013i - Subtracting Integers

Lesson 13 — Subtracting Integers

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

Learning Intentions

• To understand subtraction of an integer as addition of its opposite.
• To calculate differences between integers, including across zero.

Success Criteria

I can:

1. Rewrite any subtraction as an addition of the opposite, e.g. .
2. Calculate the difference between two integers using a number line.
3. Explain why subtracting a negative gives a larger result.
4. Find the distance between two integers on the number line.

Warmup

(6 minutes — pattern completion, mini whiteboards)

Complete each sequence and describe what you notice.

Key observation: as the number being subtracted decreases by , the answer increases by . Continuing the pattern gives and .

Prompt: So what does subtracting a negative appear to do? (It adds.) We will now see why.

Activities

Activity 1 — Explicit Instruction: Subtract = Add the opposite (12 min)

The rule:

To subtract an integer, add its opposite.

Two supporting models — use both.

Model A: taking away zero pairs. To calculate : start with yellow counters. You must remove reds, but there are none. Add three zero pairs (which changes nothing), then remove the three reds. Five yellows remain.

Model B: difference as distance and direction. asks “how far, and in which direction, do I travel from to ?” From to is steps to the right, so .

I do: Rewrite then evaluate, showing the middle line every time.

We do: , , , .

(Answers: , , , .)

You do: Rewrite as addition, then evaluate.

(Answers: , , , , , .)

Activity 2 — Distance between Integers (8 min)

Definition. The distance between two integers is always positive, and equals the larger minus the smaller.

Confirm on the number line: steps from to , then more to .

Practice: Find the distance between each pair.

1. and
2. and
3. and
4. and

(Answers: , , , .)

Discussion: Why is the distance between and only , when both numbers are “big”? (Because they sit close together on the line — size and separation are different questions.)

Activity 3 — Inquiry: Temperature Ranges (10 min)

Pairs.

A weather station records these daily minimum and maximum temperatures:

Day	Min (°C)	Max (°C)
Mon
Tue
Wed
Thu

1. Calculate the daily range (max minus min) for each day.
2. Which day had the largest range?
3. Which day was coldest overall? Is that the same as the largest range?
4. On Friday the minimum was C colder than Thursday’s minimum, and the range was C. Find Friday’s minimum and maximum.

Socratic scaffolding for Q4:

Prompt	Purpose
Understand: what are the unknowns?	Two — Friday’s minimum and Friday’s maximum.
Can you find one before the other?	Yes: the minimum depends only on Thursday’s.
What does ” colder than ” mean?	Move further left: .
Carry it out	C.
Now use the range. What does range mean?	Max min , so max min .
Carry it out	C.
Looking back	Check: ✓

Answers: 1. Mon , Tue , Wed , Thu ; 2. Wednesday; 3. Thursday was coldest (min ), but had the smallest range — coldness and range are independent; 4. Min C, max C.

Checks for Understanding

(5 minutes — exit ticket)

1. Calculate: (a) (b) (c) (d) .
2. Rewrite as an addition, then evaluate.
3. Find the distance between and .
4. Reasoning. Explain why is larger than , using either the counter model or the number line.
5. The temperature at the summit is C; at the base it is C. How much warmer is the base?

Answers: 1. , , , ; 2. ; 3. ; 4. Subtracting means removing four negatives, which leaves the total higher — equivalently, adding the opposite, ; 5. C.

Common Misconceptions

Misconception	How to pre-empt it
— treating the minus signs as cancelling into a subtraction.	Use the warmup pattern before stating the rule, so the result is discovered rather than asserted.
Applying “two negatives make a positive” to .	Emphasise that the rule concerns subtracting a negative, not adding two negatives. Contrast both on the board.
— adding instead of adding the opposite.	Require the rewriting step: . No mental shortcuts until fluent.
Giving a negative distance.	Distance is never negative. Always take larger minus smaller, and sanity-check against the number line.
Reversing the order: computing for a range.	State the formula explicitly as max min, and check the answer is positive.
Dropping brackets: writing .	Insist on throughout.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is ?

Answer

E2 (AMC Junior style). The difference between two integers is . One of them is . What are the two possible values of the other?

Answer

Either or . Both are away from .

E3 (Challenge). On a number line, point is at and point is at . Point is exactly halfway between them. Where is ?

Answer

The distance is , so half is . Starting from : . So is at . (Equivalently, the mean: .)

E4 (Challenge). Five integers have a sum of . Four of them are , , and . What is the fifth?

Answer

Homework

1. Calculate: (a) (b) (c) (d) (e) (f) .
2. Rewrite as an addition, then evaluate: (a) (b) (c) .
3. Find the distance between: (a) and (b) and (c) and .
4. Evaluate: (a) (b) (c) .
5. A mine shaft reaches m. A drilling rig at the surface extends a further m below the shaft floor. What is the new depth?
6. The highest point in Australia is Mt Kosciuszko at m; the lowest is Lake Eyre at about m. What is the difference in elevation?
7. Reasoning. Priya says “subtracting always makes a number smaller.” Give a counterexample and explain when her statement is true.
8. Challenge. and are integers with and . Find and .

Answers: Q1 — . Q2 — (a) (b) (c) . Q3 — . Q4 — (a) (b) (c) . Q5 — m. Q6 — m. Q7 — ; true only when subtracting a positive. Q8 — adding the equations gives , so and .