Math Notes and Example Problems Lesson 2.1 Integers

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1 Name Warm-up: Math Notes and Example Problems Lesson 2.1 Integers Textbook p Today s Goal: Learn to compare and order integers and to determine absolute value. The, or additive inverse, of a number is the same distance from 0 on a number line as the original number, but on the other side of 0. Zero is its own opposite. negative integers positive integers The are the set of whole numbers and their opposites. A number s is its distance from 0 on a number line. They are always or zero. Remember! The numbers are the natural numbers and zero: 0, 1, 2, 3, The symbol means is less than, and the symbol means is greater than.

2 Lesson 2.1 Example 1 Graph the integer 7 and its opposite on the number line. You Try It! Graph the integer 5 and its opposite on the number line. Lesson 2.1 Example 2 You can compare and order integers by graphing them on a number line. Integers in value as you move to the right along a number line. They decrease in value as you move to the. Compare the integers. Use < or > is farther to the right than 4, -1 is farther to the right than 5 so 4 4. so 1 5. Lesson 2.1 Example 3 Use a number line to order the integers from least to greatest. 1, -3, 4, -1, 0, 5 Write the numbers from least to greatest.

3 More Practice Use a number line to order the integers from least to greatest. -5, 4, -3, 2, -1, -2 Write the numbers from least to greatest. Lesson 2.1 Example 4 Absolute value is the distance a number is from zero. Since distance can never be negative, absolute values are never negative. They are always positive or zero. Use a number line to find each absolute value. 3 is 3 units from 0, so 3 =. Use a number line to find each absolute value. 12 is 12 units from 0, so 12 =

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5 Math Notes and Example Problems Lesson 2.2 Adding Integers Textbook p Warm-up: Today s Goal: Learn to add integers Lesson 2.2 Example 1 Use a number line to find each sum Start at 0. Move left 1 unit. Then move left 4 more units. 7 + ( 4) = Use a number line to find each sum Start at 0. Move right 3 units. Then move left 5 units = Same sign, find the Different signs, find the Lesson 2.2 Example = The signs are the same. Find the sum of the absolute values = The signs are different. Find the difference of the absolute values. Operation: Sign: Answer: Absolute Value Think Bubble Operation: Sign: Answer: Absolute Value Think Bubble = Use sign of number farthest from zero in the original problem 10 4 =

6 You Try It # = You Try It # = Operation: Sign: Answer: Operation: Sign: Answer: Lesson 2.2 Example 3 Rewrite: Evaluate a + b for a = 3, b = 15 You Try It #3 Rewrite: Evaluate x + y for x = 42, y = 71 Think Bubble Operation: Sign: Answer: Operation: Sign: Answer: You Try It #4 Rewrite: Evaluate a + b for a = 2, b = 16 You Try It #5 Rewrite: Evaluate x + y for x = 24, y = 17 Operation: Sign: Answer: Operation: Sign: Answer: Lesson 2.2 Example 4 Andrea s income from a lemonade stand was $28. Supply expenses were $9. Use integer addition to find Andrea s total profit or loss. Total = Total income + cost of supplies (expenses)

7 Math Notes and Example Problems Lesson 2.3 Subtracting Integers Textbook p Warm-up: Today s Goal: Learn to subtract integers Lesson 2.3 Example = 1 3 = Subtraction rule: ADD THE OPPOSITE! Then, you use the addition rules you already learned! Addition and subtraction are inverse operations they undo each other. Instead of subtracting a number you can add its opposite. 2 ( 4) = = Step 1: Rewrite and add the opposite of 2 nd sign: Step 1: Rewrite and add the opposite of 2 nd sign: Step 2: Same signs = Add; Different signs = Subt. Step 3: Check Sign for final answer Step 2: Same signs = Add; Different signs = Subt. Step 3: Check Sign for final answer Lesson 2.3 Example 2 7 ( 5) = Step 1: Rewrite and add the opposite of 2 nd sign: 8 2 = Step 1: Rewrite and add the opposite of 2 nd sign: Step 2: Same signs = Add; Different signs = Subt. Step 3: Check Sign for final answer Step 2: Same signs = Add; Different signs = Subt. Step 3: Check Sign for final answer

8 More Practice 2 ( 4) = Step 1: Rewrite and add the opposite of 2 nd sign: 3 2 = Step 1: Rewrite and add the opposite of 2 nd sign: Step 2: Same signs = Add; Different signs = Subt. Step 3: Check Sign for final answer Step 2: Same signs = Add; Different signs = Subt. Step 3: Check Sign for final answer Lesson 2.3 Example 3 (Show your work!) Evaluate a b for a = 4 and b = 12 Evaluate a b for a = 20 and b = 5 Lesson 2.3 Example 4 (Show your work!) The highest point in California, Mt. Whitney, is about 14,500 feet above sea level and the lowest point, Death Valley, is about 280 feet below sea level. Find the difference in elevation. More Practice #3 In Mackinaw City, Michigan the temperature rose from 5 F to 10 F. How much did the temperature increase? (hint: find the difference of greatest number minus least number)

