Math 7 Notes Unit B: Rational Numbers Teachers Before we move to performing operations involving rational numbers, we must be certain our students have some basic understandings and skills. The following notes/sections on divisibility, prime factorization, GCF and LCM should have been covered in previous grades, but are crucial to future work with rational numbers. We have determined to leave these items in the notes to emphasize that they need to be reviewed with students. Review Divisibility Before starting to work with rational numbers it is imperative that students have some basic understanding of fractions. Everyone will benefit if you take a few minutes to review or learn the Rules of Divisibility. Employing theses rules will make life a lot easier in the future; not to mention it will save you time and allow you to do problems very quickly when others are experiencing difficulty. In general if you were asked any given number was divisible by a second number you could divide them and if the remainder is 0, we would say yes the first number is divisible by the second number. For example, if asked if is divisible by we could divide to see the remainder. 44 Since the remainder is 0, we say that is divisible by. We could say is a factor of and is a multiple of. 0 What if you were asked if 97 was divisible by 4? Again we could divide and see the remainder. 4 4 97 4 57 56 Since the remainder is not 0, we say that 97 is NOT divisible by 4. We could say 4 is NOT a factor of 97 and 97 is NOT a multiple of 4. To be quite frank, you already know some of them. For instance, if I asked you to determine if a number is divisible by two, would you know the answer. Sure you do, if the number is even, then it s divisible by two. Can you tell if a number is divisible by 0? How about 5? Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page of 9 Revised 0-CCSS
Because you are familiar with those numbers, chances are you know if a number is divisible by, 5 or 0. We could look at more numbers to see if any other patterns exist that would let you know what they are divisible by, but we don t have that much time or space. So, if you don t mind, I m just going to share some rules of divisibility with you first, and then I will give the examples. Rules of Divisibility A number is divisible: by, if the number ends in 0,, 4, 6 or 8. In other words the number must be even. by 5, if the number ends in 0 or 5. by 0, if the number ends in 0. Notice here we are suggesting you teach these rules grouped together since they all involve just looking at the one s digit. by, if the sum of the digits is a multiple of. by 9, if the sum of the digits is a multiple of 9. by 6, if the number ends in 0,, 4, 6 or 8 AND the sum of the digits is a multiple of. (In other words, the number if the number is divisible by and, then it is by 6.) Notice here we are suggesting you teach these rules grouped together since they are similar they all involve finding a sum of the digits. by 4, if the last digits of the number is divisible by 4. (Remember 4= ) by 8, if the last digits of the number is divisible by 8. (Remember 8= ) Again we grouped these similar rules together. Others that are important and quick to teach are the rules for 0, 5, 50 and 00. by 0, if the number ends in 00, 0, 40, 60 or 80. by 5, if the number ends in 00, 5, 50 or 75. by 50, if the number ends in 00 or 50. by 00, if the number ends in 00. A fun one is divisibility by. Use it to challenge some of your students. The rule is wordy but the process is simple once learned. by, if the sum of the st, rd, 5 th, digits and the sum of the nd, 4 th, 6 th, digits are equal OR differ by a multiple of. Example: Is 86,0 divisible by?. the sum of the st, rd, 5 th, digits +6+=9. the sum of the nd, 4 th, 6 th, digits 8+0+=9. the sums are equal so 86,0 is divisible by. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page of 9 Revised 0-CCSS
Example: Is 806,4 divisible by?. the sum of the st, rd, 5 th, digits 8+6+=6. the sum of the nd, 4 th, 6 th, digits 0++4=5. the sums are not equal so find the difference between the sums 6-5= 4.since is a multiple of (,,, 44, 55, 66, 77, 88, 99, ) 806,4 is divisible by. Example: Is,456 divisible by?. the sum of the st, rd, 5 th, digits ++5=9. the sum of the nd, 4 th, 6 th, digits +4+6=. the sums are not equal so find the difference between the sums -9= 4. since is NOT a multiple of (,,, 44, 55, 66, 77, 88, 99, ) then,456 is NOT divisible by. Now come the examples Divisibility by Example: Is divisible by? (We could write this as ; which is read Is divisible by?.) The rule says to find the sum of the digits and if that