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NCP 501, 502, 503 -­‐LESSONS/NOTES Period____________ CRS SKILL Name_________________________________________ LEVEL Level 1 – ALL students must attain mastery at this level DESCRIPTION NCP 301 Recognize one-­‐digit factors of a number NCP 302 Identify a digit’s place value NCP 401 Exhibit knowledge of elementary number concepts including rounding and pattern identification. NCP 501 Find and use least common multiple Level 2 – MOST students will attain mastery of the focus skill in NCP 502 Order fractions NCP 503 Work with numerical factors Numbers: isolation. Concepts & Level 3 – SOME students will Properties attain mastery of focus skill with other skills NCP 501 Level 4 – SOME students will NCP 502 attain mastery of focus topics NCP 503 covered in a more abstract way Level 5 – FEW students will NCP 601 Apply number properties involving prime attain mastery of the extension factorization skill. VOCABULARY Factor, Multiple, Prime number, Composite number, Place Value, Least Common Multiple, Greatest Common Factor, Prime Factorization REQUIRED SKILLS TO MASTER Round Numbers, Order Fractions, Divisibility intuition, Factor pairs, Identifying factors and multiples Additional KHAN Skills Divisibility Tests, Prime Factorization, Divisibility, Fundamental Theorem of Arithmetic, Least Common Multiple, Greatest Common Factor Level 1 1. List all of the factors of each number: a. 24 b. 62 c. 56 1 2 2. Mr. Buteyn has 28 doughnuts. He wants to hand them out to some track stars. List the different ways he could give them out evenly. For example he could give 14track stars two doughnuts each. 3. We know that 7 × 8 =
(Select all that apply) a. 8 is a factor of 56 56 . So, which of the following statements are also true? b. 7 is a multiple of 56 c. 7 is a factor of 56 d. 56 is a factor of 8 4. List the first 5 multiples of each number below. a. 7 b. 11 c. 13 3 Use this 100 chart to find the prime numbers less than 100. 4 Divisible If: by: 2 3 5 10 Example: 5. For each number below write yes or no for each indicated factor. 73150 1365 5967 475000 22100 5 2 is a factor 3 is a factor 5 is a factor 10 is a factor 6. Write a number with the stated number of digits that is divisible by each of the given factors. Divisible by Divisible by Divisible by Divisible by 2 3 5 10 3 digits 4 digits 5 digits 6 digits 7 digits 7. Which of the following is true of the number 51? a. 51 is prime because it has no factors other than 1 and 51. b. 51 is prime because it is an odd number. c. 51 is composite because it has 3 as a factor. d. 51 is composite because it has 13 as a factor. e. It is not possible to tell whether 51 is prime or composite. 8. Identify each of the following numbers as composite or prime. If it is composite list all of its factors. Number Is it Composite or Prime If Composite list all of the factors 65 79 43 39 6 9. Identify the place value for each underlined number. millions hundred thousands ten thousands thousands hundreds tens ones . tenths hundredths thousandths ten thousandths a. 26,345,289.11234 b. 76,485,718.1459 c. 27,285.963 d. 4,527.658 ten millions number 10. What is the place value of the 6 in 9,872,461? _____________________________________ 11. The value of the 7 in the number 157,983 is _____________ times the value of the 7 in 57,962. 12. The value of the 6 in the number 64.251 is ____________times the value of the 6 in 8.675. 13. The value of the 1 in the number 73,410 is ___________ (fraction) of the value of the 1 in 19,865. 7 14. When rounded off, the amount of money that Jessie owes John is $6.00. Which of the following could be the amount that Jessie actually owes John? a. $4.68 b. $5.27 c. $5.49 d. $5.51 e. $6.51 The bar graph shows one year’s worth of expenses for a small business. Use the graph to answer the next four problems Series1, Salaries, $20,761 Expenses Series1, Supplies, $11,456 Series1, All Other Expenses, $8,496 Series1, Rent and Utilities, Series1, Series1, $7,499 Series1, Car Postage, Telephone, Expenses, $3,792 $3,652 $3,205 15. To the nearest hundred dollars, how much was spent on postage and supplies? 