Learning Activities & Assessment Strategies

Course Summary

• Module 1: Mathematics at Work: Deep Dive Into the Algorithms for Operations with Fractions

• Module 2: Models that Support A Conceptual Understanding of Operations with Fractions

• Module 3: Teaching Operations with Fractions for Conceptual Understanding 

• Apply It in Real Life: Teaching Fractions Operations to Students and Reflecting on the Results

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Learning Activities & Assessment Strategies

  Module 1:  Mathematics at Work

MODULE 1 OBJECTIVES

• By the end of this module, participants will demonstrate their understanding of how the Identity Property plays a crucial role in operations with fractions. (aligns to CLO1, CLO2)

• By the end of this module, participants will model addition and subtraction of fractions and mixed numbers with pattern blocks and connect their representations to the algorithms.  (aligns to CLO1, CLO3)

• By the end of this module, participants will model multiplication of fractions and mixed numbers with pattern blocks and connect their representations to the algorithm. (aligns to CLO1, CLO2)

• By the end of this module, participants will model division of fractions and mixed numbers with pattern blocks and connect their representations to the algorithms. (aligns to CLO1, CLO2)

Introduction

• Short video (<5 minutes) of why we, as teachers, need a deep understanding of the underlying math in operations with fractions. Video will also set the stage for the purpose and what teachers can expect from this course including articulating course learning outcomes.  (aligns to CLO1, CLO5)

• Professional Reading: Participants engage in a brief professional reading about the importance of developing conceptual understanding of fractions operations both for themselves and for their students. Specific reading to be determined.  (aligns to CLO1, CLO4, CLO5)

• Formative Assessment: Reflection Question for Discussion Board (aligns to CLO4, CLO5)

  • • Why do you think that as teachers, we benefit our students when we own the task of having a deep understanding of fractions concepts?

    • Points awarded for participation

• Self Assessment: (aligns to CLO1)

  • • Rate your comfort level and understanding of the underlying math (ie. why the algorithm works) for:

      • adding fractions and mixed numbers with unlike denominators

      • subtraction fractions and mixed numbers with unlike denominators

      • multiplying fractions and mixed numbers

      • dividing fractions and mixed numbers

      • writing context problems (word problems) for students that clearly involve multiplication or clearly involve division

      • Points awarded for participation

Part 1: Properties at Work in Fraction Addition and Subtraction

• Interaction: Intro to Pattern Blocks. Participants will use pattern blocks to identify fractional values and flexibly change the values of the pattern blocks (eg. the hexagon represents 1 whole in the first interaction, but represents 2 wholes in the next interaction). Participants will have to go through a set number of interactions but can "play" with those as long as they would like. (aligns to CLO1, CLO3)

• Short video (<5 minutes) of how the Identity Property plays a crucial role in operations with fractions. (aligns to CLO1, CLO2)

• Interaction: Participants model addition and subtraction of fractions and mixed numbers with pattern blocks and connect their representations to the algorithms.  (aligns to CLO1, CLO3)

• Short video (<5 minutes) of how the Associative, Commutative and Distributive Properties are at work in addition and subtraction of fractions and mixed numbers. (aligns to CLO1, CLO2)

• Interaction: Participants again model addition and subtraction of fractions and mixed numbers with pattern blocks and connect their representations to the properties of operations.  (aligns to CLO1, CLO3)

• Formative Assessment: Reflection Question for Discussion Board (aligns to CLO1, CLO2)

  • • How did knowing the properties of operations serve as a tool to help you make sense of the fraction addition and subtraction algorithms?

    • What are the similarities between adding and subtracting fractions? What are the differences?

    • Points awarded for participation

• Short video (<2 minutes) Reiterates how the Associative, Commutative and Distributive Properties are at work in addition and subtraction of fractions and mixed numbers. Also addresses the similarities/differences between addition and subtraction. (aligns to CLO1, CLO2)

• Summative Assessment: Knowledge Check- Quiz on Addition and Subtraction of Fractions (aligns to CLO1, CLO2). Points awarded for correct answers.

Part 2: Properties at Work in Fraction Multiplication [THIS WILL BE RESERVED FOR A FUTURE PROJECT. FOR NOW, ONLY ADDITION AND SUBTRACTION WILL BE BUILT]

• Short video (<2 minutes) overview of how the properties of operations play a crucial role as tools in multiplying and dividing with fractions and mixed numbers. (aligns to CLO1, CLO2)

• Short video (<5 minutes) Using pattern blocks to understand multiplication with fractions and mixed numbers. (aligns to CLO1, CLO2)

• Interaction: Participants model multiplication of fractions and mixed numbers with pattern blocks and connect their representations to the algorithms.  (aligns to CLO1, CLO3)

• Formative Assessment: Participants use models to identify the division equations and explain how the standard algorithm applies to the model.  (aligns to CLO3)

• Short video (<5 minutes) of how the Associative, Commutative and Distributive Properties are at work in multiplication of fractions and mixed numbers. (aligns to CLO1, CLO2)

• Interaction: Participants again model multiplication of fractions and mixed numbers with pattern blocks and connect their representations to the properties of operations.  (aligns to CLO1, CLO3)

• Summative Assessment: Reflection Question for Discussion Board (aligns to CLO1, CLO2)

  • • How did knowing the properties of operations serve as a tool to conceptually teach the fraction multiplication algorithm instead of only teaching the procedure?

