Irrational numbers: a week of ready-to-teach maths

Five lessons for Year 8 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus a square-of-area-2 kit to cut out, rational-or-irrational sort cards, number line-up cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair, and cut them before Day 1: the sort cards are for Day 1, the square-of-area-2 kit for Day 2 and the line-up cards for Day 4.
4. Collect string and round things for Day 3: cans, plates, lids, rolls of tape, a bike wheel if one is handy.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

A rational number is one that can be written as a fraction of two integers, such as 3/4, −5 or 0.333..., which is 1/3. As a decimal it either stops or settles into a block that repeats for ever. An irrational number cannot be written that way, and its decimal never stops and never repeats. Irrational numbers are not rare or strange. The side of a square of area 2 is √2, about 1.414, and the distance round any circle is π times its diameter, with π about 3.14159. The square root of a whole number is rational only when the number is a perfect square. Every irrational number still has an exact place on the number line, which closer and closer decimals pin down, and in real jobs we round it to a sensible length.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these numbers

A decimal that repeats is written with three dots after the block has shown at least twice, as in 0.454545...; the dot notation for recurring decimals comes later, in the terminating and recurring decimals unit. We write an exact answer with the root sign or π, such as √10 m or 8π cm, and a rounded answer with ≈, such as √10 ≈ 3.16. Calculators stay in the drawer on Day 1, come out for squaring decimals on Day 2 and are used freely from Day 3, always after an estimate.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 splits the decimals into those that stop or repeat and those that do neither; Day 2 meets the first irrational length, the side of a square of area 2; Day 3 finds π in every circle the class can measure; Day 4 gives each irrational number its place on the number line; and Day 5 puts exact and rounded answers to work on real jobs. It builds on Year 7, where students found square roots of perfect squares, trapped other roots between two whole numbers and wrote fractions as decimals. It prepares terminating and recurring decimals later in Year 8, the circumference and area of a circle, and Pythagoras’ theorem, whose answers are so often square roots.

Materials for the week (one trip)

• From the classroom: scissors, pencils, rulers, a pair of compasses each (Day 4 stretch), calculators from Day 2, this pack printed.
• Cut-out sheet 1, one per pair: the square-of-area-2 kit and its measuring strip (Day 2).
• Cut-out sheet 2, one per pair: 24 rational-or-irrational sort cards (Day 1) and 12 number line-up cards (Day 4).
• For Day 3: string or wool (about 2 m per pair), a tape measure for the big objects, and six or more round objects of different sizes from the classroom or home. No other equipment to buy.

Words for the word wall

rational number, irrational number, integer, fraction, terminating decimal, repeating decimal, repeating block, square root, perfect square, root sign, pi, circumference, diameter, radius, ratio, approximation, exact value, rounded value, estimate, interval, number line

Dear families

This week in maths, Year 8 is learning about irrational numbers: numbers that cannot be written as a fraction, whose decimals never stop and never repeat. Two famous ones are √2, the side of a square of area 2, and π, the number of diameters that fit round any circle. We measure real objects, compare exact answers such as √10 m with rounded ones such as 3.16 m, and learn when each kind of answer is the right one.

Try this at home

• Wrap a piece of string once round a mug or a tin, then see how many times the string fits across the top. (A little more than 3 times, however big the circle.)
• Ask: does the decimal for 1/3 ever stop? Does the one for 1/4? (No; yes, 0.25.)
• Ask: a square rug has an area of 2 m². Is each side more or less than 1.5 m? (Less: 1.5 × 1.5 = 2.25, so the side is about 1.41 m.)
• Ask: is 22/7 the same number as π? (No: 22/7 = 3.142857..., a little more than π.)

My irrational week

Fill one row a day. Find a circle or a square in something real, measure it, and write an exact and a rounded answer.

Printed from the free seegongsik Irrational numbers teaching pack · seegongsik.com/au/y8/number/AC9M8N01/pack

Two kinds of decimal

A rational number is a fraction of two integers; its decimal stops or repeats. Students divide fractions out, let the remainders decide which, and sort numbers with a reason.

