Integers on the Line and the Plane: a week of ready-to-teach maths
Five days of lessons for Year 6 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus cut-out integer and situation cards, a coordinate game board, a mini-check and every answer.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Cut out the integer and situation cards once; they are reused all week. Print one game board per child for Day 5.
- Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.
No maths background needed
This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: below zero is a real place. A negative number names a position on the other side of zero, such as a night below freezing, a diver under the surface or money owed. Integers are the whole numbers, their opposites and zero, each one fixed place on the number line. The further left, the smaller, so −9 is less than −2 even though 9 is more than 2. Two integers, read across and then up or down, fix one point anywhere on the plane.
One day, one lesson
The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.
How we say integers
Read −5 as “negative five”. In everyday talk people say “minus five degrees”, but in maths “minus” is the name of taking away, and children meet adding and subtracting integers in Year 7. Positive numbers are usually written without a sign: 5, not +5. Zero is neither positive nor negative; in a situation it is the starting place, such as the freezing point, sea level or the ground floor.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Children learn and do | On screen |
|---|---|---|---|
| 1 | Below zero is a real place | Read temperatures above and below zero, and write the integer for a depth, a floor or a sum of money owed | The mercury line |
| 2 | One fixed place on the line | Locate integers on lines that count by ones, twos and fives, and count the jumps between two integers | The number line through zero |
| 3 | Opposites and order | Name the opposite of an integer and its distance from zero, and compare two integers with < and > | Every integer has an opposite |
| 4 | In credit, overdrawn | Follow a bank balance up and down across zero, and order balances and temperatures, smallest first | When the balance falls below zero; Ordering runs left to right |
| 5 | Two numbers fix one point | Read and plot points in all four quadrants of the Cartesian plane, and name the quadrant from the signs | Two numbers fix one point |
How the week builds
Day 1 finds integers in real situations; Day 2 gives every integer a fixed place on the line; Day 3 pairs each integer with its opposite and uses the line to compare; Day 4 puts integers to work with money and orders a mixed set; and Day 5 pairs two integers to fix a point on the plane. It builds on the Year 5 number line and the first-quadrant grid, and it prepares the Year 7 work of adding and subtracting integers. This week nothing is added or subtracted as a sum: children move along a line and count the jumps.
Materials for the week (one trip)
- From the classroom: scissors, pencils, coloured pencils, rulers, this pack printed.
- For Day 2: masking tape and a marker to lay a floor number line from −10 to 10, about 4 m long with a mark every 20 cm.
- For Day 1: a real thermometer and a newspaper or website weather page with overnight lows (both optional).
- Cut out once, use all week: the integer cards and the situation cards. Print one coordinate game board per child for Day 5. No maths equipment to buy.
Words for the word wall
integer, positive, negative, zero, opposite, distance from zero, number line, greater than, less than, credit, overdrawn, balance, Cartesian plane, origin, x-axis, y-axis, quadrant, coordinates
Dear families
This week in maths, Year 6 meets integers: whole numbers, their opposites and zero. We read temperatures below zero, place integers on a number line that runs both ways, and use them for money in credit and money owed. We also plot points in all four quadrants of the Cartesian plane. On the line, further left is smaller: −8 is less than −3.
Try this at home
- Look at the overnight lows on a weather page. Find a town below zero and say the temperature as “negative two”, not “two”.
- In a lift or a car park, ask which integer each floor would be if the ground floor is zero.
- Ask which is colder, −4 °C or −9 °C, and how many degrees apart they are on a thermometer.
- Talk about a balance that goes below zero: what does −$20 on a statement mean?
My integers this week
Fill one row a day. Find a number below zero, or a place where zero is the starting point.
| Day | My integer | Where I found it | What zero means there |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Integers on the Line and the Plane teaching pack · seegongsik.com/au/y6/number/AC9M6N01/pack
Below zero is a real place
Children meet integers where they live: nights below freezing, depths below sea level, floors below the ground and money owed. They read thermometers and write the integer for each situation.
We are learning to
- read a temperature above or below zero on a thermometer,
- write the integer for a situation, with a negative sign for below zero,
- say what zero stands for in a situation.
Success criteria
- I can read a thermometer, even when the column stops below zero.
