1. Our fourth objective in Math is another part of number and number sense!! Through this objective students will continue to use problem solving, mathematical communication, mathematical reasoning, connections and representations. Objective 6.2 : The student will a. investigate and describe fractions, decimals and percents as ratios; b. identify a given fraction, decimal or percent from a representation; c. demonstrate equivalent relationships among fractions, decimals, and percents; d. compare and order fractions, decimals, and percents; e. find common multiples and factors including least common multiple and greatest common factor; f. identify and describe prime and composite numbers; and g. identify and describe the characteristics of even and odd integers. Definitions to remember and now how to use: Fraction Percent Decimal (Terminating, Repeating) Factor Prime and Composite Numbers Greatest common factor Inequality Integers, Whole Numbers Least common multiple Multiple Prime Factorization IT IS VERY IMPORTANT YOU KEEP TRACK OF TERMS YOU DO NOT UNDERSTAND, DEFINE THOSE TERMS (DEFINTIONS) AND UNDERSTAND HOW TO WORK AND USE THAT INFORMATION!! Number and Number Sense Objective 6.2

2. Math Obj 6.2 – investigate and describe fractions, decimals and percents as ratios; What is a fraction, percent and a decimal and why are they important? Fractions, percents and decimals are _____________________________________. _______________________ can be put in order, compared, and interpreted by using the relationship between the _______________________ (top number) and ______________________ (bottom number). Fractions are part or less then one. When the numerator is bigger than the denominator, then it is an improper fraction. ___________________can be represented by drawing shaded regions on grids or by finding a location on number lines. Percents are stated in relationship to 100. ________________________________________, for taxes, sales, data description, and data comparison. Some fractions with ______________________________ may also be expressed as decimals or percents in order to compare them easily. Which is greater 4 or 3 ? 7 8 The __________________ is a symbol that separates the _____________________ (1, 2, 3, 4) from the fractional part of a number (.1,.2,.3). Look at the figures and their corresponding fraction. We know that ½ is also 50%, but how do you solve the others? To convert a fraction to a decimal? 1. Divide the denominator into the numerator. Since the denominator is bigger than the numerator your number will be less than 1 For example 3 (3 divided by 4 or 4 4 goes into 3) To covert a decimal to a percent multiply. by 100

3. What is a fraction, percent and a decimal and why are they important? Fractions, percents and decimals are different ways to show the same value. Fractions can be put in order, compared, and interpreted by using the relationship between the numerator (top number) and denominator (bottom number). Fractions are part or less than one. When the numerator is bigger than the denominator, then it is an improper fraction. Percents can be represented by drawing shaded regions on grids or by finding a location on number lines. Percents are stated in relationship to 100. Percents are used in the real-life, for taxes, sales, data description, and data comparison. Some fractions with unlike denominators may also be expressed as decimals or percents in order to compare them easily. Which is greater 4 or 3 7 8 The decimal point is a symbol that separates the whole number (1, 2, 3, 4) from the fractional part of a number (.1,.2,.3).

4. Math Obj 6.2 – investigate and describe fractions, decimals and percents as ratios; Show your work here!! Look at the figures and their corresponding fraction. We know that ½ is also 50%, but how do you solve the others? To convert a fraction to a decimal? 1. Divide the denominator into the numerator. Since the denominator is bigger than the numerator your number will be less than 1 For example 3 Divide 4 into 3 4 To covert a decimal to a percent multiply by 100

5. Math Obj 6.2 – investigate and describe fractions, decimals and percents as ratios; Show your work here!! What is a fraction, percent and a decimal and why are they important? Fractions, percents and decimals are different ways to show the same value. Fractions can be put in order, compared, and interpreted by using the relationship between the numerator (top number) and denominator (bottom number). Fractions are part or less then one. When the numerator is bigger than the denominator, then it is an improper fraction. Percents can be represented by drawing shaded regions on grids or by finding a location on number lines. Percents are stated in relationship to 100. Percents are used in the real-life, for taxes, sales, data description, and data comparison. Some fractions with unlike denominators may also be expressed as decimals or percents in order to compare them easily. The decimal point is a symbol that separates the whole number (1, 2, 3, 4) from the fractional part of a number (.1,.2,.3). Look at the figures and their corresponding fraction. We know that ½ is also 50%, but how do you solve the others? To convert a fraction to a decimal? 1. Divide the denominator into the numerator. Since the denominator is bigger than the numerator your number will be less than 1 For example 3 Divide 4 into 3 =.75 4.75 4 3.000 2 8 20 To covert a decimal to a percent multiply by 100.75 * 100 = 75%

6. Fraction to Decimal to Percent Chart Fraction Decimal Percent 1 or 1 1.0 1 * 100 = 100% 1 1 1.00 15 16 7 8 2 3 4 1 2 3 8 1 3 1 4 1 5 2 7 1 6.

