Secondary 1 Integers and Number Sets Worksheet

All 10 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Absolute Value Notation — Secondary 1, 2 and 3

   Five weather stations in the Saguenay report these overnight lows, in ${}^{\circ }\text{C}$: $$-14,6,-3,0,11.$$

   1. Evaluate $|-14|$, $\left|6\right|$ and $|-3|$.
   2. Which reading is the farthest from $0^{\circ }\text{C}$, and which reading other than $0$ is the closest to it? Say how you decided.
2. Q2Adding and Subtracting Positive and Negative Numbers

   Noah's bank account is overdrawn at the start of the week, so its balance is $-35$ dollars. During the week he deposits $120 on Monday, spends $48 on Tuesday, spends $61 on Thursday and deposits $44 on Friday. Find the balance at the end of the week, showing the running total after each transaction.

3. Q3Integers (\mathbb{Z})

   Write one integer for each situation. Then give the opposite of that integer, and say what the opposite would mean in the same situation.

   1. A métro platform is $18$ metres below street level.
   2. After one game, a hockey team's goal difference is $4$ goals in its favour.
   3. Zoé owes the school store $7 for a lost calculator.
4. Q4Multiplying and Dividing Positive and Negative Numbers

   Answer each part, showing the operation on integers that you used.

   1. A snowbank loses $4$ cm of height on each of $9$ mild days. Write this as a multiplication of integers and evaluate it.
   2. A drone descends steadily and, after $8$ seconds, is $56$ m below where it started. Write a division of integers that gives its change in height per second, and evaluate it.
   3. Evaluate $(-6)\times (-5)$, $(-72)÷9$, $(-8)\times 3÷(-4)$.
5. Q5Natural Numbers (\mathbb{N})

   While organising a science fair, the committee writes down these numbers: $$27,-3,0,4.5,12,-8.$$

   1. Which of them are natural numbers?
   2. For each of the others, give the one reason it is not a natural number.
6. Q6Number Sets –

   Sort each of these numbers into exactly one of the three groups below: $$15,-6,0,3.5,-20,\frac{2}{5},-2.4$$

   1. a natural number;
   2. an integer that is not a natural number;
   3. not an integer at all.

   Then complete this sentence and give a reason: “Every number in $\mathbb{N}$ is also in $\mathbb{Z}$, but …”

7. Q7Ordering Integers

   A geologist records the position of six core samples relative to ground level, in metres: $$-12,5,-30,0,-7,18.$$

   1. Place the six values on a number line and write them in order from least to greatest.
   2. Which sample is the deepest, and which sample is nearest to ground level?
8. Q8Ordering Natural Numbers

   Over five winters, a carnival counted these numbers of visitors: $$3072,3720,3027,3702,3207.$$

   1. Write the five counts in order from greatest to least.
   2. Which count is closest to $3500$? Show the comparison you used.
9. Q9Subtracting Integers

   Answer both parts.

   1. The summit of a trail is $812$ m above sea level, and the floor of a nearby sinkhole is $43$ m below sea level. How much higher is the summit than the sinkhole floor? Write the subtraction of integers that answers this.
   2. Evaluate: $6-14$, $-9-5$, $-3-(-11)$, $0-(-8)$.
10. Q10Synthesis — drawing on several sheets in this topic

    Five checkpoints on a mountain-bike trail are labelled by their height in metres relative to the trailhead: $$A=-18,B=25,C=-40,D=0,E=12.$$

    1. Order the five checkpoints from lowest to highest.
    2. Which checkpoint is the farthest from the trailhead's height? Use absolute value to justify your choice.
    3. A rider goes from $C$ to $A$, then from $A$ to $E$. Write one expression, using subtraction of integers, that gives the total change in height for the whole ride, and evaluate it. Check your total a second way.

The 9 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Position and distance are two different questions

If you take one idea away from this topic, take this one. "Which number is bigger?" asks about position on the number line — further right is bigger. "Which number is further from zero?" asks about distance, and ignores which side you are on. For positive numbers the two questions happen to have the same answer, which is exactly why they get confused the moment negatives arrive.

