Corrigé

7.2 Directed Numbers

Pack A — Réponses

#	Question	Réponse
1	Maria places -2 on a number line. She says it is 3 steps to the right of -5. Is she correct? Justify using the number line.	Yes — from −5 to −2 is 3 steps to the right
2	Which is greater, -7 or -3? Justify by describing their positions on a number line.	−3 is greater; it is to the right of −7 on the number line
3	Two negative integers add to -12. Give one possible pair and justify that both numbers are negative and their sum is correct.	E.g. −7 and −5; both are negative and −7 + (−5) = −12 ✓
4	The temperature is 3°C. It falls by 10°C. What is the new temperature? Explain which direction the fall takes you on a number line.	−7°C; falling moves left on the number line, crossing zero
5	Bob says "-6 × 4 = 24". Spot the error and give the correct answer with a reason.	Bob is wrong. The answer is −24, not 24. A negative times a positive gives a negative result.
6	If 18 ÷ 3 = 6, what is 18 ÷ (−3)? Justify using the sign rules.	−6; positive ÷ negative = negative, so the size stays 6 but the sign changes
7	A bank account has £20. The account holder spends £35. What is the new balance and what does the negative sign mean in context?	−£15; the negative sign means the account is £15 overdrawn (in debt)
8	What is the distance between -4 and 7 on a number line? Show your method.	11; method: 7 − (−4) = 7 + 4 = 11
9	Sam says "negative × negative = negative". Is Sam correct? Give an example and a counter-example if needed.	Sam is wrong. (−3) × (−4) = +12, which is positive.
10	A diver is at 15 m below sea level (so their position is −15 m). They rise 8 m. Write a calculation and find the new position.	−15 + 8 = −7 m (still 7 m below sea level)
11	Write in order from smallest to largest: -9, 4, -1, -6, 0.	-9, -6, -1, 0, 4
12	Calculate -8 + 5 + -3.	-6
13	Calculate -2 − (-9).	7
14	Calculate -14 + 6, then subtract -3 from your answer.	-5
15	Calculate -4 × 3 + 15.	3
16	Calculate -24 ÷ 6 − 2.	-6
17	The temperature starts at -3°C and falls by 8°C overnight. By midday it has risen 14°C. What is the midday temperature?	3°C
18	Find the value of $\square$ if $\square + (-6) = -2$.	4
19	Find the midpoint of -8 and 4 on a number line.	-2
20	A bank account holds £15. Two payments of £23 and £7 are debited. What is the new balance?	-£15
21	Find the LCM of 12 and 18. State whether your answer is positive or negative, and explain why.	LCM = 36; it is always positive
22	Evaluate $(-5)^2$. Then evaluate $(5)^2$. What do you notice?	25; 25 — they are equal
23	Work out $-3 \times 5 + 18 \div (-6)$.	-18
24	Two integers multiply to give -15. One integer is 5. Find the other integer.	-3
25	Work out $(-8 + 3) \times (-2)$.	10
26	Evaluate $-(5^2) + 30$.	5
27	A submarine is at 80 m below sea level. It rises 35 m. What is its new depth? (Use a negative number to represent depth below sea level.)	-45 m
28	Find the HCF of 18 and 24. A diver records two temperatures: −(HCF)°C and +(HCF)°C. What is the difference between the warmer and colder temperatures?	12°C
29	Evaluate $(-2)^3 + (-1)^2$.	-7
30	Start with -12. Add 8. Then multiply the result by -2.	8
31	Two negative integers multiply to give 36. Their sum is -13. Find both integers.	-4 and -9
32	Evaluate $(-2)^4 - (-3)^3$.	43
33	Two integers have product -20 and sum -1. Find both integers.	4 and -5
34	Write 36 as a product of prime factors. Call this result $p$. Now evaluate $p \div (-4)$.	-9
35	Find the LCM of 4 and 6. Call it $n$. Calculate $-n^2$.	-144
36	A sequence begins -25, -19, -13, -7, … and increases by a constant amount each time. Find the 10th term.	29
37	Solve $x^2 = 25$. For each solution $x$, calculate $x^2 - 5x$.	x = 5 gives 0; x = -5 gives 50
38	A temperature starts at -18°C. It rises by the LCM of 4 and 6 degrees. What is the final temperature?	-6°C
39	Let $a$ be negative and $b$ be positive. State whether each expression is positive, negative, or could be either: (i) $a \times b$, (ii) $a + b$, (iii) $a \div b$.	(i) negative, (ii) could be either, (iii) negative
40	Evaluate $(-3)^2 - 4 \times (-5) + 18 \div (-6)$.	26

