Pre-Algebra 8 Notes Unit Four: Factors, Fractions, and Exponents

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1 Pre-Algebra 8 Notes Unit Four: Factors, Fractions, and Eponents Prime Numbers A prime number is a whole number that is greater than one and has eactly two factors, one and itself. Some eamples are,,, 7, 11, 1, 17. A composite number is a whole number greater than one and has more than two factors. Some eamples are 4, 6, 8, 9, 10, 1. Note: The number one is NOT a prime or a composite number. It has only one factor. Prime factorization is rewriting a number as a product of prime numbers. Eample: Write the prime factorization of 1. 1 is a number we are very familiar with, knowing 1 4. That s a product, but 4 is not a prime. So we rewrite 4 as. 1 4 or A factor tree allows us a systematic way of writing factors of larger numbers or numbers we are not as familiar with one step at a time. A composite number can be written as a product of primes in one and only one way. In other words, when we rewrite a number as a product of primes, there is one and only one answer. A factor tree will provide that result, but until the last step, factor trees could look different for the same prime factorization of a number. Eample: Write the prime factorization of 0. Using a factor tree: 0 The prime factorization is or 7. The standard convention for writing a number when it is factored into primes is to write the factors from smallest to largest. However, it is not wrong if you do not. McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 1 of 17 Revised 010

2 The preferred way to write 0 as a product of primes is written as7. 7, but it could have been If the factor tree for finding the prime factorization of 0 had begun with the factors 70 and, the result would have still been the same. The factor tree would look different than the one shown, but the answer would have been the same. Another way to determine the prime factorization of a composite number is to use repeated division. This is especially helpful when the composite number is unfamiliar to us, and finding a pair of factors with which to begin a factor tree is not obvious. The process starts by checking for divisibility by the smallest prime number () and dividing until you can no longer use that as a factor. Then move on to checking the net prime number and so on until you get an answer that is prime. When you are finished, all the prime factors will be lined up and grouped together, making the writing of powers much easier. Eample: Write the prime factorization of 04. Using repeated division: The prime factorization is or 7. Knowing the rules of divisibility will help with factoring. Review of Rules of Divisibility (These are important throughout the school year.) To find factors, determine if a number is divisible by: if it ends in an even number if it ends in 0 or 10 if it ends in 0 if the sum of the digits is divisible by 9 if the sum of the digits is divisible by 9 McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page of 17 Revised 010

3 6 if the number is divisible by and 4 if the last two digits of a number are divisible by 4 8 if the last three digits are divisible by 8 7 take the last digit, double it, subtract it from the rest of the number; if it is 0 or divisible by 7, then the number will also be divisible by 7 (repeat this algorithm if necessary) Eample: Is 164 divisible by 7? 4 8, ; for 08, 8 16, , which is divisible by 7 So,164 is divisible by take the last two digits, add them to the rest of the number; if it is divisible by 11, then the number will also be divisible by 11 (repeat this algorithm if necessary) Eample: Is 67,846 divisible by 11? and we know this to be divisible by11 So, 67,846is divisible by11also. Greatest Common Factor Syllabus Objective: (.1) The student will find factors, including the greatest common factor (GCF), of numbers and monomials. A common factor is a whole number that is a factor of two or more nonzero whole numbers. Eample: Find common factors of 18 and 4. Factors of 18: 1,,, 6, 9, 18 Factors of 4: 1,,, 4, 6, 8, 1, 4 Since 1,,, and 6 are factors of both of the numbers 18 and 4, they are called common factors. The greatest common factor (GCF) of two or more whole numbers is the greatest whole number that divides evenly into each number. In the first eample, the greatest common factor (GCF) of 18 and 4 is 6. There are three ways of finding the GCF. McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page of 17 Revised 010

4 Eample: Find the GCF of 4 and 6. Strategy 1: To find the GCF, list all the factors of each number. The largest factor that is in both numbers is the GCF. Factors of 4: 1,,, 4, 6, 8, 1, 4 Factors of 6: 1,,, 4, 6, 9, 1, 18, 6 The GCF is the greatest factor that is in both lists: 1. Strategy : To find the GCF, write the prime factorization of each number and identify which factors are in each number. 4 6 Each number has two s and one. So the GCF 1 Strategy : Use a Venn diagram.. Draw overlapping circles, one for each number. Write the factors for each number in its circle, placing common factors in the intersection(s). Finally, multiply all factors in the intersection. Factors of 4 Factors of 6 Multiply all factors in the intersection for the GCF: 1. Note: If there are no numbers in the intersection that means the GCF is one. That also means that the numbers are relatively prime. Two or more numbers are relatively prime if their greatest common factor is one. Eamples: and are relatively prime 8 and 9 are relatively prime 8 and 1 are NOT relatively prime; they have a GCF of 4 Eample: Find the GCF of 8 and 9. Factors of 8 Factors of 9 There are no factors in the intersection, so the GCF 1. McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 4 of 17 Revised 010

