Adding and subtracting integers: a week of ready-to-teach maths

Five lessons for Year 7 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 positive and negative counters, integer 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: counters (Days 2 and 3) and integer cards (Days 1 and 5). Real two-colour counters work as well.
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 are the whole numbers, their opposites and zero, and each has one place on the number line: further left is smaller, so −9 < −2. Adding a positive moves right and adding a negative moves left. A positive and a negative counter cancel to make zero, so −3 + 5 leaves 2. Subtracting a negative is the same as adding its opposite: taking away a $4 debt leaves you $4 better off, so 2 − (−4) = 6. The difference between two integers is the gap between them, found by subtracting the smaller from the larger.

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 integers

Read −5 as “negative five”; keep “minus” for subtracting. A negative number after an operation sign goes in brackets: 3 + (−8), 6 − (−4). Positive numbers are written without a sign. Calculators are not needed this week.

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 gives every integer a place on the line and orders them; Day 2 adds them as moves and as counters; Day 3 subtracts them, including the surprise of subtracting a negative; Day 4 reads subtraction as a difference or a change; and Day 5 uses everything in problems. It builds on the Year 6 work of placing integers on a line and on the Cartesian plane, and it prepares the integer arithmetic of algebra and the four-quadrant graphs of later years.

Materials for the week (one trip)

• From the classroom: pencils, scissors, small envelopes for the counters and cards, this pack printed.
• Cut-out sheet 1, one per pair: 20 positive and 20 negative counters (Days 2 and 3), or real two-colour counters.
• Cut-out sheet 2, one per pair: 25 integer cards from −12 to 12 (Days 1 and 5).
• No calculators and no other equipment to buy.

Words for the word wall

integer, positive, negative, zero, opposite, number line, greater than, less than, order, sum, difference, zero pair, counter, below zero, above sea level, change, rise, fall, balance

Dear families

This week in maths, Year 7 is adding and subtracting integers: positive and negative whole numbers and zero. We use number lines, counters that cancel in pairs, and real situations such as temperatures below zero and bank balances in debt.

Try this at home

• Look up tonight’s minimum and tomorrow’s maximum. How many degrees is the rise?
• In a lift, start on floor 2, go down 4 floors and up 1. Which floor are you on? (−1.)
• Ask: if you owe $10 and someone takes away your debt, are you better or worse off? Why is 0 − (−10) = 10?
• Ask: which is colder, −3 °C or −8 °C?

Integers this week

Fill one row a day with a real situation that uses an integer, and a sum or difference about it.

Printed from the free seegongsik Adding and subtracting integers teaching pack · seegongsik.com/au/y7/number/AC9M7N07/pack

Integers on the line

Every integer has one place on the number line, and further left is smaller. Students write integers for situations, compare them with < and >, and put sets in order.

We are learning to

• write the integer for a situation above or below zero,
• compare two integers with < and > by their places on the line,
• order a set of integers, smallest first.

Success criteria

• I can say why −8 < −3, even though 8 > 3.
• I can put six integers in order and check them on a line.

You need

The integer cards from cut-out sheet 2 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Ordering negatives by their digits, so −15 comes after −11. Further left is smaller.
• Forgetting where 0 goes: it sits between the negatives and the positives.
• Reading < and > the wrong way round. The point of the sign aims at the smaller number.

Answers

• Part A: −5, 120, −35, −3, 8, −12.
• Part B: −3 < 2, 0 > −1, −12 > −15, −7 < −4, −9 > −10, −20 < −2.
• Part C: −9, −6, −2, 0, 4, 7; −15, −11, −4, −1, 3, 8.
• Part D: −2 is halfway between −9 and 5, 7 steps from each; −3, −2, −1, 0, 1.

Integers on the line

A. Write the integer

B. Write < or >

C. Smallest first

4,   −6,   0,   −2,   7,   −9   ____________________________

−15,   8,   −1,   −11,   3,   −4   ____________________________

D. Stretch

Which integer is halfway between −9 and 5? ________ List every integer greater than −4 and less than 2. ____________________

Check one answer on the number line at the top and explain it to a partner.

Adding integers

Adding a positive moves right along the line and adding a negative moves left. With counters, a positive and a negative make a zero pair, and what is left after the pairs cancel is the sum.