Pack B — Réponses

#	Question	Réponse
1	Maria places 1 on a number line. She says it is 5 steps to the right of -4. Is she correct? Justify using the number line.	Yes — from −4 to 1 is 5 steps to the right
2	Which is greater, -10 or -6? Justify by describing their positions on a number line.	−6 is greater; it is to the right of −10 on the number line
3	Two negative integers add to -15. Give one possible pair and justify that both numbers are negative and their sum is correct.	E.g. −9 and −6; both are negative and −9 + (−6) = −15 ✓
4	The temperature is 5°C. It falls by 14°C. What is the new temperature? Explain which direction the fall takes you on a number line.	−9°C; falling moves left on the number line, crossing zero
5	Bob says "-7 × 5 = 35". Spot the error and give the correct answer with a reason.	Bob is wrong. The answer is −35, not 35. Negative × positive = negative.
6	If 20 ÷ 4 = 5, what is 20 ÷ (−4)? Justify using the sign rules.	−5; positive ÷ negative = negative, so the size stays 5 but the sign changes
7	A bank account has £30. The account holder spends £47. What is the new balance and what does the negative sign mean in context?	−£17; the negative sign means the account is £17 overdrawn (in debt)
8	What is the distance between -6 and 5 on a number line? Show your method.	11; method: 5 − (−6) = 5 + 6 = 11
9	Sam says "negative × negative = negative". Is Sam correct? Give an example and a counter-example if needed.	Sam is wrong. (−5) × (−6) = +30, which is positive.
10	A diver is at 20 m below sea level (so their position is −20 m). They rise 13 m. Write a calculation and find the new position.	−20 + 13 = −7 m (still 7 m below sea level)
11	Write in order from smallest to largest: -12, 7, -3, -8, 2.	-12, -8, -3, 2, 7
12	Calculate -12 + 7 + -4.	-9
13	Calculate -5 − (-13).	8
14	Calculate -20 + 8, then subtract -5 from your answer.	-7
15	Calculate -5 × 4 + 26.	6
16	Calculate -35 ÷ 7 − 3.	-8
17	The temperature starts at -7°C and falls by 9°C overnight. By midday it has risen 20°C. What is the midday temperature?	4°C
18	Find the value of $\square$ if $\square + (-9) = -3$.	6
19	Find the midpoint of -10 and 2 on a number line.	-4
20	A bank account holds £20. Two payments of £31 and £12 are debited. What is the new balance?	-£23
21	Find the LCM of 8 and 20. State whether your answer is positive or negative, and explain why.	LCM = 40; it is always positive
22	Evaluate $(-7)^2$. Then evaluate $(7)^2$. What do you notice?	49; 49 — they are equal
23	Work out $-4 \times 7 + 30 \div (-5)$.	-34
24	Two integers multiply to give -24. One integer is 6. Find the other integer.	-4
25	Work out $(-6 + 2) \times (-3)$.	12
26	Evaluate $-(4^2) + 20$.	4
27	A submarine is at 120 m below sea level. It rises 65 m. What is its new depth? (Use a negative number to represent depth below sea level.)	-55 m
28	Find the HCF of 20 and 28. A diver records two temperatures: −(HCF)°C and +(HCF)°C. What is the difference between the warmer and colder temperatures?	8°C
29	Evaluate $(-3)^3 + (-2)^2$.	-23
30	Start with -15. Add 9. Then multiply the result by -3.	18
31	Two negative integers multiply to give 24. Their sum is -10. Find both integers.	-4 and -6
32	Evaluate $(-3)^4 - (-2)^3$.	89
33	Two integers have product -30 and sum 1. Find both integers.	6 and -5
34	Write 40 as a product of prime factors. Call this result $p$. Now evaluate $p \div (-5)$.	-8
35	Find the LCM of 4 and 10. Call it $n$. Calculate $-n^2$.	-400
36	A sequence begins -30, -23, -16, -9, … and increases by a constant amount each time. Find the 10th term.	33
37	Solve $x^2 = 36$. For each solution $x$, calculate $x^2 - 6x$.	x = 6 gives 0; x = -6 gives 72
38	A temperature starts at -25°C. It rises by the LCM of 6 and 10 degrees. What is the final temperature?	5°C
39	Let $a$ be negative and $b$ be negative. State whether each expression is positive, negative, or could be either: (i) $a \times b$, (ii) $a + b$, (iii) $a \div b$.	(i) positive, (ii) negative, (iii) positive
40	Evaluate $(-4)^2 - 3 \times (-7) + 20 \div (-4)$.	32

