Comparing and Ordering Fractions

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1 Comparing and Ordering Fractions Objectives To review equivalent fractions; and to provide experience with comparing and ordering fractions. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Find equivalent fractions using a length model. [Number and Numeration Goal ] Compare fractions to the benchmarks 0, _, and. [Number and Numeration Goal ] Order fractions from least to greatest. [Number and Numeration Goal ] Solve fraction number stories using a number-line model. [Operations and Computation Goal ] Use fraction sticks to add fractions. [Operations and Computation Goal ] Key Activities Students use the Fraction-Stick Chart to find equivalent fractions and compare fractions to the benchmarks 0, _,, and _. They use the fraction-stick model to compare pairs of fractions, find equivalent fractions, and continue their exploration of fraction addition. Playing Fraction Top-It Student Reference Book, p. Fraction Cards (Math Journal, Activity Sheets ) Students practice comparing fractions by playing Fraction Top-It. Math Boxes Math Journal, p. Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. Students practice and maintain skills through Study Link activities. READINESS Making Fraction Strips strips of colored paper, each " by _ " Students use a length model to explore comparing and ordering fractions by making fraction strips. ENRICHMENT Exploring a Fraction-Stick Chart Math Masters, p. 0 Students explore relationships between the numerators and denominators of equivalent fractions. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. Students add the term equivalent fractions to their Math Word Banks. Recognizing Student Achievement Use journal page. [Number and Numeration Goal ] Informing Instruction See page 0. Key Vocabulary benchmark fraction stick equivalent fractions Materials Math Journal, pp. Study Link transparency of Math Masters, p. slate Geometry Template straightedge (optional) Advance Preparation For Part, make a transparency of the chart on Math Masters, page. For the optional Readiness activity in Part, each student will need strips of colored paper, each " by _ ". These can be prepared ahead of time or during the activity. Teacher s Reference Manual, Grades pp.,,, 0,, 0 Unit Fractions, Decimals, and Percents

2 Getting Started Mental Math and Reflexes Have students use the rulers on their Geometry Templates. Find _ inches on a ruler. How many half-inches is that? Find cm on a ruler. How many _ cm is that? Find _ inches on a ruler. How many quarter-inches is that? 0 Find _ cm on a ruler. How many _ cm is that? _ inch is what fraction of _ inches? inch is what fraction of _ inches? _ 0 Math Message Use the benchmarks 0, _, and to answer Problems on journal page. Study Link Follow-Up Allow students five minutes to compare answers and correct any errors. Have volunteers share their mixed-number problems. Teaching the Lesson Math Message Follow-Up (Math Journal, p. ) WHOLE-CLASS DISCUSSION Remind students that a benchmark is a well-known count or measure that can serve as a reference point when estimating. When working with fractions, the benchmarks 0, _, and are often used. Discuss how students applied their knowledge of numerators, denominators, and benchmarks to tell if each measurement is closest to 0, _, or. Recognizing Student Achievement Journal Page Problem Use journal page, Problem to assess students understanding of the structure of fractions. Students are making adequate progress if their explanations correctly represent the relationship between the numerator and denominator and the use of benchmarks such as 0, _, and. [Number and Numeration Goal ] Math Message Comparing and Ordering Fractions Decide whether each of these measurements is closer to the benchmark 0, _, or inch. Circle the closest benchmark.. _ inch is closest to... 0 inches. _ inch. inch.. _ inch is closest to... 0 inches. _ inch. inch. Ordering Fractions (Math Journal, p. ). _ inch is closest to... 0 inches. _ inch. inch.. _ inch is closest to... 0 inches. _ inch. inch.. Explain your solution for Problem. Sample answer: I know that _ is only _ away from _, which is _. So _ must be closest to _. On the board or a transparency, draw horizontal lines in a row to order the fractions from Problems of the journal page. Volunteers explain which of the Math Message fractions is least and which is greatest. _ is the least because it is the closest to 0; _ is the greatest because it is closest to. Write these fractions on the first and last lines, respectively. Ask: Since _ and _ are equally close to _, how do we decide where to write them? Both fractions are eighths, and is more than, which means that _ is the smaller fraction so it is closer to 0. Ordering Fractions For each problem below, write the fractions in order from least to greatest.. _, _, _,, _, _, _. _, _, _,, _, _, _. _, _, _,, _, _, _. _, _ 0, _ 0,, _ 0, _ 0, _ 0. _, _ 0, _, 0, _, _, _. _, _, _, _ 0 _, _, _, _ 0. _, _, _,, _, _, _ Math Journal, p. EMMJ_G_U0_-.indd /0/ : AM Lesson 0