9 Math Notes and Example Problems Lesson 2.4 Multiplying and Dividing Integers Textbook p Warm-up: Today s Goal: Learn to multiply and divide integers Remember! Multiplication and division are operations. They undo each other. MULTIPLYING AND DIVIDING TWO INTEGERS If the signs are: The same Different Your answer will be: Zero divided by any number is zero. For example, Why? Because We say that division by zero is. Answer on page 67 of your textbook. Lesson 2.4 Examples ( 2) = 2 3 = 5 ( 3) = 3 7 = 36 ( 4) = 20 (2) = 44 ( 11) = 25 ( 5) =

10 Lesson 2.4 Example 4 Sarah drove her police car at a constant speed down a mountain. Her elevation decreased by 200 ft over a 10-minute period. What was the change in elevation during the first minute? Helpful Hint: #34 on your book assignment: The story problem uses the word Intervals, which means breaks between sets or spans of something. How many 35-foot spans did it take for the scuba diver to get to 140 feet below sea level. 35 ft. span 140 ft below sea level More Practice Find the sum of the integers. Make a negative pile and a positive pile. Divide to find the average. Negative + Positive Numbers Numbers 29 (-7) 3 (5) 2 (-5) 2

11 Math Notes and Example Problems Lesson 2.5 Solving Equations Containing Integers Textbook p Warm-up: Today s Goal: Learn to solve one-step equations with integers. Lesson 2.5 Example 1 Use the inverse operations of Addition/Subtraction to solve the equations. m + 5 = 15 You Try It #1 n + 2 = z = 7 You Try It #2 3 + x = 9 x 10 = 97 You Try It #3 y 7 = 34 Stop

12 Lesson 2.5 Example 2 Use the inverse operation of division to solve: Multiplication d 4 = 3 You Try It #4 a 5 = 6 You Try It #5 c 5 = 20 You Try It #6 7 = e 5 Use the inverse operation of multiplication to solve: Division 205 = 5h You Try It #7 32 = 4y You Try It #8 5x = 30 You Try It #9 9n = 54 Lesson 2.5 Example 3 Prof. Burger made $2,000 cutting lawns, which turned out to be $500 more than he made last summer. How much money did Prof. burger make mowing lawns last summer? This year s profit is $500 more than Last year s profit

13 Math Notes and Example Problems Lesson 2.6 Equivalent Fractions and Decimals Textbook p Warm-up: Today s Goal: Learn to write fractions as decimals, and vice versa, and to determine whether a decimal is terminating or repeating. Decimals that come to an end are called. Decimals that repeat a pattern forever are called. Lesson 2.6 Example 1 Write each fraction as a decimal. Round to the nearest hundredth if necessary Vocabulary The decimals 0.25 and 1.8 in Example 1 are decimals because the decimals come to an end. The decimal is a decimal because the decimal repeats a pattern forever. You can also write a repeating decimal with a bar over the repeating part = = = 0.72

14 Lesson 2.6 Example You can also use mental math to convert fractions to decimals if the denominator can be easily converted to a power of ten (ex: 10, 100) You Try It # You Try It # Lesson 2.6 Example You Try It # You Try It # Lesson 2.6 Example 4 During a training session, Cart s dog Scout responded to the roll over command 16 of the 21 times he gave the command. Find Scout s success rate. Round to the nearest thousandth. Fraction Write Division Problem Success Rate in Decimal Form

15 Math Notes and Example Problems Lesson 2.7 Comparing and Ordering Rational Numbers Textbook p Warm-up: Today s Goal: Learn to compare and order fractions and decimals. A is a number that can be written as a fraction with an integer for its numerator and a nonzero integer for its denominator. Lesson 2.7 Example 1 Compare the fractions. Write < or > Compare the fractions. Write < or > Remember: The common denominator is the key! Lesson 2.7 Example 2 To compare decimals, line up the decimal points and compare digits from left to until you find the place where the digits are different. Note: Stack the decimals to compare You Try It # You Try It #

16 Lesson 2.7 Example 3 Remember: The values on a number line increase as you move from left to right. Order 4 5, 0.93, and 0.9 from least to greatest. Write as decimals with the same number of places. 4 5 = = 0. 9 = Graph the numbers on a number line < < Write the original numbers from least to greatest. You Try It! with Decimals Order 2.8, and 2.88 from least to greatest. Write as decimals with the same number of places. 2.8 = = 2.88 = Graph the numbers on a number line Write the original numbers from least to greatest.