5 Eample: Find the GCF of 18y and 8 y Writing the factors: 18y y y 8y 7 yy The common factors are y y, so the GCF is Using a Venn diagram: Factors of 18y y y 7 y. Factors of 8 y Multiply all factors in the intersection for the GCF: yy y. Don t forget to include word problems where the GCF would help students to solve them. Eample: A flower shop wants to make bouquets from types of flowers. The flowers available are 4 daisies, 7 roses, and 60 mums. What is the greatest number of identical bouquets that can be made? List what each bouquet will contain. Let s find the GCF for 4, 7 and The common factors are and, so we multiply times to get GCF = 1. We can make 1 identical bouquets. Each would contain: 4 1 daises 7 1 roses mums Equivalent Fractions Syllabus Objective: (.) The student will write rational numbers in equivalent forms. Have you ever noticed that not everyone describes the same things in the same way? For instance, a mother might say her baby is twelve months old. The father might tell somebody his baby is a year old. They re the same thing, no big deal. Well, we do the same thing in math, or in our case, with fractions. Let s look at these two cakes. McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page of 17 Revised 010

6 One person might notice that 1 out of pieces seems to describe the same amount as out of 4 in the picture above. In other words, 1. 4 If we continue this process, we would notice we have a number of different ways to epress the same thing. When two fractions describe the same amount, we say they are equivalent fractions. Equivalent fractions are fractions that have the same value. Wouldn t it be nice if we could determine when fractions were equivalent without drawing pictures? Well, if we looked at enough equivalent fractions, we would notice a pattern developing. Let s look at some. 1, 6 6, , and Do you see any relationship between the numerators and denominators in the first fraction compared to the numerators and denominators in the second? Hopefully, you might notice we are multiplying both numerator and denominator in each fraction by the same number to get the nd fraction. Eample: if you multiply both the numerator and denominator by 4 Well, you know what that means: when we see a pattern like that, we make a(n) rule, algorithm or procedure that allows us to show other people simple ways of doing problems. To generate equivalent fractions, multiply BOTH the numerator and the denominator by the SAME number. When you multiply both the numerator and the denominator by 4, you are multiplying by 4 4 or 1. When you multiply by one, the value of the original fraction does not change. Eample: Epress 6 as sitieths. 6? What did you multiply 6 by to get 60 in the 60 denominator? By 10, so we multiply the numerator by 10. So McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 6 of 17 Revised 010

7 Eample: is how many thirty-fifths? 7 7? What did you multiply 7 by to get in the denominator? By you say. We then multiply the numerator by Another way to find equivalent fractions, using cross products, will be introduced in Unit 6: Ratio and Proportion. Simplifying/Reducing Fractions Syllabus Objective: (.) The student will write rational numbers in equivalent forms. Simplifying or reducing fractions is just another form of making equivalent fractions. Instead of multiplying the fraction by one, by multiplying the numerator and denominator by the same number, we will divide the numerator and denominator by the same number. Now that we know the Rules of Divisibility, reducing fractions is going to be a piece of cake. To simplify fractions, we divide both the numerator and denominator by the same number. Eample: Simplify (reduce) Both the numbers are even, so we can divide both numerator and denominator by Eample: Simplify (reduce) Notice that the sum of the digits in 111 is and the sum of the digits in 7 is 1. Therefore, both are divisible by McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 7 of 17 Revised 010

