Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Divisibility and Factors Lesson 4-1 Pre-Algebra Objectives: 1. to use divisibility tests 2. to find factors Divisibility and Factors Lesson 4-1 Pre-Algebra New Terms: 1. 2. Tips: divisible – one integer is divisible by another if the remainder is 0 when you divide factor – one integer is a factor of another integer (not zero) if it divides that integer with a remainder zero. the product of two integers is an integer, and both integers are factors of the product. Moreover, both integers divide the product, and the product is said to be divisible by each integer. Divisibility and Factors Lesson 4-1 Pre-Algebra Is the first number divisible by the second? a. 1,028 by 2 Yes; 1,028 ends in 8. b. 572 by 5 No; 572 doesn’t end in 0 or 5. c. 275 by 10 No; 275 doesn’t end in 0. Divisibility and Factors Lesson 4-1 Pre-Algebra Is the first number divisible by the second? a. 1,028 by 3 No; 1 + 0 + 2 + 8 = 11; 11 is not divisible by 3. b. 522 by 9 Yes; 5 + 2 + 2 = 9; 9 is divisible by 9. Divisibility and Factors Lesson 4-1 Pre-Algebra Ms. Washington’s class is having a class photo taken. Each row must have the same number of students. There are 35 students in the class. How can Ms. Washington arrange the students in rows if there must be at least 5 students, but no more than 10 students, in each row? 1 • 35, 5 • 7 Find pairs of factors of 35. There can be 5 rows of 7 students, or 7 rows of 5 students. Exponents Lesson 4-2 Pre-Algebra Objectives: 1. to use exponents 2. to use the order of operations with exponents Exponents Lesson 4-2 Pre-Algebra New Terms: 1. exponents – you can use exponents to show repeated multiplication 2. power – a power has two parts, a base and an exponent Tips: the exponent is placed to the upper right of the base, and it only applies to that base. If an exponent has as its base an expression, that expression must be written in parentheses. Exponents Lesson 4-2 Pre-Algebra Write using exponents. a. (–11)(–11)(–11)(–11) (–11)4 Include the negative sign within parentheses. b. –5 • x • x • y • y • x –5 • x • x • x • y • y Rewrite the expression using the Commutative and Associative Properties. –5x3y2 Write x • x • x and y • y using exponents. Exponents Lesson 4-2 Pre-Algebra Suppose a certain star is 104 light-years from Earth. How many light-years is that? 104 = 10 • 10 • 10 • 10 = 10,000 light-years The exponent indicates that the base 10 is used as a factor 4 times. Multiply. The star is 10,000 light-years from Earth. Exponents Lesson 4-2 Pre-Algebra a. Simplify 3(1 + 4)3. 3(1 + 4)3 = 3(5)3 Work within parentheses first. = 3 • 125 Simplify 53. = 375 Multiply. b. Evaluate 7(w + 3)3 + z, for w = –5 and z = 6. 7(w + 3)3 + z = 7(–5 + 3)3 + 6 Replace w with –5 and z with 6. = 7(–2)3 + 6 Work within parentheses. = 7(–8) + 6 Simplify (–2)3. = –56 + 6 Multiply from left to right. = –50 Add. Prime Factorization and Greatest Common Factor Lesson 4-3 Pre-Algebra Objectives: 1. to find the prime factorization of a number 2. to find the greatest common factor (GCF) of two or more numbers New Terms: 1. prime number – is an integer greater than 1 with exactly two positive factors, 1 and itself 2. composite number – is an integer greater than 1 with more than two positive factors 3. greatest common factor – factors that are the same for two or more numbers or expressions are common factors. The greatest of these is called the GCF. Tips: remember a factor is a number that divides evenly into another number with a remainder of zero. Prime Factorization and Greatest Common Factor Lesson 4-3 Pre-Algebra State whether each number is prime or composite. Explain. a. 46 Composite; 46 has more than two factors, 1, 2, 23, and 46. b. 13 Prime; 13 has exactly 2 factors, 1 and 13. Prime Factorization and Greatest Common Factor Lesson 4-3 Pre-Algebra Use a factor tree to write the prime factorization of 273. 273 Prime Prime 3 • 7 • 13 273 = 3 • 7 • 13 3 • 91 7 • Start with a prime factor. Continue branching. 13 Stop when all factors are prime. Write the prime factorization. Prime Factorization and Greatest Common Factor Lesson 4-3 Pre-Algebra Find the GCF of each pair of numbers or expressions. a. 24 and 30 24 = 23 • 3 Write the prime factorizations. 30 = 2 • 3 • 5 Find the common factors. GCF = 2 • 3 Use the lesser power of the common factors. =6 The GCF of 24 and 30 is 6. b. 36ab2 and 81b 36ab2 = 22 • 32 • a • b2 Write the prime factorizations. 