Lesson 2 Comparing Fractions with Different Denominators Problem Solving: Rules for Drawing Number Lines Lesson 2 Skills Maintenance Lesson Planner Name Skills Maintenance Skills Maintenance Ordering Fractions Activity 1 Ordering Fractions, Common Multiples Order the fractions from least to greatest. 9, 2, 4, 1, 5 9 9 9 9 9 Building Number Concepts: 1 9 omparing Fractions with C Different Denominators Homework Students fill in missing fractions on a number line, order a set of fractions, and draw two number lines to compare two fractions. In Distributed Practice, students practice multiplying unit fractions by whole numbers. 114 Unit 2 • Lesson 2 9 9 greatest Activity 2 1. Write two common multiples. 4 Sample answer: 12, 24 12 2. Write two common multiples. 3 Sample answer: 12, 24 12 3. Write three common multiples. 2 Sample answer: 20, 40, 60 20 4. Problem Solving: Students will accurately draw two number lines for comparing fractions with different numerators and denominators to each other. 5 9 Write common multiples as directed in each problem. Students will compare fractions with different numerators and denominators by looking at the relative size of the numerator compared to the denominator and comparing the fractions using number lines. Objective 4 9 Common Multiples Objective Students learn two important rules for comparing fractions using number lines: the number lines must be lined up at 0 and at 1 so that the number lines represent the “same whole,” and the number lines must be divided into fair-share parts. 2 9 least In this lesson, students learn that we often compare fractions that do not have the same numerator or denominator. When students compare fractions in this way, it is important to show accurate representations of the fractions on the number line. Students will need this skill to correctly compare fractions. ules for Drawing Number R Lines Date Write four common multiples. 4 Sample answer: 12, 24, 36, 48 6 50 Unit 2 • Lesson 2 Skills Maintenance Ordering Fractions, Common Multiples (Interactive Text, page 50) Activity 1 Students order a list of fractions with the same denominator. Activity 2 Students write common multiples for two numbers. Lesson 2 Comparing Fractions with Different Denominators Problem Solving: Rules for Drawing Number Lines Building Number Concepts: Comparing Fractions with Different Denominators omparing Fractions with Different C Denominators How do we use number lines to compare fractions? When comparing fractions, consider this question, “Are the fractions far from 0 or are they close to 0?” Here is what it looks like when we compare two fractions with the same denominator. How do we use number lines to compare fractions? 5 Which fraction is greater, 2 6 or 6? (Student Text, pages 68–69) Connect to Prior Knowledge Remind students about the patterns they saw in fractions from the previous lesson. Draw four number lines on the board, one directly below the other, with the 0s and 1s lined up. Divide each number line into fourths. On the top number line, Link to Today’s Concept Tell students that we will extend our knowledge about the locations of fractions on a number line to compare fractions that do not have simple patterns. Demonstrate •Have students turn to page 68 in the Student Text. Review the process of using a number line to compare fractions with the same denominator by comparing 2 and 5. 6 6 5 2 Point out that is greater than because 6 6 5 2 is to the right of on the number line. 6 6 Also point out that when the denominator is fixed and the numerator increases, the fractions increase. 1 6 2 6 3 6 4 6 5 6 6 6 =1 0 1 6 2 6 3 6 4 6 5 6 6 6 =1 3 3 =1 2 2 =1 It is important to notice: 2 • 5 6 is greater than 6. • When the denominator stays the same and the numerator increases, the fractions get farther from 0. Now let’s look at a different pair of fractions. 1 Which fraction is greater, 1 3 or 2? 1 3 0 draw a bar from 0 to 1. On the next number 4 line, draw a bar from 0 to 2. On the third 4 number line, draw a bar from 0 to 3. And on 4 the last number line, draw a bar from 0 to 4 or 4 1. Ask students to describe the pattern. As the numerators increase, the fractions increase. 