16. Find the total amount spent, to the nearest hundred dollars, on rent and utilities and car expenses. 17. To the nearest thousand dollars, how much was spent on items other than salaries and rent and utilities? 18. To the nearest thousand dollars, how much was spent on items other than postage, supplies, and car expenses? 8 19. The speed of light is 186,282.3976 miles per second. Round this number to the nearest hundredth. 20. Which of the following statements are true? I. The difference between the numbers is 2 hundreds, 9 tens, and 7 ones. II. 534,916 < 534, 619 III. Both 534,916 and 534,619 rounded to the thousands place are 535,000 IV. The value of the 4 in both 534,996 and 534,969 is 4,000 a. I only b. I and II c. II only d. II and III e. I ,III, and IV 21. Order the following numbers from least to greatest: -­‐0.03, 0.3, -­‐ 0.3, 1.3, -­‐3.0 22. Place the appropriate symbol (<, >, =) between each two numbers. a. 345.982 ______ 345.992 b. 356.792 ______ 356.782 c. 272.81 ______ 272.18 d. 3.9082 ______ 3.9182 e. 576.91 ______ 567.91 9 LEVEL 2 23. Find the prime factorization of each number below. a. 936 b. 2550 10 c. 1740 24. Find the GCF of each of the following: 24. Find the Greatest Common Factor for each set of numbers below. Numbers Work GCF 39 and 6 54 and 45 196 and 462 4641, 30225, and 3705 11 25. Find the LCM of each of the following 25. Find the Least Common Multiple for each set of numbers below. Numbers Work LCM 22 and 16 39 and 15 196 and 462 5, 24, and 36 a.
12 26. Order the fractions from least to greatest a. b. 13 c. LEVEL 3 LCM Problems may be asking you: •
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about an event that is or will be repeating over and over. to purchase or get multiple items in order to have enough. to figure out when something will happen again at the same time. GCF Problemsmay be asking you: •
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to split things into smaller sections? to equally distribute 2 or more sets of items into their largest grouping? 27. Ben exercises every 12 days and Isabel every 8 days. Ben and Isabel both exercised today. How many days will it be until they exercise together again? 28. Samantha has two pieces of cloth. One piece is 72 inches wide and the other piece is 90 inches wide. She wants to cut both pieces into strips of equal width that are as wide as possible. How wide should she cut the strips? 29. Joanne is campaigning for class president and plans to distribute some campaign materials: 20 flyers and 16 buttons. She wants each classroom to receive an identical set of campaign materials, without having any materials left over. What is the greatest number of classrooms Joanne can distribute materials to? 30. Serena wants to create snack bags for a trip she is going on. She has 6 granola bars and 10 pieces of dried fruit. If the snack bags should be identical without any food left over, what is the greatest number of snack bags Serena can make? 31. Matthew goes hiking every 12 days and swimming every 6 days. He did both kinds of exercise today. How many days from now will he go both hiking and swimming again? 14 32. At a display booth at an amusement park, every visitor gets a gift bag. Some of the bags have items in them as shown in the table below. How often will a bag contain all three items? Items in the Gift Bags Items Bags Hat Every 2nd visitor T-­‐shirt Every 7th visitor Backpack Every 10th visitor 33. Bridget has swimming lessons every fifth day and diving lessons every third day. If she had a swimming lesson and a diving lesson on May 5, when will be the next date on which she has both swimming and diving lessons? 34. Mrs. Brenneman uses the Math Mobil Computer Cart every 3 days during period 1. Ms. Plflug uses the same cart 1st period every 5 days. Since a school year is 180 day long, how many times will they have a conflict? LEVEL 4 26. Find the Greatest Common Factor for each set of expressions below. Numbers Work 14x2 and 21x 20b and 45ab2 15 GCF Numbers 5x(x-­‐3) and 75(x-­‐3) 13x3(x+3), 39x2(x+3), and 65x(x+3) Work
