Algebraic Notation, Equality, Inequality, and Relational Thinking
This microcredential represents a teacher’s ability to understand and respond to progressions related to Algebraic Notation, Equality, Inequality, and Relational Thinking by planning and implementing instruction based on the Standards for Mathematical Practice and Effective Mathematics Teaching Practices. This involves selecting, using, and adapting mathematics curricula and teaching materials, including the integration of mathematical tools and technology, as well as being able to use an analyze formative and summative assessments to determine where students are in learning of Algebraic Notation, Equality, Inequality, and Relational Thinking. This is the first of three microcredentials in the Elementary Mathematics Endorsement: Algebraic 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 3: Instructional Practice
Element 2: Assessment Practices - Critically analyzing evidence from both formative and summative assessments to inform and adjust instruction and provide feedback to students to support learning and growth.
To earn this microcredential you will need to collect and submit three sets of evidence demonstrating your effective and consistent use of Algebraic Notation, Equality, Inequality, and Relational Thinking. You will also complete a short written or video reflective analysis.
One important, but often overlooked, part of writing expressions and equations with algebraic notation is the role of the “action” of a problem or situation. Students should be able to see a clear action happening and be able to represent that action, in sequence, using numbers, operations, and other notation.
The proficiencies contained in this microcredential include several competencies. Teachers should understand the meaning of the equal sign as relational rather than computational and recognize the difference between expressions, equations, and inequalities. Teachers should also understand equivalent expressions as equations and be able to represent unknown values with symbols and variables (e.g., informal pictures, blank boxes, letters), model mathematical and real-world problems using algebraic equations and inequalities, and use algebraic notation appropriately including the equal sign, inequality symbols, symbols for operations, letters as variables, grouping symbols (e.g., parentheses, brackets, braces). In addition, teachers should evaluate expressions and solve equations and inequalities, understand the need for and use of order-of-operations conventions, and apply and extend previous understandings of arithmetic to algebraic expressions involving variables and whole-number exponents. Finally, teachers should be able to write and evaluate numerical expressions involving whole-number exponents, write, read, and evaluate expressions in which letters represent numbers, apply the properties of operations to generate equivalent expressions, identify when two expressions are equivalent, and use the process of substitution of particular numbers into variable expressions and find the solution set of an algebraic equation or inequality.
Expressions are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between.
Equation:In algebra, an equation can be defined as a mathematical statement consisting of an equal symbol between two algebraic expressions that have the same value.
Inequality:The word inequality means a mathematical expression in which the sides are not equal to each other.
Equivalent Expression:Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value(s) for the variable(s).
Variable:Variables are placeholders for unknown values.
Algebraic Notation:Algebraic notation is a way of expressing mathematical ideas concisely using numbers, variables, and symbols.
Parentheses:Parentheses are these symbols: (). They may also be called round brackets. Parentheses are used to group numbers, operations, or variables together in math. Parentheses are also part of the order of operations in math.
Brackets/Braces:Mathematical brackets and braces are symbols, such as parentheses, that are most often used to create groups or clarify the order that operations are to be done in an algebraic expression.
Order of Operations:In math, order of operations are the rules that state the sequence in which the multiple operations in an expression should be solved.
Exponent:A quantity representing the power to which a given number or expression is to be raised (multiplied) and is expresses as a raised symbol beside the number or expression.
Mr. Rodgers wants his students to begin to use algebraic notation to represent situations where multiple operations are happening. He begins by asking students to watch a short, funny video clip. He asks his students to retell what happened in the video in the proper sequence. His students have been doing similar things in their reading classes for several years so they are very comfortable with this. Mr. Rodgers then gives students a math problem in a story context. This problem has two different operations that need to be performed. He asks them to read the problem and then, with a partner, retell what happened in the story in the proper sequence. Students are successful in this as well. As a class, Mr. Rodgers leads his class in a discussion as to what happened first in the story problem and listens to students’ responses. He then asks them how they could show what happened first using numbers and symbols rather than words. He knows that if he can get students to be aware of the action and sequence of a problem, representing these actions with algebraic notation will be easier for his students to become solid in.
Category: Preparation and Planning
Submit ONE of the evidence options below to show your preparation and planning to teach algebraic notation, equality, inequality, and relational thinking.
Submit a 7-10 minute video demonstrating how students progress in their understanding of representing situations using algebraic notation from 2nd through 6th grade. The video should include the following:
- How students progress from conceptual understanding towards procedural fluency from 2nd to 6th grade.
- Specific concepts students should understand at each grade level.
- Examples of representations students should use as they progress towards proficiency/understanding.
