Modeling Addition of Integers

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

2. Adding Real Numbers –4 –3 –2 –1 0 1 2 3 4 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. Move 5 units to the right. Start at –2. End at 3. • •

3. Adding Real Numbers –4 –3 –2 –1 0 1 2 3 4 –5 –4 –3 –2 –1 0 1 2 3 The sum can be written as –2 + 5 = 3. The sum can be written as 2 + (–6) = –4. Addition can be modeled with movements on a number line. You add a positive number by moving to the right. Model –2 + 5. Move 5 units to the right. Start at –2. End at 3. • • You add a negative number by moving to the left. Model 2 + (–6). Move 6 units to the left. End at –4. Start at 2. • •

4. Adding Real Numbers –4 + (–5) Example: 1 2 –4 –5 + = 9 The rules of addition show how to add two real numbers without a number line. RULES OF ADDITION TO ADD TWO NUMBERS WITH THE SAME SIGN: Add their absolute values. Attach the common sign. –9

5. Adding Real Numbers 1 2 –4 + (–5) –5 Example: 2 1 3 + (–9) Example: –9 3 – = 6 The rules of addition show how to add two real numbers without a number line. RULES OF ADDITION TO ADD TWO NUMBERS WITH THE SAME SIGN: Add their absolute values. Attach the common sign. –4 + = 9 –9 TO ADD TWO NUMBERS WITH OPPOSITE SIGNS: Subtract the smaller absolute value from the larger absolute value. Attach the sign of the number with the larger absolute value. –6

6. Adding Real Numbers 3 + (–2) = –2 + 3 Example: (–5 + 6) + 2 = –5 + (6 + 2) Example: 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. PROPERTIES OF ADDITION COMMUTATIVE PROPERTY The order in which two numbers are added does not change the sum. a + b = b + a ASSOCIATIVE PROPERTY The way you group three numbers when adding does not change the sum. (a + b) + c = a + (b + c)

7. Adding Real Numbers –4 + 0 = –4 Example: 5 + (–5) = 0 Example: 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. PROPERTIES OF ADDITION IDENTITY PROPERTY The sum of a number and 0 is the number. a + 0 = a PROPERTY OF ZERO (INVERSE PROPERTY) The sum of a number and its opposite is 0. a + (–a) = 0

8. Adding Three Real Numbers –5 2 3 –4 –3 –2 –1 0 1 The sum can be written as –3 + 5 + (–6) = –4. Use a number line to find the following sum. –3 + 5 + (–6) SOLUTION Start at –3. Move 5 units to the right. –3 + 5 = 2 • • • End at –4. Move 6 units to the left.

9. Finding a Sum Find the following sums. 1.4 + (–2.6) + 3.1 = 1.4 + (–2.6 + 3.1) Use associative property. Simplify. = 1.4 + 0.5 = 1.9

10. Finding a Sum 1 2 1 2 1 2 1 2 1 2 1 2 – + 3 + = – + + 3 () = – + + 3 Find the following sums. 1.4 + (–2.6) + 3.1 = 1.4 + (–2.6 + 3.1) Use associative property. Simplify. = 1.4 + 0.5 = 1.9 Use commutative property. Use associative property. Use identity property and property of zero. = 0 + 3 = 3

11. Using Addition in Real Life SCIENCE CONNECTIONAtoms 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.

12. Using Addition in Real Life SCIENCE CONNECTIONAtoms 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. 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 Al3+.

13. Finding the Total Profit January February March –$13,142.50 –$6783.16 –$4734.86 April May June $3825.01 $7613.17 $12,932.54 –289.8 +/– +/– + + + + +/– = + The display is –298.8. This means the company had a loss of $298.80. 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. SOLUTION With this many large numbers, you may want to use a calculator. 13142.50 4734.86 6783.16 3825.01 7613.17 12932.54