Developing Mathematical Thinking Institute (DMTI)

Developing Mathematical Thinking Institute (DMTI)
Jonathan Brendefur, PhD. © DMTI (2019) | Resource Materials |

© DMTI (2019) | Resource Materials | www.DMTinstitute.com
“The Developing Mathematical Thinking Institute (DMTI) is dedicated to enhancing students’ learning of mathematics by supporting educators in the implementation of research-based instructional strategies through high-quality professional development.” For more information contact Dr. Jonathan Brendefur © DMTI (2019) | Resource Materials |

About this Supplemental Module
This module can be used by teachers at many different grade levels to support students’ mental strategies for comparing fractions. There are several comparison strategies explored in throughout these lessons. The specific grade level these tasks are ideal for varies depending on current content standards and students’ familiarity and proficiency with fraction concepts. Most of these tasks are appropriate as warm-ups or mini-lesson for grades 3-6. Students in Grade 3 should have already completed at least one DMTI unit module on fractions prior to beginning any of the lessons in this supplemental Fraction Comparison module. © DMTI (2019) | Resource Materials |

Fraction Comparisons Supplemental Module © DMTI (2019) | Resource Materials |

Module Sequence Note to Teachers: This supplemental module may be used with a variety of grade levels and in many different instructional settings. It may be unnecessary to proceed from lesson to lesson depending on students’ prior knowledge or the intended purpose of using the module. Part 1: Introducing Comparison Strategies for Fractions Part 2: Comparing Fractions Less than 1 with Familiar Denominators Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators Part 4: Comparing Fractions Greater than 1 Part 5: Mixed Practice Templates © DMTI (2019) | Resource Materials |

Note to Teachers Students in some grade levels can be challenged to use the procedures for generating equivalent fractions to compare fractions. For example, when comparing and students could use the following procedures: 2 3 x > x because > 6 12 However, many students who need more proficiency with comparing fractions may not have learned these procedures and therefore the procedures will not be helpful. This supplemental module does not address the use of equivalent fraction procedures for comparing fractions as the module’s purpose is to build students’ relational thinking abilities with fractions. But, if students are familiar with these procedures teachers are encouraged to explore this as another useful strategy for fraction comparisons. © DMTI (2019) | Resource Materials |

Part 1 Introducing comparison strategies for fractions © DMTI (2019) | Resource Materials |

Part 1: Introducing Comparison Strategies for Fractions
Think about how you could decide which of these fractions is the greatest amount. 2 3 or What would models for these fractions look like? How could models help us determine which number is greater? 2. How can we compare the denominators? 3. How can we compare the numerators? 4. How can we use 1 2 as a benchmark fraction? 5. How can we compare each fraction to 1? 𝟐 𝟑 > 𝟐 𝟒 Students should share their ideas prior to any examples or guiding questions being provided. Students should write down their ideas, the equations and notations provided and draw models in math notebooks/journals. © DMTI (2019) | Resource Materials |

When we compare fractions, there are many strategies to use. Below are 5 useful strategies we just discussed. Try these strategies with other comparisons to see how helpful each strategy can be. 1. Draw models* 2. Compare denominators 3. Compare numerators 4. Compare to Compare to 1 *You have likely had many opportunities to draw models for fractions so we will not practice this strategy. © DMTI (2019) | Resource Materials |

Let’s first compare these fractions by Comparing the Denominators. Because it takes 3 ( 1 3 units) to compose 1 this means that thirds are greater than fourths. It takes 4 ( 1 4 units) to compose 1 so these units are smaller than thirds. Because we have the same number of units counted in the numerators we can tell that 2 3 > 2 4 . Students should discuss their ideas and strategies before being given the correct response comparing 2/3 to 2/6. 2/3 > 2/6 Practice: Compare the denominators to decide whether or is greater. © DMTI (2019) | Resource Materials |

Let’s now compare these fractions by Comparing the Numerators. We have counted 2 ( 1 3 units) and are comparing them to 2 ( 1 4 units) . Because thirds are greater than fourths, and we have counted the same number of each unit, 2 3 > 2 4 . Students should discuss their ideas and strategies before being given the correct response comparing 2/3 to 2/5. 2/3 > 2/5 Practice: Compare the numerators to decide whether or is greater. © DMTI (2019) | Resource Materials |

Now let’s compare these fractions by Using 1 2 as a Benchmark Fraction. It is often easy to see whether a fraction is greater than, less than or equal to Because 2 3 > 1 2 and 2 4 = 1 2 then 2 3 > 2 4 . Students should discuss their ideas and strategies before being given the correct response comparing 2/3 to 4/8. 2/3 > 4/8 Practice: Compare each fraction to to decide whether or is greater. © DMTI (2019) | Resource Materials |

Finally let’s compare these fractions by Comparing Each Fraction to 1. Because these fractions are different units, comparing to 1 may be difficult, especially because each fraction is a different number of unit fractions away from 1. But, if we were comparing 2 3 and 3 4 we could look at how each fraction is one unit away from 1 and use that to decide that 2 3 < 3 4 because the 1 4 needed to get back to 1 with 3 4 is less than the unit fraction left to make 2 3 into 1. Students should discuss their ideas and strategies before being given the correct response comparing 2/3 to 5/6. 2/3 < 5/6 Practice: Compare how close these fractions are to decide whether or is greater. © DMTI (2019) | Resource Materials |