sum is a multiple of the number is divisible by. ++=, is a multiple of, so the number is divisible by. Example: 47; (Read, Is 47 divisible by?) Adding, 4, and 7 and we. Is divisible by? If that answer is yes, that means 47 is divisible by. If you don t believe it, try dividing 47 by. Example:,6; (Read, Is,6 divisible by?) The sum is +++6=. Is a multiple of? (, 6, 9,, 5, 8,, 4, 7, 0, ) No, then,6 is NOT divisible by. Divisibility by 9 Example: Is divisible by 9? (We could write this as ;9 which is read Is divisible by 9?.) The rule says to find the sum of the digits and if that sum is a multiple of 9 the number is divisible by 9. ++=, is not a multiple of 9, so the number is NOT divisible by 9. Example: 5, 47;9 (Read, Is 5,47 divisible by 9?) 5++4+7=8 8is a multiple of 9 so yes, 5,47 is divisible by 9. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page of 9 Revised 0-CCSS
Example: 54,45;9 (Read, Is 54,45divisible by 9?) +5+4++4+5= is NOT a multiple of 9, so the number is NOT divisible by 9. Example: Divisibility by 6 Is,06 divisible by 6? (We could write this as,06;6 which is read Is,06 divisible by 6?.) The rule has two parts - the number ends in 0,, 4, 6 or 8 yes AND the sum of the digits is a multiple of. +++0+6= yes So, yes,,06 is divisible by both and so it is divisible by 6. Example : 746; 6? (This is read Is 746 divisible by 6?.) The rule has two parts - the number ends in 0,, 4, 6 or 8 yes AND the sum of the digits is a multiple of. 7+4+6=7 no So 746 is divisibile by but NOT by so it is NOT divisible by 6. Example: 48,76;6 (This is read Is,06 divisible by 6?.) The rule has two parts - the number ends in 0,, 4, 6 or 8 no AND the sum of the digits is a multiple of. Since it is not divisible by, it is NOT divisible by 6. Divisibility by 4 Example: Is,6 divisible by 4? (We could write this as,6;4 which is read Is,6divisible by 4?.) Using the rule for 4, we look at the last digits. In this example they are 6, since 6 is divisible by 4. Then,6 is divisible by 4. Example: 87,50;4 (This is read Is,6divisible by 4?.) We look at the last digits, in this case 0, which is NOT divisible by 4. Example: 96,500;4 (This is read Is 96,500divisible by 4?.) divisible by 4 Using the rule for 4, we look at the last digits. In this example they are 00, since 4 goes into 00 0 times it is divisible by 4. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 4 of 9 Revised 0-CCSS
Divisibility by 8 Example: Is 456,040 divisible by 8? (We could write this as 456,040;8 which is read Is 456,040 divisible by 8?.) Using the rule for 8, we look at the last digits. In this example the last digits are 040, since it is divisible by 8, then 456,040 is divisible by 8. Example: 85,600;8 (This is read Is 85,600divisible by 8?.) We look at the last digits, in this case 600, which is we would probably have to divide (still easier to divide 600 8 compared to 85,600 8). Once divided we see 600 8= 75 with no remainder. So yes 85,600 is divisible by 8. Example: 6,5;8 (This is read Is 6,5 divisible by 8?.) Using the rule for 8, we look at the last digits. In this example 5, since it is not even an even number we know it is NOT divisible by 8. Example: Using the rules of divisibility, determine if the following numbers are divisible by,, 4, 5, 6, 8, 9 or 0. Place a check in each box that the number IS divisible by. # By By By 4 By 5 By 6 By 8 By 9 By 0 756 9,045 6,70 86,400 70,000 Example: Write a 5 digit number that is divisible by and 4. Example: Write the smallest 6 digit number that is divisible by and 5. Example: following? What single digit value or values might the? represent in each of the a.,0? ; 4 and 6 b. 45,6?0; and 5 c. 6,?: 4 and 9?=0?=, 5 or 8?=9 Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 5 of 9 Revised 0-CCSS
Review Prime Factorization Objective: The student will find the prime factorization of composite numbers. Factor a number that is multiplied by another number to get a product. Example: 6 is a factor of 6 and is a factor of 6 Prime Number a whole number that has distinct factors. Examples: 7 7,, 5, 7,,, 7, 9 are prime numbers because they have ONLY factors one and itself. Composite Number a number that has more than distinct factors. Examples: 4 8 8 8 8 is a composite number because it has 4 factors:,,4,8. 4, 6, 8, 9, 0,, 4, 5 are composite numbers because they have more than factors. Note: The number is neither prime nor composite. It has only distinct factor, so it is not prime. Prime Factorization the process used to rewrite a composite number as a product of prime numbers. Strategy: Use a Step Diagram To determine the prime factorization of a composite number, divide the composite number by a prime factor until the quotient is. Example: Write the prime factorization of 4. Start dividing by the smallest prime number,. 4 6 OR does not divide into, so divide by the next prime number,. use a factor tree 4 6 4 The prime factorization of 4 is = Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 6 of 9 Revised 0-CCSS