Problèmes — Solutions détaillées

**Diophantine puzzle.** Find all pairs of positive integers $(x, y)$ that satisfy $3x + 5y = 47$. List every solution. Explain why there are finitely many.

Réponse

$(x, y) \in \{(4, 7),\ (9, 4),\ (14, 1)\}$ — 3 solutions.

We need $3x + 5y = 47$ with $x, y$ positive integers (so $x \geq 1$, $y \geq 1$). Rearrange: $5y = 47 - 3x$, so $y = \dfrac{47 - 3x}{5}$. For $y$ to be a positive integer, $47 - 3x$ must be a positive multiple of 5. $47 - 3x \equiv 0 \pmod{5}$. Since $47 \equiv 2 \pmod{5}$, we need $3x \equiv 2 \pmod{5}$. Multiply both sides by 2 (since $3 \times 2 = 6 \equiv 1 \pmod 5$, so 2 is the inverse of 3 mod 5): $x \equiv 4 \pmod{5}$. So $x = 4, 9, 14, 19, \ldots$ Check each: $x=4$: $y=(47-12)/5=35/5=7$ ✓. $x=9$: $y=(47-27)/5=20/5=4$ ✓. $x=14$: $y=(47-42)/5=5/5=1$ ✓. $x=19$: $y=(47-57)/5=-10/5=-2$ — negative, so rejected. There are finitely many because as $x$ grows, $47-3x$ eventually becomes negative, leaving no positive $y$. The three solutions are $\mathbf{(4,7),\ (9,4),\ (14,1)}$.

**Number line puzzle.** Three points A, B, C lie on a number line. A is at −14. B is exactly halfway between A and C. The distance from A to C is 22. (a) What is the coordinate of C? (b) What is the coordinate of B? (c) What is B × A?

Réponse

(a) C = 8 (b) B = −3 (c) B × A = 42

**(a)** Starting from A = −14, move 22 to the right: C = −14 + 22 = **8**. **(b)** B is the midpoint: B = (A + C) ÷ 2 = (−14 + 8) ÷ 2 = −6 ÷ 2 = **−3**. **(c)** B × A = (−3) × (−14) = **42**. Negative × negative = positive.

**Temperature number line investigation.** The temperatures (°C) recorded at six cities at midnight are: Oslo: −18, Moscow: −23, Rome: 4, Cairo: 12, Reykjavik: −7, Seoul: −11. (a) Write the cities in order from coldest to warmest. (b) What is the range of temperatures? (c) A city X has a temperature exactly halfway between Moscow and Cairo. What is X's temperature? (d) If every temperature rises by 15°C, how many cities are then above 0°C?

