2. Our Lesson Simplify and Equivalent Fractions

3. Confidential 2 Warm up 1) Is 8 a factor of 2832? Yes 2) List all factors of 49 1, 7 and 49 3) Solve 4x – 2y 3, where x = 2 and y =-1 10 4) Write the prime factorization of 45 5 x 3 2 5) Write the GCF of 24 and 14 2

4. Confidential 3 Lets Review Factors A whole number that divides exactly into another whole number is called a factor of that number. For example 100 / 25 = 4 So, 25 is a factor of 100 as it divides exactly into 20. 20 / 4 = 5 So, 4 is a factor of 20 as it divides exactly into 20.

5. Confidential 4 The exponent can also be referred to as the power 4 6 Base Exponent Powers and Exponents 4 6 means to multiply the base 4 by itself 6 times 4 6 = 4 x 4 x 4 x 4 x 4 x 4

6. Confidential 5 A Prime number is a positive integer >1 A Prime number is a positive integer >1 A number that has exactly two factors, 1 and itself A number that cannot be factored. Prime Numbers Example: 7 is a Prime number as it has only two factors 1 and 7

7. Confidential 6 Composite Numbers When a whole number greater than one has more than 2 factors it is called a Composite Number. 10 is a composite number as it has 1, 2, 5 and 10 as its factors

8. Confidential 7 Expressing a composite number as a product of prime numbers is called Prime Factorization Prime Factorization When we express a number as a product of prime factors, we have actually factored it completely. We refer to this process as prime factorization The number 60 is a composite number. It can be written as the product 2 x 2 x 3 x 5. Note that 2, 3 and 5 are factors of 60 and all these factors are prime numbers. We call them prime factors.

9. Confidential 8 Greatest Common Factor of two or more numbers can be defined as the greatest number that is a factor of each number Greatest Common Factor List the factors of each number. Then identify the common factors. The greatest of these common factors is the GCF. Method 1 Write the prime factorization of each number. Then identify all common prime factors and find their product. Method 2

10. Confidential 9 The least common multiple of the numbers a and b is the smallest number that is divisible by both a and b We denote the least common multiple of a and b by lcm (a, b) Least Common Multiple

11. Confidential 10 Lets get started Simplest form of Fraction A fraction is in its simplest form when the GCF of the numerator and the denominator is 1 There is no other common factor except 1 Example 2 17 5 3 31 19,,

12. Confidential 11 Steps to expressing a fraction in its simplest form 1)Find the GCF of the numerator and denominator 2)Divide the numerator and the denominator by the GCF, and write the resulting fraction

13. Confidential 12 Lets take an Example Write the 4 in its simplest form 16 Factors of 4: 1, 2, 4 Factors of 16: 1, 2, 4, 8, 16 The GCF of 4/16 is 4 4 16 = 4 16÷ 4 ÷ 4 = 1414

14. Confidential 13 Equivalent Fractions If a and c are two fractions where c m x a b d d m x b Then, a c b d = = Method 1

15. Confidential 14 Method2 If the cross products of two fractions are equal then they are equivalent fractions a c b d If ad = bc then a/b = c/d

16. Confidential 15 Equivalent fractions In this picture we have ½ of a cake because the whole cake is divided into two congruent parts and we have only one of those parts But if we cut the cake into smaller congruent pieces, we can see that 1 2 2 4 =

17. Confidential 16 Now we have 3 pieces out of 6 equal pieces, but the total amount we have is still the same Therefore, 1 2 3 2 4 6 = = Or we can cut the original cake into 6 congruent pieces,

18. Confidential 17 then we have 4 pieces out of 8 equal pieces, but the total amount we have is still the same We can generalize this to whenever n is not 0 Therefore, 1 2 3 2 4 6 = == 4848 1212 = 1 x n 2 x n If you don’t like this, we can cut the original cake into 8 congruent pieces,

19. Confidential 18 Another example: andequal? reduce This shows that these two fractions are not the same! reduce Are 24 40 30 42 24 40 24 ÷ 2 40 ÷ 2 = 12 20 12 ÷ 4 20 ÷ 4 = 3535 30 42 30 ÷ 6 42 ÷ 6 = 5757 24 x 42 = 1008 40 x 30 = 1200 24 x 42 = 40 x30 Method 2 Method 1 3 5 5 7 =

20. Confidential 19 Questions Write the following in the simplest form 1)18/20 9/10 2)4/16 ¼ 3) 36/44 9/11 4)45/105 3/7 5)848/1240 101/155

21. Confidential 20 Questions Which of these are equivalent fractions 6) 5/9 and 4/5 no 7) 4/5 and 16/20 yes 8) 9/16 and 5/9 no Fill >,< or = 9) ½ __4/8 = 10) 4/50__5/75 >

22. Confidential 21 1) Explain how can you find a fraction in which the numerator and denominator are greater than 100 and they have a common factor of 13. Multiply the numerator and denominator of a fraction by 13. (Both numbers must be greater than 7) Example 8 x 13 104 9 x 13 117 =

23. Confidential 22 2) Emily had 20 pencils, Sheena had 50 pencils and Ben had 80 pencils. After 4 months, Emily used up 10 pencils, Sheena used up 25 pencils and Ben used up 40 pencils. What fraction did each use up? Check if each has used up an equal fraction of their pencils? Emily = 10/20 = ½ Sheena = 50/25 = ½ Ben = 40/80 = ½ Yes! Each has used up equal fraction of their pencils

24. Confidential 23 3) A sitar is a South Asian instrument that has two sets of strings. One type of sitar has 6 top strings and 16 bottom strings. Express the number of top strings as a fraction of total number of strings. 6 6 3 (16 +6) 22 11 == Total number of strings

25. Confidential 24 Lets review what we have learned in this lesson Simplest form of Fraction A fraction is in its simplest form when the GCF of the numerator and the denominator is 1 There is no other common factor except 1 Example 2 17 5 3 31 19,,

26. Confidential 25 Steps to expressing a fraction in its simplest form 1)Find the GCF of the numerator and denominator 2)Divide the numerator and the denominator by the GCF, and write the resulting fraction

27. Confidential 26 Equivalent Fractions If a and c are two fractions where c m x a b d d m x b Then, a c b d = = Method 1

28. Confidential 27 Method2 If the cross products of two fractions are equal then they are equivalent fractions a c b d If ad = bc then a/b = c/d

29. Confidential 28 You had a Great lesson Today! Be sure to practice what you have learned