009i - Problem Solving with Irrational Numbers

Lesson 9 — Problem Solving and Consolidation: Irrational Numbers

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

Learning Intentions

• To consolidate understanding of rational and irrational numbers, square root estimation, and .
• To apply this understanding to solve non-routine, applied problems.

Success Criteria

I can:

1. Classify numbers as rational or irrational, with justification.
2. Estimate and compare irrational numbers without a calculator.
3. Apply to solve applied measurement problems.
4. Justify my solution and check it is reasonable.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. The square root of a whole number is irrational.
2. multiplied by a rational number gives an irrational result.
3. Two different irrational numbers can add to give a rational number.
4. An irrational number lies strictly between two consecutive integers.

Answers: 1. Sometimes — true for non-perfect squares (e.g. ), false for perfect squares (e.g. ). 2. Sometimes — true unless the rational number is , since , which is rational. 3. Sometimes — e.g. and (later) or, at this level, ; but generally two “unrelated” irrationals like stay irrational. 4. Always, for irrational numbers that are not themselves integers (all irrationals are non-integers by definition), since every real number lies between two consecutive integers, and an irrational number can never equal an integer.

Activities

Activity 1 — Mixed Consolidation Quick-fire (10 min)

Explicit instruction, then practice.

Remind students of the full toolkit from Lessons 6–8: the rational/irrational test, the bracketing-and-refining method for square roots, and the circle formulas , .

I do:

Classify , then estimate it to one decimal place.

is closer to than to (; ), so is irrational (75 is not a perfect square) and is estimated as (check: ).

You do:

1. Classify and estimate to one decimal place.
2. Find the circumference of a circle with radius cm, exact and rounded to dp.
3. Classify: , , .

(Answers: 1. Irrational, . 2. cm. 3. Rational; Rational (); Irrational.)

Activity 2 — Applied Problems (23 min)

Pairs. Every answer must carry a one-sentence justification.

Problem 1. A square courtyard has area . Estimate its side length to one decimal place, and classify whether this side length is rational or irrational.

Problem 2. A circular fountain has radius m and sits inside a square plaza of side m (the fountain touches all four sides of the square). Find the area of the plaza not covered by the fountain, to decimal place.

Problem 3. Order the following from smallest to largest without a calculator: , , , .

Problem 4 (harder — multi-step, applied). A circular garden path runs around the outside of a circular flower bed. The flower bed has radius m, and the path is m wide (so the outer edge of the path has radius m). A gardener wants to know:

(a) The exact area of just the path (not including the flower bed), in terms of .

(b) Whether the numerical value of this area (rounded to decimal place) is rational or irrational, and why this is always true for a ring-shaped (“annulus”) path area of this kind, regardless of the radii chosen.

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand the problem	The path’s area is a ring — the region between two circles, not either circle’s full area.
Devise a plan	Find the area of the larger circle, subtract the area of the smaller (flower bed) circle.
Carry out the plan	.
Carry out — evaluate	(to dp).
Is the rounded decimal rational or irrational?	The rounded decimal () is itself a terminating decimal, so technically rational — but this is only an approximation. The exact value, , is irrational, since is irrational and is a nonzero rational multiplied by it.
Looking back — does this generalise?	For any two radii , . Since is always a nonzero rational number (a difference of whole-number squares) for whole-number radii, and a nonzero rational times is always irrational, the exact path area is always irrational — only its rounded decimal approximation looks rational.

Answers: Problem 1 — m; irrational, since is not a perfect square. Problem 2 — plaza area ; fountain area ; uncovered . Problem 3 — bracketing gives , , , — order smallest to largest: . Problem 4 — (a) exactly; (b) exact value is irrational; rounded decimal only appears rational because rounding always produces a terminating decimal.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Classify as rational or irrational, and estimate it to one decimal place.
2. A circular rug has diameter m and sits inside a square room of side m. Find the uncovered floor area, to decimal place.
3. Order from smallest to largest: , , , .
4. Reasoning. Explain why is irrational, even though itself is rational.

Answers: 1. Irrational; . 2. Square area ; rug area ; uncovered . 3. , , , → order: . 4. If were rational, then would be the quotient of two rational numbers and therefore rational — but is known to be irrational, a contradiction; so must be irrational.

Common Misconceptions

Misconception	How to pre-empt it
Ordering irrational numbers by “eyeballing” the numbers under the root, rather than estimating their values.	Always require a bracketing/refining estimate before comparing or ordering irrational numbers.
Believing the rounded decimal of an irrational expression (e.g. ) proves the exact value is rational.	Distinguish clearly between the exact symbolic value and a rounded numerical approximation — only the exact value determines rationality.
Subtracting radii directly instead of subtracting areas in “ring” (annulus) problems, e.g. computing instead of .	Insist on finding each circle’s area separately first, then subtracting — never subtract inside the square/radius step.
Assuming all multiples of are “special” irrational numbers unrelated to ordinary estimation methods.	Show that for rational, nonzero can be estimated the same way as any other irrational value, using .
Treating “always/sometimes/never” statements about irrational numbers as always resolvable by a single example.	Model that a counterexample is enough to disprove “always,” but proving “always” requires a general argument, not just one example (see Warmup Q3 and Q4 discussion).

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Which is greater: , or ?

Answer

. . So is greater — confirming (as in Lesson 8 of Year 7 material) that the square root of a sum is never the sum of the square roots.

E2 (Kangaroo style). A circular pizza of radius has the same area as two smaller circular pizzas, each of radius . Verify this using the area formula.

Answer

Each small pizza has area . Two of them together: , exactly matching the large pizza’s area. ✓

E3 (Challenge). Is rational or irrational? Explain using a proof by contradiction similar to the CFU Q4 reasoning.

Answer

It is irrational. If were rational, then would be a rational number divided by an irrational number () — and such a quotient is always irrational (unless the rational numerator is ), which would make irrational anyway — a contradiction only avoided if the product itself is irrational. In short: a nonzero product of two irrational “unrelated” numbers like and is (in this case) irrational.

E4 (Investigation). A circle’s area is numerically equal to its circumference. Find its radius.

Answer

Since , . Check: area , circumference ✓ — a rational radius produces this special case, even though both area and circumference are individually irrational.

Homework

1. Classify and estimate to one decimal place: (a) (b) .
2. A circular pond of radius m sits in a square garden of side m. Find the uncovered area, to decimal place.
3. Order from smallest to largest: , , , .
4. A ring-shaped path has an inner radius of m and outer radius of m. Find its exact area in terms of , then rounded to decimal place.
5. Reasoning. Explain why the exact area of any ring-shaped path with whole-number radii is always irrational, but its rounded decimal approximation is always rational.
6. Challenge. Two circles have radii in the ratio . Their areas differ by . Find both radii.

Answers: 1(a) Irrational, . (b) Irrational, . 2. Square ; pond ; uncovered . 3. , , , → order: . 4. exactly . 5. The exact area is , a nonzero rational number () multiplied by the irrational , which is always irrational; but once rounded to a fixed number of decimal places, the result is by definition a terminating (hence rational) decimal — rounding always “manufactures” an apparent rational number, hiding the true irrational value. 6. Let radii be and . . Radii are cm and cm.