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

Five lessons for Year 9 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 family mat and number cards to sort, surd snap 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. The family mat is for Day 1; cut the 18 family cards and the 12 surd snap cards before Day 1 and keep them in envelopes.
4. Check that calculators are ready for Days 3 to 5. A spreadsheet on a laptop or tablet helps on Days 3 and 5, but every task also works on paper.
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

Every number on the number line is a real number, and the real numbers come in two families. A rational number can be written as a fraction p/q of two integers, with q not zero: whole numbers, integers, fractions, decimals that stop and decimals that repeat are all rational. An irrational number cannot be written that way, and its decimal never stops and never settles into a repeating block: √2, √13 and π are examples. A repeating decimal can always be folded back into a fraction, so it is rational however long its block. A root such as √48 can be written more simply as 4√3, with the same value. The two families are mixed so finely that between any two numbers there are always more of each. A calculator shows only a rounded part of a number, so the family has to be decided from the number itself, and a real answer is given exactly or rounded to fit the job.

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

In this pack the natural numbers are the counting numbers 1, 2, 3, ..., so 0 is an integer but not a natural number. A repeating decimal is written with three dots once its block has shown at least twice, as in 0.272727...; the dot or bar over the block, met in Year 8, means the same. An exact answer keeps its root sign or π, such as 5√2 m, and a rounded answer uses ≈, such as 5√2 ≈ 7.07. Calculators stay in the drawer on Days 1 and 2 and come out 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 names the families and nests them inside one another; Day 2 proves that every repeating decimal belongs to the rational family; Day 3 shows why a calculator screen cannot settle the question on its own; Day 4 writes irrational roots in their simplest exact form; and Day 5 shows the two families mixed all along the line and puts exact and rounded answers to work. It builds on Year 8, where students met irrational numbers such as √2 and π, wrote recurring decimals and used Pythagoras’ theorem. It prepares Pythagoras and trigonometry later in Year 9, exact against rounded answers in Year 10, and every later topic where an exact answer contains a root.

Materials for the week (one trip)

• From the classroom: scissors, envelopes for the cards, pencils, calculators from Day 3, this pack printed.
• Cut-out sheet 1, one per pair: the family mat (Day 1).
• Cut-out sheet 2, one per pair: 18 family cards (Day 1) and 12 surd snap cards (Day 4).
• If you have them: laptops or tablets with a spreadsheet for Days 3 and 5 (one per pair is plenty). No other equipment to buy.

Words for the word wall

real number, rational number, irrational number, integer, natural number, fraction p/q, terminating decimal, repeating decimal, repeating block, recurring, square root, perfect square, surd, simplest form, square factor, exact value, approximation, rounded, density, interval, digital tool, spreadsheet

Dear families

This week in maths, Year 9 is learning about the real numbers: every number on the number line. They come in two families. Rational numbers can be written as a fraction, and their decimals stop or repeat; irrational numbers, such as √2 and π, cannot, and their decimals run on for ever with no repeating block. We fold repeating decimals into fractions, simplify square roots, and use calculators and spreadsheets carefully, knowing that a screen only ever shows a rounded part of a number.

Try this at home

• Ask: is 0.272727... a fraction? (Yes: 27/99, which is 3/11. Every repeating decimal is.)
• Ask: a square floor tile is 30 cm wide. Is its diagonal a whole number of centimetres? (No: it is 30√2 cm, about 42.4 cm.)
• Ask: is there a number between 0.999... and 1? (No, because 0.999... is 1.)
• Ask: does a calculator show the whole of π? (No: it shows a rounded value, such as 3.141592654. The digits of π never end.)

My real-number week

Fill one row a day. Write a number you met today, the smallest family it belongs to, and how you know.

Printed from the free seegongsik Real numbers teaching pack · seegongsik.com/au/y9/number/AC9M9N01/pack

The real number family

Every number on the number line is real. Students nest the natural numbers inside the integers and the integers inside the rationals, set the irrationals beside them, and prove a number rational by writing it as p/q.