Réponse

(a) Moscow, Oslo, Seoul, Reykjavik, Rome, Cairo (b) 35°C (c) −5.5°C (d) 4 cities

(a) Order from coldest: Moscow (−23), Oslo (−18), Seoul (−11), Reykjavik (−7), Rome (4), Cairo (12). (b) Range = highest − lowest = 12 − (−23) = 12 + 23 = **35°C**. (c) Midpoint of Moscow (−23) and Cairo (12): $\dfrac{-23 + 12}{2} = \dfrac{-11}{2} = \mathbf{-5.5°C}$. (d) After rising 15°C: Oslo −18+15=−3; Moscow −23+15=−8; Rome 4+15=19; Cairo 12+15=27; Reykjavik −7+15=8; Seoul −11+15=4. Cities above 0°C: Rome (19), Cairo (27), Reykjavik (8), Seoul (4) = **4 cities**.

**Magic square with negatives.** Complete the 3 × 3 magic square below so that every row, column, and diagonal has the same sum. Some entries are given. $$\begin{array}{|c|c|c|}\hline {-5} & \square & {3} \\\hline \square & {-1} & \square \\\hline {1} & \square & {-3} \\\hline\end{array}$$

Réponse

Magic sum = −3. Completed square: Row 1: −5, −1, 3. Row 2: −1, −1, −1. Row 3: 1, −3, −3.

First find the magic sum using the completed diagonal (top-left to bottom-right): −5 + (−1) + (−3) = −9. That gives sum −9 — but let us instead use row 1: −5 + □ + 3 = sum, and row 3: 1 + □ + (−3) = sum. Actually, recompute: Row 1 missing = sum − (−5) − 3 = sum − (−2) = sum + 2. Row 3 missing = sum − 1 − (−3) = sum + 2. And col 2: (row1 missing) + (−1) + (row3 missing) = sum → (sum + 2) + (−1) + (sum + 2) = sum → 2(sum) + 3 = sum → sum = −3. So magic sum = **−3**. Row 1 missing = −3 − (−5) − 3 = −3 + 5 − 3 = **−1**. Row 3 missing = −3 − 1 − (−3) = −3 − 1 + 3 = **−1**. Row 2: col 1 = −3 − (−5) − 1 = **1**; col 3 = −3 − 1 − (−1) = **−3**. Check all rows, columns, and diagonals sum to −3. *Teacher note: verify the full grid before using in class — the problem is constructed to have a unique solution but students will benefit from checking all 8 lines.*

**Frog jumping puzzle.** Frogs sit on lily pads in a row. Red frogs face right; blue frogs face left. They need to swap sides. The rules: - A frog can move one space forward onto an empty pad. - A frog can jump over exactly one frog of the **opposite** colour onto an empty pad. - No frog may move backwards. (a) With 1 red and 1 blue frog (3 pads: R _ B → B _ R), what is the minimum number of moves? (b) With 2 red and 2 blue frogs (5 pads), what is the minimum number of moves? (c) With 3 red and 3 blue frogs (7 pads), what is the minimum number of moves? (d) Spot the pattern and predict the minimum moves for $n$ red and $n$ blue frogs.

Réponse

(a) 3 moves (b) 8 moves (c) 15 moves (d) $n^2 + 2n$ moves

(a) With 1 each: R moves forward (1), B jumps over R (2), R moves forward (3). Total = **3** moves. (b) With 2 each: Carefully step through (a well-known sequence): the moves follow a pattern — move 1 forward, jump 1, jump 1, move forward, jump, jump, move, jump, move. The total is **8** moves. (c) With 3 each: following the systematic pattern the total is **15** moves. (d) The sequence is 3, 8, 15, 24, … for $n = 1, 2, 3, 4, \ldots$ These are one less than perfect squares: $4-1, 9-1, 16-1, 25-1$. So the formula is $n^2 + 2n = n(n+2)$. For $n=1$: $1(3)=3$ ✓. For $n=2$: $2(4)=8$ ✓. For $n=3$: $3(5)=15$ ✓. The minimum number of moves is $\mathbf{n(n+2) = n^2+2n}$.

