MYP 3 · Maths

NUMBER

Integers

QUESTION 1 [3 marks] — Criterion A Medium
The temperature at the summit of a mountain at 6 am was $-14^\circ$C. By 2 pm it had risen by $19^\circ$C, then fell by $8^\circ$C by 8 pm.
a. Find the temperature at 2 pm.
[1]
Show Solution
$$-14 + 19 = 5^\circ\text{C}$$
b. Find the temperature at 8 pm.
[1]
Show Solution
$$5 - 8 = -3^\circ\text{C}$$
c. Find the overall change in temperature from 6 am to 8 pm.
[1]
Show Solution
$$-3 - (-14) = 11^\circ\text{C increase}$$
QUESTION 2 [4 marks] — Criterion A Medium
A submarine is at a depth of $-120$ m (120 m below sea level). It rises $45$ m, then descends $68$ m.
a. Find the submarine's new depth after both movements, showing each step.
[2]
Show Solution
After rising: $-120+45=-75$ m. After descending: $-75-68=-143$ m.
b. A second submarine starts at $-95$ m and needs to reach the same final depth as the first submarine. Find how far it must descend.
[2]
Show Solution
$$-143 - (-95) = -48 \text{ m, i.e. it must descend a further 48 m}$$
QUESTION 3 [3 marks] — Criterion A Medium
Evaluate each of the following, showing full working:
a. $(-6) + (-9) - (-14)$
[1]
Show Solution
$$-6-9+14 = -1$$
b. $(-3) \times (-4) \times (-2)$
[1]
Show Solution
$$12 \times (-2) = -24$$
c. $\dfrac{(-28)}{(-4)} + (-7)$
[1]
Show Solution
$$7 + (-7) = 0$$
QUESTION 4 [5 marks] — Criterion B Medium
Investigate the sign of the result when multiplying several negative integers together.
a. Calculate $(-2)\times(-3)$, $(-2)\times(-3)\times(-4)$, and $(-2)\times(-3)\times(-4)\times(-5)$.
[3]
Show Solution
$(-2)\times(-3)=6$ (positive). $6\times(-4)=-24$ (negative). $-24\times(-5)=120$ (positive).
b. State a rule connecting the number of negative factors to the sign of the product.
[2]
Show Solution
If the number of negative factors is even, the product is positive. If the number of negative factors is odd, the product is negative.
QUESTION 5 [5 marks] — Criterion C Medium
A classmate says: "Subtracting a negative number always makes the answer bigger."
a. Test this claim with three different examples of your own, showing full working.
[3]
Show Solution
E.g. $5-(-3)=8$ (bigger than 5). $-2-(-6)=4$ (bigger than $-2$). $-10-(-1)=-9$ (bigger than $-10$). In every case the result is bigger than the starting number.
b. Explain, in your own words, why subtracting a negative number has this effect.
[2]
Show Solution
Subtracting a negative is the same as adding its positive opposite (e.g. $-(-3)=+3$), and adding a positive number always increases the value — so the claim is correct.
QUESTION 6 [4 marks] — Criterion D Medium
A company's profit/loss (in thousands of dollars) over 4 quarters was: Q1: $-18$, Q2: $+32$, Q3: $-9$, Q4: $+41$.
a. Find the company's total profit or loss for the year.
[2]
Show Solution
$$-18+32-9+41 = 46 \text{ (i.e. a profit of \$46{,}000)}$$
b. The company needs a total annual profit of at least \$40{,}000 to avoid layoffs. Based on your answer, will layoffs be avoided? Justify your answer.
[2]
Show Solution
Yes — the total profit of \$46{,}000 exceeds the required \$40{,}000, so layoffs will be avoided, with \$6{,}000 to spare.
QUESTION 7 [6 marks] — Criterion A Hard
A submarine's depth changes are recorded (negative = below sea level): starts at $-180$m, rises $65$m, descends $340$m, then rises $95$m.
a. Find the submarine's final depth.
[2]
Show Solution
$$-180+65-340+95 = -360 \text{ m}$$
b. The submarine's hull is only rated to withstand pressure safely down to $-500$m. During the ENTIRE sequence of movements (not just the final position), find the submarine's LOWEST point reached, and determine if the hull rating was ever exceeded.