- I can write the integer for a situation and say what zero means in it.
You need
A real thermometer and a weather page with overnight lows (both optional), the situation cards from the cut-out sheet, and the worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Colder than freezing Show the board (see the box). Read the marks below zero together, counting down from zero. Ask: “Water freezes at zero degrees. On a night of negative six, is it colder or warmer than freezing, and by how much?” |
| 25 min | Above and below Children read the thermometers in Part A. Pairs match and sort the situation cards (above, at or below zero), then write the integers in Part B. Ask: “Thermometer D stops below the zero mark. Count down from zero. Which mark is it on?” |
| 15 min | What does zero mean? Children finish Parts C and D. Share a few of their own situations. Ask: “A lift button says B2. Where is zero in this building, and which integer is B2?” |
Two half-sessions instead? Split after Part A; start Session B with the situation cards.
Watch for these ideas
- Reading below zero upwards from the bottom, so −11 comes out as −9. Below zero, count down from zero: −1, −2, −3.
- Writing 6 for six degrees below zero. Without its sign, 6 is above freezing.
- Thinking zero means nothing. In most situations it is a starting place: the freezing point, sea level, the ground floor.
Answers
- Part A: A 6 °C, B −4 °C, C 13 °C, D −11 °C, E −1 °C.
- Part B: −9, −18, 25, −2, −40, 65, 0, −15.
- Part C: the point where water freezes; sea level, the surface of the sea; the ground floor; nothing saved and nothing owed. Part D: answers vary; check that the sign matches above or below, and that zero is named.
Above and below zero
A. Read the thermometers
B. Write the integer
| Situation | Integer |
|---|---|
| a frosty night, nine degrees below zero | |
| a diver 18 m below the surface of the sea | |
| a kite flying 25 m above the ground | |
| floor B2 of a car park, two floors below the ground floor | |
| owing $40 on a phone bill | |
| a balance of $65 in a bank account | |
| the surface of the sea (sea level) | |
| Kati Thanda–Lake Eyre, about 15 m below sea level |
C. What does zero mean?
In a temperature, zero is ______________________________________
For heights and depths, zero is ______________________________________
In a building, zero is ______________________________________
In a bank account, zero is ______________________________________
D. Your own
One situation above zero, with its integer: ______________________________________
One situation below zero, with its integer: ______________________________________
One fixed place on the line
Every integer is one fixed place on the number line, one jump from each neighbour. Children stand on a floor line, locate integers on lines that count by ones, twos and fives, and count the jumps between two integers across zero.
We are learning to
- locate an integer by counting jumps from zero, left for negative and right for positive,
- read a number line that counts by twos or fives through zero,
- count the jumps between two integers on a line.
Success criteria
- I can find any integer from −20 to 20 on a number line.
- I can count how many jumps apart two integers are, even when one is below zero.
You need
Masking tape and a marker for the floor line from −10 to 10, the integer cards from the cut-out sheet, and the worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | The line runs both ways Show the board (see the box). Children call each move before you press. Ask: “The flag says negative nine. Which way do we step from zero, and how many steps?” |
| 30 min | Stand on the line Lay the floor line with a mark every 20 cm and write only −10, 0 and 10 on it. Hand out integer cards: children stand on their mark and check their neighbours. Then children do Parts A and B. Ask: “Line 3 only numbers −50, 0 and 50. How big is one jump? So which integer is arrow G?” |
| 10 min | How many jumps apart? Two children stand on −3 and 5; the class counts the jumps between them, through zero. Children do Parts C and D. Ask: “From negative three to five: how many jumps to reach zero, and how many more after it?” |
Two half-sessions instead? Split after Part A; start Session B on the floor line again.
Watch for these ideas
- Counting marks instead of jumps, so −3 to 5 comes out as 9. Count the spaces between the marks.
- Reading a line that counts by twos or fives as if it counts by ones. Read two labels first and work out the size of one jump.
- Putting −8 to the right of −3 because 8 is more than 3. On the line, the further left, the smaller.
Answers
- Part A: A −7, B 3, C −2; D −14, E 6, F −4; G −35, H 15, I −5.
- Part B: from left to right, −9, −6, −1, 2, 7. Part C: 8, 5, 14, 6, 18 jumps.