We are learning to

• name the smallest family a real number belongs to, and every family that contains it,
• prove that a number is rational by writing it as a fraction p/q of two integers,
• judge statements about the families with a reason or a counterexample.

Success criteria

• I can place a number on the family mat and say which larger families also hold it.
• I can show that −2.75 and √0.81 are rational, and say why √30 is not.

You need

The worksheet; the family mat (cut-out sheet 1) and the 18 family cards (cut-out sheet 2), one set per pair. No calculators today.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the mat sort; start the second by filing the three demo numbers again.

Watch for these ideas

• A fraction sign does not make a number rational: √2/2 and π/4 are irrational (the top is not an integer).
• A number belongs to every family around its box: 7 is natural, an integer, rational and real.
• A root is not always irrational: √64 = 8 and √0.25 = 1/2.

Answers

• Part A: 15, √36: natural, integer, rational, real; −8, −√4: integer, rational, real; 4/9, 3.141414..., −0.35: rational, real; √30, −π, √2/2: irrational, real.
• Part B: −11 = −11/1; 1.6 = 8/5; 0.125 = 1/8; −2.75 = −11/4; √0.81 = 9/10.
• Part C: 1 true (n = n/1); 2 false (0.333... = 1/3); 3 false (−√16 = −4); 4 true (such as √2 and √30); 5 true (the two families fill the line); 6 false (irrational means not rational).
• Part D: √2 + √2 = 2√2 is irrational, but √2 + (−√2) = 0 and √2 × √2 = 2 are rational, so Ana is wrong: two irrational numbers can sum to a rational one.

The real number family

A. Which families?

Tick every family the number belongs to.

B. Prove it is rational

Write each number as a fraction p/q of two integers. An integer n can be written n/1.

C. True or false?

Circle T or F, then give a reason or a counterexample.

D. Stretch: adding two irrational numbers

Ana says, “Add two irrational numbers and the answer is always irrational.” Work out √2 + √2, √2 + (−√2) and √2 × √2, and say which family each answer belongs to. Is Ana right?

Repeating decimals are fractions

Year 8 folded simple repeating decimals such as 0.444... into fractions. Today the block can be long or start late, and one method handles them all: multiply so the tails line up, then subtract.

We are learning to

• choose the powers of ten that line up the repeating tails of a decimal,
• subtract to cancel the tail and write the decimal as a fraction in simplest form,
• explain why this shows that every repeating decimal is rational.

Success criteria

• I can turn a decimal such as 0.2181818... into a fraction and check it by dividing.
• I can explain why the method always works, so a repeating decimal is never irrational.

You need

The worksheet. No calculators until the check at the end of Part A.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the second demo; start the second by folding 0.555555... again.

Watch for these ideas

• Always multiplying by 10: for a two-digit block the tails of 10x and x do not line up.
• Forgetting to simplify: 63/99 is right but 7/11 is the simplest form.
• Thinking a long block means irrational: 0.135135... has a three-digit block and is 5/37.

Answers

• Part A: 0.888888... = 8/9; 0.636363... = 7/11; 0.135135... = 5/37; 1.818181... = 20/11.
• Part B: 0.3888888...: 100x − 10x = 35, so 90x = 35, x = 35/90 = 7/18; 0.58333333...: 1000x − 100x = 525, so 900x = 525, x = 525/900 = 7/12; 0.2181818...: 1000x − 10x = 216, so 990x = 216, x = 216/990 = 12/55.
• Part C: 0.0454545...: 1000x and 10x, so x = 1/22; 0.123123...: 1000x and x, so x = 41/333; 3.2777777...: 100x and 10x, so x = 59/18.
• Part D: x = 0.999999..., 10x = 9.999999..., 10x − x = 9, so 9x = 9, x = 9/9 = 1. So 0.999... is 1, written another way: there is no number between them, as Day 5 shows for any two different numbers.

Repeating decimals are fractions

A. The block starts straight away

Let x be the decimal. Multiply by 10, 100 or 1000 so the tails line up, subtract, and simplify. Then check one with a calculator.

B. The block starts late

You need two multipliers: one to move the point to the start of the block, one to the end of the first block.

C. Choose the multipliers

Write the two multiples of x you would subtract, then finish the fraction.

D. Stretch: is 0.999... equal to 1?

Let x = 0.999999... and use the same method. What does it give? Then answer: if 0.999... were less than 1, what number would sit between them?

Tails and digital tools

A calculator shows a rounded piece of a number, so its screen cannot tell the families apart. Students decide the family from the number itself, and follow a long division in a spreadsheet.

We are learning to

• read a calculator screen as a rounded value, not the whole number,
• decide whether a number is rational from what it is, not from how its decimal looks on a screen,
• use a spreadsheet (or a table) to find the repeating block of a fraction.

Success criteria

• I can explain why 5/17 is rational even though no block shows on the screen.
• I can set up a remainder table for a fraction and say how long its block is.

You need

The worksheet; calculators; a spreadsheet per pair if you have one (Part B also works on paper).

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the spreadsheet model; start the second with Part A.

Watch for these ideas

• Reading the last digit of the screen as exact: it has been rounded, up or down.
• Deciding from the screen: 5/17 shows no block in ten places, yet it is rational.
• Taking a screen that shows √2 × √2 = 2 as proof: the rule √a × √a = a proves it.

Answers

• Part A: 5/17 = 0.2941176471, R (a fraction; its 16-digit block does not fit); √17 = 4.1231056256, I (17 is not a square); 4/7 = 0.5714285714, R (a fraction of two integers); √2.25 = 1.5, R (√2.25 = 3/2); π/2 = 1.5707963268, I (half of π).
• Part B: remainders 2, 7, 5, 11, 6, 8, 2; digits 1, 5, 3, 8, 4, 6; so 2/13 = 0.153846153846..., a block of 6.
• Part C: 16 digits (0588235294117647), longer than the screen. Dividing by 17 leaves only the remainders 1 to 16, so a remainder repeats within 16 steps.
• Part D: 1.4142136² = 2.00000010642496, a little over 2, as 1.4142136 is √2 rounded up. A decimal that stops is rational, and no rational number squares to 2.

Tails and digital tools

A. What the screen shows

Write what your calculator shows for each number. Then circle the family, and give a reason that does not use the screen.

B. A spreadsheet long division: 2/13

Each row: multiply the remainder by 10, the digit is how many 13s fit, the new remainder is what is left. Stop when a remainder comes back. Column formulas: digit =INT(10*A1/13), next remainder =MOD(10*A1,13).

So 2/13 = 0.______________... and its block has ____ digits.

C. Too long for the screen

A calculator shows 1/17 as 0.0588235294. How long is the block of 1/17? (A spreadsheet or long division will tell you.) Why can its block be no longer than 16 digits?

D. Stretch: squaring a rounded root

1.4142136 is √2 rounded to seven decimal places. Square 1.4142136 exactly, by hand or with a calculator that shows every digit. Why is the answer not 2, and why can no decimal that stops square to exactly 2?

Simplifying surds

An irrational root such as √48 can be written exactly in a simpler form, 4√3, by pulling out the largest perfect square hiding inside it. The value does not change; the root just gets shorter.

We are learning to

• find the largest square factor of a number and use √(ab) = √a × √b to simplify a root,
• write a surd such as 3√5 back as one root to compare sizes,
• multiply roots and decide whether the product is rational or irrational.

Success criteria

• I can simplify √48 to 4√3 in one step, using the largest square factor.
• I can say why √3 × √12 is rational but √5 × √10 is not.

You need

The worksheet; the 12 surd snap cards (cut-out sheet 2), one set per pair; a list of the squares up to 15² on the board. No calculators until the check.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after surd snap; start the second by simplifying √50 again.

Watch for these ideas

• Using a square factor that is not the largest: √48 = 2√12 is true but not simplest; 12 still holds a 4.
• Adding under the root: √(9 + 16) = 5, but √9 + √16 = 7. Multiplying splits a root; adding does not.
• Thinking a simpler surd is a different number: 4√3 and √48 are the same point on the number line.

Answers

• Part A: √12 = √4 × √3 = 2√3; √20 = √4 × √5 = 2√5; √28 = √4 × √7 = 2√7; √45 = √9 × √5 = 3√5; √48 = √16 × √3 = 4√3; √75 = √25 × √3 = 5√3; √98 = √49 × √2 = 7√2; √200 = √100 × √2 = 10√2.
• Part B: 4√3 = √48, 2√11 = √44, 5√2 = √50, 3√5 = √45; smallest first 2√11 < 3√5 < 4√3 < 5√2.
• Part C: √3 × √12 = √36 = 6 (rational); √2 × √32 = √64 = 8 (rational); √5 × √10 = √50 = 5√2 (irrational); (√7)² = √49 = 7 (rational); 2√3 × 4√6 = 8√18 = 24√2 (irrational); √6 × √15 = √90 = 3√10 (irrational).
• Part D: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7, so roots do not split over addition. √a × √b is rational exactly when ab is a perfect square, such as √2 × √8 = √16 = 4.

Simplifying surds

A. Simplify

Find the largest square factor, split the root, and write the simplest form. The squares: 4, 9, 16, 25, 36, 49, 64, 81, 100.

B. One root again, then in order

Write each surd as a single root (3√5 = √9 × √5 = √45). Then put the four surds in order, smallest first.

Smallest first: ____________________________________________

C. Multiply

Multiply the roots, simplify, and circle whether the answer is rational (R) or irrational (I).

D. Stretch: does a root split over addition?

Work out √(9 + 16) and √9 + √16. Are they equal? Then find two whole numbers a and b, neither a perfect square, so that √a × √b is a whole number. When does that happen?

Density and real problems

Between any two real numbers there are always more rationals and irrationals. Students find some, pin down √13 by halving an interval, and answer real questions exactly and rounded.

We are learning to

• find a rational and an irrational number between any two numbers, and check them,
• trap a square root between two decimals by halving the interval,
• give an exact answer and a rounded answer to a real problem, and choose a sensible precision.

Success criteria

• I can find a rational and an irrational number between √5 and √6 and show they are in between.
• I can give the side of a square of area 50 m² exactly and to the nearest centimetre.

You need

The worksheet; calculators; a spreadsheet for Part B if you have one; the unit’s self-check quiz 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

• Thinking two close decimals have nothing between them: 1.705 sits between 1.7 and 1.71.
• Rounding too early: round only the final answer, to a precision the job can use.
• Giving only a rounded answer: 5√2 m is exact; 7.07 m is not.

Answers

• Part A, for example (check by squaring): 1.4 to 1.5: 1.45, √2.1; √5 to √6: 2.3, √5.5; 3.14 to π: 3.141, π − 0.001; 1/3 to 0.34: 0.335, √0.113.
• Part B: middles (squared) 3.5 (12.2500), 3.75 (14.0625), 3.625 (13.1406), 3.5625 (12.6914), 3.59375 (12.9150), 3.609375 (13.0276), 3.6015625 (12.9713), 3.60546875 (12.9994); both ends of the last interval round to 3.61, so √13 ≈ 3.61.
• Part C: 1 √(30² + 30²) = √1800 = 30√2 cm ≈ 42.4 cm (I); 2 √50 = 5√2 m ≈ 7.07 m (I); 3 √81 = 9 cm (R, nothing to round); 4 r² = 4/π, so r = √(4/π) m ≈ 1.13 m (I); 5 √(3² + 1²) = √10 m ≈ 3.16 m (I).
• Part D: no. Halfway between Ravi’s number and 1 is a smaller number still above 1, so there is no smallest one.