We are learning to

• divide a fraction out and say whether its decimal stops or repeats,
• write a decimal that stops or repeats as a fraction of two integers,
• sort numbers into rational and irrational, giving a reason every time.

Success criteria

• I can divide a fraction out and explain from the remainders why its decimal stops or repeats.
• I can say why √16 and 22/7 are rational but √7 and π are not.

You need

The worksheet; the sort cards (cut-out sheet 2), one set per pair. No calculators today.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the card sort; start the second by dividing 1/6 out again.

Watch for these ideas

• A long decimal is not always irrational: 1/7 = 0.142857142857... repeats.
• 22/7 and 3.14 are rational numbers close to π, not π.
• A square root is not always irrational: √16 = 4 and √0.25 = 0.5.

Answers

• Part A: 3/8 = 0.375 (stops); 2/3 = 0.666666... (repeats); 5/11 = 0.454545... (repeats); 7/20 = 0.35 (stops); 9/16 = 0.5625 (stops); 4/9 = 0.444444... (repeats).
• Part B: 0.7 = 7/10; 0.25 = 1/4; 1.6 = 8/5; 0.333333... = 1/3; 1.25 = 5/4.
• Part C: rational √16 (= 4), 22/7 (a fraction), 0.121212... (repeats: 4/33), 3.14 (stops: 157/50), −5 (= −5/1), √0.25 (= 0.5); irrational √7 (not a perfect square), π (never repeats), √20 (not a perfect square), 0.1010010001... (no block repeats).
• Part D: remainders 3, 2, 6, 4, 5, 1, back to the 1 you started with, so 142857 repeats. Dividing by 7 leaves only 1 to 6 (0 would stop it), so within 6 steps a remainder must come back.

Two kinds of decimal

A. Divide it out

Divide by short division. Write the decimal, then circle whether it stops or repeats.

B. Fractions in disguise

Write each decimal as a fraction of two integers, in simplest form. 0.333... is one you met in Year 7.

C. Rational or irrational?

Circle R (rational) or I (irrational), then give your reason.

D. Stretch: why a fraction has to repeat or stop

Divide 1 by 7 by short division, writing down each remainder as you go. Which remainders can turn up when you divide by 7? Why must the digits start repeating within six steps?

A square of area 2

The side of a square of area 2 is √2, the first irrational length. Students cut it from a 2 by 2 square, find the areas of tilted squares, sort roots into rational and irrational, and trap √2.

We are learning to

• find the area of a tilted square on a grid and write its side as a square root,
• explain why the root of a whole number is rational only for a perfect square,
• trap a square root between two decimals by squaring them.

Success criteria

• I can draw a square of area 2 on a grid and explain why its side is √2.
• I can say whether √40 is rational and which two whole numbers it lies between.

You need

The kit (cut-out sheet 1) and scissors, one per pair; the worksheet; calculators for Part C.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the tilted squares; start the second with Part B.

Watch for these ideas

• Halving instead of rooting: the side of a square of area 8 is √8, about 2.83, not 4.
• Thinking 1.41 or 1.414 is √2: squaring gives 1.9881 and 1.999396, close to 2 but not 2.
• Thinking the calculator shows all of √2: the screen runs out, not the decimal.

Answers

• Part A: A: 4 − 4 × 0.5 = 2, side √2; B: 9 − 4 × 1 = 5, side √5; C: 16 − 4 × 2 = 8, side √8; D: 16 − 4 × 1.5 = 10, side √10. Each tilted area is a² + b², where a and b are the steps along and up.
• Part B: rational √36 = 6, √64 = 8, √81 = 9, √1 = 1; irrational √35 (5 to 6), √12 (3 to 4), √90 (9 to 10).
• Part C: 1.4² = 1.96, 1.5² = 2.25, 1.41² = 1.9881, 1.42² = 2.0164, 1.414² = 1.999396, 1.415² = 2.002225. √2 lies between 1.414 and 1.415, so √2 ≈ 1.41 to two decimal places.
• Part D: 7/5, 17/12, 41/29 square to 1.9600, 2.0069, 1.9988; p² against 2q²: 49 and 50, 289 and 288, 1681 and 1682. Next: 99/70 (9801 against 9800). Always 1 apart: close to 2, never 2. No fraction squares to exactly 2; that is what irrational means.