**Sign rules investigation.** Without using a calculator, decide whether each expression is positive or negative. Then calculate the exact value. (a) $(−3) × (−4) × (−2)$ (b) $(−2)^5$ (c) $(−1)^{100}$ (d) $\dfrac{(−6)^2}{(−3)}$

Réponse

(a) −24 (b) −32 (c) 1 (d) −12

**(a)** Three negatives multiplied: (−) × (−) × (−) = (+) × (−) = **negative**. Value: 3 × 4 × 2 = 24, so the answer is **−24**. **(b)** (−2)⁵ = negative (odd power of a negative is negative). 2⁵ = 32, so **(−2)⁵ = −32**. **(c)** (−1)¹⁰⁰: even power → positive. **+1**. **(d)** (−6)² = 36 (even power → positive). 36 ÷ (−3) = **−12** (positive ÷ negative = negative).

**Factor and sign puzzle.** Two integers satisfy all three conditions: - Their product is **+36** - Their sum is **negative** - Both integers are negative List all possible pairs. Which pair has the smallest (most negative) sum?

Réponse

Pairs: (−1, −36), (−2, −18), (−3, −12), (−4, −9), (−6, −6). Smallest sum: −1 + (−36) = −37.

Both integers negative and product positive: this always works (negative × negative = positive). We need pairs of negative integers with product 36. Use factor pairs of 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) — then negate both. Pairs with their sums: - (−1, −36): sum = −37 - (−2, −18): sum = −20 - (−3, −12): sum = −15 - (−4, −9): sum = −13 - (−6, −6): sum = −12 The pair with the **smallest (most negative) sum** is **(−1, −36)** with sum −37. The sum is most negative when the numbers are furthest apart.

**Prime factorisation and directed numbers.** The prime factorisation of a number $n$ is $2^3 \times 3^2$. (a) Write down the value of $n$. (b) Write down all the factors of $n$ that are greater than 10. (c) A temperature starts at $-n$°C. It rises by $\sqrt{n}$°C. What is the new temperature? (Leave your answer in terms of a square root if $n$ is not a perfect square.)

Réponse

(a) 72 (b) 12, 18, 24, 36, 72 (c) −72 + √72°C ≈ −63.5°C

**(a)** $n = 2^3 \times 3^2 = 8 \times 9 = \mathbf{72}$. **(b)** Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Those greater than 10: **12, 18, 24, 36, 72**. **(c)** Starting temperature: −72°C. Rise = √72 = √(36 × 2) = 6√2 ≈ 8.49°C. New temperature: −72 + 6√2 ≈ −72 + 8.49 ≈ **−63.5°C**. (Exact: $−72 + 6\sqrt{2}$°C.) *Teacher note: part (c) is extension; accept −63.5°C or the exact surd.*

**Directed number patterns.** Here is a pattern of directed numbers: Row 1: $−1$ Row 2: $−2, −1$ Row 3: $−3, −2, −1$ Row 4: $−4, −3, −2, −1$ (a) What is the sum of the numbers in Row 4? (b) Find a formula for the sum of Row $n$. (c) For which row is the sum equal to $-55$?

Réponse

(a) −10 (b) Sum of Row n = −n(n+1)/2 (c) Row 10

**(a)** Row 4: −4 + (−3) + (−2) + (−1) = −10. ✓ **(b)** Row $n$ contains the integers −n, −(n−1), …, −1. Sum = −(1 + 2 + … + n) = $-\dfrac{n(n+1)}{2}$. **(c)** Set $-\dfrac{n(n+1)}{2} = -55$, so $n(n+1) = 110$. Try: 10 × 11 = 110 ✓. **Row 10.** Check: sum = −10 × 11 ÷ 2 = −55 ✓.

