UNIT 5 INTRODUCTION TO FRACTIONS

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1 UNIT INTRODUCTION TO FRACTIONS INTRODUCTION In this Unit, we will investigate fractions and their multiple meanings. We have seen fractions before in the context of division. For example, we can think of the division problem 6 as an equivalent fractional expression 6. It will be very useful to use equivalencies such as these when working with fractions. However, we will need to build up and contact multiple meanings of fractions to truly understand their meanings in numerous contexts. The table below shows the learning objectives that are the achievement goal for this unit. Read through them carefully now to gain initial exposure to the terms and concept names for the lesson. Refer back to the list at the end of the lesson to see if you can perform each objective. Learning Objective Media Examples You Try Represent fractions symbolically and with word names given various fraction language 1 Compare and contrast four models of fractions 2 Determine the unit of a fraction in context 4 Represent unit fractions in multiple ways 6 7 Represent composite fractions on a number line 8 9 Represent composite fractions using an area model Represent composite fractions using a discrete model Represent improper fractions and mixed numbers using number lines and an area model 1 14 Create equivalent fractions using an area model 1 16 Find an equivalent fraction given a fraction and a corresponding numerator or denominator Recognize the simplest form of a fraction 19 Simplify fractions using repeated division or prime factorization Compare fractions with the same denominator 22 2 Compare fractions with the same numerator 2 2 Order fractions 24 2 Use one half as a benchmark to compare fractions

2 UNIT MEDIA LESSON SECTION.1: WHAT IS A FRACTION? There are many ways to think of a fraction. A fraction can be thought of as one quantity divided by another written by placing a horizontal bar between the two numbers such as 1 where 1 is called the numerator and 2 is 2 called the denominator. Or we can think of fractions as a part compared to a whole such as 1 out of 2 cookies or 1 2 of the cookies. In this lesson, we will look at a few other ways to think of fractions as well. Officially, fractions are any numbers that can be written as a b but in this course, we will consider fractions where the numerator and denominator are integers. These special fractions where the numerator and denominator are both integers are called rational numbers. Since rational numbers are indeed fractions, we will frequently refer to them as fractions instead of rational numbers. Problem 1 MEDIA EXAMPLE Language of Fractions Each of the phrases below are one way we may indicate a fraction with words. Rewrite the phrases below in fraction form and write the fraction word name. Language 20 divided by 6 Fraction Representation Fraction Word Name 8 out of 9 A ratio of to 2 11 per 2 for every 7 In the next example, we will look at four different types of fractions in context. 1. Quotient Model (Division): Sharing equally into a number of groups 2. Part-Whole Model: A part in the numerator a whole in the denominator. Ratio Part to Part Model: A part in the numerator and a different part in the denominator 4. Rate Model: Different types of units in the numerator and denominator (miles and hours) 2

3 Problem 2 MEDIA EXAMPLE Fractions in Context: Four Models Represent the following as fractions. Determine whether it is a quotient, part-whole, part to part, or rate model. a) Three cookies are shared among 6 friends. How many cookies does each friend get? b) Four out of 6 people in the coffee shop have brown hair. What fraction of people in the coffee shop have brown hair? c) Tia won 6 games of heads or tails and lost games of heads or tails. What is the ratio of games won to games lost? d) A snail travels miles in 6 hours. What fraction of miles to hours does he travel? What fraction of hours to miles does he travel? Problem YOU-TRY - Examples of Fractions in Context Represent the following scenarios using fraction. Indicate whether the situation is a Quotient, Part to Whole, Ratio Part to Part, or Rate. a) Jorge bikes 12 miles in hours. What fraction of miles to hours does he travel? b) Callie has pairs of blue socks and 12 pairs of grey socks. What fraction of blue socks to grey socks does she have? c) Callie has pairs of blue socks and 12 pairs of grey socks. What fraction of all of her socks are blue socks?

4 Problem 4 Unit Media Lesson MEDIA EXAMPLE The Importance of the Unit When Representing Fractions Sean s family made trays of brownies. Sean ate 2 brownies from the first batch and 1 from the rd batch and shown in the image below (brownies eaten are shaded). His family disagreed on the amount of brownies he ate and gave the three answers below. Draw a picture of the unit (the amount that represents 1) that makes each answer true. Answer 1: Draw a Picture of the Unit: Answer 2: 6 Draw a Picture of the Unit: Answer : 18 Draw a Picture of the Unit: Problem YOU-TRY - The Importance of the Unit When Representing Fractions Consider the following problem and the given answers to the problem. Determine the unit you would need to use so each answer would be correct. The picture below shows the pizza Homer ate. Determine the unit that would make each answer below reasonable. Answer 1: Draw a Picture of the Unit: Answer 2: 8 Draw a Picture of the Unit: Answer : 16 Draw a Picture of the Unit: 4

5 SECTION.2: REPRESENTING UNIT FRACTIONS A unit fraction is a fraction with a numerator of 1. In this section we will develop the idea of unit fractions and use multiple representations of unit fractions. Problem 6 MEDIA EXAMPLE Multiple Representations of Unit Fractions a) Plot the following unit fractions on the number line, 1, 1, 1 Label your points below the number 2 4 line. b) Represent the fractions using the area model. The unit is labeled in the second row of the table c) Represent the unit fractions using the discrete objects. The unit is all of the triangles in the rectangle. Represent 1 4 of the triangles.