We are learning to

• add integers as moves along a number line,
• add integers with counters by cancelling zero pairs,
• say whether a sum will be positive, negative or zero before working it out.

Success criteria

• I can show −6 + 4 = −2 on a line and with counters.
• I can explain why 7 + (−7) = 0.

You need

The counters from cut-out sheet 1 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Adding the digits and keeping a sign, as in −6 + 4 = −10. Count the moves or the pairs.
• Thinking the answer takes the sign of the first number. It takes the sign of the bigger group.
• Thinking a sum is always bigger than both numbers. Adding a negative gives a smaller result.

Answers

• Part A: −5 + 9 = 4; 3 + (−8) = −5; −6 + (−2) = −8; −7 + 7 = 0; 10 + (−13) = −3; −1 + (−9) = −10.
• Part B: −4 + 6: 4 zero pairs, leaving 2; 5 + (−8): 5 zero pairs, leaving −3; −3 + (−2): 0 zero pairs, leaving −5.
• Part C: 5 + 3 = 8, 5 + 2 = 7, 5 + 1 = 6, 5 + 0 = 5, 5 + (−1) = 4, 5 + (−2) = 3, 5 + (−3) = 2; each step down by 1 in the second number gives 1 less.
• Part D: for example 4 + (−7) or 1 + (−4), any positive plus a negative 3 bigger in size, making −3; the third number is 5, as −8 + 3 = −5.

Adding integers

A. Jump along the line

Say the sign of the answer first. Then use the line.

B. Counters

Draw + and − counters, circle the zero pairs, and write what is left.

C. Continue the pattern

D. Stretch

Find a positive and a negative integer that add to −3. ____________ Three integers add to 0. Two of them are −8 and 3. What is the third? ________

Subtracting integers

Subtracting a positive moves left along the line. Subtracting a negative is the same as adding its opposite: taking away a debt leaves you better off. Students subtract both kinds and see the rule in a pattern.

We are learning to

• subtract a positive integer as a move to the left,
• subtract a negative integer by adding its opposite,
• explain the rule with a pattern or with a debt.

Success criteria

• I can work out 2 − (−4) = 6 and say why it is not −2.
• I can rewrite a subtraction of a negative as an addition.

You need

The counters from cut-out sheet 1 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Moving left for every subtraction: subtracting a negative moves right.
• Changing both signs, as in 6 − (−4) = −6 + 4. Only the operation and the second sign change.
• Thinking a − b and b − a are equal. They are opposites.

Answers

• Part A: 4 − 9 = −5; −3 − 6 = −9; 0 − 7 = −7; −8 − 2 = −10.
• Part B: 6 − (−4) = 6 + 4 = 10; −2 − (−8) = −2 + 8 = 6; −9 − (−3) = −9 + 3 = −6; 0 − (−5) = 0 + 5 = 5.
• Part C: 3 − 2 = 1, 3 − 1 = 2, 3 − 0 = 3, 3 − (−1) = 4, 3 − (−2) = 5, 3 − (−3) = 6; each answer is 1 more than the one before.
• Part D: −5 + 8 − (−2) = 5; 7 − 12 + (−3) = −8. −4 − 6 = −10 and 6 − (−4) = 10: opposites.

Subtracting integers

A. Subtract a positive

B. Subtract a negative

Rewrite each as an addition first.

C. Continue the pattern

D. Stretch

−5 + 8 − (−2) = ________

7 − 12 + (−3) = ________

Work out −4 − 6 and 6 − (−4). What do you notice? ________________________

Differences and changes

The difference between two integers is the gap between them, found by subtracting the smaller from the larger. A change is the new value minus the old one, so a fall is a negative change.

We are learning to

• find the difference between two integers as the gap on a scale,
• find a change as new minus old, with its sign,
• use differences in real contexts such as temperature and height.

Success criteria

• I can find the difference between 6 °C and −5 °C: 11 degrees.
• I can say why a change from 10 to −4 is −14.

You need

The worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Subtracting the digits, so the gap from −3 to 8 is 5. Across zero, the distances add.
• Writing a change as old minus new, which flips its sign.
• Thinking a difference can be negative. A gap is a distance; a change can be negative.