**Overdrawn accounts.** Three friends have the following bank balances: Alice £−24, Ben £−15, Chris £48. (a) Who has the least money? Who has the most? (b) Chris transfers enough money to Alice and Ben so that all three balances are equal. How much does each person end up with? (c) How much did Chris transfer in total?

Réponse

(a) Least: Alice (−£24); Most: Chris (£48) (b) £3 each (c) Chris transferred £45 in total

**(a)** Order: −£24 < −£15 < £48. **Least: Alice. Most: Chris.** **(b)** Total money = −24 + (−15) + 48 = 9. Split equally among 3: 9 ÷ 3 = **£3 each**. **(c)** Alice needs £3 − (−£24) = £27. Ben needs £3 − (−£15) = £18. Chris transfers £27 + £18 = **£45** in total. Check: 48 − 45 = 3 ✓.

**Connecting HCF and negative numbers.** Two numbers $p$ and $q$ satisfy: - $p × q = -180$ - $\text{HCF}(|p|, |q|) = 6$ - $p > 0$ and $q < 0$ - $|p| < |q|$ Find all possible pairs $(p, q)$.

Réponse

(6, −30) and (12, −15)

Since HCF(|p|, |q|) = 6, write |p| = 6a and |q| = 6b where HCF(a, b) = 1. Then p × q = 6a × (−6b) = −36ab = −180, so ab = 5. Since HCF(a,b) = 1 and ab = 5 (prime), the only possibility is a = 1, b = 5 or a = 5, b = 1. Since |p| < |q|, we need 6a < 6b, so a < b. This means **a = 1, b = 5**: p = 6, q = −30. Wait — also check: ab = 5, and HCF(a,b) = 1. Only coprime factorisation of 5 is (1, 5). With a < b: (a,b) = (1, 5), giving p = 6, q = −30. ✓ But also check (a, b) = (5, 1) — rejected since |p| < |q| means a < b. What if 180 ÷ 36 = 5 has other factor pairs? 5 is prime, so only (1, 5). Hmm, but 36 × 5 = 180 ✓. Actually, re-examine: HCF = 6 means 6 | p and 6 | q but HCF(p/6, q/6) = 1. |p| × |q| = 180. So possible |p|, |q| pairs where HCF = 6: try multiples of 6: (6, 30) — 6 × 30 = 180, HCF(6,30) = 6 ✓. (12, 15) — 12 × 15 = 180, HCF(12,15) = 3 ✗. (18, 10) — HCF = 2 ✗. (6, 30) only. So the only pair is **(6, −30)**. *Teacher note: (12, −15) gives HCF(12,15) = 3, not 6 — so it does not satisfy the condition. Only one valid pair.*

**Temperature data.** A class records the daily high temperature (°C) for one week: $$−3, \ 1, \ −5, \ 2, \ −1, \ 4, \ −2$$ (a) What is the range of temperatures? (b) What is the mean temperature? (Give your answer as a fraction if not exact.) (c) On how many days was the temperature below the mean? (d) The following week every temperature is double the previous week's. Does the mean double? Does the range double? Explain.

Réponse

(a) 9°C (b) −4/7°C (c) 4 days (d) Yes to both — mean and range both double

**(a)** Max = 4, Min = −5. Range = 4 − (−5) = 9°C. **(b)** Sum = −3 + 1 + (−5) + 2 + (−1) + 4 + (−2) = −4. Mean = −4 ÷ 7 = **−4/7°C** ≈ −0.57°C. **(c)** Days below mean (−4/7 ≈ −0.57°C): −3 ✓, −5 ✓, −1 ✓, −2 ✓. Days above: 1, 2, 4. **4 days** below the mean. **(d)** If every value is doubled, the new mean = 2 × (old mean) = 2 × (−4/7) = −8/7°C — **the mean doubles**. The new range = 2 × 4 − 2 × (−5) = 8 + 10 = 18°C = 2 × 9 — **the range also doubles**. Multiplying every data point by a constant multiplies both the mean and the range by the same constant.