Review Greatest Common Factor Objective: The student will find the greatest common factor of two or more whole numbers. Greatest Common Factor (GCF) the greatest whole number that divides evenly into each number. Strategy To find the GCF, list all the factors of each number. The largest factor that is on both lists is the GCF. Example: Find the GCF of 4 and 6. Factors of 4:,,, 4, 6, 8,, 4 Factors of 6:,,, 4, 6, 9,, 8, 6 GCF is, the greatest factor on both lists. * This strategy is laborious, but shows what factors are. Strategy To find the GCF, write the prime factorization of each number and identify which factors are common. Then multiply the common factors. Example: Find the GCF of 4 and 6. Prime Factorization of 4 = Prime Factorization of 6 = The common factors are,, and. The GCF is = This can also be done in a Venn Diagram as shown below. 4 6 The GCF is shown in the intersection of the diagram. So for 4 and 6, you multiply the common factors. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 7 of 9 Revised 0-CCSS
Note that in this model if no numbers are listed in the intersection then the GCF = and we could say the numbers are relatively prime. For example, if asked to find the GCF for 5 and 6, the diagram would look like this: 5 6 5 Strategy To find the GCF, create a factor tree for each number. Start with the GREATEST factor common to each number. Then multiply the common factors. Example: Find the GCF of 4 and 6 4 6 OR 4 6 6 4 6 6 4 6 4 9 GCF = 6()= GCF = 4()= Multiple strategies are illustrated, but only use the ones you believe are appropriate for your students. Helpful Hints: If the smaller number divides evenly into the larger number, the smaller number is the GCF. Example: the GCF of 9 and 7. Since 9 divides evenly into 7, then 9 is the GCF of 9 and 7. If the numbers do not share any factors greater than, then the GCF is. SBAC Example: Two of these statements are true in all cases: Statement : The greatest common factor of any two distinct prime numbers is. Statement : The greatest common factor of any two distinct composite numbers is. Statement : The product of any two integers is a rational number. Statement 4: The quotient of any two integers is a rational number. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 8 of 9 Revised 0-CCSS
Part A: Which two statements are true in all cases? Part B: For both statements that you did not choose in Part A, provide one clear reason and/or example for each statement that proves the statement can be false. Statement Reason/example Statement Reason/example Solution: Part A: Which two statements are true in all cases? Statement Statement Part B: For both statements that you did not choose in Part A, provide one clear reason and/or example for each statement that proves the statement can be false. Statement Reason/example Statement is not true because the GCF of and 6 is 4. Statement 4 Reason/example Statement 4 is not true because 0 is not a rational number. Review Least Common Multiple Objective: The student will find the least common multiple of two or more whole numbers. Multiple the product of a number and a nonzero whole number. Least Common Multiple (LCM) the common multiple with the least value. Strategy * To find the LCM, list the multiples of each number. The multiple with the least value on each list is the LCM. Example: Find the LCM of 0 and 5. Factors of 0: 0, 0, 0, 40, 50, 60 Factors of 5: 5, 0, 45, 60 LCM is 0, the least value on both lists. * This strategy is laborious, but shows what multiples are. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 9 of 9 Revised 0-CCSS
Strategy To find the LCM, write the prime factorization of each number and identify the greatest power of each prime factor. Then multiply the factors. Example: Find the LCM of 0 and 5. Prime Factorization of 0 = 5 Prime Factorization of 5 = 5 The prime factors are s, s and 5 s. The greatest power of each prime factors are The LCM is 5 = 0. Example: Find the LCM of 4 and 6. Prime Factorization of 4 = Prime Factorization of 6 = The prime factors are s and s. The greatest power of each prime factors are The LCM is = = 7.,, and 5. and These examples can also be done in a Venn Diagram as shown below. The LCM is found by multiplying all the factors shown.. 0 5 5 The LCM (for 0 and 5) = 5 0. 4 6 The LCM (for 4 and 6) = 7. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 0 of 9 Revised 0-CCSS