Answers

• Part A: 8 − (−3) = 11 degrees; −2 − (−9) = 7 degrees; 15 − (−4) = 19 degrees; 0 − (−12) = 12 degrees.
• Part B: A temperature: 3 °C − (−6 °C) = +9; A bank balance: −$15 − $40 = −55; A diver’s depth: −30 m − (−12 m) = −18; A temperature: −5 °C − (−20 °C) = +15.
• Part C: 2228 − (−15) = 2243 m.
• Part D: 2 and −5: 2 − (−5) = 7 and 2 + (−5) = −3.

Differences and changes

B. Changes: new minus old

C. High and low

Mt Kosciuszko, the highest point on mainland Australia, is 2228 m above sea level. The lowest point of Kati Thanda–Lake Eyre is about 15 m below sea level. Write both as integers and find the difference in height. ________________________

D. Stretch

Two integers have a difference of 7 and a sum of −3. What are they? ________________________

Integer problems

Problems with money, lifts, temperatures, golf and depths put the week together. Students write each situation as an integer sum or difference, work it out, and check with a line or a scale.

We are learning to

• write a situation as a sum or difference of integers,
• solve problems with several steps,
• check an answer with a line, a scale or the story.

Success criteria

• I can turn “$30, spend $45, earn $20” into 30 − 45 + 20 = 5.
• I can say whether an answer is sensible from the story.

You need

The worksheet, the integer cards for the game, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Writing the steps in the wrong order. Follow the story from start to finish.
• Losing the sign of a depth or a debt. Below sea level and owing money are negative.
• Thinking a lift can stop at floor −0. Zero has no sign.

Answers with working

1. $25 − $60 = −$35, then + $50 = $15.
2. 4 − 7 + 2 = −1, 1 floor below ground.
3. 13 − (−4) = 17 degrees.
4. −2 + 3 + −1 + 0 = 0, level par.
5. −120 + 45 − 70 = −145 m.
6. −7 − (−10) = 3 and −10 − (−7) = −3; the first is greater by 6.
7. rows: 3, −4, 1 / −2, 0, 2 / −1, 4, −3; every line adds to 0.
8. no: −2 − (−9) = 7 is positive; −9 − (−2) = −7 is negative; it depends on which is bigger.

Integer problems

Write each story as a sum or difference first. Then work it out and check that it makes sense.

Problems (7 and 8 are the stretch)

1. A bank account has $25. The owner spends $60, then deposits $50. What is the balance now?
2. A lift starts on floor 4, goes down 7 floors and up 2. Floor 0 is the ground floor. Which floor is it on?
3. The overnight low was −4 °C and the day’s top was 13 °C. How much did the temperature rise?
4. A golfer’s scores against par for four holes are −2, +3, −1, 0. What is the total against par?
5. A submarine is at −120 m. It rises 45 m, then dives 70 m. Where is it now?
6. Which is greater, −7 − (−10) or −10 − (−7)? By how much?
7. Stretch. Complete the magic square below with the integers from −4 to 4 so that every row, column and diagonal adds to 0.

   3
   	0
   	4

8. Stretch. Is the difference between two negative integers always negative? Give examples.

Positive and negative counters

Cut out the counters. A positive and a negative counter together make a zero pair.

Positive counters

Negative counters

If you print without colour, the signs alone tell the counters apart.

Integer cards

Cut out the 25 cards, −12 to 12. Day 1: deal eight and order them. Day 5: Target zero, turning three cards and adding or subtracting them to get as close to 0 as you can.

What we know: Adding and subtracting integers

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

1. Put in order, smallest first: 3,   −8,   0,   −2,   5,   −11. ______________________
2. Write < or >: −6 ____ −9     −1 ____ 1
3. −7 + 4 = ________
4. −5 + (−6) = ________
5. 3 − 9 = ________
6. −4 − (−10) = ________
7. At 6 am the temperature is −7 °C. By noon it is 5 °C. How much has it risen? ________
8. A diver at −18 m rises 7 m, then dives 11 m. Where is the diver now? ________

Answers and marking guide

Answers with working

1. −11, −8, −2, 0, 3, 5.
2. −6 > −9; −1 < 1.
3. −3: 4 zero pairs cancel, leaving 3 negative.
4. −11: two negatives combine; nothing cancels.
5. −6: move 9 left from 3.
6. 6: −4 + 10.
7. 5 − (−7) = 12 degrees.
8. −18 + 7 − 11 = −22 m.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and can check an answer on a number line.

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: on the line, adding, subtracting, differences, and problems.