8 If you don t know the rules of divisibility, you would have to try and reduce the fractions by trying to find a number that goes into both numerator and denominator. That s too much guessing, so spend a few minutes and commit the rules of divisibility to memory. The rules of divisibility and the GCF are tools we can use to help us to simplify fractions. We use those same ideas to simplify rational epressions in algebra. Eample: Simplify 10y. 1y 10y y 1y y y y Least Common Multiple/Denominator A multiple is the product of a number and any nonzero whole number. Eample: 1 is a multiple of because 1 Eample: Three multiples of 1 are 1, 4, and 6. The least common multiple (LCM) of two or more numbers is the common multiple with the least value. It s a number all the other numbers will divide into. For that to occur, the LCM has to contain all of the factors of each of the numbers. There are a number of ways of finding the LCM. Eample: Find the LCM 10 and 1. Strategy 1: List the multiples of each number and find the multiple with the least value on each list. Multiples of 10: Multiples of 1: 10, 0, 0, 40, 0, 60, 1, 0, 4, 60, Both these numbers have 0 as the least common multiple; the LCM is 0. McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 8 of 17 Revised 010

9 Strategy : Write the numbers as a fraction, reduce, and then find the cross product in this proportion. This is often called the reducing method Find the cross product:10 1 ; the LCM is 0. Strategy : Write the prime factors of each number and use each only once in a product You can only use each common prime factor once, 0. Strategy 4: Use a Venn diagram. Draw overlapping circles, one for each number. Write the factors for each number in its circle, placing common factors in the intersection(s). Finally, multiply all factors in the diagram. Factors of 10 Factors of 1 Let s try a few more eamples. Eample: Find the LCM of 6 and 4. Multiply all factors in diagram for the LCM: 0. Writing the prime factors, 6 or 4 or Using the common factors only once: 180. Use a Venn diagram. Factors of 6 Factors of 4 Multiply all factors in diagram for the LCM: 180. McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 9 of 17 Revised 010

10 Eample: Find the LCM of a and 16a. One way is to write the prime factors, a a 4 a a a 16 Using each common prime factor only once, we have 4 a a 16a a or 80 a. The LCM of a and 16a is 80a. Or, using the Reducing Method, we rewrite the terms as a fraction and reduce: a 16a 16a, Find the cross product: 16 a 80 a. The LCM of a and 16a is 80 a. The least common denominator (LCD) of two or more fractions is the least common multiple of the denominators. We can use this to compare and order fractions. Eample: Last year 10 students attended the back-to-school dance, including 6 students new to the school. This year, the dance had 17 students in attendance, including 4 new students. In which year was the fraction of new students greater? First, write the fractions and simplify. number of new students 6 Last year: total number of students number of new students 4 9 This year: total number of students 17 9 The least common multiple of and is 70, so this is the LCD. 10 Writing equivalent fractions, and We can now compare the numerators to find students was greater last year. 1 18, so the fraction of new McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 10 of 17 Revised 010

11 Eample: Order the numbers from least to greatest: 1 8,, 6 We will need to rewrite the mied number as a fraction, find the LCD and then write equivalent fractions Comparing the numerators, we have, so we would list the numbers from least to greatest as 1, 8, 6 An alternative way to do the problem would be to convert the numbers to mied numbers, and then compare the fractions Comparing the fractions (with common 1 0 denominators) we have, so we would list from least to greatest (the original fractions) as 1, 8, 6 CCSD teachers please note: The net two sections regarding eponents are not included in the benchmarks. However, we have chosen to include these sections because: (1) Students have been asked to reduce algebraic fractions including eponential terms. They have been introduced to the concept of rewriting the epressions without eponents and cancelling. This logically leads to the net step of having them discover the rules for multiplying and dividing epressions with eponents (and to make the simplification process shorter). () The CCSD syllabus objectives (and our state standards) do include the concept of scientific notation with negative eponents. It seems to make sense to have students look at the concept of negative eponents before we use it in scientific notation. McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 11 of 17 Revised 010

12 Rules of Eponents Review with students that an eponent is the superscript which tells how many times the base is used as a factor. eponent base In the number, read to the third power or cubed, the is called the base and the is called the eponent. Eamples: To write an eponential in standard form, compute the products. i.e. Since we will be addressing powers of 10 in scientific notation, emphasize this in your eamples. Eamples: What pattern allows you to find the value of an eponential with base 10 quickly? Answer: The number of zeroes is equal to the eponent! Caution: If a number does not have an eponent visible, it is understood to have an eponent of ONE! Let s practice writing numbers in eponential form. Eamples: Write 81 with a base of. 81, 81, therefore 81? 4 Write 1 with a base of. 1, 1, therefore 1? McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 1 of 17 Revised 010