Note that in this model below, if no numbers are listed in the intersection then we could say the numbers are relatively prime and we could simply multiply the given numbers to find the least common multiple. The LCM for 5 and 6, the diagram would look like this: 5 6 5 The LCM = 5 = 5 6 = 40 Strategy ** Simplifying Method Write the numbers as a fraction, simplify, then find the cross product (either one is sufficient). Find the LCM of 0 and 5. Example: 0 the cross products are 0 = 5 = 0 LCM =0 5 Find the LCM of 4 and 6. Example: 4 the cross products are 4 = 6 = 7 LCM = 7 6 Find the LCM of 5 and 6. Example: 5 6 Since the fraction does not simplify, just multiply the given numbers. LCM =5 6 40 ** This strategy is very simple, and reinforces the skill of cross-products, but it is less intuitive of the meaning of a common multiple. Helpful Hints: If the smaller number divides evenly into the larger number, the larger number is the LCM. Example: 9, 7 Since 9 divides evenly into 7, then 7 is the LCM of 9 and 7. If the numbers do not share any factors greater than, then the LCM is the product of the numbers. NOTE Did you know that the product of the given numbers should equal the product of the GCF times the LCM? Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page of 9 Revised 0-CCSS
From the previous examples we found that for the numbers: 0 and 5, the GCF is 5 and the LCM is 0. 4 and 6, the GCF is and the LCM is 7. 5 and 6, the GCF is and the LCM is 40. 05 50 50 50 46 7 864 864 56 40 40 40 Equivalent Fractions and Mixed Numbers Objective: To identify, write, and convert equivalent fractions and mixed numbers. Equivalent Fractions fractions that have the same value. Examples: 6 0 8 4 0 4... 4 6 8 Mixed Number A number that contains both a whole number and a fraction. 5 Examples:, 4 5 6 Improper Fraction a fraction in which the numerator is greater than or equal to the denominator. The value of an improper fraction is greater than or equal to. 4 7 Examples:, 7 Proper Fraction a fraction in which the numerator is less than the denominator. The value of a proper fraction is less than. Examples: 5 5,, 8 7 Equivalent Fractions To make equivalent fractions, multiply the fraction by one by multiplying both numerator and denominator by the same number. Examples: Find two fractions equivalent to. 5 6 8 4 5 0 5 8 40 Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page of 9 Revised 0-CCSS
Examples: Find the next three equivalent fractions to 4. 7 4 8 4 4 4 6 7 4 7 7 4 8 4 8 6 Solution: 7 4 8 Examples: Determine whether the fractions in each pair are equivalent. 6 9 and Method 8 6 6 9 9 8 8 4 4 Since both initial fractions equal, yes, they are equivalent. 4 Method 6 8? 9 6 7 and 8 9 7 Since the cross products are equal, yes they are equivalent fractions. 8 5 8 8 6 5 5 5 5 and Method 5 0 5 5 5 0 0 5 4 Since 6 5, the initial fractions are not equivalent. 5 4 8 5 Method? 08 60 and 5 5 75 5 0 Since the cross products are not equal, they are not equivalent fractions. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page of 9 Revised 0-CCSS
Simplifying Fractions To simplify fractions (another way to make equivalent fractions), divide by one by dividing both numerator and denominator by the same number. Be sure to apply divisibility rules. Examples: Simplify 4 4 8 8 4 Examples: Simplify 7 04 75 Simplify 5 75 5 5 5 5 0 Simplify 4 0 5 7 0 6 7 OR 4 57 9 4 6 9 For many students looks to be prime. Most students will see the numbers are even. Remind them that ++= and ++=6 But again, 5 and 57 look prime to many so each are divisible by. students. Remind them 5+=6 and 5+7= so each are divisible by. Converting Between an Improper Fraction and Mixed Number To convert an improper fraction to a mixed number, divide the numerator by the denominator. Examples: 4 4 844 4 4 4 8 8 4 To convert a mixed number to an improper fraction, multiply the denominator by the whole number and add that to the numerator. Use that result as the numerator of the improper fraction. Examples: Convert 5 to an improper fraction. 5 7 5 Equivalent Fractions and Decimals Syllabus Objective:. The student will translate among various forms of equivalent numbers including fractions, decimals, and percents. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 4 of 9 Revised 0-CCSS
