006i - Rational and Irrational Numbers

Lesson 6 — Rational and Irrational Numbers

Strand: Number | Descriptor: AC9M8N01 | Duration: 45 minutes

Learning Intentions

• To understand the distinction between rational and irrational numbers.
• To recognise irrational numbers in applied contexts, including square roots and .

Success Criteria

I can:

1. Define a rational number as one that can be written as a fraction of two integers, (with ).
2. Define an irrational number as one whose decimal expansion never terminates and never recurs, and which cannot be written as a fraction of integers.
3. Classify given numbers as rational or irrational, justifying my decision.
4. Explain why the square root of a non-perfect square is irrational.

Warmup

(5 minutes — mini whiteboards, sort and justify)

Sort each number as rational or irrational. Be ready to justify one choice you are unsure about.

Teacher note: Do not resolve or yet — flag them as today’s focus. Most students will correctly classify , , , and as rational from their fraction/decimal form alone.

Activities

Activity 1 — Explicit Instruction: Rational Numbers (10 min)

I do: Define a rational number and demonstrate the three equivalent forms it can take.

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

Say aloud: “A number is rational if its decimal either stops (terminates) or repeats forever in a pattern (recurs). Both of these can always be written exactly as a fraction of integers.”

We do: Together classify , , and as rational, writing each as a fraction of integers.

You do: Write each as a fraction : , , , .

(Answers: ; ; ; .)

Activity 2 — Explicit Instruction: Irrational Numbers (10 min)

I do: Introduce using a calculator display and highlight that its decimal expansion never settles into a repeating pattern and never stops.

An irrational number is a number that cannot be written as a fraction of two integers. Its decimal expansion goes on forever without ever repeating in a fixed pattern.

Explain why is irrational (informally): there is no whole number that, squared, gives exactly — and , so sits strictly between and , and no fraction of integers squares to give exactly either.

Key rule: The square root of a non-perfect square (like ) is always irrational. The square root of a perfect square (like ) is always rational — it is a whole number.

We do: Together decide whether , , and are rational or irrational, and justify each.

You do: Classify: , , , .

(Answers: rational; irrational (5 is not a perfect square); irrational (50 is not a perfect square); rational.)

Activity 3 — Inquiry Task: Sorting into a Venn Diagram (14 min)

Pairs.

Draw two overlapping regions labelled Rational and Irrational (they should not overlap — every number is one or the other, never both). Sort the following numbers into the diagram, writing a one-line justification beside each:

Socratic scaffolding for the tricky cases ( and ):

Prompt	Purpose
Understand: what makes different from ?	One has a growing gap of zeros between the s (never settling into a fixed repeating block); the other repeats the exact block "" forever.
Devise a plan	Check: does the decimal repeat the same fixed group of digits forever? If yes → rational. If the pattern keeps changing → irrational.
Carry out the plan	never repeats a fixed block (the run of zeros keeps growing), so it is irrational. repeats "" forever, so it is rational.
What about ?	It is written as a fraction of two integers, so by definition it is rational — even though it is a common decimal approximation for , it is not equal to .
Looking back	Check — a recurring decimal, confirming it is rational, and clearly different from , which never repeats.

Answers: Rational: , , , , , . Irrational: , , , .

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Classify as rational or irrational, with a reason: .
2. Classify as rational or irrational, with a reason: .
3. Explain why is rational even though it involves a square root symbol.
4. A student says ” is rational because is a fraction that equals it exactly.” Explain the error in this statement.

Answers: 1. Irrational — is not a perfect square, so cannot be written as a fraction of integers and its decimal never terminates or recurs. 2. Rational — it is a recurring decimal, which can always be written as a fraction of integers. 3. is a perfect square (), so , a whole number, which is rational. 4. is only an approximation to (accurate to two decimal places); ‘s true decimal expansion never terminates or recurs and is not exactly equal to any fraction of integers.

Common Misconceptions

Misconception	How to pre-empt it
”All square roots are irrational.”	Directly contrast (rational) with (irrational) — the perfect square test is the deciding factor, not the square root symbol itself.
”A decimal that looks long or messy must be irrational.”	Show has many digits but is rational, because it repeats a fixed block forever; contrast with , which looks similar but never settles into a fixed repeat.
” exactly, so is rational.”	Compare decimal expansions side by side to the 6th or 7th decimal place to show they diverge; reinforce that is a convenient rational approximation.
”Irrational numbers can’t be placed on a number line since we can’t write them exactly.”	Preview Lesson 7: irrational numbers can be estimated and placed accurately on a number line, even without an exact decimal or fraction form.
Believing every fraction with a large denominator, like , must be irrational.	Emphasise the definition: if it is written as (or can be written as) integer over integer, it is rational by definition, regardless of how “ugly” its decimal expansion looks.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Which of the following is irrational: , , , ?

Answer

is irrational, since is not a perfect square (, ). The others are all rational: , is already a fraction, and is a recurring decimal.

E2 (Kangaroo style). How many whole numbers between and (inclusive) have an irrational square root?

Answer

Perfect squares from to : — that’s numbers with a rational (whole number) square root. So numbers have an irrational square root.

E3 (Challenge). Is the sum of a rational number and an irrational number always irrational? Test with and explain your reasoning generally.

Answer

Yes, always. If is rational and is irrational, and were rational, then would be the difference of two rational numbers — which is always rational. This contradicts being irrational. So must be irrational. For example, , which never terminates or recurs.

E4 (Investigation). Is the product of two irrational numbers always irrational? Test and .

Answer

No — not always. (rational), and (rational). Sometimes two irrationals multiply to give a rational result, so “product of two irrationals” is not always irrational — a useful counterexample to remember.

Homework

1. Classify each as rational or irrational, with a one-line reason: (a) (b) (c) (d) (e) .
2. Write each recurring decimal as a fraction of integers (do not simplify further): (a) (b) .
3. A student says , so the sum of two irrational numbers must be rational. Explain what is wrong with this argument.
4. Reasoning. Explain why every integer is a rational number, using the definition .
5. Challenge. Between which two consecutive whole numbers does lie? Use this to explain why cannot be a whole number, and therefore must be irrational.

Answers: 1(a) Rational — . 1(b) Irrational — is not a perfect square. 1(c) Rational — recurring decimal (). 1(d) Irrational — the gaps of zeros keep growing, never settling into a fixed repeating block. 1(e) Rational — already a fraction of integers. 2(a) . 2(b) . 3. is a special calculation error — adding correctly gives , which is irrational, not ; the student incorrectly combined the surds. 4. Any integer can be written as , which is a ratio of two integers with a nonzero denominator, satisfying the definition of a rational number. 5. and , and , so lies between and ; since it is not exactly or , and is not a perfect square, cannot be a whole number, so it must be irrational.