    •  Points awarded for participation

Part 3: Properties at Work in Fraction Division [THIS WILL BE RESERVED FOR A FUTURE PROJECT. FOR NOW, ONLY ADDITION AND SUBTRACTION WILL BE BUILT]

• Short video (<5 minutes) Using pattern blocks to understand division with fractions and mixed numbers. (aligns to CLO1, CLO2)

• Interaction: Participants model division of fractions and mixed numbers with pattern blocks and connect their representations to the algorithms.  (aligns to CLO1, CLO3)

• Formative Assessment: Participants use models to identify the division equations and explain how the standard algorithm applies to the model. 

• Short video (<5 minutes) of how the Associative, Commutative and Distributive Properties are at work in division of fractions and mixed numbers. (aligns to CLO1, CLO2)

• Interaction: Participants again model division of fractions and mixed numbers with pattern blocks and connect their representations to the properties of operations.  (aligns to CLO1, CLO3)

• Summative Assessment 1: Reflection Question for Discussion Board (aligns to CLO1, CLO2)

  • How did knowing the properties of operations serve as a tool to conceptually teach the fraction division algorithm instead of only teaching the procedure? 

  • Reply, then read and reply to other people's posts

  • Points awarded for participation

• Summative Assessment 2: Knowledge Check Quiz (aligns to CLO1, CLO2) Points awarded for correct answers)

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  Module 2:  Models that Support Conceptual Understanding of Operations with Fractions

  MODULE 2 OBJECTIVES

• By the end of this module, participants will explain the strengths and limitations of the pattern blocks as a model. (aligns to CLO3)

• By the end of this module, participants will use and describe a variety of models that are more universal than pattern blocks. (aligns to CLO3, CLO4)

Introduction

• Short video (<7 minutes) Why patterns blocks are a great introduction for students (benefits) but also have limitations. This video introduces a need for other models and strategies. It also leads to an understanding as to why algorithms are the most efficient while stressing the importance of having students derive the algorithms for themselves through the models, just as teachers did in the previous module. Also provides the "why" this it is worth investing instructional time in the classroom in models and deriving algorithms versus jumping right into teaching algorithms. (aligns to CLO3, CLO4)

• Professional Reading: Participants engage in a brief professional reading about the use of models to support conceptual understanding. Specific reading to be determined.  (aligns to CLO3)

• Formative Assessment 1: Reflection Question for Discussion Board (aligns to CLO3, CLO4)

  • Why should we spend time helping students model operations with fractions? How does it benefit students to spend time helping them learn to model fractions operations? Why do we not want to teach the model for the sake of teaching a model, but rather teach students how to model the mathematics?

  • Reply, then read and reply to other people's posts

  • Points awarded for participation

Part 1: Modeling Addition and Subtraction

• Short video (<3 minutes) Beyond pattern blocks: other models to support addition and subtraction with fractions and mixed numbers and why they are needed after pattern block fractions. (aligns to CLO3, CLO4)

• Interaction: Participants practice addition and subtraction with newly introduced models and connect their representations to the algorithms.  (aligns to CLO1, CLO3)

• Formative Assessment: Performance Task (aligns to CLO3)

  • Rubric provided at the start.

Revisit your work from the last interaction with addition and subtraction using area models, tape diagrams, and number lines. 

Choose one of your work samples and take a photo of it. Upload the photo along with your responses to the following reflection questions:

1. What do you notice is similar and different about these models as compared to the pattern blocks?

2. What benefits do these models have that the pattern blocks do not?

3. Describe where you see the properties of operations in your work with these 3 models. Cite a specific example.

Part 2: Modeling Multiplication and Division [THIS WILL BE RESERVED FOR A FUTURE PROJECT. FOR NOW, ONLY ADDITION AND SUBTRACTION WILL BE BUILT]

• Short video (<3 minutes) Other models to support multiplication with fractions and mixed numbers and why they are needed after pattern block fractions. (aligns to CLO3, CLO4)

• Interaction: Participants practice multiplication with newly introduced models and connect their representations to the algorithms.  (aligns to CLO1, CLO3)

• Short video (<3 minutes) Other models to support division with fractions and mixed numbers and why they are needed after pattern block fractions. (aligns to CLO3, CLO4)

• Interaction: Participants practice division with newly introduced models and connect their representations to the algorithms.  (aligns to CLO1, CLO3)

• Summative Assessment 1: Reflection Question for Discussion Board (aligns to CLO3)

  • What are the the strengths and limitations of the pattern blocks as a model?  Why are other models needed? What is the benefit of starting with pattern blocks and then moving to other models?