Part 2 Comparing fractions less than 1 with familiar denominators © DMTI (2019) | Resource Materials |

Part 2: Comparing Fractions Less than 1 with Familiar Denominators
For each pair of fractions given, use the Strategy Menu to help you decide different ways you can decide which fraction is greater or which fraction is less. If you have a different strategy you think will work, try your own strategy and see whether it produces the same answer as one from the Strategy Menu. Always use a strategy you are comfortable with unless you are asked to use a specific strategy. Use the symbols =, < and > to write down your comparisons. Students should record their work in math notebooks/journals. © DMTI (2019) | Resource Materials |

Strategy Menu Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > > © DMTI (2019) | Resource Materials |

Strategy Menu Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > < © DMTI (2019) | Resource Materials |

Strategy Menu Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > = © DMTI (2019) | Resource Materials |

Part 3 Comparing fractions less than 1 with less familiar denominators © DMTI (2019) | Resource Materials |

Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators For each pair of fractions given, use the Strategy Menu to help you decide different ways you can decide which fraction is greater or which fraction is less. If you have a different strategy you think will work, try your own strategy and see whether it produces the same answer as one from the Strategy Menu. Always use a strategy you are comfortable with unless you are asked to use a specific strategy. Use the symbols =, < and > to write down your comparisons. © DMTI (2019) | Resource Materials |

Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators Strategy Menu Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > > © DMTI (2019) | Resource Materials |

Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators Strategy Menu Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > < © DMTI (2019) | Resource Materials |

Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators Strategy Menu Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > = © DMTI (2019) | Resource Materials |

Part 4 Comparing fractions greater than 1 © DMTI (2019) | Resource Materials |

Part 4: Comparing Fractions Greater than 1
So far we have been comparing fractions that were either greater than or less than 1. However, we can also compare fractions that are greater than 1. Sometimes these fractions are improper fractions and sometimes they can be mixed numbers. To complete these following tasks, students will need background knowledge on improper fractions and mixed numbers. Refer to DMTI modules for Grades 4 and 5 to see more foundational lessons on these number concepts. © DMTI (2019) | Resource Materials |

If the numbers are improper fractions, you can use the strategies you have already learned about comparing fractions less than 1. Strategy Menu Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use any of the strategies from the menu to decide which fraction is greater. Both fractions are one unit fraction greater than 1. Because thirds are greater than fourths, that means…. > > © DMTI (2019) | Resource Materials |

If the numbers are mixed numbers, you can first compare the whole numbers and then use the strategies you have already learned about comparing fractions. > > © DMTI (2019) | Resource Materials |

Strategy Menu Comparing Whole Numbers Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > < © DMTI (2019) | Resource Materials |

Strategy Menu Comparing Whole Numbers Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > > © DMTI (2019) | Resource Materials |

Strategy Menu Comparing Whole Numbers Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > = © DMTI (2019) | Resource Materials |

Strategy Menu Comparing Whole Numbers Compare Denominators Compare Numerators Use as a Benchmark Compare Each Fraction to 1 Use what you know about comparing fractions to decide which fraction is greater. Some times the first fraction is greater, sometimes it is less. In some cases the fractions will be equal. Strategies will vary. Students can discuss their strategies in pairs, small groups or as a class. Students should record their work in math notebooks/journals. Students should complete their work prior to being presented with the correct answer. Correct equations/inequalities should be written down in math notebooks/journals. > < © DMTI (2019) | Resource Materials |

Part 5: Mixed Practice Templates
Complete the Mixed Practice Worksheet by comparing the fractions using the >, < or = symbols. Then write a brief explanation to describe your thinking. Note to teachers: Worksheet 1.1 has a template designed for fractions less than or equal to 1. Worksheet 1.2 has a template for both the same type of fractions or improper fractions and another section for mixed numbers. Replace the letters with appropriate numbers/digits. © DMTI (2019) | Resource Materials |

Fraction Comparison Worksheet 1.1: Use the >, < or = symbols to describe each pair of numbers. Name:__________________________ Symbolic Notation Explanation I know is greater than because you have counted more units in than in 3 4 ___ 2 4 𝑎 𝑏 ___ 𝑐 𝑑 > 𝑎 𝑏 ___ 𝑐 𝑑 𝑎 𝑏 ___ 𝑐 𝑑 Fraction Comparison Worksheet 1.1 𝑎 𝑏 ___ 𝑐 𝑑 𝑎 𝑏 ___ 𝑐 𝑑 © DMTI (2019) | Resource Materials |

Fraction Comparison Worksheet 1.2: Use the >, < or = symbols to describe each pair of numbers. Name:__________________________ Symbolic Notation Explanation I know is greater than because units of are greater than units of and both numbers have 6 units. I know is less than because the whole number 2 is less than the whole number 3. 6 4 ___ 6 5 ___ > 𝑎 𝑏 ___ 𝑐 𝑑 𝑁 𝑎 𝑏 ___ M 𝑐 𝑑 Fraction Comparison Worksheet 1.1 𝑁 𝑎 𝑏 ___ M 𝑐 𝑑 𝑎 𝑏 ___ 𝑐 𝑑 © DMTI (2019) | Resource Materials |

Brendefur and Strother (2019)
For more information contact Dr. Brendefur at © DMTI (2019) | RESOURCE MATERIALS | © DMTI (2019) | Resource Materials |

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