6 Problem 7 YOU-TRY Multiple Representations of Unit Fractions a) Plot the following unit fractions on the number line 1, 1. Label your points below the number line. 4 b) Represent the fractions using the area model. The unit is labeled in the second row of the table c) Represent the unit fractions using the discrete objects. The unit is all of the triangles in the rectangle. Represent 1 of the triangles. 6

7 SECTION.: COMPOSITE FRACTIONS In this section, we will use unit fractions to make composite fractions. Composite fractions are fractions that have a numerator that is an integer that is not 1 or 1. We will look at both proper and improper fractions. Proper fractions are fractions whose numerator is less than their denominator. Improper fractions are fractions whose numerator is greater than or equal to its denominator. Problem 8 MEDIA EXAMPLE Cut and Copy: Composite Fractions on the Number Line a) Plot the following composite fractions on the number line line. 2 4,. Label your points below the number b) Plot the following composite fractions on the number line, 8. Label your points below the number 2 line. c) Plot the following composite fractions on the number line 12, 8. Label your points below the 6 4 number line. 7

8 Problem 9 YOU-TRY Cut and Copy: Composite Fractions on the Number Line Plot the following composite fractions on the number line,,, 12. Label your points below the number line. Problem 10 MEDIA EXAMPLE Cut and Copy: Composite Fractions and Area Models Represent the composite fractions using an area model. A single rectangle is the unit. An additional rectangle is given in each problem for the fractions which may require it. a) Represent 4 with a rectangle as the unit. copies of (unit fraction) b) Represent 7 4 with a rectangle as the unit. copies of (unit fraction) 8

9 Problem 11 Unit Media Lesson MEDIA EXAMPLE Cut and Copy: Composite Fractions Using Discrete Models Represent the composite fractions using the discrete objects. The unit is all of the triangles in the rectangle. a) Represent 6 of the triangles. Drawing of associated unit fraction: copies of (unit fraction) b) Represent of the triangles. Drawing of associated unit fraction: copies of (unit fraction) Problem 12 YOU-TRY - Cut and Copy: Composite Fractions and Area and Discrete Models a) Represent the composite fractions using an area model. A single rectangle is the unit. An additional rectangle is given in each problem for the fractions which may require it. Represent 8 with a rectangle as the unit. copies of (unit fraction) b) Represent the composite fractions using the discrete objects. The unit is all of the triangles in the rectangle. Represent 4 of the triangles. Drawing of associated unit fraction: copies of (unit fraction) 9

10 SECTION.4: IMPROPER FRACTIONS AND MIXED NUMBERS Improper fractions are fractions whose numerators are greater or equal to their denominators. You may have noticed that these fractions are greater than equal to 1. We can also represent improper fractions as mixed numbers. A mixed number is the representation of a number as an integer and proper fraction. In this section, we will represent and rewrite improper fractions as mixed numbers and vice versa. Problem 1 MEDIA EXAMPLE Improper Fractions and Mixed Numbers a) Represent the unit) 7 with a rectangle as the unit. Then rewrite it as a mixed number. (A single rectangle is Mixed Number: b) Represent 8 on the number line. Then rewrite it as a mixed number. Mixed Number: Problem 14 a) Represent the unit) 8 7 YOU-TRY Improper Fractions and Mixed Numbers with a rectangle as the unit and then rewrite it as a mixed number. (A single rectangle is Mixed Number: b) Represent 7 on the number line and then rewrite it as a mixed number. Mixed Number: 10

11 SECTION.: EQUIVALENT FRACTIONS At some point in time, you have probably eaten half of something, maybe a pizza or a cupcake. There are many ways you can have half of some unit. A pizza (the unit) can be cut into 4 equal pieces and you have 2 of these pieces, or 2 4. Or maybe a really big pizza is cut into 100 equal pieces and you have 0, or the amount you have is equivalent to In either case, because you ate one for every two pieces in the unit. In this section we will investigate the idea of equivalent fractions and learn to find various equivalent fractions. Problem 1 MEDIA EXAMPLE Creating Equivalent Fractions a) Create two fractions equivalent to the given fraction by cutting the given representations into a different number of equal pieces. Given Fraction: 2 2 is equivalent to the fraction: 2 is equivalent to the fraction: b) Create two fractions equivalent to the given fraction by grouping the total number of pieces into a smaller number of equal pieces. Given Fraction: is equivalent to the fraction: 8 12 is equivalent to the fraction: Problem 16 YOU-TRY - Creating Equivalent Fractions Create two fractions equivalent to the given fraction by grouping the total number of pieces into a smaller number of equal pieces. Given Fraction: is equivalent to the fraction: 11