A square of area 2

A. Tilted squares

For each square: the area of the big grid square, take away the four shaded corner triangles. Then write the side of the tilted square as a square root.

B. Rational or irrational root?

Write the exact value of each rational root. For the others, write the two whole numbers the root lies between.

C. Trapping √2 (calculator)

√2 lies between ______ and ______. To two decimal places, √2 ≈ ______.

D. Stretch: fractions that nearly square to 2

Square 7/5, 17/12, 41/29 as fractions. For each p/q, compare p² with 2q². Each next fraction is (p + 2q)/(p + q): find the one after 41/29. Can any fraction square to exactly 2?

π in every circle

Circumference ÷ diameter is the same number, π, for every circle. Students measure round objects, pool the class results and compare the fractions people use for π with π itself.

We are learning to

• measure round objects and find circumference ÷ diameter,
• explain why every circle gives about 3.14 and why no measurement can give π exactly,
• write a circumference exactly in terms of π, then rounded.

Success criteria

• I can measure a circle, divide, and explain why my answer is close to π but not equal to it.
• I can say that a circle of diameter 6 cm has circumference 6π cm, about 18.85 cm.

You need

String and a ruler per pair, a tape measure, six or more round objects, the worksheet, calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the class table; start the second with Part B.

Watch for these ideas

• Thinking π changes with the size of the circle: big and small objects give the same ratio, and only the measuring error changes.
• Measuring the radius instead of the diameter, which doubles the ratio to about 6.3.
• Thinking 22/7 or 3.14 is exact: 22/7 = 3.142857..., which repeats, so it is rational; π is not.

Answers

• Part A (example class, can to wheel): 3.15, 3.15, 3.13, 3.15, 3.16, 3.13; mean ≈ 3.15.
• Part B: 3 is off by about 0.14; 3.14 is off by about 0.0016; 22/7 is off by about 0.0013; 355/113 is off by about 0.00000027. Closest: 355/113. All four are fractions, so none is π.
• Part C: diameter 10 cm: C = 10π cm ≈ 31.42 cm; diameter 2.5 m: C = 2.5π m ≈ 7.85 m; radius 4 cm: C = 8π cm ≈ 25.13 cm.
• Part D: the can gives 3.04 to 3.28; the wheel moves by only about 0.02, so big objects measure π better. A measurement shows the ratio is near 3.14, never that its decimal never repeats: that is proved.

π in every circle

A. Circumference ÷ diameter

Work out C ÷ d to two decimal places for the example class, then measure three objects of your own.

Mean of the example class’s six ratios: ______

Why are they not all the same? ______________________________________________

B. How good are the stand-ins for π?

π = 3.14159265... Use a calculator to write each as a decimal and find how far it is from π.

C. Exact, then rounded

C = πd. Write each circumference exactly in terms of π, then to two decimal places.

D. Stretch: can a measurement prove π is irrational?

The drink can was measured as C = 20.5 cm and d = 6.5 cm. Work out C ÷ d if the diameter was really 6.25 cm or 6.75 cm. What does this say about using measurements to find π? Could any measurement show that the decimal for π never repeats?

Irrationals on the number line

Every irrational number has one exact place on the number line. Students estimate roots and π to two decimal places, mark them, order mixed lists and line up a set of cards.

We are learning to

• estimate a square root to two decimal places by squaring, and mark it on a number line,
• order a list that mixes fractions, decimals, square roots and π,
• find a rational number between two irrationals, and an irrational between two rationals.

Success criteria

• I can mark a root such as √7 in the right tenth of a number line and say how I knew.
• I can order a list such as 2.2, √5 and 9/4, showing a decimal for each number.