3 0.. Which is larger: or?. Which is closest to: 0 or or? Fraction-Stick Chart 0.. Which is larger: or?. Which is closest to: 0 or or?.. Which is larger: or?. Which is closest to: 0 or or?. Which is larger: or? Which is closer to : or? Math Journal, p. 0 A whole stick is worth... Fill in the blanks. Circle the correct answer Adding with Fraction Sticks. Use the fraction sticks to find equivalent fractions. a. halves quarters b. c. d. e.. Use the fraction sticks to add fractions with the same denominator. Example: a. eighths sixteenths 0 Explain that examining numerators and denominators is the first step when comparing and ordering fractions. Refer students to journal page. Examples: Ask students to describe the fractions in Problem. The denominators are the same. So we know that all the unit fractions are the same size. Only the number of pieces (the numerators) need to be compared. Ask students to describe the fractions in Problem. The numerators are the same. Now what do we know? There are the same number of pieces for each fraction. So only the size of the pieces (the denominators) needs to be compared. Remind students that the smaller the denominator is, the larger the piece is. Ask students to describe the fractions in Problem. There are two different denominators; there are two unit fractions. First compare the unit fractions _ and _. The smaller denominator is the larger fraction. The remaining fractions are within one piece of that is, _ is _ away from, and _ is _ away from. Since _ is greater than _, that makes _ farther away from. Ask partners to complete the problems on the journal page. When most students are finished, discuss their strategies for Problems, including the following: Problem : The least fraction is _. That leaves the other three to compare. Change each to an equivalent fraction with a denominator of 0. Problem 0: One of these fractions is close to 0, one is close to, and the other two are close to _. Problem : One way to compare a fraction to _ is to see if the numerator is more or less than _ of the denominator. Two of the fractions are greater than _. Their order is determined because _ is close to _ and _ is close to. Of the two fractions 0 that are less than _, the denominators are close to each other, and one numerator is times the numerator of the other. Problem : One fraction equals _ (the numerator is _ of the denominator). The others are close to _, and _ is less than _. The other two are greater than _ but only by _ of a piece each that is, _ is _ of a seventh greater than _, and _ is _ of a fifth greater than _. Because fifths are greater than sevenths, _ is greater than _. b. c.. Use the fraction sticks to add fractions with different denominators. a. b. c. d., or, or 0,, or Introducing the Fraction-Stick Chart (Math Journal, p. 0; Math Masters, p. ) WHOLE-CLASS Use a transparency of the chart from Math Masters, page to demonstrate how to use the Fraction-Stick Chart. Math Journal, p. 0 Unit Fractions, Decimals, and Percents

4 . Skip-Counting with Fractions Please refer students to journal page 0. Explain that a fraction stick is a model for the whole, or the ONE, that shows unit fractions for the interval between 0 and. Each row of the Fraction-Stick Chart combines fraction sticks to show the interval from 0 to, divided into unit fractions for a particular denominator. Example: The third row shows two sticks, each divided into thirds. There are pieces in this row, each labeled _. The pieces can be used to count by thirds.. Finding Equivalent Fractions The Fraction-Stick Chart on Math Masters, page can be used to find equivalent fractions. Survey the class for examples of pairs of fractions that name the same part of a whole. Emphasize that these are equivalent fractions and list the examples on the board. For example, _ is equivalent to _. Example: Find equivalent fractions for _. Step : The denominator is, so use the thirds stick to locate the fraction _. Count the pieces from left to right. The right edge of the second piece is _. Step : Place one edge of a straightedge at _, that is, along the right edge of the second _ piece. The straightedge should be parallel to the sides of the Fraction-Stick Chart. Now look for other fraction sticks on the chart along the straightedge. On the sixths stick, the straightedge touches the right edge of a piece. Count the sixths-stick pieces from left to right. The straightedge is at the end of the fourth piece, which is _. So _ is equivalent to _. On the ninths stick, the straightedge touches the end of the sixth piece, which is _. So _ = _. On the twelfths stick, the straightedge touches the end of the eighth piece, which is _. So _ = _. The fractions _, _, _, and _ are equivalent. They name the same distance on the Fraction- Stick Chart. On the other sticks, the straightedge cuts through some of the pieces, so _ cannot be written as an equivalent fraction using the denominator on those sticks.. Comparing Fractions The Fraction-Stick Chart on Math Masters, page can be used to compare fractions, for example, compare _ and _. (See margin.) Step : The denominator of the first fraction is, so use the ninths stick to locate _. Count the pieces from left to right. The right edge of the fourth piece is _. Place the straightedge along this edge. 0 NOTE A fraction stick is a narrow rectangle divided into pieces that represents fractions. Sometimes it is helpful to make physical fraction sticks, as in Part of this lesson. eighths ninths The third row of the Fraction-Stick Chart 0 0 Informing Instruction Watch for students who confuse the fraction labels with the actual end of the fraction stick. Remind students that the labels are in the center of a portion, not at its end. Have students draw a light pencil line along the straightedge and use the line instead of the straightedge to identify equivalent portions. 0 Lesson 0

