Numerical Expressions 1 and Factors 11.1.1 WWholehole NNumberumber OOperationsperations 11.2.2 PPowersowers aandnd EExponentsxponents 11.3.3 OOrderrder ooff OOperationsperations 11.4.4 PPrimerime FFactorizationactorization 11.5.5 GGreatestreatest CCommonommon FFactoractor 1.61 Least Common MultipleMultiple x1 “So, why isn’t x2 called x-lined?” “And you say that 3 x-squared.” x is called “Dear Sir: Youx-cubed.” say that is called “My sign on adding fractions with unlike denominators is keeping “See,“See, it’s it’s working.” working.” the hyenas away.” What You Learned Before “Because 6 is composite, you can have 2 piles of 3 blocks, 3 piles of 2 blocks, or...” Example 1 Determine whether 26 is prime or composite. Because the factors of 26 are 1, 2, 13, and 26, it is composite. Example 2 Determine whether 37 is prime or composite. Because the only factors of 37 are 1 and 37, it is prime. Determine whether the number is prime or composite. 1. 5 2. 14 3. 17 4. 23 5. 28 6. 33 7. 43 8. 57 9. 64 3 1 Example 3 Find 2 — + 4 — . 5 5 3 1 2 ⋅ 5 + 3 4 ⋅ 5 + 1 2 — + 4 — = — + — Rewrite the mixed numbers as improper fractions. 5 5 5 5 13 21 =— + — Simplify. 5 5 13 + 21 =— Add the numerators. 5 34 4 =—, or 6 — Simplify. 5 5 Add or subtract. 1 7 1 6 7 3 10. 4 — + 2 — 11. 6 — + 3 — 12. 3 — + 4 — 9 9 11 11 8 8 8 2 1 3 1 5 13. 5 — − 1— 14. 7 — − 3 — 15. 4 — − 2 — 13 13 4 4 6 6 1.1 Whole Number Operations How do you know which operation to choose when solving a real-life problem? 1 ACTIVITY: Choosing an Operation Work with a partner. The double bar graph shows the history of a citywide cleanup day. City Cleanup Day 7000 6390 Trash 6000 Recyclables 4970 5000 3975 4000 3000 2130 2000 10951095 Amount collected (pounds) 1000 732 555 183 0 2010 2011 2012 2013 Year ● Copy each question below. ● Underline a key word or phrase that helps you know which operation to use to answer the question. State the operation. Why do you think the key word or phrase indicates the operation you chose? ● Write an expression you can use to answer the question. ● Find the value of your expression. COMMON CORE a. What is the total amount of trash Whole Numbers collected from 2010 to 2013? In this lesson, you will b. How many more pounds of ● determine which operation to perform. recyclables were collected in 2013 ● divide multi-digit numbers. than in 2010? Learning Standard c. How many times more recyclables MACC.6.NS.2.2 were collected in 2012 than in 2010? d. The amount of trash collected in 2014 is estimated to be twice the amount collected in 2011. What is that amount? 2 Chapter 1 Numerical Expressions and Factors 2 ACTIVITY: Checking Answers Math Work with a partner. Practice a. Explain how you can use estimation to check the reasonableness of the value of your expression in Activity 1(a). Communicate Precisely b. Explain how you can use addition to check the value of your expression in What key words Activity 1(b). should you use so c. Explain how you can use estimation to check the reasonableness of the that your partner understands your value of your expression in Activity 1(c). explanation? d. Use mental math to check the value of your expression in Activity 1(d). Describe your strategy. 3 ACTIVITY: Using Estimation Work with a partner. Use the map. Explain how you found each answer. a. Which two lakes have a combined area of about Lake Superior 33,000 square miles? 31,698 mi2 b. Which lake covers an area about three times greater than the area of Lake Huron Lake Erie? 23,011 mi2 c. Which lake covers an Lake Ontario Lake Michigan 7320 mi2 area that is about 16,000 22,316 mi2 square miles greater than Lake Erie the area of Lake Ontario? 9922 mi2 d. Estimate the total area covered by the Great Lakes. 4. IN YOUR OWN WORDS How do you know which operation to choose when solving a real-life problem? 5. In a magic square, the sum of the numbers in ? 9 2 each row, column, and diagonal is the same ? 