Modeling Addition of Integers

+ – + –

+ –

To model addition problems involving positive and negative integers, you can

use tiles labeled and . Each represents positive 1, and each

represents negative 1. Combining a with a gives 0. You can use

algebra tiles to find the sum of –8 and 3.

1 Model negative 8 and positive 3 using algebra tiles.

+– – – – – – – – + +

2 Group pairs of positive and negative tiles. Count the remaining tiles. +

– – – – – – – –

+ +

3 The remaining tiles show the sum of –8 and 3.

–8 + 3 = –5

3

Each pair has a sum of 0.Each pair has a sum of 0.

– – – – –

–80

The sum can be written as –2 + 5 = 3.

Addition can be modeled with movements on a number line.

You add a positive number by moving to the right.Model –2 + 5.

1–3 –2 –1 0 32– 4 4

Move 5 units to the right.Start at –2. End at 3.

• •

Adding Real Numbers

The sum can be written as –2 + 5 = 3.

Addition can be modeled with movements on a number line.

You add a positive number by moving to the right.

You add a negative number by moving to the left.

Model –2 + 5.

1–3 –2 –1 0 32– 4 4

0– 4 –3 –2 –1 21–5 3

Move 5 units to the right.

Move 6 units to the left.

Model 2 + (–6).

Start at –2. End at 3.

• •

Start at 2.End at – 4.

••

Adding Real Numbers

The sum can be written as 2 + (–6) = – 4.

The rules of addition show how to add two real numbers without a number line.

Adding Real Numbers

RULES OF ADDITION

TO ADD TWO NUMBERS WITH THE SAME SIGN:

1 Add their absolute values.

2 Attach the common sign.

Example: – 4 + (–5) – 4 –5+ = 9 –9

The rules of addition show how to add two real numbers without a number line.

Adding Real Numbers

RULES OF ADDITION

TO ADD TWO NUMBERS WITH THE SAME SIGN:

1 Add their absolute values.

2 Attach the common sign.

Example: – 4 + (–5) – 4 –5+ = 9 –9

TO ADD TWO NUMBERS WITH OPPOSITE SIGNS:

1 Subtract the smaller absolute value from the larger absolute value.

2 Attach the sign of the number with the larger absolute value.

Example: 3 + (–9) –9 3– = 6 –6

The rules of addition on the previous slide will help you find sums of positive and negative numbers. It can be shown that these rules are a consequence of the following Properties of Addition.

Adding Real Numbers

PROPERTIES OF ADDITION

COMMUTATIVE PROPERTY

The order in which two numbers are added does not change the sum.

a + b = b + a Example: 3 + (–2) = –2 + 3

ASSOCIATIVE PROPERTY

The way you group three numbers when adding does not change the sum.

(a + b) + c = a + (b + c) Example: (–5 + 6) + 2 = –5 + (6 + 2)

The rules of addition on the previous slide will help you find sums of positive and negative numbers. It can be shown that these rules are a consequence of the following Properties of Addition.

Adding Real Numbers

PROPERTIES OF ADDITION

IDENTITY PROPERTY

The sum of a number and 0 is the number.

a + 0 = a Example: – 4 + 0 = – 4

PROPERTY OF ZERO (INVERSE PROPERTY)

The sum of a number and its opposite is 0.

a + (–a) = 0 Example: 5 + (–5) = 0

Adding Three Real Numbers

Use a number line to find the following sum.

–3 + 5 + (– 6)

SOLUTION

0– 4 –3 –2 –1 21–5 3

Move 6 units to the left.

–3 + 5 = 2

End at – 4.

••Start at –3. Move 5 units to the right.

The sum can be written as –3 + 5 + (–6) = – 4.

•

Finding a Sum

Find the following sums.

1.4 + (–2.6) + 3.1 Use associative property.= 1.4 + (–2.6 + 3.1)

Simplify.= 1.4 + 0.5

= 1.9

Finding a Sum

Find the following sums.

1.4 + (–2.6) + 3.1 Use associative property.= 1.4 + (–2.6 + 3.1)

Simplify.= 1.4 + 0.5

= 1.9

– + 3 + 12

12

= – + + 312

12

– + + 312

12( ) =

= 0 + 3 = 3

Use commutative property.

Use associative property.

Use identity property and property of zero.

Using Addition in Real Life

SCIENCE CONNECTION Atoms are composed of electrons, neutrons, and protons. Each electron has a charge of –1, each neutron has a charge of 0, and each proton has a charge of +1. The total charge of an atom is the sum of all the charges of its electrons, neutrons, and protons. An atom is an ion if it has a positive or a negative charge. If an atom has a charge of zero, it is not an ion. Are the following atoms ions?

Aluminum: 13 electrons, 13 neutrons, 13 protons

SOLUTION The total charge is –13 + 0 + 13 = 0, so the atom is not an ion. In chemistry, this aluminum atom is written as Al.

Using Addition in Real Life

SCIENCE CONNECTION Atoms are composed of electrons, neutrons, and protons. Each electron has a charge of –1, each neutron has a charge of 0, and each proton has a charge of +1. The total charge of an atom is the sum of all the charges of its electrons, neutrons, and protons. An atom is an ion if it has a positive or a negative charge. If an atom has a charge of zero, it is not an ion. Are the following atoms ions?

Aluminum: 13 electrons, 13 neutrons, 13 protons

Aluminum: 10 electrons, 13 neutrons, 13 protons

SOLUTION The total charge is –10 + 0 + 13 = 3, so the atom is an ion. In chemistry, this aluminum atom is written as Al

3 +.

SOLUTION The total charge is –13 + 0 + 13 = 0, so the atom is not an ion. In chemistry, this aluminum atom is written as Al.

Finding the Total Profit

A consulting company had the following monthly results after comparing income and expenses. Add the monthly profits and losses to find the overall profit or loss during the six-month period.

$12,932.54$7613.17$3825.01

JuneMayApril

–$4734.86–$6783.16–$13,142.50

MarchFebruaryJanuary

SOLUTION With this many large numbers, you may want to use a calculator.

13142.50 6783.16 4734.86

3825.01 7613.17 12932.54

+/– +

=

+/– + +/–

+ + + –289.8

The display is –298.8. This means the company had a loss of $298.80.