Algebraic Properties and Conjectures
This microcredential represents a teacher’s ability to understand and respond to progressions related to algebraic properties and conjectures by planning and implementing instruction based on the Standards for Mathematical Practice and Effective Mathematics Teaching Practices. It involves 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 of algebraic properties and conjectures. This is the second 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 2: Learning Progression - Demonstrating a comprehensive understanding of where students have been, where they are now and where they are going using strategically sequenced learning experiences aligned within and across grade levels. -
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 two sets of evidence demonstrating your understanding of algebraic properties and conjectures. You will also complete a short written or video reflective analysis.
In algebra, students need opportunities to experience math in context and to notice patterns as they occur. Once they notice these patterns, students should have the opportunity to make and prove conjectures. These experiences help students become mathematical thinkers and investigators rather than just “doers.” Proving these conjectures as either true or false allows students to make connections to and find meaning for existing rules. The proficiencies contained in this microcredential includes several competencies.
Teachers should understand:
- How properties of operations may be used in computations and to deduce the correctness of algorithms (i.e., commutative, associative, distributive, multiplicative identity of one, additive identity of zero),
- That conjectures arise when one notices a pattern that holds true for many cases,
- When a conjecture is rigorously proved it becomes a rule, property, or definition (e.g., when adding positive numbers, the sum is larger than each individual addend, when multiplying by positive fraction less than one, the product will be less than the other factor),
- Make conjectures about classes of numbers (e.g., odd/even, prime/composite, factors: patterns for numbers being divisible by 2, 4, 5, 10, etc.),
- Make conjectures about rules for carrying out specific computational procedures (e.g., what happens when multiplying by powers of ten),
- Make conjectures about inverse operations and other relationships among addition, subtraction, multiplication, and division. Teachers should also,
- Connect conjectures to existing rules, properties, and definitions 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 specific numbers into variable expressions and find the solution set of an algebraic equation or inequality.
A conjecture is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many cases. However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases. Conjectures must be proved for the mathematical observation to be fully accepted. When a conjecture is rigorously proved, it becomes a theorem.
Ms. Blaisdell and her fourth-grade students are working with multiplication. She wants them to understand that using the commutative property can make solving multiplication problems easier. She knows, however, that if she just tells her students about the commutative property, this limits their understanding. Instead, she starts to give them activities to notice the commutative property. As the students are working on the activities, one student notices that 3 x 5 has the same product as 5 x 3. This student says that “If you flip the order of the numbers, the answer is the same.” Mrs. Blaisdell asks the students if this is always true in multiplication. She and the students set about trying to prove whether this conjecture always works. They try many different scenarios and determine that the order of the factors in multiplication doesn’t affect the product. They are comfortable with this conclusion. Mrs. Blaisdell connects what students discover to the commutative property of multiplication.
Category: Implementation
Submit the TWO evidence options below to demonstrate your effective use of algebraic properties and conjectures in your instruction.
Observe a teacher in your building teaching a math concept where conjectures will be formed and tested, leading to student understanding of a rule or property. 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) Background information about 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) What misconceptions and barriers the teacher 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) 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.
Submit a video demonstrating your implementation of a lesson focused on helping students understand a mathematical rule or property (the commutative property of addition, for example). Your lesson should clearly show how students were given opportunities to form and test a conjecture about the desired rule and how their conclusion was connected to the actual rule or property.
Showcase teacher facilitation of helping students form and test conjectures to understand a mathematical property or rule. Video requirements:
- Showcase student interaction with the forming of conjectures and testing them to understand a rule or property.
- Showcase the connection between the conjectures students made and their process of proof and the actual rule or property.
- 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 properties and conjectures.
Criterion 3: Evidence demonstrates the understanding of and use of effective teaching practices designed to help students understand algebraic properties and conjectures.
When you provide students with the opportunity to form conjectures and prove/disprove these conjectures, what effect does this have on their long-term mastery of the rule/property being explored? