Operations with rational 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 quarter-marked number line, order cards, a domino loop, 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 number line and order cards are for Day 1, the dominoes for Day 5.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

Integers, fractions and decimals, positive and negative, are all rational numbers, and they all live on one number line. The same four operations work on all of them. Adding moves right and subtracting moves left; subtracting a negative undoes a move left, so it moves right. When multiplying or dividing, the same signs give a positive answer and different signs a negative one. Fractions add over a common denominator, multiply across, and divide by multiplying by the reciprocal. Brackets come first, then multiplication and division, then addition and subtraction. Choosing a tidy order, such as −5 × 2 before × 17, is part of the skill.

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 negatives

We print negatives with a true minus sign and put brackets round a negative that follows an operation sign: 5 + (−2), −7 × (−3). Say “negative 2” for the number and “minus” for the operation. On a calculator the negative key and the subtract key are different keys: Day 5 shows why (−3)² and −3² are different.

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 puts every number on one line; Day 2 adds and subtracts by moving along it; Day 3 builds the sign rules from patterns rather than handing them out; Day 4 carries all four operations over to fractions and decimals; and Day 5 puts them together in the right order. It builds on Year 7, where students added and subtracted integers and worked with positive fractions and decimals. It prepares algebra with negative coefficients in Years 8 and 9 and solving equations whose answers are negative or fractional.

Materials for the week (one trip)

• From the classroom: scissors, pencils, glue or tape, this pack printed.
• Cut-out sheet 1, one per pair: a number line from −4 to 4 in quarters and 12 order cards (Day 1).
• Cut-out sheet 2, one per class or group: 11 dominoes for the loop game (Day 5).
• Calculators for checking on Days 4 and 5. No other equipment to buy.

Words for the word wall

integer, rational number, positive, negative, opposite, number line, sum, difference, product, quotient, sign rule, reciprocal, common denominator, mixed number, order of operations, brackets, efficient strategy, estimate, check

Dear families

This week in maths, Year 8 is using all four operations with negative numbers, fractions and decimals. Some surprises: subtracting a negative number adds (5 − (−3) = 8), and multiplying two negatives gives a positive ((−4) × (−6) = 24). We also practise doing operations in the right order and choosing a quick way, such as working out a quarter of 36 instead of 0.25 × 36 the long way.

Try this at home

• Look up tonight’s forecast minimum and tomorrow’s maximum in a cold town. How many degrees does the temperature rise?
• Ask: what is −8 + 3? (−5: start at −8 and move 3 to the right.)
• Ask: what is (−2) × (−5)? (10: two negatives make a positive.)
• Ask: what is half of a quarter, as a fraction? (1/2 × 1/4 = 1/8.)

My numbers this week

Fill one row a day. Find a negative number or a fraction in real life (temperatures, bank balances, recipes, sport scores) and do one calculation with it.

Printed from the free seegongsik Operations with rational numbers teaching pack · seegongsik.com/au/y8/number/AC9M8N04/pack

One number line for every number

Integers, fractions and decimals, positive and negative, are all rational and share one line. Students place them, order mixed lists and write each number as a fraction of two integers.

We are learning to

• place positive and negative fractions and decimals on a number line,
• order a list that mixes integers, fractions and decimals, using a decimal for each,
• write any integer, terminating decimal or mixed number as a fraction of two integers.

Success criteria

• I can explain why −3/4 is to the left of −2/3 on the number line.
• I can show that −2 and 1.6 are rational by writing them as fractions.

You need

The worksheet; the number line and order cards (cut-out sheet 1), one set per pair.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Ordering negatives by size of digits: −3/4 = −0.75 is less than −0.7.
• Thinking only whole numbers can be negative: −1/4 sits between −1 and 0.
• Thinking integers are not fractions: −3 = −3/1, so every integer is rational.

Answers

• Part A: −5/2 at −2.50; 3/4 at 0.75; −1.25 at −1.25; 7/3 at 2.33; −0.5 at −0.50; −3 1/2 at −3.50.
• Part B: −1 < −3/4 < −0.7 < −2/3 < 0.5; −1.25 < −6/5 < 1 1/8 < 1.2 < 5/4.
• Part C: −3 = −3/1; 0.25 = 1/4; −1.5 = −3/2; 2 1/3 = 7/3; −0.04 = −1/25.
• Part D: for example −0.45 (or −9/20); there are infinitely many, because between any two rationals their average is another.
• Order cards: −2.5, −7/4, −1.2, −2/3, −0.5, −1/4, 0, 0.3, 3/8, 0.75, 4/3, 1.5.

One number line for every number

A. Place them

Mark each number on the line with a dot and its label. The small marks are quarters.

B. In order, smallest first

Write a decimal under each number first.

C. As a fraction of two integers

D. Stretch: always another one between

Find a rational number between −0.5 and −0.4. Then find one between your number and −0.4. Could you keep going for ever? Explain.

Adding and subtracting as movement

Adding moves right and subtracting moves left, and subtracting a negative undoes a move left, so it moves right. Students add and subtract integers on the line, then use them in temperature and money problems.

We are learning to

• add and subtract integers by moving along a number line, including across 0,
• explain why subtracting a negative number gives the same result as adding its opposite,
• write and solve temperature and money problems with negative numbers.

Success criteria

• I can work out −3 + 5 and 2 − (−4) and show each as a move.
• I can say how many degrees warmer 3 °C is than −5 °C.

You need

The worksheet; the cut-out number line from Day 1; a coloured pencil for arrows.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the debt story; start the second with Part A.

Watch for these ideas

• Moving the wrong way for a negative: −6 + (−6) = −12, not 0.
• Treating every − sign as subtraction: 12 − (−3) = 15.
• Finding a temperature difference by subtracting the wrong way, or getting a negative difference.

Answers

• Part A: −7 + 4 = −3; 3 − 9 = −6; −5 − (−8) = 3; −6 + (−6) = −12; 12 − (−3) = 15; −10 + 15 = 5; −4 − 6 = −10; 0 − (−7) = 7.
• Part B: each subtraction of a negative rewrites as adding its opposite, for example −5 − (−8) = −5 + 8.
• Part C: −6 + 9 = 3 °C; 5 − 11 = −6 °C; −15 + 40 = $25; 4 − (−9) = 13 degrees.
• Part D: every line adds to 0: 1, −4, 3 / 2, 0, −2 / −3, 4, −1.

Adding and subtracting as movement

A. Move and land

Work each out. Sketch the move on your number line when it crosses 0.

B. Rewrite as an addition

Rewrite every subtraction from Part A as an addition of the opposite. Check you get the same answer.

C. Temperatures and money

Write a number sentence, then the answer with its unit.

D. Stretch: a magic square with negatives

Every row, column and diagonal of this square adds to 0. Fill the gaps, then check all eight lines.

The sign rules for multiplying and dividing

The sign rules are not arbitrary: they keep the patterns of the times table going. Students fill a table that crosses into the negatives, then multiply and divide integers and predict the sign of longer products.

We are learning to

• extend multiplication patterns past 0 to find the sign of a product,
• multiply and divide integers using the sign rules,
• predict the sign of a product of several numbers by counting the negatives.

Success criteria

• I can use a pattern to explain why (−4) × (−2) = 8.
• I can say at once whether (−1) × (−2) × (−3) is positive or negative.

You need

The worksheet, one per student. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Dropping a sign: (−7) × (−3) is 21, not −21.
• Thinking two negatives always give a positive, even when adding: −3 + (−3) = −6.
• Getting the size right and the sign wrong in division: 36 ÷ (−9) = −4.

Answers

• Part A: each cell is the row number times the column number; the positive cells are the top-left and bottom-right blocks.
• Part B: −6 × 4 = −24; −7 × (−3) = 21; 36 ÷ (−9) = −4; −48 ÷ (−6) = 8; 5 × (−8) = −40; −45 ÷ 5 = −9; −12 × (−5) = 60; −63 ÷ (−7) = 9.
• Part C: (−2) × (−3) × (−1) = −6, 3 negatives; (−1) × (−1) × (−1) × (−1) = 1, 4 negatives; (−2) × 5 × (−3) × 2 = 60, 2 negatives; (−1) × 2 × (−3) × 4 × (−5) = −120, 3 negatives.
• Part D: −1 × (1 + (−1)) = −1 × 0 = 0. Expanded, that is (−1 × 1) + (−1 × −1) = −1 + (−1 × −1), which must be 0, so −1 × −1 = 1.

The sign rules for multiplying and dividing

A. The sign table

Each cell is row × column. The top-left block is done. Follow the patterns into the negatives.

B. Multiply and divide

C. Count the negatives

Predict the sign first, then work out the product.

D. Stretch: why must −1 × −1 be 1?

Work out −1 × (1 + (−1)) in two ways: do the bracket first, then expand the bracket. What must −1 × −1 equal for both ways to agree?

Fractions and decimals, positive and negative

The fraction methods and the sign rules work together. Students add and subtract over a common denominator, multiply across, divide by the reciprocal, carry signs through, and look for an efficient way before starting.

We are learning to

• add, subtract, multiply and divide fractions, giving answers in simplest form,
• carry the sign rules into fraction and decimal calculations,
• choose an efficient strategy before calculating, and estimate to check.

Success criteria

• I can work out −1/3 + 1/2 and −3/4 × 2/5 with the right sign.
• I can explain why dividing by 1/2 doubles a number.

You need

The worksheet, one per student; calculators to check Part C only.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Adding across: 2/3 + 1/4 is not 3/7; use twelfths.
• Flipping the wrong fraction when dividing: only the divisor turns over.
• Losing the sign in a decimal: 1.2 ÷ (−0.3) = −4.

Answers

• Part A: 2/3 + 1/4 = 11/12; 5/6 − 1/3 = 1/2; 3/5 × 10/9 = 2/3; 4/7 ÷ 2/3 = 6/7.
• Part B: −1/2 + 3/4 = 1/4; 1/3 − 5/6 = −1/2; −2/3 × 3/5 = −2/5; −3/4 ÷ (−1/2) = 3/2; −5/8 − (−1/4) = −3/8.
• Part C: −2.5 × 0.4 = −1; 1.2 ÷ (−0.3) = −4; −0.6 + 1.25 = 0.65; −3.2 − (−1.7) = −1.5.
• Part D: 24 × 3/8 = 9 (24 ÷ 8 = 3, then 3 × 3); 0.25 × 36 = 9 (0.25 is a quarter, and a quarter of 36 is 9); −5 × 17 × 2 = −170 (−5 × 2 = −10 first); 3/4 + 5/8 + 1/4 = 13/8 (3/4 + 1/4 = 1 first).

Fractions and decimals, positive and negative

A. Positive fractions

Give each answer in simplest form.

B. Signed fractions

Decide the sign, then the size.

C. Signed decimals

Estimate first, then work it out. Check with a calculator.

D. Stretch: find the quick way

Work each out in your head, then write the step that made it quick.

Working in the right order

With negatives and fractions in play, the order of operations matters even more. Students work lines step by step, place brackets to hit a target, check a calculator’s order, then play the domino loop and the quiz.

We are learning to

• apply the order of operations to expressions with integers and fractions,
• place brackets to change the value of an expression,
• use a calculator’s negative key and brackets correctly.

Success criteria

• I can work out 8 − 2 × (−3) step by step.
• I can explain why (−3)² = 9 but −3² = −9.

You need

The worksheet; the dominoes (cut-out sheet 2), one set per group of up to eleven; calculators; the quiz on the unit page.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Working left to right: −3 + 4 × (−2) is −11, not −2.
• Squaring only the 3 in (−3)², or squaring the sign in −3².
• Typing the subtract key for a negative number on a calculator.

Answers

• Part A: −3 + 4 × (−2) = −3 + (−8) = −11; (−6 + 2) × 3 = −4 × 3 = −12; 20 ÷ (−4) − 3 = −5 − 3 = −8; −2 × (5 − 9) + 1 = −2 × (−4) + 1 = 8 + 1 = 9; 1/2 + 1/4 × 6 = 1/2 + 3/2 = 2; (−3)² − 10 = 9 − 10 = −1.
• Part B: −2 × (3 + 4) = −14; 12 ÷ (−2 + 4) = 6; (−5 + 1) × −3 = 12.
• Part C: (−3)² = 9 and −3² = −9: the square is done before the sign is applied. A calculator that follows the order gives the same.
• Part D: for example (−2 + 5) × 4 = 12, −2 × (5 + 4) = −18, −2 + 5 × 4 = 18; there are others.

Working in the right order

A. Step by step

Write one step per line until you reach the answer.

B. Place the brackets

Add one pair of brackets to make each line true.

C. What does the calculator do?

Type (−3)² and then −3² into your calculator. Write both answers. Explain the difference using the order of operations.

D. Stretch: three numbers, many answers

Using −2, 5 and 4 once each, with any operations and brackets, make as many different answers as you can. What is the biggest? The smallest?

Number line and order cards

For Day 1, one set per pair (keep the line for Day 2)

Cut out the number line along the dashes. Lay each order card above the line where its number belongs.

Twelve order cards

Teacher note: the cards run −2.5, −7/4, −1.2, −2/3, −0.5, −1/4, 0, 0.3, 3/8, 0.75, 4/3, 1.5 from left to right.

Domino loop: I have, who has?

For Day 5, one set per group of up to eleven

Deal one domino to each player (some take two in a small group). The holder of “I have −12” starts.

Teacher note: the loop runs −12 (−20 ÷ (−5) = 4) → 4 (3 − 8 = −5) → −5 (−3/2 × 3 = −9/2) → −9/2 (−6 ÷ (−2) = 3) → 3 (1/4 − 3/4 = −1/2) → −1/2 (−7 + 5 = −2) → −2 (−15 + 25 = 10) → 10 (−9 + 5 = −4) → −4 (−3 − 1/2 = −7/2) → −7/2 (−9 ÷ 3 = −3) → −3 (−8 − 4 = −12) and back to −12.

What we know: Operations with rational numbers

Work on your own, without a calculator. Show your working.

1. Put in order, smallest first: −0.4,   −1/2,   1/3,   −1 ______________________
2. Work out −8 + 13. ______________________
3. Work out −4 − (−9), and rewrite it as an addition. ______________________
4. Work out −9 × (−4) and 42 ÷ (−6). ______________________
5. Work out −2/5 + 1/2. ______________________
6. Work out −3/4 ÷ 3/8. ______________________
7. Work out −1.5 × (−0.6). ______________________
8. Work out −4 + 6 ÷ (−2) × 3, one step per line. ______________________

Answers and marking guide

Answers with working

1. −1 < −1/2 < −0.4 < 1/3 (as decimals −1, −0.5, −0.4, 0.33).
2. −8 + 13 = 5: start at −8 and move 13 right.
3. −4 − (−9) = −4 + 9 = 5.
4. −9 × (−4) = 36; 42 ÷ (−6) = −7.
5. −4/10 + 5/10 = 1/10.
6. −3/4 × 8/3 = −2: different signs, so negative.
7. −1.5 × (−0.6) = 0.9: same signs, so positive.
8. −4 + 6 ÷ (−2) × 3 = −4 + (−3) × 3 = −4 + (−9) = −13.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and decides the sign before the size.

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 number line, adding and subtracting, multiplying and dividing, fractions and decimals, and the order of operations.