  • Reply, then read and reply to other people's posts

  • Points awarded for participation

• Summative Assessment 2: Performance Task (aligns to CLO3, CLO4)

  • Rubric provided at the start.

  • Participants engage in an interaction or a simulation to solve a variety of fractions operations problems using any 2 models of their choice for each problem.

  • Participants explain how each strategy they chose supports their conceptual understanding of fractions operations and explain how they might use these models with students. 

  • Course instructor reviews against the rubric and then awards based on how the participant met the criteria on the rubric. 

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  Module 3:  Teaching Operations with Fractions for Conceptual Understanding

  MODULE 3 OBJECTIVES:

• By the end of this module, participants will explain why students need to understand the underlying math behind fractions operations instead of memorizing the steps of the algorithms. (aligns to CLO4, CLO5)

• By the end of this module, participants will describe how strong fractions operations knowledge will support students in middle school and high school math. (aligns to CLO5)

• By the end of this module, participants will describe how strong fractions operations knowledge will support students in daily life and in their careers. (aligns to CLO4, CLO5)

• Short video - TEDx about the language of math. Participants should think about how thinking of math as a language will help students in their long term understanding. (aligns to CLO4, CLO5)

• Interaction: Participants practice middle school and high school types of problems with the familiar models and properties in module 2 and connect their representations to the algorithms.  (aligns to CLO1, CLO2, CLO3, CLO5). 

• Formative Assessment: Reflection Question for Discussion Board (aligns to CLO3)

  • Why do students need a deep understanding of fractions operations in elementary school? What benefits does it provide to their middle and high school math careers? What benefits does it provide for students in their daily lives beyond school?

  • Reply, then read and reply to other people's posts

  • Points awarded for participation

• Interaction: Participants practice middle school and high school types of problems with the familiar models and properties in module 2 and connect their representations to the algorithms.  (aligns to CLO4, CLO5). 

• Summative Assessment: Performance Task (aligns to CLO4, CLO5)

  • 1. Rubric provided at the start.

    2. Participants engage in a simulation in which a student is struggling with a certain fractions operation. Work sample is provided and student verbal explanation is provided. Participant decides what they might do in that situation.

    3. Participants explain how their choice of what to do with the student supports their conceptual understanding of fractions operations and explains what models they chose to use and why 

    4. Course instructor reviews against the rubric and then awards based on how the participant met the criteria on the rubric. 

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   Apply It In Real Life:  Instructional Coaching Support in the Mathematics Classroom

  COACHING OBJECTIVES:

• By the end of one coaching cycle, participants will use a variety of models with students and describe their observed benefits and drawbacks. (aligns to CLO3, CLO4)

• By the end of two coaching cycles, participants will analyze student work in PLCs (Professional Learning Communities) and discuss as a team how student understanding was increased as a result of teaching students the underlying mathematics of fractions operation. (aligns to CLO4)

• By the end of the school year, participants will have filmed one classroom lesson with fractions operations and discuss with the instructional coach where there is evidence of student strengths and limitations in understanding. Participants will develop a plan for "next steps" in instruction with the guidance of the instructional coach. (aligns to CLO4 and CLO5)

• Activities for this section will be developed at a later time since it is the synchronous part of this mini-course.  Activities for this part will be developed as a guide for the instructional coach. This section is intended to align with CLOs 2, 3, 4, and 5 as educators apply their learning from the 3 modules into classroom instruction with 3rd, 4th and/or 5th grade students. 

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Justification of Chosen Learning Activities

• Lecture and presentation of underlying fundamental mathematics behind fraction operations in the form of short videos (2-3 minutes).  Lectures may also include interactive elements such as flip books that let the learner "flip" between content as much as they would like. This is especially beneficial when content is challenging because flip books make small changes from page to page to comprehensively show what's happening in a concept. (Addresses CLO Objective 1)

• Hands-on interactions, simulations, and performance tasks will follow videos to  provide participants with a chance to apply the mathematics in a safe space that allows them multiple attempts to be successful. Interactions might demonstrate the math while also allowing educators to experiment with the ways in which they might teach it or address students' misconceptions. (Addresses CLOs 2 , 3 and 4)

• Socratic questioning will be used to help educators think deeply and articulate about how the math is applied in later grades. Later, their thoughts from the socratic questioning will be applied to discussion or debate boards will also help educators share and gather ideas. (Addresses CLOs 4 and 5)

• Game based activities such as points or badges may motivate educators to keep trying. When these points are attached to quizzes, it helps with both the accountability and the motivation to "get it right" as well as provides me with data about how well learners are understanding the material. Providing a game, such as a game that matches an equation with fractions to a model that represents that equation might also be motivating for educators and help them analyze how models support the math (addresses CLO 3).  Providing a certificate at the end of the course that documents the number of hours spent in the course helps educators when they need to submit documentation of their professional learning hours for state licensure requirements.