0 2 3 1 2 0 It is important to notice: 1 •1 2 is greater than 3. • When the numerator stays the same and the denominator increases, the fractions get closer to 0. 6868 Unit 2 • Lesson 2 •Now have students look at the number lines for 1 and 1. Have them compare the positions of 3 2 0 and 1 on the number lines. Point out that the distance between 0 and 1 is the same on the number lines. So when we compare the fractions like this, the fractions use the same whole, and we can clearly see the fair share for 1 is smaller than the fair share for 1. 2 3 •Finally ask students which fraction is farther to the right on the number line. Because 1 is to 2 1 1 1 the right of , is closer to 1, and is closer to 3 2 3 0. These are all important things for students to notice. Unit 2 • Lesson 2 115 Lesson 2 Lesson 2 Not all fractions being compared have the same numerator or the same denominator. In this case, comparing the fractions becomes a bit more difficult. How do we use number lines to compare fractions? (continued) 5 Consider the fractions 6 8 and 6. In both fractions, the numerator is close to the denominator. So we know both fractions are close to 1. Which fraction is closer to 1? Let’s use a number line to find out. Demonstrate •Have students look at page 69 of the Student Text. In Example 1 , students will Example 1 or 5? Which fraction is greater, 6 8 6 Begin by showing each fraction on a number line. compare 6 and 5. Before comparing these 0 fractions, discuss the magnitude of each fraction by discussing the relative size of each numerator as compared to the 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 8 =1 6 6 =1 6 1 6 2 6 3 6 4 6 5 6 6 5 6 Because 5 6 is to the right of 8, it is closer to 1. So 6 is greater than 8. denominator. Think about the fraction 6. 8 How does the numerator compare to the denominator? Because 6 is close to 8, we know that 6 is close to 1. Think about 8 5 the fraction . How does the numerator 6 compare to the denominator? Because 5 is close to 6, we know that 5 is also close to 1. 6 Because these fractions are both close to 1, a number line can be used to compare the fractions. •Discuss all the things we can observe from these number line models about the fractions 6 and 5. We can see that 5 is to 8 6 6 the right of 6. So we know 5 > 6. This is the 8 6 8 way we have been comparing numbers on number lines for a long time. Apply Skills Reinforce Understanding Turn to Interactive Text, page 51. Use the Unit 2 Lesson 2 Teacher Talk Tutorial to review lesson concepts. Unit 2 • Lesson 2 Ask: We will compare the fractions 1 and 7. How does 5 8 1 the numerator of compare to its denominator? 5 (The numerator is much smaller than the denominator.) Is 1 closer to 0 or closer to 1? (closer to 0) 5 Check for Understanding Engagement Strategy: Think, Think Ask students the questions that follow about comparing fractions. Tell them that one of them will be called on to answer a question. Ask them to listen for their name to be called. After each question, allow time for students to think of the answer. Then call on a student. 116 Unit 2 • Lesson 2 How does the numerator of 7 compare to its 8 denominator? (The numerator is very close to the denominator.) Is 7 closer to 0 or closer to 1? (closer to 1) 8 Which fraction is greater? ( 7 ) 8 Reinforce Understanding Remind students that they can review lesson concepts by accessing the online Unit 2 Lesson 2 Teacher Talk Tutorial. 69 69 Lesson 2 Apply Skills Name Date Apply Skills Strategies for Comparing Fractions Apply Skills Activity 1 (Interactive Text, pages 51–52) Unit 2 Follow the directions for completing the number lines. 1. Show 1 4 on the number line. Have students turn to pages 51 and 52 in the Interactive Text, which provide an opportunity to compare fractions on a number line. 1 4 0 2. Show 3 2 on the number line. 0 3. Activity 1 4. 3 2 1 1 2 Show 1 5 on the number line. 1 5 0 Students are shown a fraction on a number line and they find the location of a different fraction with the same denominator. 