= = = -1

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1 MA.7.A.3.1 Use and justify the rules for adding, subtracting, multiplying, and dividing, and finding the absolute value of integers. Integers less than zero are negative integers. Integers greater than zero are positive integers. Zero is neither positive nor negative. NEGATIVE INTEGERS POSITIVE INTEGERS The absolute value of an integer is the distance the number is from zero on a number line. Two vertical bars are used to represent absolute value. The symbol for absolute value of 3 is 3. Evaluate Evaluate On the number line, the point is also units away from 0 so = Evaluate each expression. On the number line, the point is units away from 0 so = Zero Pairs: Suppose you had $ in your piggy bank. That would be represented by +. Your mother comes in and says that you owe her $. That would be represented by. How much money do you have? You have 0 dollars since you had to give your $ to your mother. + and are called zero pairs Any number and its opposite are called zero pairs because they cancel each other out. They equal zero. Adding Integers: You can use counters to add integers. Positive Negative = = = = -1 Notice, when the signs are the same, you add the numbers and keep the sign. Notice, when the signs are different, you actually need to subtract the numbers and use the sign from the largest absolute value. You can also use number lines to add integers. + means move to the right. means move to the left or go in the opposite direction. 3 + Start at 3 and move places to the right 3 + = 5

2 3 + Start at 3 and move places to the left 3 + = Start at 3 and move places to the left 3 + = Start at 3 and move places to the right 3 + = 1 Subtracting Integers You can use number lines to subtract integers. + means move to the right. means move to the left or go in the opposite direction. 3 Start at 3 and move places to the left 3 = 1 3 Start at 3 and move places to the left 3 = 5 3 Start at 3. The minus sign says go left but then the negative sign says turn around and go right. 3 = 5 3 Start at 3. The minus sign says go left but then the negative sign says turn around and go right. 3 = 1 You can also use counters to subtract integers. You can use counters to add integers. Positive Negative 3 Start with 3 positives and remove of them. 3 = 1 3 Start with 3 negatives and remove of them. 3 = 1 3 Start with 3 negatives. You need to remove positives. You don t have any positives to remove so add some zero pairs until you have positives. (Remember, adding zero does not change the original number.) Now when you remove the positives, you have 5 negatives left. 3 = 5 3 Start with 3 positives. You need to remove negatives. You don t have any negatives to remove so add some zero pairs until you have negatives. (Remember, adding zero does not change the original number.) Now when you remove the negatives, you have 5 positives left. 3 = 5 Subtraction Rule: Add the opposite of the second number. 3 + = = = = 3 + +

3 Multiplying Integers Let blue chips stand for positive integers and red chips stand for negative integers. To show 3 x 4 - The positive 3 means you put in three groups. - The positive 4 tells you that each group put in will contain 4 blue chips. To show 3 x - 4 = 1 - The positive 3 means you put in three groups. - The negative 4 tells you to use four red chips in each group. To show -3 x 4, use three groups of four blue chips and three group so four red chips - Start with no chips - The negative 3 means you take away three groups. Since there are no chips to take away, we add enough zeroes so that there are enough blue chips to take away. - The positive 4 tells you the groups being removed will each contain four blue chips. - Once the three groups of four blue chips are taken away, 1 red chips will remain. To show -3 x -4 use three groups of four blue chips and three groups of four red chips. - The negative 3 means you take away three groups. Since there are no chips to take away, we add enough zeroes so that there are enough red chips to take away. - The negative 4 tells you the groups being removed will each contain 4 chips. - Once the three groups of four red chips are taken away, 1 blue chips will remain. Multiplication Rule 3 x 4 = 1 When both numbers have 3 x 4 = 1 3 x 4 = 1 the same sign, the answer will be positive. 3 x 4 = 1 When each number has a different sign, the answer will be negative. Division Rule Since 3 x 4 = 1, then we can go backwards to see that 1 4 = 3 Since 3 x 4 = 1, then we can go backwards to see that 1 4 = 3 When both numbers have the same sign, the answer will be positive. Since 3 x 4 = 1, then we can go backwards to see that 1 4 = 3 When each number has a different sign, the answer will be negative. Since 3 x 4 = 1, then we can go backwards to see that 1 4 = 3 FL.MA.7.A.3. Add, subtract, multiply, and divide integers, fractions, and terminating decimals, and perform exponential operations with rational bases and whole number exponents including solving problems in everyday contexts.