[4]
Show Solution
Tracking each stage: start $-180$, after rise: $-115$, after descent: $-455$, after final rise: $-360$. The lowest point reached was $-455$m (after the descent stage), which is within the $-500$m safety rating — so the hull rating was never exceeded, even though the descent came close.
QUESTION 8 [8 marks] — Criterion B Hard
Investigate whether the parity (odd/even) of the sum of $n$ consecutive integers depends on the starting number, or only on $n$ itself — for the case where $n$ is EVEN.
a. Find the sum of the 2 consecutive integers starting at 5 (i.e. 5,6), and separately the sum starting at 10 (i.e. 10,11). State the parity of each sum.
[2]
Show Solution
$5+6=11$ (odd). $10+11=21$ (odd). Both sums are odd, despite starting from very different numbers.
b. Now find the sum of 4 consecutive integers starting at 3 (3,4,5,6), and separately starting at 7 (7,8,9,10). State the parity of each.
[2]
Show Solution
$3+4+5+6=18$ (even). $7+8+9+10=34$ (even). Both sums are even, again regardless of the different starting points.
c. Based on parts (a) and (b), conjecture a rule: for an EVEN count $n$ of consecutive integers, does the sum's parity depend on the starting number? Prove your conjecture algebraically, using $n$ consecutive integers starting at $a$: $a, a+1, \ldots, a+n-1$.
[4]
Show Solution
Conjecture: for even $n$, the sum's parity does NOT depend on the starting number $a$ — it depends only on $n$. Proof: the sum is $S=\frac{n}{2}(2a+n-1)$. Since $n$ is even, $n-1$ is odd, so $2a+n-1$ is always ODD (even plus odd is odd), regardless of $a$. Since an odd number times anything has the same parity as that 'anything', $S$ has the same parity as $\frac{n}{2}$ — which depends only on $n$, never on $a$. This proves the starting number never affects the sum's parity when $n$ is even.
QUESTION 9 [5 marks] — Criterion C Hard
A student says: 'A negative number times a negative number is positive, so negative numbers basically don't exist in the final answer — they always disappear.'
a. Define clearly: under what SPECIFIC condition does multiplying two negative numbers give a positive result, and give a counterexample showing negative numbers do NOT always 'disappear'.
[3]
Show Solution
Two negative numbers multiplied together give a POSITIVE result specifically because $(-a)\times(-b) = ab$ for positive $a,b$ (the two sign flips cancel). However, this is only true for MULTIPLICATION of exactly two negatives — e.g. $(-3)+(-5)=-8$ (addition of two negatives stays negative), and $(-3)\times(-3)\times(-3)=-27$ (three negatives multiplied gives a negative result), clearly disproving the student's overgeneralization.
b. State a precise, general rule for when a product of several negative numbers is positive versus negative.
[2]
Show Solution
A product of several negative numbers is POSITIVE if there is an EVEN number of negative factors, and NEGATIVE if there is an ODD number of negative factors.
QUESTION 10 [6 marks] — Criterion D Hard
A company's quarterly profit/loss (in thousands of dollars) over 2 years (8 quarters) was: $-45, 62, -18, 71, -30, 55, -12, 68$.
a. Find the company's total profit or loss over the full 2 years.
[2]
Show Solution
$$-45+62-18+71-30+55-12+68=151 \text{ (a profit of \$151{,}000)}$$
b. The company's board wants to identify a TREND: are losses (negative quarters) getting smaller in magnitude over time, suggesting improving stability? List the magnitude of EACH loss quarter in order, and evaluate whether there's a clear improving trend, being explicit about the limitations of drawing conclusions from only 4 data points.
[4]
Show Solution
Loss magnitudes in order: $45, 18, 30, 12$. There is a GENERAL downward trend in loss size (45?18?30?12), though it isn't perfectly smooth (30 is a slight increase from 18). With only 4 loss-quarters of data, this is too small a sample to confidently confirm a genuine long-term improving trend — more quarters of data would be needed to rule out random fluctuation versus a real underlying pattern.
QUESTION 11 [6 marks] — Criterion A Hard
A submarine's depth changes are recorded (negative = below sea level): starts at $-180$m, rises $65$m, descends $340$m, then rises $95$m.
a. Find the submarine's final depth.
