Comparing Fractions With Different Denominators

Part 1: Lesson Description

Lesson Title

Comparing fractions with different denominators 

Abstract

This lesson focuses on ordering fractions in ways that encourage deeper understanding of ‘number sense’and extend learners understanding of equivalent fractions by supporting learners to consider different techniques for ordering and comparing fractions. The three techniques, aligned with the CCRS 4.NF.2, covered in this lesson are those used to compare fractions with like numerators or denominators,fractions with unlike numerators and denominators and creating fractions with a common numerator or denominator by finding least common multiples as well as comparing using a benchmark (example one half - ½).

  The lesson used real world context, including sales discounts and understanding school score reports, about fractions for instructor to demonstrate, learners to practice and in the evaluation exercise and also includes  demonstration and practice activities to compare fractions using a number line. This lesson focused on the two latter techniques while  the technique for comparing fractions with like numerators or denominators serves as an introduction to the lesson.

Learner Audience / Primary Users

This  lesson is designed for instructors of adult learners at Grade 4 Level C of Career and College Readiness Standards who are seeking to bridge knowledge and skills gap to be life or college ready and to pass the High School Equivalency Test.

Educational Use

• Curriculum / Instruction

Language

English

Material Type

• Instructional Material
• Assignment

Keywords

• Designers for Learning
• Adult Education
• Number Fractions
• Ordering fractions
  
• Equivalent fractions
  
• Real world problems
• Sales discounts
• Cooking measurements
• 4.NF.2
  

Time Required for Lesson

60 minutes

Targeted Skills

Key skills covered in this lesson include:

•  College readiness skills – CCRS standard expectations that learners are able to apply number sense in ordering fractions;
• Life skills- using fractions to make sense of sales discounts, cooking measurements and simple survey results data which included making sense of school score reports.
• Employability skills – As indicated on MyNextMove[1] – Arithmetic skills for a career as a cashier

Learning Objectives

By the end of this lesson, the learners should be able to:

• Compare two fractions at a time using the symbol >, =, or <
• Recognize that fractions can only be compared when they have common numerators or denominators
• Determine when to order a fraction set as it is or need to create fractions with common numerators and denominators using the lowest common multiple before ordering fractions with unlike numerators or denominators or by using a benchmark of one half to compare with great independence
• Use number lines as visual fraction model to order fractions 

College & Career Readiness Standards (CCRS) Alignment

• Level : Adult Education
• Grade Level:  CCRS Level C
• Subject : CCRS Mathematics
• Domain : Number and Operations: Fractions
• Standards Description: Extend understanding of fraction equivalence and ordering.
• “Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <

Prior Knowledge

It is expected that the intended learners for this lesson have prior knowledge of fractions representation, be able to describe and recall the name for part of a whole for numerator and the whole as denominator. Given a fraction, learners should be able to change it to other equivalent fractions. They should also be familiar with number line.

Required Resources

Prior to the lesson, the instructor should print or copy provided fractions on cardstock to use for warm up activity and prepare different number lines ( third, fifths, sixths, eighths and twelfths and fifteenths for ordering on a number line activities) in order to maximize lesson time for instruction. It is highly recommended to bring sample sale discount offer available in magazines or weekly grocery ads to the lesson so they serve as concrete examples. 

Access to computer, internet access and a printer to print out materials.

Lesson Author & License

• Lesson Author: Rasheedat Yahaya

• License: Creative Commons CC BY 4.0 licence

Part 2: Lesson

Instructional Strategies and Activities

Warm-Up

Time: 5 minutes

 Forced choice: Instructor should place 4 pairs of fractions cards around the class with right and wrong descriptors and labels for numerator and denominators; right and wrong equivalent fractions. Then ask student to examine them, one at a time starting with the labels and ending with the equivalent fractions, and then stand by their choice.  Learners standing by the same choice explain their reasoning to each other. Instructor should walk around to hear discussions and provide feedback if any. This activity also serves to energize kinesthetic learners. (Note - Forced choice for labels and descriptors shoud take a minute each while the remaining minutes is used to make their choice about equivalent fractions which are placed in the front of class.)

Introduction

Time: 10 minutes

Instructor should connect learners’ prior knowledge about sales discounts by asking them to consider a fractional discounts on grocery store weekly ads. Given the same price for two different products but one says ‘Get ¼ off’ and the second ‘Get ⅔ off’. Ask learners, which fraction discount will leave more money back into their pockets? Record their answer to review after demonstration.