- Part D: −1, −3, −5; −3, 0, 3; −4, −8, −12; −15, −10, −5, 0.
Find it on the line
A. Which integer is at each arrow?
Line 1: A ______ B ______ C ______
Line 2: D ______ E ______ F ______
Line 3: G ______ H ______ I ______
B. Mark and label each integer on the line
−6, 2, −9, 7, −1
C. How many jumps apart?
D. Keep the pattern going
5, 3, 1, ______, ______, ______
−12, −9, −6, ______, ______, ______
8, 4, 0, ______, ______, ______
−25, −20, ______, ______, ______, ______
Opposites and order
Fold the number line at zero and every integer lands on its opposite, the same distance from zero on the other side. Children find opposites, then compare integers on the line with < and >.
We are learning to
- name the opposite of an integer and its distance from zero,
- compare two integers by their places on the line,
- write a comparison with < or >.
Success criteria
- I can say the opposite of any integer, and why zero is its own opposite.
- I can say which of two integers is smaller, and write it with < or >.
You need
A paper strip with a number line from −10 to 10 for each pair (or the floor line from Day 2), and the worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Fold the line Show the board (see the box). Pairs fold their strip at zero and read the marks that meet. Ask: “Six and negative six land on each other when we fold. What is the same about them, and what is different?” |
| 25 min | Which is smaller? Compare −9 and −4 on the line: −9 is further left, so −9 < −4. Children do Parts A and B, checking each pair on their strip. Ask: “Nine is more than four. So why is negative nine less than negative four?” |
| 15 min | True or not? Children decide Part C, explaining each answer with the line. Ask: “Which is closer to zero, negative eight or five? How do you know without counting every mark?” |
Two half-sessions instead? Split after Part A; start Session B by folding the strips again.
Watch for these ideas
- Comparing the digits and ignoring the sign: −9 > −4 because 9 > 4. On the line −9 is further left, so −9 < −4.
- Thinking the opposite of 7 is 1, or 70. It is the same distance from zero on the other side: −7.
- Mixing up < and >. The point aims at the smaller number and the open side faces the larger.
Answers
- Part A: 7: opposite −7, distance 7; −12: opposite 12, distance 12; 0: opposite 0, distance 0; −25: opposite 25, distance 25; 40: opposite −40, distance 40; −1: opposite 1, distance 1.
- Part B: −9 < −4, 3 > −8, −15 < −12, −1 < 0, 0 > −20, 11 > −11, −30 < −3, −6 < 6.
- Part C: 1 true (further left); 2 false (negatives are left of zero); 3 false (−30 is further left); 4 true (both 15 from zero); 5 true (zero sits between them); 6 false (8 jumps against 5).
Opposites and order
A. Opposites and distance from zero
| Integer | Its opposite | Distance from zero |
|---|---|---|
| 7 | ||
| −12 | ||
| 0 | ||
| −25 | ||
| 40 | ||
| −1 |
B. Write < or > between each pair
C. True or false? Explain with the line.
| Statement | True or false? | Because |
|---|---|---|
| 1. −10 is smaller than −2. | ||
| 2. −3 is greater than 1. | ||
| 3. −20 < −30 | ||
| 4. The opposite of −15 is 15. | ||
| 5. Every negative integer is smaller than every positive integer. | ||
| 6. −8 is closer to zero than 5. |
In credit, overdrawn
A balance above zero is credit, money held; spend past it and the balance falls below zero, money owed. Children follow a bakery’s balance up and down a balance line, then order balances and temperatures, smallest first.
We are learning to
- say whether a balance is in credit or overdrawn, and by how much,
- follow a balance up and down a balance line as money comes in and goes out,
- order a mixed set of integers, most negative first.
Success criteria
- I can read a balance such as −$20 as $20 owed.
- I can put a set of integers in order from smallest to largest.