[2]
Show Solution
$$-180+65-340+95 = -360 \text{ m}$$
b. The submarine's hull is only rated to withstand pressure safely down to $-500$m. During the ENTIRE sequence of movements (not just the final position), find the submarine's LOWEST point reached, and determine if the hull rating was ever exceeded.
[4]
Show Solution
Tracking each stage: start $-180$, after rise: $-115$, after descent: $-455$, after final rise: $-360$. The lowest point reached was $-455$m (after the descent stage), which is within the $-500$m safety rating — so the hull rating was never exceeded, even though the descent came close.
QUESTION 12 [8 marks] — Criterion B Hard
Investigate whether the parity (odd/even) of the sum of $n$ consecutive integers depends on the starting number, or only on $n$ itself — for the case where $n$ is EVEN.
a. Find the sum of the 2 consecutive integers starting at 5 (i.e. 5,6), and separately the sum starting at 10 (i.e. 10,11). State the parity of each sum.
[2]
Show Solution
$5+6=11$ (odd). $10+11=21$ (odd). Both sums are odd, despite starting from very different numbers.
b. Now find the sum of 4 consecutive integers starting at 3 (3,4,5,6), and separately starting at 7 (7,8,9,10). State the parity of each.
[2]
Show Solution
$3+4+5+6=18$ (even). $7+8+9+10=34$ (even). Both sums are even, again regardless of the different starting points.
c. Based on parts (a) and (b), conjecture a rule: for an EVEN count $n$ of consecutive integers, does the sum's parity depend on the starting number? Prove your conjecture algebraically, using $n$ consecutive integers starting at $a$: $a, a+1, \ldots, a+n-1$.
[4]
Show Solution
Conjecture: for even $n$, the sum's parity does NOT depend on the starting number $a$ — it depends only on $n$. Proof: the sum is $S=\frac{n}{2}(2a+n-1)$. Since $n$ is even, $n-1$ is odd, so $2a+n-1$ is always ODD (even plus odd is odd), regardless of $a$. Since an odd number times anything has the same parity as that 'anything', $S$ has the same parity as $\frac{n}{2}$ — which depends only on $n$, never on $a$. This proves the starting number never affects the sum's parity when $n$ is even.
QUESTION 13 [5 marks] — Criterion C Hard
A student says: 'A negative number times a negative number is positive, so negative numbers basically don't exist in the final answer — they always disappear.'
a. Define clearly: under what SPECIFIC condition does multiplying two negative numbers give a positive result, and give a counterexample showing negative numbers do NOT always 'disappear'.
[3]
Show Solution
Two negative numbers multiplied together give a POSITIVE result specifically because $(-a)\times(-b) = ab$ for positive $a,b$ (the two sign flips cancel). However, this is only true for MULTIPLICATION of exactly two negatives — e.g. $(-3)+(-5)=-8$ (addition of two negatives stays negative), and $(-3)\times(-3)\times(-3)=-27$ (three negatives multiplied gives a negative result), clearly disproving the student's overgeneralization.
b. State a precise, general rule for when a product of several negative numbers is positive versus negative.
[2]
Show Solution
A product of several negative numbers is POSITIVE if there is an EVEN number of negative factors, and NEGATIVE if there is an ODD number of negative factors.
QUESTION 14 [6 marks] — Criterion D Hard
A company's quarterly profit/loss (in thousands of dollars) over 2 years (8 quarters) was: $-45, 62, -18, 71, -30, 55, -12, 68$.
a. Find the company's total profit or loss over the full 2 years.
[2]
Show Solution
$$-45+62-18+71-30+55-12+68=151 \text{ (a profit of \$151{,}000)}$$
b. The company's board wants to identify a TREND: are losses (negative quarters) getting smaller in magnitude over time, suggesting improving stability? List the magnitude of EACH loss quarter in order, and evaluate whether there's a clear improving trend, being explicit about the limitations of drawing conclusions from only 4 data points.
[4]
Show Solution
Loss magnitudes in order: $45, 18, 30, 12$. There is a GENERAL downward trend in loss size (45?18?30?12), though it isn't perfectly smooth (30 is a slight increase from 18). With only 4 loss-quarters of data, this is too small a sample to confidently confirm a genuine long-term improving trend — more quarters of data would be needed to rule out random fluctuation versus a real underlying pattern.