Solution (For Instructor to use when providing feedback or for learners who need a refresher or find this concept new.)

Comparing fractions with common numerators or denominators.

Example 1

 Using the example fractions ‘Get ¼ off’, ‘Get ½ off’, ‘Get ⅕ off’, instructor should demonstrate how to compare this fraction by using 3 steps procedure.

Solution:

 First: Ask ‘are the numerators the same?’

 Next: If yes, we order as they are.

 Last: Compare the fractions ONLY two at a time. Since the fractions have a common numerator of 1, we will compare their denominators. When two fractions have the same numerator, the larger the denominator the smaller is the fraction.

So ¼ is less than ½ .  Think of it as when you get 1 part of the price splitted into four is less than when you get a half of the price off.

Mention that less than is represented with a symbol <, therefore we will rewrite the relationship as  ¼ <1/2

Instructor should demonstrate this using a number line to provide a visual reinforcement.

Example 2

A pair of shoes sells at $60 at two different stores. B&T store offers a ‘⅓’ discounts off and Tom store offers a ‘⅔’ off. Which store offers a greater discount?

Solution

  Again, remind learners that to start they must first determine whether the fractions have different numerator and denominators or not. By asking:

First: Are the numerators and denominators the same?

Next: Yes, they have a common  denominator 3,  so order as they are.

Last: Compare two fractions at a time. Since we have only two fractions, it is easier to compare.

When fractions have common denominator, the larger the numerator of a fraction the bigger the fraction.

Since the numerator 2 is larger than 1, the fraction ⅔ is bigger. So ⅔ > ⅓.

Agree or Disagree Matrix: Then instructor should copy or display sets of fraction discounts with like and unlike denominators or numerators on the board. Then poll learner to agree or disagree with the statements “I can compare the fractions as they are”  or “I cannot compare the fractions as they are” for each set. Record the poll on a matrix and keep  to compare to another poll taken at the end of the lesson.

At this time, the instructor should state or display the learning objectives on a projector slide or  flip chart.

Presentation / Modeling / Demonstration

Time: 25 minutes

Description Activity: Instructor should begin the lesson to explain to learners that given a set of fractions, the first thing they need to do is  to determine whether they can compare them as they are or not. To do this, tell them  that fractions with common numerators or denominators such as ‘Get ¼ off’, ‘Get ½ off’, ‘Get ⅕ off’  can be compared as they are because they have a common numerator which in these fractions is 1.  But fraction discounts like, ‘Get ¼ off’ and  ‘Get ⅔ off’ cannot be compared as they are because both did not have common numerators nor denominators. In this case, to compare fractions they must first change  the  fractions to new fractions so that the fractions have the same or like denominator or numerator.

Demonstration

Comparing fractions with unlike numerators or denominators.

Example 3

10+2 Activity: Going back to the previous problem of ‘Get 1/5 off’ and  ‘Get ⅔ off’. Explain that we notice these fractions have different numerators and denominators. Then explain that we say they have unlike numerators and denominators. To compare these discounts we will use the 3 steps procedure.

Solution

First: Are the denominators or numerators the same?

Next: No, continue to explain that since the fractions have different numerators and denominators it is difficult to compare them as they are. We need to create new fractions so that we can change these fractions to fractions that have common or like denominator or numerator. Mention that you like to create new fractions with a common denominator.

To do this, we will find the least common multiple (LCM) of 5 and 3 ( the denominators).

Write out the LCM (good idea is to write one multiple at a time for both numbers so learners can see why they have to stop when they find a common mutiple)

multiples of  5 = 5, 10, 15

Multiples of 3 = 3, 6, 9, 12, 15

We found it, there 15 is the least common multiple so it will become the least common denominator for the new fractions.

[Or just multiply the denominators 5 X 3 = 15 if the learners find this a better method. Review LCM if some learners need a refresher.]

Now we write each fraction in terms of fifteenths.  This means we create an equivalent fraction to 1/5 in terms of fifteenths and an equivalent fraction to 2/3 in terms of fifteenths.

Write the euivalent now as 1/5 = {}/15, so we need to find the new numerator.

