Secondary 2 Integers and Number Sets Worksheet

10 of the 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 10 of the 12 questions are printed below. The other 2 are built on a diagram or a table of values that does not translate to the page, so they are in the free PDF — marked below where they would have come.

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

   At a diving site on Lac Memphrémagog, every position is measured in metres from the water surface, which counts as $0$. A position above the surface is written with a $+$ sign and a position below it with a $-$ sign.

   1. A gull flies at $+12$, a diver hangs at $-8$, a marker on the anchor line sits at $-3.5$ and a float rests at $0$. Write the absolute value of each of these four positions.
   2. Explain, in the words of this situation, what the absolute value of a position tells you.
   3. Which of the four is farthest from the surface?
   4. Give a position, other than the diver's, that is exactly $8$ m from the surface.
2. Q2Arithmetic

   Arithmetic is the branch of mathematics that studies numbers and the rules for calculating with them. Here are five tasks a Secondary 2 student might be given.

   1. List all the divisors of $60$.
   2. Find the perimeter of a hexagonal patio.
   3. Work out $15-(-7)$.
   4. Draw a bar graph of the class's favourite winter sports.
   5. Write $0.6$ in fractional notation.

   Say which of the five belong to arithmetic. For each of the other two, name the branch of mathematics it belongs to instead, and say in one sentence what that branch studies.

3. Q3Ascending and Descending Order

   This question is built around a diagram or a table of values. Open it in the PDF.

4. Q4Expressing Numbers Using Different Types of Notation

   Three notations are in play here: fractional notation in lowest terms, decimal notation, and percentage. Each number below is written in one of the three; write it in the other two.

   1. $\frac{3}{8}$
   2. $0.35$
   3. $6\%$
   4. $2\frac{1}{4}$
5. Q5Integers (\mathbb{Z})
   1. Write the integer that describes each situation: three floors below street level in a parking garage; a gain of $12$ points in a video game; a temperature of eight degrees below zero; the level of the water surface itself.
   2. Give the opposite of $-12$, of $7$ and of $0$.
   3. Explain why $4.5$ is not an integer, while $\frac{12}{4}$ is one.
6. Q6Natural Numbers (\mathbb{N})

   The set of natural numbers, written $\mathbb{N}$, is the set of counting numbers $\{0,1,2,3,4,\dots \}$.

   1. Which of these quantities are always counted with natural numbers: the number of goals a team scores; the temperature in degrees Celsius; the number of students on a bus; the balance of a bank account; the number of pages in a book? Justify the ones you leave out.
   2. Which of $0$, $-4$, $\frac{12}{4}$, $7.5$ and $91$ are natural numbers?
   3. Explain in one sentence why every natural number is also an integer.
7. Q7Number Sets –

   The symbol $\subset$ means “is contained in”.

   1. Give an integer that is not a natural number, and say in one sentence what puts it outside $\mathbb{N}$.
   2. True or false, with a reason: every natural number is an integer.
   3. Complete the statement $\mathbb{N}\underset{―}{}\mathbb{Z}$ with the symbol $\subset$, and write out in words what the completed statement says.
8. Q8Number Systems

   Our everyday number system is positional and uses base $10$: the value of a digit depends on the column it sits in.

   1. Write $3407$ in expanded form, showing the place value of each digit.
   2. A computer stores a small counter as $10110$ in base $2$, where the columns are worth $16$, $8$, $4$, $2$ and $1$. What is this number in base $10$?
   3. Our clocks use base $60$: an hour is $60$ minutes and a minute is $60$ seconds. How many seconds are there in $2$ h $35$ min $40$ s?
9. Q9Operations

   An operation combines numbers, called the operands, using an operator symbol, and the result has a name of its own.

   1. For each of $24+9=33$, $40-15=25$, $7\times 6=42$ and $45÷9=5$: name the operation and give the correct name for the result.
   2. $12÷5=2.4$. Is $2.4$ still called a quotient, even though it is not a whole number? Explain.
10. Q10Subtracting Integers

    Subtracting a number is the same as adding its opposite.