You need

The worksheet; the line-up cards (cut-out sheet 2), one set per group of three or four; calculators; compasses for Part D.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the line-up; start the second with Part A.

Watch for these ideas

• Ordering by the number under the root sign: √10 is about 3.16, not 10.
• Treating 22/7 as π when ordering: 22/7 = 3.1428... is a little bigger than π = 3.1415...
• Marking √8 at 2.8: 2.8² = 7.84, so √8 sits just past 2.8, at about 2.83.

Answers

• Part A: √3 ≈ 1.73 (irrational); 5/2 = 2.5 (rational); √8 ≈ 2.83 (irrational); π ≈ 3.14 (irrational); 1.5 (rational); √5 ≈ 2.24 (irrational).
• Part B: 1.7 < 1.73 < √3 < 7/4; π < 22/7 < √10 < 3.2.
• Part C, one answer each (accept any that checks): 1.5 (1.5² = 2.25); π or √10 (9 < 10 < 16); 3.142 (π = 3.14159..., 22/7 = 3.14285...).
• Part D: the arc lands at about 1.41 (√2) and, for the 2 by 1 rectangle, at about 2.24 (√5).
• Line-up cards: √2, 1.5, √3, 7/4, √5, 2.25, 5/2, √8, 2.9, π, 22/7, √10.

Irrationals on the number line

A. Estimate and mark

Estimate each number to two decimal places (square your estimate to check). Then mark it on the line with a dot and its name.

B. In order

Write a decimal under each number (three decimal places), then write the list from smallest to largest.

C. Find one between

Give one example each, and show how you know it fits.

1. A rational number between √2 and √3: ________   because ______________________

2. An irrational number between 3 and 4: ________   because ______________________

3. A rational number between π and 22/7: ________   because ______________________

D. Stretch: build √2 and √5 with compasses

Draw a number line. Build a 1 by 1 square on it from 0 to 1, put the compass point on 0 and swing the diagonal down onto the line. Where does it land? Do the same with a 2 by 1 rectangle to find √5.

Measuring a real garden

Real lengths often come out irrational. Students give exact answers, round them the way the job needs (nearest centimetre to build, up to the whole metre to buy) and play the self-check quiz as a class game.

We are learning to

• give an exact answer with a root sign or π, and a rounded answer to a sensible accuracy,
• decide whether a job needs an exact value, a rounded value or a value rounded up,
• round only at the end, and check a rounded side by squaring it.

Success criteria

• I can give the side of a 40 m² square bed exactly and rounded, and the edging to buy.
• I can say why a carpenter rounds to the millimetre but edging is bought in whole metres.

You need

The worksheet, one per student; calculators; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Rounding in the middle: 4 × 3.16 = 12.64, but 4 × √10 = 12.65, so round at the end.
• Rounding down when buying: the edging ends up short.
• Thinking a rounded answer is wrong, or that an exact one is useless: the job decides.

Answers

• Part A (side, perimeter, buy): 20 m²: √20 ≈ 4.47 m, 17.89 m, 18 m; 30 m²: √30 ≈ 5.48 m, 21.91 m, 22 m; 45 m²: √45 ≈ 6.71 m, 26.83 m, 27 m; 50 m²: √50 ≈ 7.07 m, 28.28 m, 29 m.
• Part B (C, buy): round table top: 1.2π ≈ 3.77 m, 4 m; round pond: 3π ≈ 9.42 m, 10 m; circular garden bed: 2.4π ≈ 7.54 m, 8 m.
• Part C: 1. exact: 5π cm (any decimal would be rounded); 2. nearest millimetre (a saw cuts to about a millimetre); 3. rounded up to whole metres (rounding down leaves a gap); 4. roughly, to about 100 m (nobody walks an exact circle).
• Part D: the square on the diagonal has twice the gate’s area, 2 × 1.2² = 2.88 m², so the brace is √2.88 m ≈ 1.697 m: cut it at 1697 mm.

