Method 1 — divide by the HCF (from Lesson 7):

Method 2 — divide repeatedly by any common factor:

Both reach the same answer. Method 1 is faster; Method 2 is safer if the HCF is not obvious.

Prime factorisation connection. For a stubborn fraction, factorise both parts and cancel:

You do — simplify to lowest terms:

(Answers: ; ; ; ; ; .)

You do — fill the gap:

(Answers: ; ; .)

Activity 3 — Inquiry: the Equivalence Triangle (10 min)

Pairs.

Complete this table. Each row shows one quantity in three forms.

Fraction (simplest)	Decimal	Percentage

Socratic scaffolding for the last row:

Prompt	Purpose
What does mean as a division?	.
Carry it out. What happens?	— it never terminates.
How do we write that?	, with a dot over the repeating digit.
Now the percentage. What do you multiply by?	, giving .
Is correct?	Not exactly — it is an approximation, slightly less than a third.
Why do some rows terminate and others not?	Denominators built only from s and s terminate, because . Thirds and sevenths do not.
Looking back	The fraction is the exact form here; the decimal is only ever an approximation.

Completed table:

Fraction	Decimal	Percentage

Extension — which denominators terminate? Test against . (Terminating exactly when the simplified denominator has only s and s as prime factors — a direct application of Lesson 6.)

Checks for Understanding

(5 minutes — exit ticket)

1. Simplify: (a) (b) (c) .
2. Fill the gap: .
3. Write as a decimal and a percentage.
4. Reasoning. Explain why cannot be written exactly as a terminating decimal.
5. Is a rational number? Justify.

Answers: 1. (a) (b) (c) ; 2. ; 3. and ; 4. A decimal terminates only when the denominator’s prime factors are s and s. Since is neither, the division never ends — it recurs as ; 5. Yes — it equals , a fraction of two integers.

Common Misconceptions

Misconception	How to pre-empt it
Adding the same number to numerator and denominator to get an equivalent fraction.	Test it: and are clearly different. Only multiplying or dividing preserves value.
Believing a simplified fraction is a different number.	and are the same quantity in different notation. Show both on a number line.
Treating as exactly .	Multiply by to get , not .
Thinking a larger denominator means a larger fraction.	Compare and .
Believing percentages are not numbers.	is — a number written in a particular way.
Simplifying by cancelling digits rather than factors.	is by coincidence, not by cancelling the s. Discuss briefly as a caution.

Enrichment — Competition-Style Problems

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

Answer

Convert all to decimals: , , . The largest is .

E2 (AMC Junior style). Simplify to lowest terms.

Answer

E3 (Challenge). For how many whole numbers from to does give a terminating decimal?

Answer

Only when ‘s prime factors are s and s: — 8 values.

E4 (Challenge). A fraction equals . Its numerator and denominator add to . Find the fraction.

Answer

The parts total , so one part is . The fraction is . (Check: , and ✓)

E5 (Challenge). Show that equals exactly.

Answer

One argument: , so , giving .

This surprises students, and is worth discussing rather than rushing. The two notations denote the same number.

Homework

1. Simplify to lowest terms: (a) (b) (c) (d) (e) .

2. Fill the gaps: (a) (b) (c) (d) .

3. Write three fractions equivalent to (a) (b) .

4. Complete the table:

   Fraction	Decimal	Percentage

5. State whether each is rational, with a reason: (a) (b) (c) (d) (e) .

6. Order from smallest to largest: , , , .

7. Reasoning. Explain why adding to both the numerator and the denominator of does not give an equivalent fraction.

8. Reasoning. Explain how you can tell, from its denominator, whether a fraction gives a terminating decimal.

9. Challenge. A fraction equals , and its numerator and denominator differ by . Find the fraction.

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) (b) (c) (d) . Q4 — ; ; ; . Q5 — all rational; each can be written as a fraction of integers. Q6 — , , , — so that order. Q7 — ; equivalence needs multiplication or division, not addition. Q8 — simplify first, then check whether the denominator’s only prime factors are and . Q9 — parts differ by , so one part is ; the fraction is .