Addition and Subtraction with Rational Numbers
This microcredential represents a teacher’s ability to understand and respond to progressions related to addition and subtraction with rational numbers, including conceptualizations, properties, strategies, and representations, by planning and implementing instruction based on the Standards for Mathematical Practice and Effective Mathematics Teaching Practices. It includes selecting, using, and adapting mathematics curricula and teaching materials, including the integration of mathematical tools and technology, as well as using and analyzing formative and summative assessments to determine where students are in learning addition and subtraction with rational numbers. This is the fifth of seven microcredentials in the Elementary Mathematics Endorsement: Rational Numbers and Proportional Reasoning. These microcredentials can be earned in any order.
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Utah Effective Teaching Standards > Standard 2: Instructional Design Clarity
Element 1: Content - Demonstrating a comprehensive understanding of Utah Core Standards, communicating relevance of content, communicating clear pathways to student mastery and designing learning experiences aligned to clear learning intentions and success criteria. -
Utah Effective Teaching Standards > Standard 2: Instructional Design Clarity
Element 3: Instructional Planning - Planning high quality, personalized instructional activities that are informed by student progress data, provide multiple opportunities for students to reflect upon and assess their own growth and allow multiple opportunities and means for demonstration of competency. -
Utah Effective Teaching Standards > Standard 3: Instructional Practice
Element 1: Instructional Strategies - Using appropriate academic language and evidence-based strategies to stimulate higher-level thinking, discourse and problem solving and to scaffold learning experiences to meet the needs of all students. -
Utah Effective Teaching Standards > Standard 3: Instructional Practice
Element 3: Relevance - Providing relevant learning opportunities that value students’ interests and backgrounds and allow learner agency and choice in accessing learning and demonstrating competency.
To earn this microcredential you will need to collect and submit three sets of evidence demonstrating your effective and consistent use of appropriate instructional strategies for teaching addition and subtraction with rational numbers. You will also complete a short written or video reflective analysis.
Students benefit from opportunities to work with addition or subtraction of rational numbers in real world context. As they work through different representations of addition and subtraction of rational numbers, students determine when procedural thinking is appropriate and how numerical representations and procedures have meaning rather than simply being a disassociated set of steps to follow.
Addition and subtraction with rational numbers includes the following:
- Teachers should apply and extend whole number strategies to rational numbers using a comprehensive repertoire of representations (visual, physical, symbolic, contextual, verbal), properties of addition and common ways they can be applied, and a variety of methods to solve addition and subtraction problems,
- Teachers should understand the relationship between addition and subtraction to add and subtract rational numbers,
- Teachers should develop context for adding and subtracting rational numbers, fractions, and decimals through reasoning, visual models, number lines, and mental strategies, as well as understanding that same-sized pieces must be used to combine and separate pieces. This may include finding common denominators and aligning decimals points, and
- Teachers should develop context for developing and analyzing algorithms for adding and subtracting with rational numbers.
A rational number is any number that can be expressed as a fraction or ratio of two integers. For example, 3/4, 8.75, 2, and -6 are all considered rational numbers. Note that integers, or "whole numbers", are rational numbers. This is because they can be expressed as fractions.
Mrs. Spencer is ready for her students to start to think about adding and subtracting rational numbers. She presents them with a problem to find a solution to. After a party, Bob had 2/8 of a pan of brownies left over in one pan and ¼ of a pan of brownies left over in a pan. He put the brownies together in one pan. How much of a pan of brownies were left over from his party? Students recognized from the context of the story that the two fractional amounts of brownies needed to be combined. Many saw this action as addition. Students tried many different ways to combine the brownies, but the one thing that continued to elude them is how they could “name” the total amount of brownies in the pan. Finally, one student noticed that she could cut the larger piece in half (¼ = 2/8) and then all of the pieces would be the same size. Mrs. Spencer was pleased and plans to pursue the idea of making same sized pieces over the next few days.
Category: Preparation and Planning
Submit TWO of the evidence options to demonstrate your effective and consistent preparation and planning for instruction on addition and subtraction of rational numbers.
Design a unit that could be used in 5th or 6th grade to help students deeply understand the concepts of fraction addition and subtraction. This unit plan should include:
- Grade level
- Unit title
- Selected standards for the unit, including one or more practice standards.
- Common misconceptions
- Required student prior knowledge.
- How ideas will be sequenced and grouped.
- Written objectives aligned to standards for each lesson of the unit, including how long each lesson is expected to take.
- Resources (tasks, activities, etc.) for each day of the unit that are aligned to the objective of that particular day.
- Description of end-of-unit assessment and how it will assess standards for the whole unit.
- Any additional summative or formal formative (collected and evaluated) assessments for the unit.
