011i - Integers on the Number Line

Lesson 11 — Integers on the Number Line: Comparing and Ordering

Strand: Number | Descriptor: AC9M7N07 | Duration: 45 minutes

Learning Intentions

• To understand that integers extend the number line in both directions from zero.
• To compare and order positive and negative integers.

Success Criteria

I can:

1. Place positive and negative integers correctly on a number line.
2. Use and to compare any two integers.
3. Order a set of integers from smallest to largest.
4. Explain what the absolute value (distance from zero) of an integer means, and why .

Warmup

(6 minutes — context brainstorm, whole class)

Where do negative numbers appear in real life? Collect on the board:

• Temperature below zero
• Bank balances and debt
• Altitude below sea level
• Golf scores relative to par
• Floors below ground in a lift
• Time before a rocket launch ( minus )

Focus question: In Canberra one winter morning the temperature was C. In Melbourne it was C. Which city was colder? How do you know?

Teacher note: Expect some students to say is warmer because ” is bigger than “. Do not correct it yet — capture it and return in Activity 1.

Activities

Activity 1 — Explicit Instruction: the Extended Number Line (12 min)

Draw a horizontal number line from to on the board, large and clearly labelled.

Key principles, stated explicitly:

1. Zero is the origin. Positive integers extend to the right; negative integers to the left.
2. The further right, the larger. This single rule settles every comparison.
3. is read “negative five”, not “minus five” — minus is an operation, negative is a sign.

Resolving the warmup. Mark and on the line. sits further left, therefore . Canberra was colder.

Use the thermometer model — rotate the number line to vertical — for students who find “left/right” abstract. Colder is lower is smaller.

I do: Order by plotting each point, then reading left to right.

We do: Insert or between each pair.

(Answers: , , , .)

You do: Order from smallest to largest.

(Answers: ; ; .)

Activity 2 — Distance from Zero and Opposites (10 min)

Definitions.

• Two integers are opposites if they are the same distance from zero on opposite sides: and .
• The absolute value of an integer is its distance from zero, written .

Critical distinction to model: because , but because lies further left. Distance and value are different questions.

Practice.

1. Evaluate: , , , .
2. Write the opposite of: , , .
3. Which is further from zero, or ?
4. Which is larger, or ?

(Answers: 1. ; 2. , , ; 3. (distance ); 4. .)

Activity 3 — Inquiry: the Temperature Table (10 min)

Pairs.

Overnight minimum temperatures recorded at five weather stations:

Station	Temperature (°C)
Alpine Ridge
Barra Flats
Coldwell
Dunmore
Eastvale

1. Order the stations from coldest to warmest.
2. How much colder is Coldwell than Barra Flats?
3. Which two stations have the smallest temperature difference?
4. By morning, every station had risen by C. Write the new temperatures.

Socratic scaffolding for Q2:

Prompt	Purpose
Understand: what is being asked?	A difference — a gap on the number line, not a temperature.
Draw a number line and mark both points.	Makes the gap visible rather than abstract.
How would you count the gap?	From up to is steps; from up to is more.
Devise a plan	Add the two parts of the journey.
Carry it out	.
Looking back	Does C seem sensible for the gap between and ? Check on the line.
Generalise	Crossing zero, the total distance is the sum of the two absolute values.

Answers: 1. Coldwell , Alpine Ridge , Dunmore , Eastvale , Barra Flats ; 2. C; 3. Dunmore and Eastvale, a difference of C; 4. .

Checks for Understanding

(5 minutes — exit ticket)

1. Order from smallest to largest: .
2. Insert or : (a) (b) (c) .
3. Evaluate .
4. Reasoning. Sam says ” is bigger than because is bigger than .” Explain his error using a number line.
5. A lift is at level (three floors below ground). It travels up floors. Which level is it on now?

Answers: 1. ; 2. (a) (b) (c) ; 3. ; 4. Sam is comparing distance from zero, not position. lies further left than , so it is smaller — it is colder, deeper, or a bigger debt; 5. Level .

Common Misconceptions

Misconception	How to pre-empt it
” because .”	Confront it directly in the warmup. Always resolve comparisons by position on the line, never by ignoring the sign.
Ordering negatives ascending as .	Have students physically plot before ordering, and read the finished line left to right.
Reading as “minus five” and confusing it with subtraction.	Model the language: “negative five” for the number, “minus” only for the operation.
Believing is negative, or that is positive.	Zero is neither. It is the boundary point.
Confusing absolute value with the number itself: $	-8
Thinking there is a “smallest” integer.	Ask: what is the smallest negative number? The line continues forever in both directions.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). How many integers lie strictly between and ?

Answer

From to inclusive: — 11 integers.

E2 (AMC Junior style). The temperature at midnight was C. It rose by C each hour for four hours, then fell by C. What was the final temperature?

Answer

E3 (Challenge). List all integers with . How many are there?

Answer

— 7 integers.

E4 (Challenge). Two integers have a sum of and are apart on the number line. Find both.

Answer

Let them be and with and . Adding: , so and . Check: ✓ and ✓

Homework

1. Order from smallest to largest: (a) (b) (c) .
2. Insert or : (a) (b) (c) (d) .
3. Evaluate: (a) (b) (c) .
4. Write the opposite of: (a) (b) (c) .
5. The lowest recorded temperature in Australia is about C (Charlotte Pass, NSW); in Antarctica it is about C. How much colder is the Antarctic record?
6. Reasoning. Explain why is greater than , even though is much greater than .
7. Challenge. A submarine at m rises m, then dives m. What is its final depth? Write the calculation as a single expression.

Answers: Q5 — C colder. Q7 — m.