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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
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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. 1–3–2–1032– 4– 44 Move 5 units to the right. Start at –2. End at 3. Adding Real Numbers
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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. 1–3–2–1032– 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.
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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
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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
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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)
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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
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Adding Three Real Numbers Use a number line to find the following sum. –3 + 5 + (– 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 –3 + 5 + (–6) = – 4.
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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
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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 + 1212 1212 = – + + 3 1212 1212 – + + 3 1212 1212 ( ) = = 0 + 3 = 3 Use commutative property. Use associative property. Use identity property and property of zero.
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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.
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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.
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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. 017613. 1712932. 54 +/–+= + +++ –289.8 The display is –298.8. This means the company had a loss of $298. 80.
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