Measuring a real garden

A. Square garden beds

Edging is sold by the whole metre. Round only at the end.

B. Round things to edge

C. What does the job need?

For each job write: exact, nearest millimetre, rounded up, or roughly. Say why in a few words.

D. Stretch: the gate brace

A square gate is 1.2 m by 1.2 m. Its diagonal brace is the side of the square drawn on the diagonal (think of Day 2’s tilted squares). Find the area of that square, then the length of the brace, exact and to the nearest millimetre.

The square-of-area-2 kit

For Day 2, one kit per pair

Cut out each grid square along its solid outside edge, then cut along the dashed tilted square inside it. From kit square 1, the four shaded corners fit together, right angles meeting in the middle, to make a second square just like the tilted one. Lay the strip along a side of each tilted square to measure it in grid units.

Measuring strip (cut it out too)

Teacher note: kit square 1 has area 4, its tilted square 2 (side √2, about 1.41 units); kit square 2 has area 9, its tilted square 5 (side √5, about 2.24 units). The strip is printed at the kit’s own scale, so the readings hold however the sheet is scaled.

Rational-or-irrational sort and line-up cards

Rational or irrational? (Day 1): 24 cards

Number line-up (Day 4): 12 cards

Groups lay the twelve cards in a line, smallest on the left, writing a decimal on the back of each card.

Teacher note: the rational cards are 3/4, √25, 0.333..., −2, 22/7, 0.75, √0.49, 3.14, 0, 0.121212..., 1.5, −0.6, 2/9; the irrational cards are √2, 0.1010010001..., π, √10, √0.5, 2π, √7, √12, √50, 1 + √2, √3. The line-up runs √2, 1.5, √3, 7/4, √5, 2.25, 5/2, √8, 2.9, π, 22/7, √10.

What we know: Irrational numbers

Work on your own. Use a calculator only for Questions 4 to 7. Show your working.

1. Write 7/8 and 5/12 as decimals. Which one stops and which one repeats? ______________________
2. Circle the irrational numbers, then give a reason for one of them:   √49     √50     0.777...     π     3.14159     2/3
3. Show that 0.45 and −2.5 are rational by writing each as a fraction of two integers. ______________________
4. Between which two whole numbers is √60? Square decimals to give √60 to one decimal place. ______________________
5. A square courtyard has an area of 18 m². Give its side exactly, and to the nearest centimetre. ______________________
6. A round garden bed has a diameter of 4 m. Give its circumference exactly and to two decimal places. How many whole metres of edging should be bought? ______________________
7. Put in order, smallest first:   3.14,   √10,   22/7,   π,   3.1   ______________________
8. Ella says, “22/7 is π, because both start 3.14.” Is Ella right? Explain. ______________________

Answers and marking guide

Answers with working

1. 7/8 = 0.875, which stops; 5/12 = 0.41666666..., which repeats.
2. Irrational: √50 (50 is not a perfect square), π (its decimal never stops or repeats). Rational: √49 (√49 = 7), 0.777... (it repeats: 7/9), 3.14159 (it stops, so it is 314159/100000), 2/3 (a fraction of two integers).
3. 0.45 = 9/20; −2.5 = −5/2.
4. 7² = 49 < 60 < 64 = 8², so √60 is between 7 and 8; 7.7² = 59.29, 7.8² = 60.84, 7.75² = 60.0625, so √60 ≈ 7.7.
5. Side = √18 m ≈ 4.24 m (check: 4.24² = 17.9776).
6. C = 4π m ≈ 12.57 m; buy 13 m, rounding up so the edging reaches all the way round.
7. 3.1 < 3.14 < π < 22/7 < √10 (3.1, 3.14, 3.1416, 3.1429, 3.1623).
8. No: 22/7 = 3.142857142857... repeats, so it is rational, while π = 3.14159... never stops or repeats. They differ from the third decimal place, by about 0.0013.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, and gives an exact and a rounded answer where a question asks for both.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: rational or irrational, square roots, π in circles, the number line, and exact and rounded answers.