NEW CCSS 7.NS.d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. NEW CCSS 7.EE.- Convert between forms of rational numbers as appropriate. A decimal is a special fraction whose denominator is a power of 0. The numerator is the number to the right of the decimal point. The power of ten in the denominator is determined by the number of digits to the right of the decimal point. Strategy If the denominator will divide evenly into a power of 0, then make equivalent fractions and use the definition of a decimal to write the decimal numeral..75 Examples: 75 0.75 OR 4 00 4.00 8 0 0 4 0.4 5 0 OR.4 5.0 0 Strategy If the denominator contains prime factors other than s or 5 s, the decimal will repeat, and the best way to determine the decimal equivalent is by dividing the numerator by the denominator. Example: Convert : to a decimal. 0..0000 0... = 0. The bar over the digit (called a vinculum) means that digit is repeating over and over and over to infinity. Example: Convert : 6 to a decimal..44 6.0000 0.444... 0.4 55 50 44 60 55 50 44 6 Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 5 of 9 Revised 0-CCSS
7 Example: Convert : to a decimal. Since has prime factors other than or 5, the decimal equivalent will be nonterminating. Divide the numerator 7, by the denominator. 0.58 7.00000 0.58 Notice here the 5 and the 8 are NOT repeating, therefore the vinculum does not go over the digits 5 and 8. Example: Convert : to a decimal. 7.48574857 7.000000000000 =.4857 Here the pattern starts to repeat with the digits 4857. Be sure to warn students that they must divide until they are sure of the repeating digit(s). CCSD and CCSS may require knowledge of converting any fraction to a decimal. To write a decimal as a fraction, write the decimal as the numerator, and the proper power of 0 as the denominator. Then simplify. Examples: 4 7 0.4 00 50 56 4 7 0.056 000 50 5 Syllabus Objective:.4 The student will explain the relationship among fractions, decimals, and percents. Students should be able to state that decimals are special fractions and that both fractions and decimals represent a part of a whole unit. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 6 of 9 Revised 0-CCSS
Comparing and Ordering Rational Numbers Syllabus Objective:. The student will order rational numbers expressed as fractions, decimals, and percents. Rational Number any number that can be written as a ratio of two integers, written in form of a, whereb 0. b To compare or order rational numbers, rewrite the numbers in the same form (generally decimals). Example: Order 7, 0.75,,, 0. in ascending order (least to greatest). 8 5 7 0.875, 0.75,.4,.6, 0. Convert to decimals. 8 5.4, 0., 0.6, 0.75, 0.875 Order the decimals. 7, 0.,, 0.75, Rewrite in original form. 5 8 Example: Order 8 5 0.58,, 0.5 in descending order (greatest to least). 5, 8, 8 5.6, 0.58,.6666..., 0.65, 0.5 Convert to decimals. 5 8.6, 0.65, 0.58, 0.5,.6666 Order the decimals. 8 5,, 0.58, 0.5, Rewrite in original form. 5 8 Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 7 of 9 Revised 0-CCSS
Negative Exponents Syllabus Objective. The student will translate written and oral expressions including ratios, proportions, exponents, radicals, scientific notation, and positive and negative numbers to numerical form. Note: Scientific Notation with positive exponents was taught in Unit. While teaching integers, this would be a time to review that and extend to include negative exponents and scientific notation involving negative exponents. In Holt, it is an extension at the end of Chapter. In McDougal Littel, it is an extension in Chapter 6 for negative and zero exponents. Pattern development is a very effective way to introduce the concept of negative exponents. Consider the following pattern that students should have seen previously. 4 6 8 4 0 As we review this pattern, students should see that each time the exponent is decreased by, the expanded form contains one less factor of and the product is half of the preceding product. 4 6 8 4 0 Following this pattern, is. Continuing this pattern, 4 So, and 4. Looking at powers of 0, 0 0 0 Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 8 of 9 Revised 0-CCSS
0 0 00 00 0 0 00 0,000 Scientific Notation Using Negative Exponents Now we can learn to write very small numbers in scientific notation using powers of ten with negative exponents. Example: Write 0.0049 in scientific notation. 0.0049 4.9 0 since we moved the decimal point places to the right. Example: Write 0.0087 in scientific notation. 0.0087.087 0 since we moved the decimal point places to the right. Example: Write 7.0 4 in standard form. 4 7.0 0.0007 since we move the decimal point 4 places to the left. Example: Write.4 0.4 0 0.04 since we move the decimal point places to the left. Math 7, Unit 0B: Rational Numbers Holt: Chapter Section B Page 9 of 9 Revised 0-CCSS