In a separate section of the unit plan, cite the sources you used to develop your explanations. See the Resources section below for examples of sources to cite.
Submit an 8-10 minute video demonstrating your understanding of addition and subtraction of fractions in either 4th or 5th grade across all three domains of understanding (conceptual, representational, procedural). The video should include the following pieces:
- What are the specific things each student should understand about the addition and subtraction of fractions?
- What representations should students become familiar with in order to understand the addition and subtraction of fractions?
- Each representation should be modeled in the video. 4. Each representation should be narrated as you are modeling so that the pictures and actions of each representation are clear.
- How do the concepts and representations shown earlier in the video connect to the procedures for the addition and subtraction of fractions?
Submit an 8-10 minute video demonstrating your understanding of addition and subtraction of decimals in either 5th or 6th grade across all three domains of understanding (conceptual, representational, procedural). The video should include the following pieces:
- What are the specific things each student should understand about addition and subtraction of decimals?
- What representations should students become familiar with in order to understand addition and subtraction of decimals?
- Each representation should be modeled in the video. 4. Each representation should be narrated as you are modeling so that the pictures and actions of each representation are clear.
- How do the concepts and representations shown earlier in the video connect to the procedures for addition and subtraction of decimals?
Category: Implementation
Submit ONE of the evidence options to demonstrate your effective and consistent implementation of appropriate pedagogical practices for instruction on addition and subtraction with rational numbers.
Submit a lesson plan that demonstrates how you have helped students become proficient in addition and subtraction of fractions or addition and subtraction of decimals. The lesson plan should clearly illustrate how the instruction effectively helps students progress from conceptual understanding towards procedural fluency. The lesson plan should include the following:
- Grade level
- Relevant standard(s)
- Intended learning outcome(s)
- Instructional materials (manipulatives, paper, etc.)
- Instructional activities (tasks, etc.)
- Written plan explaining how you will help students progress using the activities you chose.
- Assessment techniques
In a separate section of the lesson plan, include citations for research supporting your instructional approach. (See the resources section for examples to cite.)
Teach a lesson focused on some aspect of fraction or decimal addition and/or subtraction. Using student work as a formative tool, find a misconception that needs to be addressed. In your submission:
- Include student work from the first lesson focused on a specific misconception.
- Design the next day’s lesson based on the collected data.
- After teaching the next lesson, include student work from the same students showing their growth and improvement.
- Include a write up explaining the following: (a) A description of the initial lesson (1 paragraph). (b) A diagnosis of the misconception from selected students’ work (1 paragraph). (c) A description of the 2nd lesson (1 paragraph). (d) A summary of student learning based on student work from the 2nd lesson (1 paragraph). (e) Next steps based on student work from 2nd lesson (1 paragraph).
Review Criteria
Criterion 1: Evidence demonstrates an understanding of a clear learning trajectory that helps students move from conceptual understanding to procedural fluency.
Criterion 2: Evidence demonstrates the candidate's understanding of fraction and decimal concepts.
Criterion 3: Evidence demonstrates the understanding of and use of effective teaching practices designed to help students understand fraction and decimal concepts.
What effect does a deficit in addition and subtraction of rational numbers have on students’ math success in future grade levels? Include at least one example standard and how that standard is affected by concepts.
Reflect on the changes you will make in your practice based on your engagement with this microcredential.
Review Criteria
Criterion 1: The reflection indicates that the educator understands concepts founded in addition and subtraction of rational numbers.
Criterion 2: The reflection indicates that the educator understands the impact/importance that adding and subtracting rational numbers has on future success in mathematics.
https://www.schools.utah.gov/file/f0e86540-5617-4166-a701-fea403f2f848
The Utah effective Teaching Standards articulate what effective teaching and learning look like in the Utah public education system.
NCTM Effective Mathematics Teaching Practices that Support Learning For All Students: A Focus on Elementary School
https://www.nctm.org/uploadedFiles/Conferences_and_Professional_Development/Institues/Pre-K_Grade_12_Common_Core_Series/Huinker-ElementaryKeynote_presentation.pdf
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Elementary and Middle School Mathematics: Teaching Developmentally John Van de Walle
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Utah Core Standards: Mathematics Core Guides
https://www.schools.utah.gov/curr/mathematics/core?mid=4514&tid=2
The Utah State Board of Education adopted the K-12 Utah Core Standards for Mathematics in January 2016. Core guides provide a description of the Core Standards, including concepts and skills to master, critical background knowledge and academic vocabulary. The course overview documents show the major work of the grade level and the coherence of content across grade levels.
Teaching Student-Centered Mathematics Grades K-3
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Progressions for the Common Core State Standards in Mathematics
http://ime.math.arizona.edu/progressions/
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