Developing Mathematical Thinking Institute DMTI Jonathan Brendefur Ph
Developing Mathematical Thinking Institute (DMTI) Jonathan Brendefur, Ph. D. jonathan@dmtinstitute. com © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 1
“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 jonathan@dmtinstitute. com © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 2
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 | WWW. DMTINSTITUTE. COM 3
Fraction Comparisons SUPPLEMENTAL MODULE © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 4
Module Sequence 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 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. © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 5
Note to Teachers © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 6
Part 1 INTRODUCING COMPARISON STRATEGIES FOR FRACTIONS © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 7
Part 1: Introducing Comparison Strategies for Fractions © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 8
Part 1: Introducing Comparison Strategies for Fractions *You have likely had many opportunities to draw models for fractions so we will not practice this strategy. © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 9
Part 1: Introducing Comparison Strategies for Fractions © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 10
Part 1: Introducing Comparison Strategies for Fractions © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 11
Part 1: Introducing Comparison Strategies for Fractions © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 12
Part 1: Introducing Comparison Strategies for Fractions © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 13
Part 2 COMPARING FRACTIONS LESS THAN 1 WITH FAMILIAR DENOMINATORS © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 14
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. © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 15
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 16
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 17
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. < > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 18
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. = > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 19
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. < > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 20
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 21
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 22
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. = > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 23
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 24
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 25
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. = > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 26
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 27
Part 2: Comparing Fractions Less than 1 with Familiar Denominators 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. < > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 28
Part 3 COMPARING FRACTIONS LESS THAN 1 WITH LESS FAMILIAR DENOMINATORS © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 29
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 | WWW. DMTINSTITUTE. COM 30
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 31
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 32
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. < > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 33
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. = > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 34
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. < > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 35
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 36
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 37
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. = > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 38
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 39
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 40
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. = > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 41
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 42
Part 3: Comparing Fractions Less than 1 with Less Familiar Denominators 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. < > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 43
Part 4 COMPARING FRACTIONS GREATER THAN 1 © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 44
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. © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 45
Part 4: Comparing Fractions Greater than 1 If the numbers are improper fractions, you can use the strategies you have already learned about comparing fractions less than 1. any of the strategies Use 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 | WWW. DMTINSTITUTE. COM 46
Part 4: Comparing Fractions Greater than 1 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 | WWW. DMTINSTITUTE. COM 47
Part 4: Comparing Fractions Greater than 1 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 | WWW. DMTINSTITUTE. COM 48
Part 4: Comparing Fractions Greater than 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. < > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 49
Part 4: Comparing Fractions Greater than 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. < > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 50
Part 4: Comparing Fractions Greater than 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 51
Part 4: Comparing Fractions Greater than 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 52
Part 4: Comparing Fractions Greater than 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. = > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 53
Part 4: Comparing Fractions Greater than 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 54
Part 4: Comparing Fractions Greater than 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. = > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 55
Part 4: Comparing Fractions Greater than 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 56
Part 4: Comparing Fractions Greater than 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. < > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 57
Part 4: Comparing Fractions Greater than 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. > > © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 58
Part 5 MIXED PRACTICE TEMPLATES © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 59
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 | WWW. DMTINSTITUTE. COM 60
Fraction Comparison Worksheet 1. 1: Use the >, < or = symbols to describe each pair of numbers. Symbolic Notation > Explanation Name: _____________ Explanation © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 61
Fraction Comparison Worksheet 1. 2: Use the >, < or = symbols to describe each pair of numbers. Symbolic Notation > Explanation Name: _____________ Explanation © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 62
Brendefur and Strother (2019) For more information contact Dr. Brendefur at jonathan@dmtinstitute. com © DMTI (2019) | RESOURCE MATERIALS | WWW. DMTINSTITUTE. COM 63
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