Remember earlier when we talk about how to create equal fractions? Point to the correct equivalent fraction placed in front of the class. We multiplied the numerator and denominator by the same number to get an equivalent fraction.

To change 1/5 to a fraction that have denominator 15. We multiply 5 by 3 to get 15 therefore we will multiply the numerator 1 by 3 too. So 1 x 3/ 5 X3 = 3/15.  So 1/5 is equivalent to 3/15. (Review that equivalent means different fractions with the same value.)

To change ⅔ to a fraction that have a denominator of 15, we will multiply both numerator and denominator by 5.

So, 2 x 5/ 3 x 5 = 10/15. Therefore  ⅔ is equivalent to 10/15.

Last: Compare the fractions now with the common denominator. Copy the new fractions 3/15 and 10/15.

Explain that since both fractions have the same denominator of 15,  3/15 is less than 10/15 because 10 is greater than 3. Think of it like getting 10 parts off a price is bigger than getting 3 parts off.

We can write with the symbol as 3/15 < 10/15

  So, 1/5 off is less than ⅔ off. Write the comparion with the less than symbol as 1/5 < 2/3.

Instructor should also demonstrate the technique for comparing fractions using a 1/2 benchmark.

Display a half, fifths, and thirds number lines for comparison. Emphasise to learners, that the number line must be divided into the equal parts of a fraction shown by the denominator.

Shade in the number line up to 1/5 on the fifths number line

Then do the same for the third number line up to 2/3.

Now compare the shaded parts on these number lines to the shaded parts on the 1/2 benchmark.

We see that 1/5 < 1/2 but 2/3 > 1/2

So 1/5 < 2/3

 Example 4: Copy or display this question.

Question: Crystalle's daughter, Maxime,  brought home her Grade 6 End-of-course score report. In the English language Arts (ELA) result section she scored 3 out of 4 points for language usage and organization and 2 out of 3 in narrative writing. Help Crystalle find which section Maxime score higher.

Solution 1 - Creating fractions to have a common denominator

Begin by writing the scores in fraction form as language usage and organization = 3/4 and narrative writing = 2/3. Using the 3 steps procedure,  point out

First: We ask, are the numerators and denominators the same?

Next: No,  therefore, we need to create new fractions that  a common denominator by finding the least common multiples of the denominators.

[Mention that the LCM of the denominators is 3 x  4 =12. Encourage them to use this method in particular when the denominators are small numbers that are easy to multiply mentally.]

Or write the multiples

3 = 3, 6, 9, 12

4 = 4, 8, 12

Now we need to create new fractions for each in terms of twelfths.

Again, we multiplied the numerator and denominator by the same number to get an equivalent fraction.

Since, 3/4= {}/12, we have changed the denominator to 12 which is 4 times 3. Therefore, weneed to multiply thenumeratorby the same number 3.

So 3 x 3/ 4 x 3 = 9/12  

and

2/3 = {}/12. So  2 x 4/ 3 x 4 =8/12.

Conclude that 9/12 is greater than the 8/12 since both have same denominator so the bigger the numerator the larger the fraction.

Solution 2 - Creating fractions with common numerators

Explain to learners we can also find out which fraction is greater, by changing the fractions so they have a common numerator.  Using the 3 steps procedure, explain that

First: We ask, are the numerators and denominators the same?

Next: No,  therefore, we need to create new fractions that have a common numerator using LCM.

Mention that the LCM of the numerators is 3 x  2 = 6

Therefore, 3/4 becomes 3 x 2/4 x 2 = 6/8

and

2/3 becomes 2 x 3/ 3 x 3 = 6/9.

Conclude that now that the fractions have the same numerator it is easy to compare them. 6/8 is greater than the 6/9 becasue 6/8 has a smaller parts we are dividing the whole to compare to 6/9 where we have to divide the whole even more into nine.

Learners should think of it like having to share a bottle of juice among 8 people means each person will have more juice than to share it among 9 people.

Written using symbol, 6/8 > 6/9.  Therefore 3/4 > 2/3.

Show students how to write using symbol less than if the question had been to find the smallest score.

Using the less than symbol, 6/9 is less then 6/8 is written as 6/9 < 6/8 so, 2/3 <3/4

Number line: Instructor should use a twelfth and fifteenth number lines to provide a visual representation of the solutions and compare the fractions to 1/2 (6/12 or 7 1/2 ) benchmark.

