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1. Adding and subtracting signed numbers

Warm-up

Three quick questions from earlier lessons.

Discover

It is $-3$ °C and the temperature rises 5 degrees. What is it now?

$2$ °C
$-8$ °C

Key idea

Add by moving on the number line; subtract by adding the opposite.

Adding a positive number moves right; adding a negative moves left. With different signs, subtract the sizes and keep the sign of the larger one. Subtracting is adding the opposite:

a - b = a + (-b)

, so

a - (-b) = a + b

Worked example

$-4 - (-9)$ .

-4 + 9

Add the opposite.

5

Different signs: 9 − 4, sign of 9.

Your turn, guided

Your turn: $-6 + (-7)$ .

-13

Both left.

Practice ★

Compute $7 + (-16)$ .
Compute $9 + (-20)$ .

Compute $-9 - 10$ .

Watch out!

Practice ★★

Compute $2 - (-5)$ .
Compute $8 - (-5)$ .

Exit ticket

Three questions to check you have got it.

Compute $7 + (-16)$ .
Compute $9 + (-20)$ .
Compute $8 + (-15)$ .

Summary card

Adding and subtracting signed numbers

Move on the number line; subtract = add the opposite.

Watch out: minus a negative is plus.

2. Multiplying and dividing signed numbers

Warm-up

Three quick questions from earlier lessons.

Discover

A tank loses 3 liters every minute (a change of $-3$ per minute). What was the change over the last 4 minutes?

$-12$ liters
$12$ liters

Key idea

Signs for products and quotients.

Same signs give a positive product or quotient; different signs give a negative one.

(-a)(-b) = ab

(-a) \cdot b = -ab

, and division follows the same rules.

Worked example

$(-6)(-5) \div (-3)$ .

(-6)(-5) = 30

Same signs.

30 \div (-3) = -10

Different signs.

Your turn, guided

Your turn: $(-48) \div (-6)$ .

8

Same signs.

Practice ★

Compute $(-6) \cdot 7$ .
Compute $(-7) \cdot 11$ .

Compute $(-4) \cdot (-6)$ .

Watch out!

Practice ★★

Compute $(-144) \div 12$ .
Compute $(-120) \div 12$ .

Boss problem. At midnight it is 5 °C. The temperature changes by $-6$ °C every hour (it drops 6 degrees per hour).

Exit ticket

Three questions to check you have got it.

Compute $(-6) \cdot 7$ .
Compute $(-7) \cdot 11$ .
Compute $(-11) \cdot 10$ .

Summary card

Multiplying and dividing signed numbers

Same signs: positive. Different signs: negative.

Watch out: negative times negative is positive.

3. Solve linear equations in one variable

Warm-up

Three quick questions from earlier lessons.

Discover

A pizza party costs 40 dollars plus 12 dollars per pizza, and the total is 100 dollars. Which equation fits?

$40 + 12p = 100$
$12 + 40p = 100$

Key idea

Equivalent equations have the same solutions.

To solve a linear equation, make equivalent equations: add or subtract the same amount on both sides, multiply or divide both sides by the same non-zero number, distribute, collect like terms. Stop when

x

is alone.

Worked example

Solve $5x + 3 = 2x + 15$ .

3x + 3 = 15

Subtract 2x.

3x = 12

Subtract 3.

x = 4

Divide by 3.

Your turn, guided

Your turn: $3(x + 2) = 21$ . Find $x$ .

x = 5

Divide by 3: x + 2 = 7.

Practice ★

Solve $3x + 3 = -18$ . $x = \;?$
Solve $8x + 14 = -26$ . $x = \;?$

Solve $4x + 8 = 3x + 9$ . $x = \;?$

Watch out!

Practice ★★

Solve $6(x + 4) = 30$ . $x = \;?$
Solve $3(x + 4) = 33$ . $x = \;?$

Exit ticket

Three questions to check you have got it.

Solve $3x + 3 = -18$ . $x = \;?$
Solve $8x + 14 = -26$ . $x = \;?$
Solve $4x + 4 = 32$ . $x = \;?$

Summary card

Linear equations

Do the same to both sides until $x$ is alone.

Watch out: distribute to every term.

Answers: Rational number arithmetic

1. Adding and subtracting signed numbers

$-9$
$-11$

$-19$

$7$
$13$

$-9$
$-11$
$-7$

2. Multiplying and dividing signed numbers

$-42$
$-77$

$24$

$-12$
$-10$

$17$

$-42$
$-77$
$-110$

3. Solve linear equations in one variable

$-7$
$-5$

$-7$
$-5$
$7$