1 3 4 1 3 5 Show 1 3 on the number line. 1 3 0 1 2 3 Activity 2 Students use two number lines to compare fractions with different denominators. Monitor students’ work as they complete these activities. Watch for: Unit 2 • Lesson 2 51 •Can students use the unit fraction to find the other fraction? Lesson 2 Apply Skills •Can students find the unit fraction given the Name location of another fraction with the same denominator? Date Activity 2 Follow the directions for completing the number lines. Then write the inequality. •Can students compare two fractions with different denominators by locating them on number lines? 2 Use the number lines to compare 1 2 and 3. 1 2 0 •Can students complete the inequality to Model 1 3 0 compare the fractions? The inequality is: Reinforce Understanding Remind students that they can review lesson concepts by accessing the online Unit 2 Lesson 2 Teacher Talk Tutorial. 1. 1 2 2 3 2 3 . < 1 3 Use the number lines to compare 1 4 and 8. 1 4 0 0 1 8 2 4 2 8 The inequality is: 2. 1 3 8 1 4 < 4 8 3 8 3 4 5 8 1 6 8 7 8 1 . 2 Use the number lines to compare 3 6 and 5. 0 0 1 6 2 6 1 5 The inequality is: 3 6 2 5 2 5 < 4 6 3 5 3 6 5 6 4 5 1 1 . Reinforce Understanding Use the Unit 2 Lesson 2 Teacher Talk Tutorial to review lesson concepts. 52 Unit 2 • Lesson 2 Unit 2 • Lesson 2 117 Lesson 2 Lesson 2 Problem Solving: Rules for Drawing Number Lines What are some common mistakes we make when we compare fractions using number lines? Problem Solving: We have to remember two important rules when we use number lines to compare fractions. Rules for Drawing Number Lines Rule 1: The distance between 0 and 1 has to be the same on both number lines. (the same whole) Rule 2: The fractional parts on each number line are fair shares. (equalsized parts) What are some common mistakes we make when we compare fractions using number lines? 1 We know 1 3 < 2. The next two examples show incorrect comparisons of these fractions because they fail to follow one of the rules above. Example 1 shows us what happens if the distance between 0 and 1 on the number lines is not the same. (Student Text, pages 70–71) Example 1 What happens when the number lines are not the same length between 0 and 1? Demonstrate •Have students turn to page 70 in the Student Text. Tell them that there are some very common mistakes that we can make when drawing number lines. One error is to use two different number lines that do not line up. The other mistake is failing to divide the number line into fair shares. 0 0 importance of these two rules for drawing number lines: Number lines must be lined up at 0 and 1. Number lines must be divided into fair shares. ■ •Have students look at Example 1 . They are shown one of these common errors. Point out where the 1s are located on each number line. Explain that if the 0s and the 1s are not aligned, we are not using the same whole for the fractions. We can only compare fractions from a whole that has the same length. 118 Unit 2 • Lesson 2 1 1 3 ERROR 2 3 1 These number lines show that 1 2 < 3 because the distance between 0 and 1 is not the same for each number line. The number line for thirds is longer, so it makes the fair-share segments much longer than they should be. •Make sure students understand the ■ 1 2 7070 Unit 2 • Lesson 2 1 Lesson 2 Demonstrate •Next have students look at Example 2 . Students are shown the second common error. Here the number lines are not divided into fair shares. Point out on the top number line that the distance from 0 to 1 is less 2 than the distance between 1 and 1. This 2 number line does not show fair shares. Example 2 shows us what can happen if we do not draw fair shares on the number line. Example 2 What happens when the fractional parts on each number line are not fair shares? 0 1 2 0 1 1 3 2 3 ERROR 1 1 These number lines show that 1 2 < 3. Look at the way each number line is partitioned. The segments are not fair shares. That is why 1 2 looks closer to 0 than 1 is an easy mistake to make, especially with 3.1This 1 fractions such as 8 or 10. •Likewise, point out on the bottom number line that the distance between 0 and 1 3 is much greater when compared to the distance between the next two parts. This number line also does not show fair shares. •Be sure students understand that we cannot trust our comparison of two fractions if we use number lines that do not follow these important rules. Problem-Solving Activity Turn