7. Fraction to Decimal to Percent Chart Fraction Decimal Percent 1 or 1 1.0 1 * 100 = 100% 1 1 1.00 15 16 7 8 2 3 4 1 2 3 8 1 3 2 7 1 4 1 5 1 6 16 15.0000..875.6666 can be written as.6 or.67 (terminating decimal) (repeating decimal).75.50.375.3333 can be written as.3 or.33.28571428571 =.29.25.20.1666 can be written as.17 (repeating decimal).9375 93.75% 87.5% 67% 75% 50% 37.5% 33% 29% 25% 20% 17%

8. To covert a decimal to a percent multiply by 100 For example.75 * 100 = 75% Shortcut move the decimal point 2 places to the right, and add the "%" sign. Convert the following decimals to %.6945 = 69.45% 1..745 2..6789 3..25 To covert a percent a decimal divide % by 100 For example 75% divide by 100 =.75 Shortcut move the decimal point 2 places to the left and remove the % Convert the following % to decimals 42.5 % =.425 4. 25.67% 5. 78% 6. 13.45% What is a benchmark? A benchmark ________________________________________________________ ___________________________________________________________________ 0 1 1 1 2 3 1 4 3 2 3 4 Write as a Decimal % Whole numbers, fractions, and decimals may be positioned along a conventional number line. A number to the left has a lesser value than a number to its right. -.50 -.25 0.25.50..75 1 0 1 1 1 2 3 1 4 3 2 3 4 Benchmark fractions on a number line

9. To covert a decimal to a percent multiply by 100 For example.75 * 100 = 75% Shortcut move the decimal point 2 places to the right, and add the "%" sign. Convert the following decimals to %.6945 = 69.45% 1..745 2..6789 3..25 To covert a percent a decimal divide % by 100 For example 75% divide by 100 =.75 Shortcut move the decimal point 2 places to the left and remove the % Convert the following % to decimals 42.5 % =.425 4. 25.67% 5. 78% 6. 13.45% What is a benchmark? A benchmark is a standard measurement that can be used as a reference especially when making comparisons. 0 1 1 1 2 3 1 4 3 2 3 4 Write as a Decimal % Whole numbers, fractions, and decimals may be positioned along a conventional number line. A number to the left has a lesser value than a number to its right. -.50 -.25 0.25.50..75 1 0 1 1 1 2 3 1 4 3 2 3 4 Benchmark fractions on a number line.25.3333.50.666.75 1 25% 33% 50% 67% 75% 100%

10. Place the following numbers on the number line below ¾, ½. 5/8. 2/7,.85,.67,.20 The numbers should be placed in proportion (1/2 the line should be.50) 0 1 Comparison between fractions, decimals, or whole numbers can be described using the mathematical symbols: < (“is less than”) < (“is less than or equal to”) > (“is greater than”) > (“is greater than or equal to”) The symbols >, >, <, and < are called inequality symbols. Statements formed by placing an inequality symbol between two unequal expressions are called inequalities. Use the correct inequality symbol to solve the problems below. 1. 3 1 2. 4 3 3. 2 5 4. 1 2 4 2 7 8 7 11 4 5 Some additional information: You should have noticed that the symbol * can be used in place of × to indicate multiplication. Benchmark fractions can used as a reference. For example you should know that 4 is less than 1 (4/8 would equal ½) 7 2 Fractions and ratios work the same way with ONE exception!! 6 as a fraction is improper and converts to 1 1. As a ratio is stays as 6 or 6 to 4 or 4 2 4 6:4

11. Place the following numbers on the number line below ¾, ½. 5/8. 2/7,.85,.45,.20 The numbers should be placed in proportion (1/2 the line should be.50) 0 1 Comparison between fractions, decimals, or whole numbers can be described using the mathematical symbols: < (“is less than”) < (“is less than or equal to”) > (“is greater than”) > (“is greater than or equal to”) The symbols >, >, <, and < are called inequality symbols. Statements formed by placing an inequality symbol between two unequal expressions are called inequalities. Use the correct inequality symbol to solve the problems below. 1. 3 1 2. 4 3 3. 2 5 4. 1 2 4 2 7 8 7 11 4 5 Some additional information: You should have noticed that the symbol * can be used in place of × to indicate multiplication. Benchmark fractions can used as a reference. For example you should know that 4 is more than 1 (the numerator of 4 is more than ½ of the denominator, so it 7 2 will be greater than 50%) Fractions and ratios work the same way with ONE exception!! 6 as a fraction is improper and converts to 1 1. As a ratio is stays as 6 or 6 to 4 or 4 2 4 6:4 ><<> 1 2 (.50) 3 4 (.75) 5 8 (.625) 2 7 (.29).20.45.85

