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. 1Model negative 8 and positive 3 using algebra tiles. +––––––––++ 2 Group pairs of positive and negative tiles. Count the remaining tiles. + –––––––– ++ 3The remaining tiles show the sum of –8 and 3. –8 + 3 = –5 3 Each pair has a sum of 0. ––––– –8 0

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 – –3–2–1032– 4– 44 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 – –3–2–1032– 4– 440– 4– 4–3–2–121–53 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: 1Add their absolute values. 2Attach the common sign. Example: – 4 + (–5)– 4– 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: 1Add their absolute values. 2Attach the common sign. Example: – 4 + (–5)– 4– 4–5+=9–9 TO ADD TWO NUMBERS WITH OPPOSITE SIGNS: 1Subtract the smaller absolute value from the larger absolute value. 2Attach 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 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. – (– 6) SOLUTION 0– 4– 4–3–2–121–53 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 – (–6) = – 4.

Finding a Sum Find the following sums (–2. 6) Use associative property. = (– ) Simplify. = = 1.9

Finding a Sum Find the following sums (–2. 6) Use associative property. = (– ) Simplify. = = 1.9 – = – – ( ) = = = 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 – = 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 – = 3, so the atom is an ion. In chemistry, this aluminum atom is written as Al 3 +. SOLUTION The total charge is – = 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, $ $ JuneMayApril –$ –$ –$13, MarchFebruaryJanuary SOLUTION With this many large numbers, you may want to use a calculator /–+= –289.8 The display is – This means the company had a loss of $