    1. $-6-9$
    2. $4-(-11)$
    3. $-13-(-5)$
    4. A survey drone hovers $45$ m above the ground, and a gallery of an old mine runs $78$ m below the ground. What is the vertical distance between the drone and the gallery? Write the subtraction you used.
11. Q11Writing Numbers

    The same quantity can be written in several ways, and one way is often more useful than another. A recipe calls for $0.75$ L of milk.

    1. Write this quantity in fractional notation, in lowest terms.
    2. Write it as a percentage of a litre.
    3. Write it in words.
    4. A measuring jug is marked with graduations. Which of the three forms would you print beside the graduations, and why?
12. Q12Synthesis — drawing on several sheets in this topic

    This question is built around a diagram or a table of values. Open it in the PDF.

The 8 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.

Subtracting an integer: add the opposite, every time

One rule covers every case on this sheet, and it is worth writing out rather than remembering as a slogan about signs: subtracting a number is the same as adding its opposite.

The same rule gives you distances. The gap between something 45 m above ground and something 78 m below it is 45 − (−78) = 123 m, which the addition confirms: 45 m down to ground level, then 78 m further. When a subtraction of integers produces a number that surprises you, walking it along the line is the check.

Order runs backwards on the negative side

Comparing −3 and −8 by their digits gets it wrong. The farther a negative number sits from zero, the smaller it is: −20 is a worse debt than −3, and it is the smaller number. Of two negative integers, the greater is the one whose absolute value is smaller — the one closer to zero.

"Below −10" means to the left of −10, so −9 does not qualify however it looks. When ordering a column of temperatures, sketch the number line and place them on it; the ordering then reads off left to right, and the negatives cannot go backwards.

Absolute value is a distance, and distance is not position

|−8| = 8 because −8 sits eight units from zero. It is never negative, it forgets which side you were on, and it answers a different question from "which number is smaller".

Natural numbers, integers, and why one sits inside the other

The natural numbers are the counting numbers 0, 1, 2, 3 and so on. The integers add their opposites. Every natural number is therefore an integer, which is what ℕ ⊂ ℤ says in symbols, and the reverse fails on any negative number you like.

Two things trip people up. First, how a number is written is not what it is: 12⁄4 looks like a fraction and is the integer 3, while 4.5 sits between two integers and is neither. Second, choosing the set is a modelling decision — goals scored, students on a bus and pages in a book are counts and belong to ℕ; a temperature and a bank balance can go negative and can carry decimals, so they do not.

Place value, in ten and in other bases

Our system is positional: a digit's worth depends on the column it stands in, and each column to the left is worth ten times the one before. Writing 3407 out in expanded form shows why the zero is not decoration — it holds the tens column open so the 3 and the 4 keep their places.

Change the multiplier and the same machinery gives another base. In base two the columns are 16, 8, 4, 2, 1; in base five they are 25, 5, 1 and no digit 5 exists, because five ones become one group of five and carry — the same rule that turns ten ones into "10" in base ten. Clocks are the everyday example: hours, minutes and seconds are base sixty, which is why two hours is 2 × 3600 seconds, exactly as two hundreds is 2 × 100 units.

The vocabulary counts too

Sum, difference, product, quotient — the name of a result comes from the operation that produced it, not from the kind of number that comes out. The quotient of 12 by 5 is 2.4, and it is still a quotient. Only when a question asks for a whole-number division does one speak of a quotient of 2 with a remainder of 2. A sheet question also asks you to sort tasks by branch: working with the numbers themselves is arithmetic; measuring a figure is geometry; organising and displaying data is statistics.

Getting the most out of it

Draw the number line, and use it

Every ordering question and every subtraction on this sheet can be checked by walking along a line: right for adding, left for subtracting. It takes ten seconds and it is the check that catches a sign error even when the rule has been half-remembered.

Rewrite each subtraction as an addition before computing

Put −13 − (−5) = −13 + 5 on the page before you work out anything. The whole difficulty of subtracting integers lives in that one rewrite, and doing it in your head is how the error gets made.

Ask "position or distance?" before answering

Coldest, lowest and smallest are questions about position. Farthest, deepest below and how far apart are questions about distance, which is where absolute value belongs. Naming which one is being asked is most of the work in the harder questions here.

Common questions