Density and real problems

A. Find one of each between

Write a rational and an irrational number between each pair, and show how you checked.

B. Halve the interval for √13

Square the middle. If the square is less than 13, the low end moves up to the middle; if not, the high end moves down. Stop when both ends round to the same two decimal places.

So √13 ≈ ______ to two decimal places.

C. Real problems: exact and rounded

Give an exact answer, then a rounded one that suits the job. Say whether the exact answer is rational or irrational.

D. Stretch: the smallest number above 1

Ravi says, “The smallest number bigger than 1 is 1.000...0001, with a lot of zeros.” Is there a smallest number bigger than 1? Explain.

The real number family mat

For Day 1, one mat per pair. Place each family card in the smallest box it belongs to: a card in the natural numbers box is also an integer, rational and real.

Family cards and surd snap cards

Family cards (Day 1): 18 cards

Surd snap (Day 4): 12 cards

Lay the cards face down. Turn two over; keep them if one is a root and the other is its simplest form.

Teacher note: the keys are with the mini-check answers, so this sheet can go to pairs whole.

What we know: Real numbers

Work on your own. Use a calculator only for Questions 6 to 8. Show your working.

1. Name the smallest family each number belongs to:   −√49,   5/12,   √12,   0.222...   ______________________
2. Write 0.474747... as a fraction in simplest form. Show the step that cancels the tail. ______________________
3. Write 0.2333333... as a fraction in simplest form. ______________________
4. Simplify √72 and √300. ______________________
5. Is √6 × √24 rational or irrational? Show why. ______________________
6. A calculator shows 2/17 as 0.1176470588, with no block in sight. Is 2/17 rational? Explain. ______________________
7. Circle every number that lies between √3 and 2:   1.7,   √3.5,   1.8,   √2.9,   7/4,   √4.1. Which of the numbers you circled is irrational? ______________
8. A square window has an area of 2.5 m². Give its side exactly, and to the nearest centimetre. ______________________

Answers and marking guide

Answers with working

1. −√49: integer (so also rational, real); 5/12: rational (so also real); √12: irrational (so also real); 0.222...: rational (so also real).
2. x = 0.474747...; 100x = 47.474747...; 100x − x = 47, so 99x = 47; x = 47/99.
3. x = 0.2333333...; 10x = 2.333333...; 100x = 23.333333...; 100x − 10x = 21, so 90x = 21; x = 21/90 = 7/30.
4. √72 = √36 × √2 = 6√2; √300 = √100 × √3 = 10√3.
5. √6 × √24 = √144 = 12, which is rational: the product is a perfect square under one root.
6. Yes: 2/17 is a fraction of two integers, so it is rational. Its block is 16 digits long, longer than the screen, which is why no repeat shows.
7. √3 ≈ 1.732, so the numbers between √3 and 2 are √3.5 (1.871), 1.8 (1.800), 7/4 (1.750); the irrational one is √3.5.
8. Side = √2.5 m ≈ 1.58 m (check: 1.58² = 2.4964, just under 2.5).

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. Fraction check for Questions 2 and 3: 47/99 and 7/30.

Card keys

Family cards (Day 1): natural 7, √64, 20/4; integer −5, 0, −√36, −18/6; rational 3/8, −0.6, 0.454545..., √0.25, 22/7; irrational √11, π, −√5, 0.1010010001..., π/4, 2 + √3. Surd snap (Day 4): √24 = 2√6, √32 = 4√2, √63 = 3√7, √80 = 4√5, √108 = 6√3, √150 = 5√6.

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: the families, folding repeating decimals, reading screens, simplifying surds, and density with real answers.