Comparing to a benchmark

Using the number lines as visual, the half way on a number line should be marked and the fractions from each  demonstrations above can be placed around to compare each fraction to a 1/2  benchmark.

Guided Practice

Time: 5 minutes

1.  Give learners one similar problem to examples 1 and 2 to compare fractions with like numerators or denominators using the 3 steps procedure. ( available in required resources for print out)
2. Give learners two similar problems to examples 3 and 4 to compare fractions with unlike numerators or denominators using the 3 steps procedure.  (available in required resources for print out)

Note: Instructor should circulate around the class so as to provide oral feedback and support to students who may need it.

Evaluation

Time: 10 minutes

Instructor should use the provided real-world problems involving fractional sales discounts, health and cosmetics survey data, and recipe measurements for learners to compare in order for them to apply the procedure learnt and skills gained to compare fractions using the following activities:

Air drawing: Ask learners to draw the procedures in air before solving by starting with determining which technique to use.

Think Ink Pair share: After air drawing activity, ask learners to write their solutions and use the  symbols >, = or <. When they are done they should share their solutions with a partner. Remind learners to consider wheteher their partner compares fractions only when they have or have been changed to fractions with the same whole (denominator).

Acting out a problem: Instructor can either put students in groups of 4 or use  whole class activity depending on the class size. Give each learner a fraction card and  have learners hold their fraction card close to chest and then stand in order. Use questioning to guide them to form a number line in increasing or decreasing order.

( 5 questions,  all available in required resources for print out)

Application

Time: 5 minutes

Discussion: Instructor should  ask two volunteers, one  to summarize the lesson objective and the other to explain how to determine whether a set of fraction should be ordered as they are or when they need to create fractions with common denominators or numearators before ordering.

[Answer hint: learners should be able to explain that before comparing two fractions they must have common or like numerator or denominator. If not, they need to change the fractions to equivalent fractions that have a common or like denominator or numerator.]

Agree or disagree matrix: Instructor returned to the previous poll and poll learner again to see if they have mastered how to determine when to order a fraction as it or they will need to create fractions with common numerator or denominator.

Assignment: Instructor should ask learners to think of other real life situations that they would apply the new skills and knowledge. Conclude the class by linking this lesson future lessons where they calculate actual sales discounts, know what to do when a recipe with fractionl quantities calls for less or more than their intended servings as well as be able to connect this lesson to decimal and percentages. Then handout Marge’s Dinner Party worksheet by Alyysa Petrino. (Instructor should cancel "or use any manipulative in the classroom" and replace with "or use a number line" on the worksheet. Check and provide feedback in next lesson.)

[I need to insert document for the print material in required resources!]

Key Terms and Concepts

Comparing fractions: To find out which fraction is larger or smaller which is done only when the fractions refer to the same whole. E.g., ⅗, ⅕, or ⅔, and 2/10

Greater than: This is the larger of two fractions and its represented witht the symbol >. Example, 1/2 > 1/4

Less than: This is used to show which of two fractions is smaller. and its represented with the symbol <. Example, 1/4 < 1/2

Equivalent fraction: They are fractions that look different but have exactly the same amount or of the same value e.g., ½, 3/6, 5/10, 50/100

Lowest common denominator - This is the smallest denominator that can be used when creating a common denominator from fractions with unlike denominators.

Increasing order - Arrange from smallest to largest

Decreasing order - Arrange from largest to smallest

Part 3: Supplementary Resources & References

Supplementary Resources

Comparing fractions - https://www.khanacademy.org/math/arithmetic/fraction-arithmetic/arith-review-comparing-fractions/v/comparing-fractions-with-greater-than-and-less-than-symbols

www. reference.com - List of benchmark fractions

BBC SkillsWise

References

See citation at the end of the resource.

Attribution Statements

Content created by Mountain Heights Academy  originally published at http://openhighschoolcourses.org/mod/book/view.php?id=64&chapterid=150 under a Creative Commons Attribution 4.0 International Licence.

Content created by Alyssa Petrino of Millstone River School for TES originally published at https://alyssapetrino.wikispaces.com/Comparing+and+Ordering+Fractions+Fraction+Packet+and+Fraction+Strips under a Creative Commons Attribution Share-Alike 3.0 Portions not contributed by visitors are Copyright 2016 Tangient LLC