5 ? and each number from 1 to 9 is used only once. Complete the magic square. Explain how you 8 ? ? found the missing numbers. Use what you learned about choosing operations to complete Exercises 8−11 on page 7. Section 1.1 Whole Number Operations 3 1.1 Lesson Lesson Tutorials Recall the four basic operations: addition, subtraction, multiplication, and division. Operation Words Algebra Addition the sum of a + b Subtraction the difference of a − b Multiplication the product of a × b a ⋅ b a Division the quotient of a ÷ b — b )‾ a b EXAMPLE 1 Adding and Subtracting Whole Numbers The bar graph shows the attendance at a three-day art festival. a. What is the total attendance for the art festival? Art Festival Attendance You want to fi nd the total attendance for the three days. 111 4500 2570 3876 In this case, the phrase total attendance indicates you 4000 3145 need to fi nd the sum of the daily attendances. 3500 3145 + 3876 3000 2570 Line up the numbers by their place values, then add. 9591 2500 2000 The total attendance is 9591 people. 1500 1000 b. What is the increase in attendance from Day 1 to Day 2? Number of people 500 0 You want to fi nd how many more people attended on Day 2 than 123 on Day 1. In this case, the phrase how many more indicates you need Day to fi nd the difference of the attendances on Day 2 and Day 1. 10 Line up the numbers by their place values, then subtract. 2014 3145 The increase in attendance from Day 1 to Day 2 − 2570 is 575 people. 575 EXAMPLE 2 Multiplying Whole Numbers A school lunch contains 12 chicken nuggets. Ninety-fi ve students buy the lunch. What is the total number of chicken nuggets served? Study Tip You want to fi nd the total number of chicken nuggets in 95 groups of In Example 2, you can 12 chicken nuggets. The phrase 95 groups of 12 indicates you need to use estimation to check fi nd the product of 95 and 12. the reasonableness of your answer. 12 12 × 95 ≈ 12 × 100 × 95 = 1200 60 Multiply 12 by the ones digit, 5. Because 1200 ≈ 1140, 108 Multiply 12 by the tens digit, 9. the answer is 1140 Add. reasonable. There were 1140 chicken nuggets served. 4 Chapter 1 Numerical Expressions and Factors Find the value of the expression. Use estimation to check your answer. Exercises 12–20 1. 1745 + 682 2. 912 − 799 3. 42 × 118 EXAMPLE 3 Dividing Whole Numbers: No Remainder You make 24 equal payments for a go-kart. You pay a total of $840. How much is each payment? You want to fi nd the numbermber of groups of 24 in $840. TThehe phrase groups of 24 in $840840 indicates you need to fi nndd the quotient of 840 and 224.4. Use long division to fi ndd the quotient.uotient Decide where to write thehe fi rst ddigitigit ooff tthehe quotient. ? 24 )‾ 840 Do not use the hundreds place because 24 is greater than 8. ? 24 )‾ 840 Use the tens place because 24 is less than 84. So, divide the tens and write the fi rst digit of the quotient in the tens place. 3 24 )‾ 840 Divide 84 by 24: There are three groups of 24 in 84. − 72 Multiply 3 and 24. 12 Subtract 72 from 84. Next, bring down the 0 and divide the ones. 35 24 )‾ 840 Divide 120 by 24: There are fi ve groups of 24 in 120. Remember − 72 120 dividend Check Find the = quotient − 120 Multiply 5 and 24. divisor product of the quotient 0 Subtract 120 from 120. So, quotient × divisor and the divisor. = dividend. The quotient of 840 and 24 is 35. 35 quotient × 24 divisor So, each payment is $35. 140 70 840 dividend ✓ Section 1.1 Whole Number Operations 5 Find the value of the expression. Use estimation to check your answer. Exercises 21–23 986 4. 234 ÷ 9 5. — 6. 840 ÷ 105 58 7. Find the quotient of 9920 and 320. When you use long division to divide whole numbers and you obtain a remainder, you can write the quotient as a mixed number using the rule remainder dividend ÷ divisor = quotient + — . divisor EXAMPLE 4 Real-Life Application A 301-foot-high swing at an amusement park can take 64 people on each ride. A total of 8983 people ride the swing today. All the rides are full except for the last ride. How many rides are given? How many people are on the last ride? To fi nd the number of rides given, you need to fi nd the number of groups of 64 people in 8983 people. The phrase groups of 64 people in 8983 people indicates you need to fi nd the quotient of 8983 and 64. Divide the place-value positions from left to right. 140 R23 64 )‾ 8983 There is one group of 64 in 89. − 64 258 There are four groups of 64 in 258. Do not stop here. You − 256 must write a 0 in the ones 23 There are no groups of 64 in 23.