Comparison and Equivalence with Rational Numbers
This microcredential represents a teacher’s understanding of comparison and equivalence with rational numbers and their ability to understand and respond to progressions related to comparison and equivalence with rational numbers 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 comparison and equivalence with rational numbers. This microcredential is the fourth of seven in the Elementary Mathematics Endorsement: Rational Numbers and Proportional Reasoning Stack. 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 2: Instructional Design Clarity
Element 4: Engagement - Designing lessons and activities that actively engage students in their learning and use a variety of effective tools and strategies.
To earn this microcredential you will need to collect and submit two sets of evidence demonstrating your effective and consistent use of appropriate instructional strategies for teaching comparison and equivalence with rational numbers. You will also complete a short written or video reflective analysis.
Comparison and equivalence of rational numbers starts as early as 3rd grade as students start to compare fractions with similar numerators or similar denominators by reasoning about the size of each piece (comparison) and break fractions into smaller pieces to see that though the size of the pieces have changed, the amount being considered hasn’t change (equivalence). Students expound on this idea of comparison in 4th grade by using benchmarks and start to see how equivalent fractions can be made by using algorithms. These ideas continue through middle school and junior high standards as students compare decimals to other forms of rational numbers.
Comparison and equivalence with rational numbers includes:
- Knowing relative sizes of fractions using benchmark fractions.
- Recognizing that comparisons are valid only when the two fractions refer to the same whole.
- Attending to units (e.g., small pizza vs. large pizza, measurement units).
- Comparing fractions by reasoning about the size of the fractions.
- Using visual models.
- Finding common numerators or denominators.
- Recognizing and generate equivalent fractions.
- Converting flexibly between numerical and visual representations of fractions, decimals, and percents (e.g., represent a fraction and decimal on a number line, convert between fractions and decimals, identify most appropriate representations for a given quantity and context).
- Comparing and order rational numbers.
- Using place-value and scientific notation to order numbers, estimate, and represent order of magnitude.
Refers to the gradual shift from more concrete mathematical thinking and representations (e.g., manipulatives) to less concrete representations and thinking (e.g., pictures) to abstract mathematical representations thinking and (e.g., numbers, symbols, equations, etc.).
Fraction:A fraction is used to represent a portion of a whole when the whole has been split into a number of equal shares or a portion of a set.
Denominator:The convention of expressing a fraction, whether the fraction of a whole or fraction of a set, to denote the total amount of pieces in the whole or total number of items in the set.
Numerator:The convention of expressing a fraction, whether the fraction of a whole or fraction of a set, to denote the number of parts of the whole or parts of the set being indicated.
Benchmark Fractions:Common fractions that can be used to measure or judge another fraction(s) against, when measuring or comparing.
Decimal:A number whose whole number part and the fractional part is separated by a decimal point.
Percent:An amount figured or expressed on the basis of a rate or proportion per hundred. The word “percent” means parts per hundred.
Rational Number:Any number that can be represented with a ratio made of whole numbers.
Equivalent Fractions:Fractions with different numerators and denominators that represent the same value or proportion of the whole.
Scientific Notation:Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It may be referred to as scientific form or standard index form.
Mr. Craig is starting to help his students think about equivalent fractions. He knows that last year in 3rd grade students started thinking about equivalent fractions and the understanding that even though the number of pieces of a fraction changed the fractional amount being considered was still the same. He knows that as he continues to help students think about equivalent fractions he can leverage that thinking by helping students see that if he cuts the total amount of pieces into 2 times as many pieces the number of pieces being considered will also be 2 times as many. Through this process his students will make a strong conceptual connection between their visual models to an algorithm for creating equivalent fractions.
Category: Preparation and Planning
Submit TWO of the evidence options to demonstrate your effective and consistent preparation and planning for instruction on comparison and equivalence with rational numbers.
Submit a 7-10 minute video demonstrating your understanding of equivalence of rational numbers across grade levels. The video should include the following:
- Discuss how students progress from conceptual understanding towards procedural fluency in grades 3 - 6, specifically:
(a) This progression should include specific concepts students should understand at each grade level.
(b) This progression should include examples of representations students should use as they progress towards proficiency/understanding.
(c) As you transition from concept to concept, reference the standard(s) connected to each concept.
Submit a 7-10 minute video demonstrating your understanding of comparison of rational numbers across grade levels. The video should include the following:
- Discuss how students progress from conceptual understanding towards procedural fluency in grades 3 - 6, specifically:
(a) This progression should include specific concepts students should understand at each grade level.
(b) This progression should include examples of representations students should use as they progress towards proficiency/understanding.
(c) As you transition from concept to concept, reference the standard(s) connected to each concept.
Design either a comparison or equivalence unit that could be used to help students deeply understand the magnitude of grade-level appropriate numbers. 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.
Category: Implementation
Submit the evidence option to demonstrate your effective and consistent implementation of appropriate pedagogical practices for instruction on comparison and equivalence with rational numbers.
Observe one teacher in your building teaching a comparison or equivalence of rational numbers concept. This focused observation will take place in three parts: before, during, and after.
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Before the observation, preconference with the cooperating teacher. The discussion should cover: (a) What proficiency in the standard being observed looks like for students. (b) A background on what the teacher has already done to help students gain conceptual understanding. (c) How the teacher is planning to help students form and test conjectures during the lesson. (d) Ask the teacher what misconceptions and barriers he/she anticipates students having. (e) Include these interview notes in your submission.
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During the lesson, do the following: (a) Take notes on what students are doing. (b) Take notes on what conjectures students are making. (c) Take notes on what students are understanding. (d) Take notes on what students don’t seem to be understanding. (e) Take notes on what students are doing to test the conjecture they’ve made. (f) Include these notes in your submission.
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After the lesson, complete a written reflection on the lesson. Specifically include thoughts about the following: (a) Pieces of the lesson that helped students develop understanding. (b) Some ideas on how you might be able to help students who aren’t understanding. (c) How his grade level's standard connects to your standards. (d) Include these notes in your submission.
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 comparison and/or equivalence of rational numbers concepts.
Criterion 3: Evidence demonstrates the understanding of and use of effective teaching practices designed to help students understand comparison and/or equivalence of rational numbers concepts.
Reflect on why students often struggle with comparison of fractions and decimals even though they are proficient at comparing whole numbers. Include ideas on how this can be remedied.
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 comparison and/or equivalence of rational number concepts.
Criterion 2: The reflection indicates that the educator understands the impact/importance that comparison and/or equivalence of rational number concepts 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.
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Progressions for the Common Core State Standards in Mathematics
http://ime.math.arizona.edu/progressions/
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