12 Problem 17 Unit Media Lesson MEDIA EXAMPLE Rewriting Equivalent Fractions with One Value Given Rewrite the given fractions as equivalent fractions given the indicated numerator or denominator. a. Rewrite 7 with a denominator of 21. b. Rewrite with a numerator of 120. c. Rewrite 8 60 with a denominator of 12. d. Rewrite 6 2 with a numerator of 9. Problem 18 YOU-TRY - Rewriting Equivalent Fractions with One Value Given Rewrite the given fractions as equivalent fractions given the indicated numerator or denominator. a. Rewrite 8 with a denominator of 2. b. Rewrite 18 with a numerator of 6. SECTION.6: WRITING FRACTIONS IN SIMPLEST FORM Definition: The simplest form of a fraction is the equivalent form of the fraction where the numerator and denominator are written as integers without any common factors besides 1. Problem 19 MEDIA EXAMPLE What is a Simplified Fraction? a) Write the fraction number for each diagram below the figure using one circle as the unit. b) What do the fractions have in common? c) Which fraction do you think is the simplest and why? d) Divide the numerators and denominators of the second and fourth fractions by 2. What do you notice? e) Rewrite the last three fractions below by writing their numerators and denominators in terms of their prime factorizations. Do you see any patterns? f) Simplify your fractions in part e by cancelling out all of the common factors (besides 1) that the numerators and denominators share. 12

13 Problem 20 MEDIA EXAMPLE Simplifying Fractions by Repeated Division and Prime Factorization We can use two different methods to simplify a fraction; repeated division or prime factorization. 1. Repeated Division: Look for common factors between the numerator and denominator and divide both by the common factor. Continue this process until you are certain the numerator and denominator have no common factors. 2. Prime Factorization: Write the prime factorizations of the numerator and denominator and cancel out any common factors. Simplify the given fractions completely using both the repeated division and prime factorization methods. In each case, state which you think is easier and why. a) b) c) 84 6 Problem 21 YOU-TRY Simplifying Fractions by Repeated Division and Prime Factorization Simplify the given fractions completely using both the repeated division and prime factorization methods. In each case, state which you think is easier and why. a) 6 8 b)

14 SECTION.7: COMPARING FRACTIONS In this section, we will learn to compare fractions in numerous ways to determine their relative size. Problem 22 MEDIA EXAMPLE Comparing Fractions with Same Denominator a) Shade the following areas representing the fractions using the rectangles below., 6, b) Order the numbers from least to greatest by comparing the amount of the unit area shaded. Plot the following fractions on the number line,, 1, 2, 1. Label your points below the number line. c) Using the number line, order the numbers from least to greatest. d) Develop a general rule for ordering fractions with the same denominator. i. If two fractions have the same denominator and the fractions are positive, then the fraction with the numerator is greater. ii. If two fractions have the same denominator and the fractions are negative, then the fraction with the numerator is greater. iii. If one fraction is positive and the other is negative, then the _ fraction is greater. 14

15 Problem 2 MEDIA EXAMPLE Comparing Fractions with Same Numerator a) Identify the fractions represented by area shaded in the rectangles below. b) Order the numbers from least to greatest by comparing the amount of the unit area shaded. c) Plot the following fractions on the number lines below. Label your points below the number lines ,,,,, 8 8 d) Develop a general rule for ordering fractions with the same numerator. i. If two fractions have the same numerator and the fractions are positive, then the fraction with the denominator is greater. ii. If two fractions have the same numerator and the fractions are negative, then the fraction with the denominator is greater. iii. If one fraction is positive and the other is negative, then the _ fraction is greater. 1

16 Problem 24 MEDIA EXAMPLE Comparing Fractions with Equal Numerators or Denominators Order the fractions from least to greatest and justify your answer. a) 7 1 0,,, Ordering: Justification: b),,, Ordering: Justification: c) 2,, Ordering: Justification: 8 d) 2,, Ordering: Justification: Problem 2 YOU-TRY Comparing Fractions with Equal Numerators or Denominators Order the fractions from least to greatest and justify your answer. a) 1 4,, Ordering: Justification: b) 7,, Ordering: Justification: 16

17 Problem 26 MEDIA EXAMPLE The Fraction One Half as a Benchmark a) Each of the fractions below are equivalent to one half. Write the numeric representation in terms of the number of equally shaded pieces below each image. (Note: the dashed lines represent a unit fraction that has been cut in half) a) Using the images above, determine whether the following fractions are less than, equal to or greater than one half. Use the symbols, <, =, or > b) Use the information from part b to compare the fractions. Use the symbols, <, =, or > c) Give an example when you cannot use one half as a benchmark to order fractions. d) Develop a general rule for ordering fractions using one half as a benchmark. e) Develop a general rule for ordering fractions using one half as a benchmark. 17

18 Problem 27 YOU-TRY The Fraction One Half as a Benchmark a) Determine whether the following fractions are less than, equal to or greater than one half. Use the symbols, <, =, or > b) Use the information from part a to compare the fractions. Use the symbols, <, =, or >