GCF
27. Find the Least Common Multiple for each set of expressions below. Numbers Work 6ab and 10b 14x2 and 16x3y 3(x-­‐2) and 5x(x-­‐2) 16 LCM Numbers 5, 2(y+5), and 15y 6x3yz, 18xyz, and 24x3y2z2 Work
GCF
35. What pair of numbers shares a greatest common factor of 3 and a least common multiple of 108? LEVEL 5 36. What is the smallest integer value of x such that 50x is a perfect square? 37. What is the smallest integer value of x such that 550x is a perfect cube? 38. Find the first two integer values of y such that 35 y is a rational number. 17 PRIMES AND MENTAL ARITHMETIC Awareness of prime divisors can greatly simplify mental arithmetic problems. Combining factors of 2 and 5 together is always useful: 15 × 14 = (3 x 5) x (7 x 2) = (3 x 7) x (2 x 5) = 21 x 10 = 210
825 ÷ 25 = (825 x 2) ÷ (2 x 25)= 1650÷50= (2 x 1650)/(2 x 50) =3300 ÷ 100 = 33. In other situations, dealing separately with different prime divisors can be helpful: 17 × 33 = (17 x 3 ) x 11=51 × 11 = 561
a 616 ÷ 28 = (616÷2) ÷14= (308 )÷ 14 = (308÷2)÷7= 154 ÷ 7 = 22. Powers of 2 and powers of 5 are particularly easy to work with: 8 × 17 = 4 × 34 = 2 × 68 = 136 (when multiplying by 8 double , double, and double again) 496 ÷ 8 = 248 ÷ 4 = 124 ÷ 2 = 62 (When dividing by 8, divide by two three times.) 112 × 125 = (2 x2 x 2 x 14) x 125= 56 × 250 = 28 × 500 = 14 × 1000 = 14000 39. Calculate mentally using primes: 75 × 12,
18 714 ÷ 21,
17 × 55,
1325 ÷ 25,
1072 ÷ 16. LISTING FACTORS OF A NUMBER Once the prime factorization of a number has been obtained, all its factors can quickly be written down. For example, 60 = 22 × 3 × 5; so the complete list of factors of 60 is 2
22 × 3 × 5 = 60 1 × 3 × 5 = 30 22 × 1 × 5 = 20 21 × 1 × 5 = 10 2
2 × 3 × 1 = 12 21 × 3 × 1 = 6 22 × 1 × 1 = 4 21 × 1 × 1 = 2 and the total number of factors is 1 × 3 × 5 = 15 1 × 1 × 5 = 5 1 × 3 × 1 = 3 1 × 1 × 1 = 1 (2 + 1) × (1 + 1) × (1 + 1) = 12, because the power of 2 in the factor can be 20, 21 or 22 (3 choices), the power of 3 can be 30 or 31 (2 choices), and the power of 5 can be 50 or 51 (2 choices). 40. List all the factors of 256 and 210. 41. Find a site to play Kenken online. This puzzle requires on to think about all the ways to factor a given number. 19 PRIME FACTORISATION AND THE GCF AND LCM The GCF and LCM of two whole numbers can be found from their prime factorisations. For example, 784 = 24 × 30 × 50 × 72 and 210 = 21 × 31 × 51 × 71. The GCF is found by taking the smaller index of each prime, and the LCM by taking the larger index of each prime, so GCF(784, 210) = 21 × 30 × 50 × 71 = 14 LCM(784, 210) = 24 × 31 × 51 × 72 = 11 760. ‘When a prime is missing from one prime factorization, exclude the prime power from the GCF, but include the prime power in the LCM.’ THREE GENERAL RESULTS ABOUT THE GCF AND LCM OF TWO WHOLE NUMBERS a Find the prime factorisations of 540 and 792 b Hence find the LCM and GCF of 540 and 792. 20 A CONCISE LAYOUT FOR FINDING THE GCF AND LCM AND BOTH PRIME FACTORISATIONS The working on the right below shows a concise way to lay out the working for find the GCF and LCM of two numbers, using the same numbers 336 and 840 as in the example above: • At each step divide by a common prime factor, placed on the left. • Stop when the two quotients are relatively prime (have no factors in common), • The product 23 × 3 × 7 = 168 of the primes on the left is the GCF. • The product 168 × 2 × 5 = 1680 of the GCF and the numbers along the bottom is the LCM. We have also incidentally found the prime factorisations of 336 and 840, 336 = 24 × 3 × 7
and
840 = 23 × 3 × 5 × 7. 42. Using the method above find the GCF and LCM of 7920 and 2376. Also show the prime factorization of each number. 21