You need
The integer cards from the cut-out sheet (a handful per pair), and the worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Below the zero line Show the bank balance on the board (see the box). Ask: “The balance reads negative fifty dollars. Does the account hold fifty dollars, or owe fifty dollars?” |
| 25 min | Follow the balance Model the first row of Part A on the balance line: from $30, paying $50 moves down five jumps of $10, past zero to −$20. Children finish Part A, moving a finger along the line for each row. Ask: “The balance is negative ten dollars and $70 comes in. How far up the line do we move, and where do we land?” |
| 15 min | Smallest first Show the four cards on the board. Pairs order a handful of integer cards, then do Parts B and C. Ask: “Who owes more, someone at negative thirty-five dollars or someone at negative twenty? So whose balance is lower?” |
Two half-sessions instead? Split after Part A; start Session B with the four cards on the board.
Watch for these ideas
- Reading −$30 as having $30. The negative sign means owed.
- Ordering negatives by their digits: −2, −5, −9. The most negative is the smallest: −9, −5, −2.
- Stopping at zero: from $10, spending $30 goes past zero, two more jumps of $10, to −$20.
Answers
- Part A: −$20 overdrawn; $20 in credit; −$40 overdrawn; −$10 overdrawn; $60 in credit.
- Part B: −$40, −$15, −$5, $0, $25, $60; −15 °C, −8 °C, −1 °C, 3 °C, 12 °C.
- Part C: Ali owes more, so Ali’s balance is lower (−$35 is below −$20). −$45 is further from zero: 45 against 30.
Follow the balance
The bakery’s account starts at $30. Above the zero line is money in the bank (in credit); below it is money owed (overdrawn).
A. Follow the balance on the line
| What happens | New balance | In credit or overdrawn? |
|---|---|---|
| Pays $50 for flour | ||
| Sells bread for $40 | ||
| Pays $60 to fix the oven | ||
| Sells pies for $30 | ||
| Sells cakes for $70 |
B. Put each set in order, smallest first
Balances: $25, −$40, $0, −$5, $60, −$15
______________________________________________________
Temperatures: 3 °C, −8 °C, −1 °C, 12 °C, −15 °C
______________________________________________________
C. Think it through
Ali’s balance is −$35 and Bea’s is −$20. Who owes more? Whose balance is lower?
______________________________________________________
Which balance is further from zero, −$45 or $30? How do you know?
______________________________________________________
Two numbers fix one point
The Cartesian plane is two number lines crossing at zero, the x-axis across and the y-axis up. Children read and plot points in all four quadrants, across first and then up or down, and name the quadrant from the signs.
We are learning to
- read the coordinates of a point in any of the four quadrants,
- plot a point from its coordinates, across first, then up or down,
- name the quadrant, or the axis, from the signs of the coordinates.
Success criteria
- I can plot (−3, 2) and (4, −1) and say which quadrant each is in.
- I can explain why (2, −5) and (−5, 2) are different points.
You need
The coordinate game board from the cut-out pages (one per child), coloured pencils, and the worksheet, one per child.
Lesson flow (about 50 minutes)
| 10 min | Across, then up or down Show the plane on the board (see the box). Children call each press before you make it. Ask: “The ring is at negative three, two. Which button first, and how many presses?” |
| 25 min | Read and plot Read point A together: from the origin, 4 right, then 3 up, so (4, 3). Children do Parts A to D. Ask: “Point E sits on the y-axis. What is its x-coordinate, and why?” |
| 15 min | Hidden reef Pairs play the game on the cut-out board. Finish with the quiz on the board as a class game. Ask: “You hit the reef at (−2, 3). Which four points next door would you call next, and what are their coordinates?” |
Two half-sessions instead? Split after Part A; start Session B with the game, then Parts B to D.
Watch for these ideas
- Reading up first, so (−3, 2) lands at (2, −3). Across first, then up or down: along the corridor, then up the stairs.
- Counting squares instead of lines. Points sit where the grid lines cross, not inside the squares.
- Putting a point with a zero in a quadrant. (0, −3) is on the y-axis and (−6, 0) on the x-axis.
Answers
- Part A: A (4, 3), B (−5, 2), C (−2, −4), D (3, −5), E (0, −3), F (−6, 0), G (5, −1), H (−1, 5). Part B: a sailing boat on the water.
- Part C: (−7, 3) second quadrant; (5, −2) fourth quadrant; (−1, −1) third quadrant; (8, 6) first quadrant; (0, −4) on the y-axis; (−3, 0) on the x-axis.
- Part D: (−4, −3), third quadrant; (−3, 6), second quadrant.
