Pre-Algebra Workbook Grade 6, 7, 8

The number line is a visual model of numbers arranged in order. It is the foundation for understanding integers, fractions, decimals, and the relationships between numbers, and it is the model used throughout pre-algebra to compare values, measure distance, and graph inequalities.

A number line is a straight horizontal line with numbers placed at equal intervals. Arrows at each end show that it extends infinitely in both directions. Zero sits at the center, positive numbers extend to the right, and negative numbers extend to the left.

Term	Meaning
Origin	Thepointthatrepresentszero thereferencepointfromwhicheveryothernumberismeasured ,
Positivenumbers	Locatedtotherightofzero(1 2 3 4 );thefartherright thegreaterthevalue , , , , … ,
Negativenumbers	Located to the left of zero and written with a minus sign ( −1 −2 −3 …); the farther left the smaller the , , , ,
	value
Integers	All whole numbers and their opposites: { −3 −2 −1 0 1 2 3 …} Zero is an integer but is neither …, , , , , , , , .
	positivenornegative

To plot a number, start at zero. For a positive number, count that many units to the right; for a negative number, count that many units to the left; then mark the location with a dot. To plot 5, move five units right. To plot −3, move three units left. To plot zero, mark the origin. Fractions and decimals have exact locations too: 1/2 lies halfway between 0 and 1, and 2.5 lies halfway between 2 and 3. Negative fractions and decimals follow the same rule on the left side of zero.

To compare two numbers, locate them on the line: the number farther to the right is greater. The number 7 is greater than 4 because it lies to the right of 4, and −2 is greater than −5 because −2 lies to the right of −5. The same principle applies to integers, fractions, and decimals.

The absolute value of a number is its distance from zero, regardless of direction. Because distance is never negative, absolute value is always positive or zero. The absolute value of n is written |n|: |6| = 6, |−6| = 6, and |0| = 0. Absolute value is useful for comparing the size (magnitude) of numbers without regard to sign; −10 and 10 have the same absolute value because each is ten units from zero.

Two numbers are opposites when they are the same distance from zero on opposite sides of the number line. The opposite of 4 is −4, the opposite of −7 is 7, and the opposite of zero is zero. On the number line, opposites are mirror images with zero as the line of reflection. Adding a number and its opposite always gives zero, which is the additive inverse property: a + (−a) = 0.

Absolute value bars are not parentheses. |−2| = 2, never −2. However, −|2| = −2, because the negative sign is outside the bars and is applied after the absolute value is taken.