Grade 7 Mathematics Study Guide

Count the money in your wallet. The number you get — \(R47\) or \(R250\) or \(R3\,472\) — is a whole number. This chapter is about how whole numbers work, how we write them, and how we multiply, divide, add and subtract them correctly.

What is a whole number?

A whole number is any number with no fractions or decimals. Examples: \(0\), \(5\), \(47\), \(1\,000\,000\).

A natural number (or counting number) is a positive whole number — the numbers you count with: \(1, 2, 3, 4, \ldots\). The number \(0\) is a whole number, but it is NOT a natural number, because you do not count starting at zero.

Place value

Each digit in a number has a value that depends on WHERE it stands. In \(3\,472\):

• The \(3\) is in the THOUSANDS place, so it is \(3 \times 1\,000 = 3\,000\).
• The \(4\) is in the HUNDREDS place, so it is \(4 \times 100 = 400\).
• The \(7\) is in the TENS place, so it is \(7 \times 10 = 70\).
• The \(2\) is in the UNITS place, so it is \(2 \times 1 = 2\).

Add them up: \(3\,000 + 400 + 70 + 2 = 3\,472\). This is called the expanded form of the number.

Factors and multiples

A factor of a number is any number that divides into it exactly (with no remainder).

Example: the factors of \(12\) are \(1, 2, 3, 4, 6, 12\). Because \(12 \div 1 = 12\), \(12 \div 2 = 6\), \(12 \div 3 = 4\), and so on.

A multiple of a number is what you get when you multiply by that number.

Multiples of \(5\): \(5, 10, 15, 20, 25, \ldots\) — it is simply the \(5\) times table.

Prime numbers

A prime number has EXACTLY TWO factors: \(1\) and itself.

First prime numbers: \(2, 3, 5, 7, 11, 13, 17, 19, 23, \ldots\)

• \(7\) is prime — its factors are only \(1\) and \(7\).
• \(9\) is NOT prime — its factors are \(1, 3, 9\). (Three factors.)
• \(1\) is NOT prime — it has only one factor (itself).
• \(2\) is the only EVEN prime number.

Order of operations

If a calculation has more than one operation, do them in this order:

1. Brackets first.
2. Exponents second.
3. Multiply and divide third — from left to right.
4. Add and subtract last — from left to right.

Remember: multiply and divide are on the SAME level — work from left to right. The same goes for add and subtract.

Example: calculate \(8 + 4 \times (15 - 7)\).

Step 1 (brackets): \(15 - 7 = 8\), so the calculation becomes \(8 + 4 \times 8\).
Step 2 (no exponents).
Step 3 (multiply): \(4 \times 8 = 32\), so the calculation becomes \(8 + 32\).
Step 4 (add): \(8 + 32 = 40\).

Answer: \(\boxed{40}\)

Without the order you might do \(8 + 4 = 12\), then \(12 \times 8 = 96\) — wrong! That is why order matters so much.

Rounding

Sometimes we want to make a number simpler. Rounding brings it closer to a round number.

Rule: look at the digit JUST RIGHT of where you are rounding.

• Is it \(0, 1, 2, 3,\) or \(4\)? Round DOWN.
• Is it \(5, 6, 7, 8,\) or \(9\)? Round UP.

Example: round \(3\,478\) to the nearest hundred.

The hundreds digit is \(4\). Just right of it is \(7\). Seven is \(\ge 5\), so round UP: \(3\,478 \to 3\,500\).

Remember this

• Whole numbers = numbers with no fractions or decimals.
• Natural numbers = positive whole numbers (\(1, 2, 3, \ldots\)).
• Place value: each digit's value depends on its position.
• Factors divide in exactly; multiples come from the times table.
• Prime numbers have exactly two factors.
• Order of operations: Brackets, Exponents, Multiply/Divide, Add/Subtract.

Two streets cross each other. The point where they meet forms four ANGLES. \(90^\circ\) — a RIGHT angle. Angles are everywhere in your surroundings. This chapter teaches you to identify, measure, and calculate them.

What is an angle?

An angle is formed where two lines, rays, or segments meet at a common point (the vertex).

We measure angles in degrees. Symbol: \(^\circ\). A full turn about a point = \(360^\circ\).

Types of angles

Five main categories:

• Acute angle: smaller than \(90^\circ\). Short, sharp.
• Right angle: exactly \(90^\circ\). Like the corner of a square.
• Obtuse angle: between \(90^\circ\) and \(180^\circ\). Wider than a right angle.
• Straight angle: exactly \(180^\circ\). A straight line.
• Reflex angle: between \(180^\circ\) and \(360^\circ\). The "other way round" of an obtuse angle.

A full turn is \(360^\circ\).

Angles on a straight line

Important fact: angles on a straight line ADD UP TO \(180^\circ\).

Example: if one angle is \(60^\circ\), what is the other angle on the line?

\(180 - 60 = 120^\circ\).

Example 2: three angles on a straight line are \(40^\circ\), \(75^\circ\), and \(x\). Find \(x\).

\(40 + 75 + x = 180\)
\(115 + x = 180\)
\(x = 65^\circ\)

Angles around a point

Important fact: angles around a point ADD UP TO \(360^\circ\).

Example: four angles around a point: \(90^\circ, 80^\circ, 100^\circ, x\). Find \(x\).

\(90 + 80 + 100 + x = 360\)
\(270 + x = 360\)
\(x = 90^\circ\)

Vertically opposite angles

When two lines cross, they form FOUR angles. The angles OPPOSITE each other (across the crossing) are EQUAL.

So if one pair of angles is \(70^\circ\), the other opposite pair is also \(70^\circ\). And the other two angles must be \(110^\circ\) each (because \(70 + 110 = 180\) on a straight line).

Parallel lines and a transversal

Two parallel lines never touch each other. Write it as \(L_1 \parallel L_2\).

A transversal is a line that crosses both parallel lines. It forms \(8\) angles.

Three important facts:

• Corresponding angles (the same position at each parallel line) are EQUAL.
• Alternate angles (cross-over, between the lines) are EQUAL.
• Co-interior angles (on the same side, between the lines) ADD UP TO \(180^\circ\).

These let you calculate unknown angles quickly without measuring them.

Angle sum in a triangle

CRITICAL fact: the three angles inside a triangle ADD UP TO \(180^\circ\).

Example: a triangle has angles of \(60^\circ\) and \(70^\circ\). What is the third?

\(180 - 60 - 70 = 50^\circ\).

Angle sum in a quadrilateral

Fact: the four angles inside a quadrilateral add up to \(360^\circ\).

(Why? Because a quadrilateral can be divided into TWO triangles, and \(2 \times 180 = 360\).)

Remember this

• Angle types: acute (\(<90^\circ\)), right (\(=90^\circ\)), obtuse (\(90^\circ\)–\(180^\circ\)), straight (\(=180^\circ\)), reflex (\(>180^\circ\)).
• Angles on a straight line: add up to \(180^\circ\).
• Angles around a point: add up to \(360^\circ\).
• Vertically opposite angles (across a crossing) are equal.
• Triangle angles: \(180^\circ\). Quadrilateral angles: \(360^\circ\).