5 Fraction Number Stories Shade the fraction sticks to help you solve these fraction number stories. Write a number model for each story.. Chris made pizza dough with cup of white flour and cup of whole wheat flour. a. How much flour did he use in all? cup b. Number model: Step : Locate _ (the right edge of the third piece on the eighths stick). Since _ is to the left of _, it is less than _. Conversely, _ is to the right of _, so it is greater than _. Ask partners to complete journal page 0. Ask volunteers to explain their solutions.. Sheryl s puppy weighed pounds when it was born. After two weeks, the puppy had gained pounds. a. How much did the puppy weigh after two weeks? pounds b. Number model: Adding with Fraction Sticks (Math Journal, p. ). Shade the fraction sticks to solve the number model. Then write a fraction number story that fits the number model. a. = b. Number story:. Make up your own fraction number story. Draw and shade fraction sticks to solve it. Write a number model for your story. Answers vary. a. Number story: Answers vary. The five fraction sticks at the top of journal page provide a length model for fraction work. Note that the denominators are restricted to,,,, and. This model will be used to formally introduce the addition of fractions. All these problems can be solved visually. The correct amount of shading for any fraction can be decided by using the appropriate fraction stick at the top of the page. b. Number model: c. Solution: Math Journal, p. Solving Fraction Number Stories (Math Journal, p. ) INDEPENDENT PROBLEM SOLVING Students solve the fraction number stories on journal page. Circulate and assist. Encourage students to use the benchmarks 0, _, and when explaining their solutions and assessing the reasonableness of answers. Discuss the answers and write the strategies on the board. Ongoing Learning & Practice Math Boxes. Write a 0-digit numeral that has in the tens place, in the millions place, in the billions place, in the hundred-millions place, in the thousands place, and in all other places.,,, Write the numeral in words.. Where students vacationed,... a. what fraction of the students traveled within their state? b. what fraction traveled to Europe? c. what fraction traveled to Canada or Mexico? Five billion, seven hundred sixty-three million, six hundred sixty-one thousand, six hundred ninety-six. Divide. a. R0 b. R, or, or. Write each fraction as a whole number or a mixed number. a. b. c. d. e. Vacation Travel within state Canada or Mexico Europe another state stayed home. Find the following landmarks for this set of numbers:,,,,,,,, 00,,, 0, a. Minimum: b. Maximum: c. Mode: d. Range: e. Median: 00 Playing Fraction Top-It (Student Reference Book, p. ; Math Journal, Activity Sheets ) Students practice comparing fractions by playing Fraction Top-It. Students use the Fraction Cards that were stored in Lesson -. Math Boxes (Math Journal, p. ) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson -. The skill in Problem previews Unit content. Writing/Reasoning Have students write a response to the following: Explain how you converted the fractions to mixed numbers in Problem. Sample answer: For the whole number part, I found how many groups of the denominator were in the numerator. The fraction part was what was left. Math Journal, p. 0 Unit Fractions, Decimals, and Percents

6 Study Link (Math Masters, p. ) INDEPENDENT Home Connection Students find equivalent fractions, add fractions, and solve fraction number stories using fraction sticks. Name STUDY LINK Fraction-Stick Problems Shade the fraction sticks to help you find equivalent fractions.... Shade the fraction sticks to help you solve the addition problems... Study Link Master Differentiation Options READINESS Making Fraction Strips SMALL-GROUP 0 Min To explore comparing and ordering fractions using a length model, have students make fraction strips. Students cut strips of paper, each " by _ " and then fold and label them to represent halves, thirds, fourths, fifths, sixths, and eighths. (See margin.) To fold into thirds, fold one end in so the doubled parts and the single part are the same size. To fold into sixths, fold a strip into thirds and then in half. For fifths, fold the two outside edges of the strip in toward the middle but not together. Fold them so the doubled parts and the space between them look like three equal parts. Now fold the doubled parts back the other way at the point where the folded ends first came down. Have students label the fractions on the folds. As they finish, ask students to fold two strips so only the unit fraction shows and then make a comparison statement. For example, _ is less than _. ENRICHMENT Exploring a Fraction-Stick Chart (Math Masters, p. 0) 0 Min To apply students understanding of fractions, have them explore the relationships between the numerators and denominators of equivalent fractions. When they finish, have them share one of their discoveries. ELL SUPPORT SMALL-GROUP Building a Math Word Bank (Differentiation Handbook, p. ) 0 Min To provide language support for fractions, have students use the Word Bank Template found on Differentiation Handbook, page. They write the term equivalent fraction, draw pictures related to the term, and write other related words. See the Differentiation Handbook for more information.. Shade the fraction sticks to help you solve the fraction number stories.. Joe was baking a cake. He added cup of white sugar and cup of brown sugar. How much sugar did he use in all? (unit). On the back of this page, write a number story using fractions. Then write a number model to show how you solved it. Practice, or cups. 0. R. R. R Math Masters, p. When folded into thirds, the edge of the folded end divides the rest into halves. When folded into fifths, the two folded ends are equal in size to the space in the middle. Name Teaching Master Fraction-Stick Chart Using the Fraction-Stick Chart, list all the fractions that are equivalent to. a. What pattern do you notice in the numerators for these fractions? Numerators increase by until the last one, which increases by. b. What pattern do you notice in the denominators for these fractions? Denominators increase by until the last one, which increases by. No c. Are the patterns complete? d. What fraction is missing that would make the pattern complete?. Using the Fraction-Stick Chart, list all the fractions that are equivalent to.,, a. What pattern do you notice in these fractions?,,, 0,, Numerators increase by and denominators increase by. b. Use this pattern to find the next fractions that are equivalent to.,, Math Masters, p. 0 Lesson 0

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