to Interactive Text, page 53. Reinforce Understanding Remind students that they can review lesson concepts by accessing the online Unit 2 Lesson 2 Problem Solving Teacher Talk Tutorial. Reinforce Understanding Use the Unit 2 Lesson 2 Problem Solving Teacher Talk Tutorial to review lesson concepts. Unit 2 • Lesson 2 Unit 2 • Lesson 2 71 71 119 Lesson 2 Problem-Solving Activity Name Problem-Solving Activity (Interactive Text, page 53) Have students turn to page 53 in the Interactive Text, which provides students an opportunity to draw number lines in order to compare the two fractions 5 and 1. Remind them to line the 8 4 number lines up at 0 and 1 and make sure that the parts are fair shares. Monitor students’ work as they complete these tasks. Watch for: •Can students line up the two number lines Date Problem-Solving Activity Drawing Number Lines and Comparing Fractions Draw number lines to use to compare the given fractions. Be sure to follow the rules for drawing accurate number lines. Then write the inequality. 1 Use the number lines to compare 5 8 and 4. 0 1 8 2 8 3 8 1 4 0 The inequality is: fair shares? •Can students see that 14 < 58 from the number lines they drew? 5 8 6 8 2 4 1 4 5 8 < 3 4 7 8 1 1 . Challenge Problem Error Analysis 5 Marilyn compared 2 3 and 6 using the two number lines below. From 2 her drawing, she decided that 5 6 < 3. 1 3 0 correctly? •Can students divide up the number lines into 4 8 Unit 2 Lesson 2 0 1 6 2 6 2 3 3 6 4 6 5 6 3 =1 3 6 =1 6 Do you agree with Marilyn? (circle one) YES or NO Explain your reasoning. Sample answer: Although the number lines line up at 0 and 1, the shares in the number line for sixths are not fair shares because the distances between the tick marks are not equal. •Can students make other observations about the fractions in terms of magnitude by comparing the numerator to the denominator? (e.g., Five is relatively large when compared to 8, and 1 is relatively small when compared to 4.) Challenge Problem Error Analysis Students are shown another student’s answer to a fraction comparison problem and they are to determine whether the answer is correct or incorrect. Then they are to explain their reasoning. 120 Unit 2 • Lesson 2 Unit 2 • Lesson 2 53 Monitor students’ work as they complete this task. Watch for: •Can students see that the first number line has been drawn correctly? •Can students see that the 0s and 1s are lined up correctly? •Can students see that the second number line has not been divided into fair shares? •Do students realize that 23 < 56? Reinforce Understanding Remind students that they can review lesson concepts by accessing the online Unit 2 Lesson 2 Problem Solving Teacher Talk Tutorial. Lesson 2 Homework Activity 1 Write the fractions for the letters on the number lines. Homework (b) 1. Go over the instructions on page 72 of the Student Text for each part of the homework. 0 (a) 1 2 1 0 (f) 1 4 (g) 2 4 0 (l) 1 3 (m) 2 3 2. Activity 1 3. Students fill in missing fractions on number lines labeled by letters. 2 2 (d) (e) 5 2 (h) 1 (k) 5 4 1 (p) (r) 5 3 (c) 3 2 4 2 4 (j) 4 3 4 (n) 3 3 4 3 Activity 2 Order the fractions from least to greatest. 1, 1, 1, 1, 1 4 3 7 5 2 1 7 Activity 2 1 5 1 4 1 3 1 2 least Students order a list of fractions with different denominators from least to greatest. greatest Activity 3 Compare the fractions by drawing two number lines. Be sure that the distance between 0 and 1 is the same on both number lines and the number lines are divided into fair shares. See number lines below. 3 2 1. Which fraction is less: 4 or 5? Activity 3 2 1 2. Which fraction is less: 3 or 6? Students compare two fractions by drawing two number lines. Remind students to line up the number lines and divide them carefully into fair shares. 2 is less 5 1 is less 6 Activity 4 • Distributed Practice Solve. 1. 3 × 1 4 3 4 2. 8 × 1 5 8 5 3. 4 × 1 9 4 9 4. 2 × 1 3 2 3 5. 6 × 1 2 6 2 6. 9 × 1 6 9 6 Activity 4 • Distributed Practice Students practice basic multiplication of unit fractions by whole numbers to find other fractions. 7272 Unit 2 • Lesson 2 Additional Answers Activity 3 1. 1 4 0 1 5 0 2. 2 5 1 6 2 6 3 4 3 5 1 3 0 0 2 4 4=1 4 4 5 5=1 5 2 3 3 6 4 6 3=1 3 5 6 6=1 6 Unit 2 • Lesson 2 121