12. Fraction to Decimal to Percent Chart Example 2 11 Example 62.5% 75% 3 7 5 11 decimal%fraction.22.35 86% 87.5% 12%

13. Convert Percent to Fraction To convert a Percent to a Fraction follow these steps: Step 1: Write down the percent divided by 100. Step 2: If the percent is not a whole number, then multiply both top and bottom by 10 for every number after the decimal point. (For example, if there is one number after the decimal, then use 10, if there are two then use 100, etc.) Step 3: Simplify (or reduce) the fractionSimplify

14. Example 2: Express 75% as a fraction Step 1: Write down: Step 2: The percent is a whole number, so no need for step 2. Step 3: Simplify the fraction: ÷ 25 75 = 3 1004 75 100 75% as a fraction = 3 4 ÷ 25

15. Example 3: Express 62.5% as a fraction Step 1: Write down: 62.5 100 Step 2: 62.5 is not a whole number. Multiply both top and bottom by 10 (because there is 1 digit after the decimal place) Step 3: Simplify the fraction (this took me two steps, you may be able to do it one!) : Answer = 5/8 x 10 62.5 625 = 100 1000 x 10 ÷ 25 ÷ 5 625 25 5 1000 40 8 ÷ 25 ÷ 5

16. ____________________ 2, 4, 6, 8, 10 or numbers that end in those numbers ____________________ 1, 3, 5, 7, 9 or numbers that end in those numbers A ________________ is a natural number that has exactly two different factors, 1 and the number itself. _______________________________ A ________________cannot be divided by any other number. Students need to know prime numbers up to 100. List Prime numbers up to 100 (2 is the only prime even number) 2 3 ___________________________________________ _________________________________________________ A ________________ is a natural number that has more than two different factors. _______________________________ ____ only one factor, itself. Although zero has an infinite number of factors, Zero is not a natural number. The ___________________ of a number is a representation of the number as the product of its prime factors. For example, the prime factorization of 4 is 2 * 2 10 is 5 * 2 14 is 7 * 2 18 is 2 * 3 * 3.

17. Even numbers 2, 4, 6, 8, 10 or numbers that end in those numbers Odd numbers 1, 3, 5, 7, 9 or numbers that end in those numbers A prime number is a natural number that has exactly two different factors, 1 and the number itself. 0 and 1 are not prime. A prime number cannot be divided by any other number. Students need to know prime numbers up to 100. List Prime numbers up to 100 (2 is the only prime even number) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 A composite number is a natural number that has more than two different factors. 0 and 1 are neither prime nor composite. 1 only one factor, itself. Although zero has an infinite number of factors, Zero is not a natural number. The prime factorization of a number is a representation of the number as the product of its prime factors. For example, the prime factorization of 4 is 2 * 2 10 is 5 * 2 14 is 7 * 2 18 is 2 * 3 * 3.

18. A multiple of a number is the product of the number and any natural number. 4, 8, 12, 16, 20 The least common multiple of two or more numbers is the smallest common multiple of two or more numbers (other than 0). 4 and 6 4, 8, 12, 16, 20, 24 6, 12, 18, 24, 32 Find the least common multiple (LCM) 4 and 9 5 and 6 8 and 9 10 and 12 A factor of a number is an integer that divides evenly into that number. In other words, it is a divisor of the number. A common factor of two or more numbers is a divisor that all of the numbers share. The greatest common factor of two or more numbers is the largest of the common factors that all the numbers share. 24 and 36 1, 2, 3, 4, 6, 8, 12, 24 GCM = 12 1, 2, 3, 4, 6, 6, 9, 12, 36 Find the greatest common factor (GCM) 15 and 45 14 and 49 12, 16 and 18 15, 45 and 90

19. Who cares about Prime Numbers? A lot of energy has been expended by mathematicians (ever since the time of Pythagoras) on studying prime numbers. Recently, a very important branch of mathematics to emerge is encryption, where sensitive information is hidden from others when it is transmitted electronically (e.g. when we send credit card numbers over the Internet or by mobile phone). Encryption works by coding the message using very large prime numbers. The device receiving the message decodes the message using the same very large prime numbers. The larger the numbers used, the better the encryption. The largest known prime currently is 2 43,112,609 − 1. (This is huge - it has over 13 million digits).