81b = 34 • b Find the common factors. GCF = 32 • b Use the lesser power of the common factors. = 9b The GCF of 36ab2 and 81b is 9b. Simplifying Fractions Lesson 4-4 Pre-Algebra Objectives: 1. to find equivalent fractions 2. to write fractions in simplest for New Terms: 1. Tips: simplest form – a fraction is in simplest form when the numerator and the denominator have no common factors other than 1. always check your answers and the steps involved in finding the answers Simplifying Fractions Lesson 4-4 Pre-Algebra Find two fractions equivalent to a. 18 21 18 . 21 18 • 2 = 21 • 2 36 = 42 b. 18 21 = 18 ÷ 3 21 ÷ 3 = 6 7 The fractions 6 36 18 and are both equivalent to . 7 42 21 Simplifying Fractions Lesson 4-4 Pre-Algebra You learn that 21 out of the 28 students in a class, or 21 , buy their lunches in the cafeteria. Write this fraction in 28 simplest form. The GCF of 21 and 28 is 7. 21 = 21 ÷ 7 28 28 ÷ 7 = 3 4 Divide the numerator and denominator by the GCF, 7. Simplify. 3 of the students in the class buy their lunches in the cafeteria. 4 Simplifying Fractions Lesson 4-4 Pre-Algebra Write in simplest form. p a. 2p p p1 = 1 2p 2p = 1 2 Divide the numerator and denominator by the common factor, p. Simplify. Simplifying Fractions Lesson 4-4 Pre-Algebra (continued) b. 14q2rs3 8qrs2 14q2rs3 8qrs2 = 2•7•q•q•r•s•s•s 2•2•2•q•r•s•s Write as a product of prime factors. = 21 • 7 • q1 • q • r 1 • s 1 • s 1 • s 21 • 2 • 2 • q1 • r1 • s1 • s1 Divide the numerator and denominator by the common factors. = 7•q•s 2•2 Simplify. = 7•q•s 4 Simplify. = 7qs 4 Problem Solving Strategy: Solve a Simpler Problem Lesson 4-5 Additional Examples Pre-Algebra Aaron, Chris, Maria, Sonia, and Ling are on a class committee. They want to choose two members to present their conclusions to the class. How many different groups of two members can they form? Problem Solving Strategy: Solve a Simpler Problem Lesson 4-5 Pre-Algebra Additional Examples (continued) First, pair Aaron with each of the four other committee members. Next, pair Chris with each of the three members left. Since Aaron and Chris have already been paired, you don’t need to count them again. Repeat for the rest of the committee members. Each successive tree has one less branch. Aaron Chris Maria Sonia Ling Chris Maria Sonia Ling Maria Sonia Ling Sonia Ling There are 10 different groups of two committee members. Rational Numbers Lesson 4-6 Pre-Algebra Objectives: 1. to identify and graph rational numbers 2. to evaluate fractions containing variables New Terms: 1. Tips: rational number – is any number you can write as a fraction the quotient of two integers with the same sign is positive Rational Numbers Lesson 4-6 Pre-Algebra Write two lists of fractions equivalent to 2. 3 2 = 4 = 6 = … Numerators and denominators are positive. 3 6 9 2 = –2 = –4 = … Numerators and denominators are negative. 3 –3 –6 Rational Numbers Lesson 4-6 Additional Examples Graph each rational number on a number line. 3 a. – 4 b. 0.5 c. 0 d. 1 3 Pre-Algebra Rational Numbers Lesson 4-6 Pre-Algebra A fast sports car can accelerate from a stop to 90 ft/s in 5 seconds. What is its acceleration in feet per second per second (ft/s2)? Use the formula a = f – i , where a is t acceleration, f is final speed, i is initial speed, and t is time. a= = f–i t Use the acceleration formula. 90 – 0 5 Substitute. 90 = 5 Subtract. = 18 Write in simplest form. The car’s acceleration is 18 ft/s2. Exponents and Multiplication Lesson 4-7 Pre-Algebra Objectives: 1. to multiply powers with the same base 2. to find a power of a power Exponents and Multiplication Lesson 4-7 Pre-Algebra Tips: when in doubt, write it out. Exponents and Multiplication Lesson 4-7 Pre-Algebra Simplify each expression. a. 52 • 53 52 • 53 = 52 + 3 Add the exponents of powers with the same base. = 55 = 3,125 Simplify. b. x5 • x7 • y2 • y x5 • x7 • y2 • y = x5 + 7 • y2 + 1 = x12y3 Add the exponents of powers with the same base. Simplify. Exponents and Multiplication Lesson 4-7 Pre-Algebra Simplify 3a3 • (–5a4). 3a3 • (–5a4) = 3 • (–5) • a3 • a4 Use the Commutative Property of Multiplication. = –15a3 + 4 Add the exponents. = –15a7 Simplify. Exponents and Multiplication Lesson 4-7 Pre-Algebra Simplify each expression. a. (23)3 (23)3 = (2)3 • 3 Multiply the exponents. = (2)9 Simplify the exponent. = 512 Simplify. b. (g5)4 (g5)4 = g5 • 4 = g20 Multiply the exponents. Simplify the exponent. Exponents and Division Lesson 4-8 Pre-Algebra Objectives: 1. To divide expressions containing exponents 2. To simplify expressions with integer exponents Exponents and Division Lesson 4-8 Pre-Algebra Tips: Sometimes students think an expression with a negative exponent makes the expression negative, that is not true. A negative exponent makes the number smaller (unless the base is less than one but greater than zero). Exponents and Division Lesson 4-8 Pre-Algebra Simplify each expression. a. 412 48 412 48 b. = 412 – 8 Subtract the exponents. = 44 Simplify the exponent. = 256 Simplify. w18 w13 w18 = w18 w13 = w5 – 13 Subtract the exponents. Simplify the exponent. Exponents and Division Lesson 4-8 Pre-Algebra Simplify each expression. 73 a. (–12)73 (–12) (–12)73 73 = (–12) 73 (–12) = (–12)0 =1 b. 8s20 32s20 8s20 32s20 1 = 4 s0 1 = 4 •1 1 = 4 – 73 Subtract the exponents. Simplify. Subtract the exponents. Simplify Simplify s0. Multiply. 8 . 32 Exponents and Division Lesson 4-8 Pre-Algebra Simplify each expression. 12 a. 614 6 612 = 612 614 – 14 Subtract the exponents. = 6–2 1 62 1 = 36 = Write with a positive exponent. Simplify. 4 b. z15 z z4 = z4 – 15 z15 Subtract the exponents. = z–11 = 111 z Write with a positive exponent. Exponents and Division Lesson 4-8 Pre-Algebra a2b3 Write without a fraction bar. ab15 a2b3 2 – 1b3 = a 15 ab = ab–12 – 15 Use the rule for Dividing Powers with the Same Base. Subtract the exponents. Scientific Notation Lesson 4-9 Pre-Algebra Objectives: 1. to write and evaluate numbers in scientific notation 2. to calculate with scientific notation New Terms: 1. 2. Tips: scientific notation – a way to write numbers using powers of 10 standard notation – write the number using all digit and decimal places be careful with which way you move the decimal, think if the number is suppose to be smaller or larger Scientific Notation Lesson 4-9 Pre-Algebra About 6,300,000 people visited the Eiffel Tower in the year 2000. Write this number in scientific notation. 6,300,000 Move the decimal point to get a decimal greater than 1 but less than 10. 6 places 6.3 6.3 x 106 Drop the zeros after the 3. You moved the decimal point 6 places. The original number is greater than 10. Use 6 as the exponent of 10. Scientific Notation Lesson 4-9 Pre-Algebra Write 0.00037 in scientific notation. 0.00037 Move the decimal point to get a decimal greater than 1 but less than 10. 4 places 3.7 3.7 x 10–4 Drop the zeros before the 3. You moved the decimal point 4 places. The original number is less than 1. Use –4 as the exponent of 10. Scientific Notation Lesson 4-9 Pre-Algebra Write each number in standard notation. a. 3.6 x 104 3.6000 36,000 b. 7.2 x 10–3 007.2 0.0072 Write zeros while moving the decimal point. Rewrite in standard notation. Write zeros while moving the decimal point. Rewrite in standard notation. Scientific Notation Lesson 4-9 Pre-Algebra Write each number in scientific notation. a. 0.107 x 1012 0.107 x 1012 = 1.07 x 10–1 x 1012 = 1.07 x 1011 Write 0.107 as 1.07 10–1. Add the exponents. b. 515.2 x 10–4 515.2 x 10–4 = 5.152 x 102 x 10–4 = 5.152 x 10–2 Write 515.2 as 5.152 102. Add the exponents. Scientific Notation Lesson 4-9 Pre-Algebra Order 0.035 x 104, 710 x 10–1, and 0.69 x 102 from least to greatest. Write each number in scientific notation. 0.035 x 104 3.5 x 102 710 x 10–1 0.69 x 102 7.1 x 10 6.9 x 10 Order the powers of 10. Arrange the decimals with the same power of 10 in order. 6.9 x 10 7.1 x 10 Write the original numbers in order. 0.69 x 102, 710 x 10–1, 0.035 x 104 3.5 x 102 Scientific Notation Lesson 4-9 Pre-Algebra Multiply 4 x 10–6 and 7 x 109. Express the result in scientific notation. (4 x 10–6)(7 x 109) = 4 x 7 x 10–6 x 109 Use the Commutative Property of Multiplication. = 28 x 10–6 x 109 Multiply 4 and 7. = 28 x 103 Add the exponents. = 2.8 x 101 x 103 Write 28 as 2.8 101. = 2.8 x 104 Add the exponents. Scientific Notation Lesson 4-9 Pre-Algebra In chemistry, one mole of any element contains approximately 6.02 x 1023 atoms. If each hydrogen atom weighs approximately 1.67 x 10–27 kg, approximately how much does one mole of hydrogen atoms weigh? Multiply number of atoms by (6.02 x 1023)(1.67 x 10–27) weight of each. = 6.02 x 1.67 x 1023 x 10–27 10.1 x 1023 x 10–27 Use the Commutative Property of Multiplication. Multiply 6.02 and 1.67. = 10.1 x 10–4 Add the exponents. = 1.01 x 101 x 10–4 Write 10.1 as 1.01 101. = 1.01 x 10–3 Add the exponents. One mole of hydrogen atoms weighs approximately 1.01 x 10–3 kg.