4 DEFINITIONS You can use powers to shorten how you represent repeated multiplication, such as. A power has two parts, a base and an exponent. The exponent tells you how many times to use the base as a factor. You read the power 3 as two to the third power or two cubed. You read 5 as five to the second power or five squared. You simplify a numerical expression when you replace it with its single numerical value. For example, the simplest form of 8 is 16. The simplify a power, you replace it with its simplest name. For example, the simplest form of 3 is 8. 3 is called exponential form while 8 is called standard form. EXAMPLES Example 1 Write each expression using exponents. a = 7 4 The number 7 is a factor 4 times. So, 7 is the base and 4 is the exponent. b. y y x y x y y x y x = y y y x x Commutative Property = (y y y) (x x) Associative Property = y 3 x Definition of exponents To evaluate a power, perform the repeated multiplication to find the product. Example Evaluate (-6) 4. (-6) 4 = (-6) (-6) (-6) (-6) Write the power as a product. = 1,96 Multiply.

5 1 Example 3 Evaluate = 1 1 Write the power as a product. 3 3 = 9 1 Multiply the numerators Multiply the denominators Example 4 Evaluate = Write the power as a product = 8 15 Multiply the numerators (remember that multiplying Multiply the denominators 3 negatives results in a negative answer) The order of operations states that exponents are evaluated before multiplication, division, addition, and subtraction. Example 3 Evaluate m + (n m) 3 if m = -3 and n =. m + (n m) 3 = (-3) + ( (-3)) 3 Replace m with -3 and n with. = (-3) + (5) 3 Perform operations inside parentheses. = (-3-3) + (5 5 5) Write the powers as products. = or 134 Add.

6 MA.7.A.3.1 Use and justify the rules for adding, subtracting, multiplying, and dividing, and finding the absolute value of integers. Evaluate each expression Solve using either a number line or draw chips (-8) (-8) (-7) (-7) (-7) Solve by drawing chips (-3). (4) 3. - (5) 4. - (5) 5. - (-3) Solve by using the rule (-8) 3. ( 5) 4. (5) Evaluate each expression if r = -4, s = 10, and t = r 7. t s 3. t r 4. r s Evaluate each expression if g = -5, h = -3, and k = ghk. -3hk 3. h 4. gh Evaluate each expression if m = -3, n =, and p = p m Solve.

7 1. A balloon rises 340 feet into the air. Then it descends 130 feet. How high is the balloon?. A roller coaster begins at 90 feet above ground level. Then it descends 105 feet. Find the height of the coaster after the first descent. 3. Suppose the temperature outside is dropping 3 degrees each hour. How much will the temperature change in 8 hours? 4. During a 5-day workweek, the stock market decreased by 65 points. Find the average daily change in the market for the week. 5. Use a set of integer chips to model one method for evaluating (-). Explain your work. 6. Which of the following best represents the method for finding the value of ? 7. Which of the following best represents the method for finding the value of (-4). a) b) c) d)

8 MA.7.A.3. Add, subtract, multiply, and divide integers, fractions, and terminating decimals, and perform exponential operations with rational bases and whole number exponents including solving problems in everyday contexts. Evaluate each exponent ( 3) Solve each word problem. 7. Shiangtai s office has a budget of about $10,000,000. Write this amount in exponential form. 8. The Africa bush elephant is the largest land animal and weighs about 8 tons. Write this amount in exponential form. 9. To find the volume of a rectangular box, you multiply the length times the width times the height. In a cube, all sides are the same length. If the cube has length, widths, and height of 6 inches, write the volume as a product. Then write it in exponential form. 10. A certain type of cell doubles every hour. If you start with one cell, at the end of one hour you would have cells, at the end of two hours you have 4 cells, and so on. The expression x x x x tells you how many cells you would have after five hours. Write this expression in exponential form; then evaluate it. 11. Many prefixes are used in mathematics and science. The prefix giga in gigameter represents 1,000,000,000 meters. Write this number as a power of ten. 1. The school library contains 9 4 books. How many library books are in the school library? 13. The concession stand at the county fair sold 6 3 hot dogs on the first day. How many hot dogs did they sell? 14. What is the value of the expression 14 5²? Evaluate the expression (98 51)

9 16. You ve won! For a door prize, you get to choose between the two options shown. What is the better prize? Why? PRIZE 1 PRIZE You win $60 immediately! You get $1 the first day. Then, each day for the next five days, you get twice the previous day s amount. 17. On a cold day, Rupert measured the outside temperature and discovered it was 13ºF. Each hour after that, Rupert measured the outside temperature and discovered it was 3ºF colder than the previous hour s temperature. At these rates, how many hours would it take for the temperature to reach 17ºF? A. 4 Hours B. 9 Hours C. 10 Hours D. 30 Hours