13 In algebra, we often have to find the products and quotients of algebraic epression. For eample, what is the product of the problem below?? Caution: Many students will jump to an answer of 6, which is incorrect. Watch for this error! Have students rewrite each term in epanded form, and then convert it back to eponential form. Since and, ( ) ( ) or. We do not multiply the eponents as we might suspect: we add them! Let s try a few more problems to verify our conjecture: Eamples: ( ) ( ) or 4 7 ( ) or 6 ie, or ie, or 1 6 We are now ready to state the rule for multiplying eponential epressions with the same base. When multiplying powers with the same base, add their eponents; a b a b that is,. What might we suspect about the rule for division? Since division is the inverse of multiplication, and multiplying eponential epressions involves the addition of eponents, what would division of eponential epressions involve? (Note how confusing this all seems!) We might suggest subtraction is the key here; we can show this to be true with a few eamples: Eample: ie, or Eample: 6 6 ie, or 61 McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 1 of 17 Revised 010

14 Let s now state the rule: When dividing powers with the same base, subtract their eponents (subtract the eponent in the denominator from the eponent in the numerator); a ab that is, b Emphasize with students to be careful with their integer operations now they may have the tendency to add when they should multiply. A simple eample is to look at the following 4 7 product: 4. The answer is 8, of course. Common errors are to multiply all numbers 1 involved, arriving at the incorrect answer of 8 ; or to add all numbers, arriving at the 7 incorrect answer of 6. Look at the following eample for other errors to watch for. Eample: subtractingeverything: 4 ; but watch for answersof dividingeverything: Eample: 7 1 ; but watch for answersof a a a adding everything: 8a multiplying everything: 1a 7 10 At this point, students will probably be feeling that math rules do not always make sense! Emphasize that they can always go back to epanding the epression to notation without eponents to arrive at the answer. Negative and Zero Eponents Pattern development is a very effective way to introduce the concept of negative and zero eponents. Consider the following pattern that students should have seen previously McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 14 of 17 Revised 010

15 As we review this pattern, students should see that each time the eponent is decreased by 1, the epanded form contains one less factor of and the product is half of the preceding product Following this pattern, 1 is Continuing this pattern, 4 So 1 1, and Looking at powers of 10, or or ,000 or Applying Eponents: Scientific Notation Syllabus Objective: (.) The student will represent numbers using scientific notation. n A number is written in scientific notation if it has the form 10 where 1 10 and n is an integer. An eample of a number written in scientific notation is To convert a number written in scientific notation to standard form, the eponent tells you how many times to move the decimal point to the right or left. Eample: Write 4.10 in standard form. The eponent indicates you move the decimal point places to the right ,000 McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 1 of 17 Revised 010

16 Eample: Write Why does this work? Rewrite the problem using product form, and then do the multiplication ,000 4, in standard form. The eponent 7 indicates you move the decimal point 7 places to the right ,000,000 Eample: Write in standard form. The eponent 4 indicates you move the decimal point 4 places to the left Why does this work? Rewrite the problem using product form, and then do the multiplication Eample: Write.4 10 in standard form. The eponent indicates you move the decimal point places to the left To convert a number in standard form to scientific notation, rewrite the number as a product of a power of 10 and a number greater than or equal to 1 but less than 10. Eample: Write 60 in scientific notation.? Rewrite 60 in product form: How many places do you need to move the decimal point to get 60? places to the right, so Eample: Write 7,000 in scientific notation.? Rewrite 7,000 in product form: How many places do you need to move the decimal point to get 7,000? places to the right, so 7, Eample: Write in scientific notation.? Rewrite in product form: How many places do you need to move the decimal point to get ? places to the left, so Eample: Write in scientific notation McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 16 of 17 Revised 010

17 Applying Scientific Notation (Please note that this concept is not included in the CCSD benchmarks.) Rewriting numbers in scientific notation will allow us to multiply very large or very small numbers quickly. It also will help to avoid errors that occur with so many zeroes. Look at the following eamples. Eample: The distance from Earth to the Moon is about 80,000 km. Suppose a space capsule travels this distance 000 times. How many kilometers does the capsule travel? We know how to rewrite the numbers in scientific notation: 80, To find the total distance travelled, we must multiply: 80, ,000,000 The total distance travelled is 760,000,000 km. Eample: The smallest flowering plant in the world weighs about g. How much would one million of these plants weigh? Rewrite the numbers using scientific notation: million 1,000, To find the total weight, we must multiply: million The total weight of the plants would be 10 g. McDougal Littell, Chapter 4 Pre-Algebra 8, Unit 04: Factors, Fractions, and Eponents Page 17 of 17 Revised 010