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Learning Targets · I can use a number line to graph and order real numbers. · I can identify properties of and use operations with real numbers. Algebra 2 Chapter 1 Lessons 1.1 Real Number and Number Operations 1.1 Real Numbers • • • • Rational Numbers Can be written as a quotient of integers. Can be written as decimals that terminate or repeat. Irrational Numbers Cannot be written as quotients of integers. Cannot be written as decimals that terminate or repeat. Real Numbers Natural (Counting) numbers: N = {1, 2, 3, …} Whole numbers: W = {0, 1, 2, 3, …} Integers: Z = {0, 1, 2, 3, …} Rational Numbers: Any number that can be written as a fraction where the numerator and denominator are both integers and the denominator doesn’t equal zero Irrational Numbers: Any number that isn’t a rational number Real Numbers Irrational Numbers 7 Rational Numbers Integers .34 1 -2 -1 -5 Whole Numbers Natural Numbers e - 2 1 2 3 0 2 3.7 Learning Target #1. Example 1 I can use a number line to graph and order real numbers. Graph the real numbers – 5 and 3 on a number line. 4 SOLUTION Note that – 5 = –1.25. Use a calculator to approximate 4 3 to the nearest tenth: 3 1.7. (The symbol means is approximately equal to.) 5 between –2 and –1, and graph 3 between 4 1 and 2, as shown on the number line below. So, graph – EXAMPLE 2 Standardized Test Practice SOLUTION From lowest to highest, the elevations are – 408, –156, –86, – 40, –28, and –16. ANSWER The correct answer is D. GUIDED PRACTICE for Examples 1 and 2 1. Graph the numbers – 0.2, 7 , –1, 2 , and – 4 on a 10 number line. ANSWER –4 –4 –1 – 0.2 –3 –2 –1 0 7 10 2 1 2 3 4 GUIDED PRACTICE for Examples 1 and 2 2. Which list shows the numbers in increasing order? – 0.5, 1.5, – 2, – 0.75, 7 – 0.5, – 2, – 0.75, 1.5, 7 – 2, – 0.75, – 0.5, 1.5, 7 7 , 1.5, – 0.5 , – 0.75, – 2 ANSWER The correct answer is C. 1.1 Properties of Addition and Multiplication Let a, b, and c be real numbers. Property Addition Example Multiplication Example Closure a + b is a real number 5 + -6 = -1 ab is a real number ½ (4) = 2 Commutative a+b=b+a -3+7=7+-3 ab=ba -4(3)=3(-4) Associative (a+b)+c=a+(b+c) (2+6)+1=2+(6+1) (ab)c=a(bc) (2*7)1=2(7*1) Identity a+0=a, 0+a=a -2+0=-2, 0+-2=-2 a*1=a, 1*a=a ¾ *1= ¾ 1* ¾ = ¾ Inverse a+(-a)=0 .5+-.5=0 a* 1/a = 1, a≠0 2* ½ = 1 Distributive a(b+c)=ab+ac (combines adding & multiplying) 2(1+4)=2*1+2*4 EXAMPLE 3 Identify properties of real numbers Identify the property that the statement illustrates. a. 7+4=4+7 SOLUTION Commutative property of addition b. 13 1 = 1 13 SOLUTION Inverse property of multiplication EXAMPLE 4 Use properties and definitions of operations Use properties and definitions of operations to show that a + (2 – a) = 2. Justify each step. SOLUTION a + (2 – a) = a + [2 + (– a)] Definition of subtraction = a + [(– a) + 2] Commutative property of addition = [a + (– a)] + 2 Associative property of addition =0+2 Inverse property of addition =2 Identity property of addition for Examples 3 and 4 GUIDED PRACTICE Identify the property that the statement illustrates. 3. (2 3) 9 = 2 (3 9) SOLUTION Associative property of multiplication. 4. 15 + 0 = 15 SOLUTION Identity property of addition. GUIDED PRACTICE for Examples 3 and 4 Identify the property that the statement illustrates. 5. 4(5 + 25) = 4(5) + 4(25) SOLUTION Distributive property. 6. 1 500 = 500 SOLUTION Identity property of multiplication. GUIDED PRACTICE for Examples 3 and 4 Use properties and definitions of operations to show that the statement is true. Justify each step. 7. b b) = 4 when b = 0 (4 SOLUTION b (4 b) = b (4 (1 b = (b 1 ) b =1 4 =4 =b 1) b Def. of division 4) Comm. prop. of multiplication 4 Assoc. prop. of multiplication Inverse prop. of multiplication Identity prop. of multiplication GUIDED PRACTICE for Examples 3 and 4 Use properties and definitions of operations to show that the statement is true. Justify each step. 8. 3x + (6 + 4x) = 7x + 6 SOLUTION 3x + (6 + 4x) = 3x + (4x + 6) Comm. prop. of addition = (3x + 4x) + 6 Assoc. prop. of addition = 7x + 6 Combine like terms. EXAMPLE 5 a. Use unit analysis with operations You work 4 hours and earn $36. What is your earning rate? SOLUTION 36 dollars = 9 dollars per hour 4 hours b. You travel for 2.5 hours at 50 miles per hour. How far do you go? SOLUTION (2.5 hours) 50 miles = 125 miles 1 hour EXAMPLE 5 c. Use unit analysis with operations You drive 45 miles per hour. What is your speed in feet per second? SOLUTION 45 miles 1 hour 1 hour 60 minutes = 66 feet per second 1 minute 60 seconds 5280 feet 1 mile EXAMPLE 6 Use unit analysis with conversions Driving Distance The distance from Montpelier, Vermont, to Montreal, Canada, is about 132 miles. The distance from Montreal to Quebec City is about 253 kilometers. a. Convert the distance from Montpelier to Montreal to kilometers. b. Convert the distance from Montreal to Quebec City to miles. EXAMPLE 6 Use unit analysis with conversions SOLUTION 1.61 kilometers 1 mile a. 132 miles b. 1 mile 253 kilometers 1.61 kilometers 213 kilometers 157 miles GUIDED PRACTICE for Examples 5 and 6 Solve the problem. Use unit analysis to check your work. 9. You work 6 hours and earn $69. What is your earning rate? SOLUTION 11.50 dollars per hour 10. How long does it take to travel 180 miles at 40 miles per hour? SOLUTION 4.5 hours GUIDED PRACTICE for Examples 5 and 6 Solve the problem. Use unit analysis to check your work. 11. You drive 60 kilometers per hour. What is your speed in miles per hour? SOLUTION about 37 mph GUIDED PRACTICE for Examples 5 and 6 Perform the indicated conversion. 12. 150 yards to feet SOLUTION 450 ft GUIDED PRACTICE for Examples 5 and 6 Perform the indicated conversion. 13. 4 gallons to pints SOLUTION 32 pints GUIDED PRACTICE for Examples 5 and 6 Perform the indicated conversion. 14. 16 years to seconds SOLUTION 504,576,000 sec Classwork Pair- share on #2 to #54 page 7 to 8 (Even Nos Only)