012i - Adding Integers

Lesson 12 — Adding Integers

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

Learning Intentions

• To add integers using the number line and the zero-pair model.
• To recognise when a sum of integers will be positive, negative or zero.

Success Criteria

I can:

1. Model an integer addition on a number line, starting at the first number and moving by the second.
2. Use zero pairs to simplify an addition.
3. Add two integers of the same sign and of different signs.
4. Predict the sign of a sum before calculating it.

Warmup

(5 minutes — number line moves, mini whiteboards)

Start at the given number and follow the instruction. Where do you land?

1. Start at , move right.
2. Start at , move left.
3. Start at , move right.
4. Start at , move left.
5. Start at , move right.

(Answers: , , , , .)

Bridging question: Which direction does “adding a positive” move you? What about “adding a negative”?

Activities

Activity 1 — Explicit Instruction: the Number Line Model (10 min)

The rule, stated once and used throughout:

Start at the first number. Adding a positive moves right. Adding a negative moves left.

I do: Model each on a large number line, narrating the movement.

Bracket convention: write , not . The brackets keep the sign attached to its number.

We do: , , , .

(Answers: , , , .)

Activity 2 — The Zero-pair Model (10 min)

Use two-colour counters — red for negative, yellow for positive.

Core idea: one positive and one negative cancel to make zero. This is a zero pair.

I do: Model .

• Place red counters and yellow counters.
• Pair off reds with yellows — three zero pairs, removed.
• reds remain, so .

Deriving the general rules — do not simply state them. After several counter examples, ask what pattern the class notices, then record:

Case	What happens	Sign of the answer
Same signs	Distances add	Keeps the shared sign
Different signs	Distances subtract	Sign of the number further from zero
Opposites	Everything cancels	Zero

You do: Predict the sign first, then calculate.

(Answers: , , , , , .)

Activity 3 — Inquiry: the Bank Account (10 min)

Pairs.

Maya’s account starts at 40$. Over one week the following happen, in order:

deposit 25$70$30$50$.

1. Write each transaction as a positive or negative integer.
2. Track the balance after each transaction.
3. What is the final balance?
4. At what point (if any) is the account overdrawn?
5. What single deposit at the start would have kept the account from ever going below zero?

Socratic scaffolding for Q5:

Prompt	Purpose
Understand: what is the unknown?	An extra starting amount.
What is the condition?	The running balance never drops below zero.
Which moment matters most?	The lowest point the balance reaches.
Find that lowest point.	Track the running total and read off the minimum.
Devise a plan	Add enough at the start to lift that minimum to zero.
Carry it out	Minimum balance is , so an extra 35$ is needed.
Looking back	Re-run the whole week with the extra 35$ and check no balance is negative.

Working:

Answers: 1. ; 2. ; 3. 15$35$.

Checks for Understanding

(5 minutes — exit ticket)

1. Calculate: (a) (b) (c) (d) .
2. Without calculating, state whether is positive or negative. Explain.
3. Draw a number line to show .
4. Reasoning. A student writes . Identify the error and give the correct answer.
5. The temperature is C and rises by C. What is the new temperature?

Answers: 1. , , , ; 2. Negative — the signs differ and , so the answer takes the sign of ; 3. Start at , six steps right, landing on ; 4. The student added the distances but dropped the sign; both are negative, so the answer is ; 5. C.

Common Misconceptions

Misconception	How to pre-empt it
— treating two negatives as making a positive.	That rule belongs to multiplication, not addition. Model with counters: seven red counters are clearly .
— adding distances when signs differ.	Return to zero pairs. Adding a positive must move you towards zero, so the answer gets smaller in size.
Losing a sign when copying, e.g. writing for .	Insist on brackets around every negative term.
Believing the answer always takes the sign of the first number.	Contrast with .
Assuming adding always makes things bigger.	Adding a negative makes the result smaller. Anchor to the bank-account model.
Confusion when zero appears: .	Zero is the starting point; you simply move left to .

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is the value of ?

Answer

Pair the terms: .

E2 (AMC Junior style). The sum of five consecutive integers is . What is the largest of them?

Answer

The middle number is , so the integers are . The largest is .

E3 (Challenge). Fill each box with a different integer from so that the sum is as small as possible, then as large as possible. What are the two totals?

Answer

All four must be used, so both totals are the same: . A useful reminder that addition is commutative — order does not change a sum.

E4 (Challenge). In a magic square using the integers through exactly once, every row, column and diagonal has the same sum. What must that sum be?

Answer

The nine integers total . Three rows share this total equally, so each row sums to .

Homework

1. Calculate: (a) (b) (c) (d) (e) (f) .
2. Fill in the missing number: (a) (b) (c) .
3. Evaluate: (a) (b) (c) .
4. A diver at m ascends m. Write the calculation and state the new depth.
5. Overnight the temperature fell from C by C. Write this as an integer addition and state the new temperature.
6. Reasoning. Explain why for any integer , using the zero-pair model.
7. Challenge. A game starts at points. You gain , lose , gain , lose , then gain . What is the final score, and what was the lowest score during the game?

Answers: Q1 — . Q2 — (a) (b) (c) . Q3 — (a) (b) (c) . Q4 — m. Q5 — C. Q7 — running scores ; final , lowest .