Print the chapter

Every lesson of the chapter, ready for paper. Exercise sets are lettered A, B, C… and answered at the end.

Include answers

To save a PDF, choose “Save as PDF” in the print dialog.

1. Ordering signed numbers; absolute value

Warm-up

Three quick questions from earlier lessons.

Discover

It is $-8$ °C in one town and $-3$ °C in another. Which is warmer?

$-3$ °C
$-8$ °C

Key idea

Order on the number line; absolute value is distance from 0.

Numbers increase from left to right on the number line, so

-8 < -3

. The absolute value

|a|

is the distance from

a

to 0, never negative:

|-5| = 5

. The distance between numbers on opposite sides of 0 is the sum of their absolute values.

Worked example

Order $-2$ , $4$ , $-6$ , $0$ .

-6 < -2 < 0 < 4

Left to right.

Your turn, guided

Your turn: $|-9| + |-1|$ .

10

9 + 1.

Practice ★

Which number is greater: $-5$ or $-12$ ?
1. $-5$
2. $-12$
3. They are equal
Which number is greater: $-5$ or $-10$ ?
1. $-5$
2. $-10$
3. They are equal

Compute $|-4| + |10|$ .

Watch out!

Practice ★★

How far apart are $-7$ and $6$ on the number line?
How far apart are $-15$ and $1$ on the number line?

Exit ticket

Three questions to check you have got it.

Compute $|-4| + |10|$ .
Compute $|-2| + |10|$ .
Compute $|-8| + |3|$ .

Summary card

Ordering and absolute value

Right is greater; |a| is the distance to 0.

Watch out: −8 is less than −3.

2. The coordinate plane: quadrants and distances along grid lines

Warm-up

Three quick questions from earlier lessons.

Discover

The point $(3, -2)$ : where is it?

3 right, 2 down
2 right, 3 down

Key idea

Four quadrants; distances along a grid line.

A point

(x, y)

x

right (left if negative) and

y

up (down if negative). The axes split the plane into quadrants I–IV counterclockwise from upper right. Two points with the same $x$ (or same

y

) are

|y_1 - y_2|

apart; across an axis, add the absolute values.

Worked example

$(-3, 4)$ and $(5, 4)$ .

Horizontal distance

Same y.

3 + 5 = 8

3 to 0, then 5.

Your turn, guided

Your turn: distance between $(2, -6)$ and $(2, 1)$ .

7

6 + 1.

Practice ★

In which quadrant is the point $(-9, 5)$ ?
1. Quadrant II (left, up)
2. Quadrant I (right, up)
3. Quadrant IV (right, down)
In which quadrant is the point $(-6, 2)$ ?
1. Quadrant II (left, up)
2. Quadrant I (right, up)
3. Quadrant IV (right, down)

What is the distance between $(9, 1)$ and $(9, -5)$ ?

Watch out!

Practice ★★

What is the distance between $(-2, 1)$ and $(9, 1)$ ?
What is the distance between $(-6, -4)$ and $(2, -4)$ ?

Boss problem. A rectangle has vertices $(-2, 2)$ , $(3, 2)$ , $(3, -3)$ and $(-2, -3)$ .

Exit ticket

Three questions to check you have got it.

What is the distance between $(9, 1)$ and $(9, -5)$ ?
What is the distance between $(9, 5)$ and $(9, -6)$ ?
What is the distance between $(7, 2)$ and $(7, -3)$ ?

Summary card

The coordinate plane

x right/left, y up/down; same coordinate: subtract or add across 0.

Watch out: x comes first.

3. Greatest common factor and least common multiple

Warm-up

Three quick questions from earlier lessons.

Discover

Buses leave every 4 minutes and trains every 6 minutes. Both leave now. When do they next leave together?

In 12 minutes
In 2 minutes

Key idea

GCF splits into equal groups; LCM lines things up.

The greatest common factor (GCF) is the largest number that divides both numbers. The least common multiple (LCM) is the smallest number that both divide. Example: GCF(12, 18) = 6, LCM(12, 18) = 36.

Worked example

12 and 20.

GCF = 4

Factors in common: 1, 2, 4.

LCM = 60

Multiples of 20: 20, 40, 60.

Your turn, guided

Your turn: the GCF of 18 and 24.

6

Largest common factor.

Practice ★

What is the greatest common factor of 12 and 30?
What is the greatest common factor of 20 and 28?

Juice boxes come in packs of 5 and straws in packs of 7. What is the smallest number of each you can buy so that the numbers match?

Watch out!

Practice ★★

What is the least common multiple of 18 and 30?
What is the least common multiple of 9 and 15?

Exit ticket

Three questions to check you have got it.

What is the greatest common factor of 12 and 30?
What is the greatest common factor of 20 and 28?
What is the greatest common factor of 14 and 10?

Summary card

Common factors and multiples

GCF: largest common divisor. LCM: smallest common multiple.

Watch out: do not swap them.

Answers: Rational numbers

1. Ordering signed numbers; absolute value

$-5$
$-5$

$14$

$13$
$16$

$14$
$12$
$11$

2. The coordinate plane: quadrants and distances along grid lines

Quadrant II (left, up)
Quadrant II (left, up)

$11$
$8$

$20$

$6$
$11$
$5$

3. Greatest common factor and least common multiple

$35$

$90$
$45$