- As you transition from concept to concept, reference the standard(s) connected to each concept.
Submit a lesson plan that demonstrates how you have helped students become proficient in representing situations using algebraic notation. 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.)
Category: Implementation
Submit TWO of the evidence options below to show your implementation of instruction focused on algebraic notation, equality, inequality, and relational thinking.
The learner work sample should highlight how a student progressed in their understanding of the concept you have chosen to observe. The samples should include:
- Work from 1 student that shows their progression in ideas across each day of the unit. In other words, there should be an example of the student’s work for each day of the unit (at least 4).
- A written analysis of a student’s understanding for each artifact, based on the work sample collected for that day. This analysis should include:
(a) The lesson taught that day.
(b) The concepts explored during that lesson.
(c) The representations used to enhance or demonstrate understanding.
(d) An evaluation of how well the student is understanding the concept at this point of the unit. - Repeat steps a-d for each day’s artifact.
These work samples should show how conceptual understanding was achieved.
Submit a video demonstrating your implementation of a lesson focused on a coordinate plane concept. Requirements for the video:
- Showcase teacher facilitation of common misconceptions and students’ ways of thinking about algebraic notation concepts.
- Showcase student interaction with the common misconceptions and ways of thinking about algebraic notation concepts.
- Showcase the connection between visual and/or verbal representations of a situation and the algebraic notation for the same situation.
- Showcase student work from the lesson.
- No longer than 45 minutes.
- Contain no more than 2 edits.
Review Criteria
Criterion 1: Evidence demonstrates a clear content understanding that helps students move from conceptual understanding to procedural fluency.
Criterion 2: Evidence demonstrates the candidate's understanding of algebraic notation (equations, inequalities, notation, etc.).
Criterion 3: Evidence demonstrates the understanding of and use of effective teaching practices designed to help students understand algebraic notation (equations, inequalities, notation, etc.).
Describe the effect that a deficit in understanding, writing, and solving problems having or using algebraic notation can have in future grade levels (include at least one example standard and how that standard is affected).
Describe the effect that focusing mainly on procedural fluency when teaching algebra concepts would have on student understanding and their ability to use these measures in future grade levels as well as real life?
Looking ahead, what changes could you make to your practice to help students become proficient in your standards focused on algebraic notation, equality, inequality, and relational thinking?
Review Criteria
Criterion 1: The reflection indicates that the educator understands algebraic notation, equality, inequality, and relational thinking concepts and the application of these measures to real-life contexts.
Criterion 2: This reflection discusses changes the educator plans to make to future instruction to help students reach proficiency in grade-level standards focusing on algebraic notation, equality, inequality, and relational thinking concepts.
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
These eight mathematics teaching practices provide a framework for strengthening the teaching and learning of mathematics. This research-informed framework of teaching and learning reflects the current learning principles as well as other knowledge of mathematics teaching that has accumulated over the last two decades. In essence, these teaching practices represent a core set of high-leverage practices and essential teaching skills necessary to promote deep learning of mathematics.
Elementary and Middle School Mathematics: Teaching Developmentally John Van de Walle
Elementary and Middle School Mathematics: Teaching Developmentally illustrates how children learn mathematics, and then shows teachers the most effective methods of teaching PreK-8 math through hands-on, problem-based activities.
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
Teaching Student-Centered Mathematics Grades K-3 provides practical guidance along with proven strategies for practicing teachers of kindergarten through grade 3. This volume offers brand-new material specifically written for the early grades.
Mathematics for Elementary Teachers with Activities Sybilla Beckman
This book connects the foundations of teaching elementary math and the “why” behind procedures, formulas, and reasoning so students gain a deeper understanding to bring into their own classrooms. Through her text, Beckmann teaches mathematical principles while addressing the realities of being a teacher.
Progressions for the Common Core State Standards in Mathematics
http://ime.math.arizona.edu/progressions/
The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics.
Elementary Mathematics is Anything but Elementary: Content and Methods From a Developmental Perspective Damon L. Bahr and Lisa Ann DeGarcia
Elementary Mathematics is Anything but Elementary: Content and Methods From a Developmental Perspective is a comprehensive program that delivers both a content and a methods text. Serving as a professional development guide for both pre-service and in-service teachers, this text''s integrated coverage helps dissolve the line between content and methods--and consequently bolsters teachers'' confidence in their delivery of math instruction. A strong emphasis on the National Council of Teachers of Mathematics five core standards provides key information common to most state curricula relative to NCTM standards for pre-K through sixth grade. In addition, text content is based on thorough elementary mathematical scope and sequences that have been shown to be an effective means for guiding the delivery of curriculum and instruction.
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