Points on the plane
A. Write the coordinates of each point
A ( ______ , ______ )
B ( ______ , ______ )
C ( ______ , ______ )
D ( ______ , ______ )
E ( ______ , ______ )
F ( ______ , ______ )
G ( ______ , ______ )
H ( ______ , ______ )
B. Plot the points and join them in order
Hull: (−4, 1) → (4, 1) → (2, −2) → (−2, −2) → (−4, 1)
Mast: (0, 1) → (0, 5)
Sail: (0, 5) → (3, 2) → (0, 2)
What did you draw? ______________
C. Which quadrant, or which axis?
D. Mystery points
My x is the opposite of 4, and my y is 3 below zero. I am at ( ______ , ______ ).
My x is 2 jumps right of −5, and my y is the opposite of −6. I am at ( ______ , ______ ).
Integer cards and situation cards
Integer cards (Days 2 and 4)
Cut out the 25 cards. On Day 2 each child stands on the floor line at their card; on Day 4 pairs take a handful and order them, smallest first.
Situation cards (Day 1)
Cut out the eight cards. Each one matches one integer card above: pairs find the match, then sort the situations into above zero, at zero and below zero.
Teacher note: the matches are three floors below the ground floor: −3; a $7 debt: −7; a frosty morning, two degrees below zero: −2; a submarine 10 m below the surface: −10; a balloon 9 m above the ground: 9; sea level: 0; a cool day, twelve degrees above zero: 12; a $15 overdraft: −15.
Hidden reef
A game for two. Print one board for each child and keep it hidden from your partner.
- On My sea, secretly mark three reef points where grid lines cross. Do not use the axes.
- Take turns to call a point, such as (−2, 3). Your partner answers “hit” or “miss”.
- Record each call on My partner’s sea: a cross for a miss, a filled dot for a hit, and write it in the table.
- The first to find all three of the other player’s reef points wins.
| Point I called | Hit or miss | Point I called | Hit or miss |
|---|---|---|---|
Teacher note: after a hit, good players call the four points next door, one jump left, right, up and down. Ask them to say each point before calling it.
What we know: Integers on the Line and the Plane
Work on your own. Show your thinking if you can.
- Write the integer: nine degrees below zero ______ a balance with $35 owed ______
- What temperature does the thermometer show? ______ °C
- Which integers are at the arrows? A ______ B ______
- How many jumps apart are −4 and 6 on the number line? ______
- Write < or > : −7 ____ −3−2 ____ 10 ____ −9
- Put these in order, smallest first: 4, −6, −11, 0, −2. ______________________________
- A balance is $15 and $40 is spent. What is the new balance? ________ Is the account in credit or overdrawn? ______________
- Write the coordinates: P ( ____ , ____ ) Q ( ____ , ____ ). In which quadrant is (−3, −5)? ______________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers
- −9 and −35.
- −7 °C (seven marks below zero).
- A −6, B 4 (one jump is 2).
- 10 jumps (4 to zero, then 6 more).
- −7 < −3; −2 < 1; 0 > −9.
- −11, −6, −2, 0, 4.
- −$25: $15 down to zero, then $25 more below it. The account is overdrawn, with $25 owed.
- P (−3, 4), Q (2, −5); (−3, −5) is in the third quadrant.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Integers in situations (Q1, Q2) | writes the number but drops the sign | writes the integer for a situation and reads a thermometer below zero | says what zero stands for in each situation |
| The number line (Q3, Q4) | counts the marks instead of the jumps | locates integers on lines by ones and twos and counts jumps across zero | reads a line that counts by fives or tens from two labels |
| Comparing and ordering (Q5, Q6) | orders negatives by their digits | compares with < and > and orders a mixed set, smallest first | explains a comparison with distance from zero and opposites |
| Money and the plane (Q7, Q8) | reads a negative balance as money held, or plots up first | follows a balance across zero and reads points in all four quadrants | names the quadrant from the signs alone |
Eight questions, four ideas. A child at standard answers most questions and can say why, pointing to the place on the line or the plane that decides.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Reads below zero | Locates on a line | Opposites and < > | Balances and order | Plots four quadrants |
|---|---|---|---|---|---|
The five columns are the five days: below zero, the number line, opposites and comparing, money and ordering, and the plane.