Q1 asks the distance question and Q7 asks the position question, on data that looks the same in both. A reading of −14 °C is the furthest from zero of the five in Q1; in Q7 the sample at −30 m is the smallest of the six, precisely because it is the furthest from zero on the left. Same fact, two different sentences, and the marks are in saying which one you answered.

Ordering integers: read the number line, do not read the digits (Q7)

Line the values up on a number line and order becomes something you can see rather than something you have to argue about. Negatives come first, then zero, then positives; and among the negatives, the one with the biggest digits is the smallest number. So −30 < −12 < −7, which looks backwards until you remember that −30 is thirty metres down.

Adding and subtracting: keep a running total (Q2, Q9)

Q2 is a bank balance that starts overdrawn, and the instruction to show the total after each transaction is not busywork — it is the method. Deposits are positive, withdrawals are negative, and you add them to the balance one at a time. Doing it in one long chain is where a sign gets dropped; doing it line by line means you can see the account cross back below zero on the way to a positive finish.

Then check it a second way. Add up all the changes on their own, and add that single total to the starting balance. Two routes to the same number is the cheapest error-check available on this topic, and it takes one line.

Q9 is where subtraction earns its reputation. The summit is at +812 and the sinkhole floor at −43, and the gap between them is 812 − (−43) = 855 m. Subtracting a negative adds, and the picture says why: the gap covers everything above sea level plus everything below it. If the rule ever feels arbitrary, draw the two heights and measure.

Multiplying and dividing: where the sign rule comes from (Q4)

A negative times a positive is a repeated loss, which is how Q4 opens: nine mild days, four centimetres of snowbank lost each time, 9 × (−4) = −36. Division is the same situation read backwards — a drone 56 m below its starting point after 8 seconds is changing height by (−56) ÷ 8 = −7 m each second.

Two negatives giving a positive is the part that has to be learned rather than pictured, and the useful version is a counting rule: pair the negative factors up, each pair cancels to a positive, and the sign of the whole product depends only on whether one factor is left over. Even number of negatives, positive; odd number, negative.

N, Z, and answering the question that was actually asked (Q3, Q5, Q6)

Three questions here are about the words rather than the arithmetic. Q3 asks you to turn a situation into an integer and to say what its opposite would mean back in the situation — a platform 18 m below street level, a debt of seven dollars, a goal difference — because an integer is only useful when you know what zero is. Q5 sorts a list into natural numbers and non-members, with one reason each. Q6 sorts the same kind of list three ways and then asks you to complete the sentence "Every number in N is also in Z, but …".

Place value still decides ordering (Q8)

Q8 hands you five counts built from the same four digits, so the only way through is to compare place value by place value from the left: same thousands, so look at the hundreds; same hundreds, so look at the tens. Its second part asks which count is nearest to 3 500, and "nearest" is a distance question again — subtract both ways and compare the two gaps rather than trusting the eye.

Q10: the synthesis question

Five checkpoints on a trail, labelled by height relative to the trailhead, and three parts that use three different ideas on one set of numbers: order them (position), say which is furthest from the trailhead's height using absolute value (distance), and find the total change in height over a two-leg ride (subtraction). The third part rewards writing each leg as arrival − departure and then checking the total against the direct difference between the starting and finishing checkpoints. When the two agree, the middle height cancelled — which is the whole reason the check works.

Getting the most out of it

Draw the number line, every time

A line with zero marked and the values placed on it settles almost every ordering question on this sheet without an argument, and it makes "further from zero" and "smaller" visibly different things. It costs ten seconds and it is the picture the whole topic is built on.

Say which question you are answering

Before you write, decide out loud whether the question is about position ("bigger", "colder", "deeper", "in order") or about distance ("furthest from", "closest to", absolute value). Write the words into your answer. Most lost marks here are a correct calculation attached to the wrong one of those two sentences.

Check a second way

Every running total can be checked by adding the changes separately; every subtraction can be checked by adding the answer back. The questions on this sheet are built so both routes are short, and getting into the habit now is what makes signed arithmetic reliable later.

Common questions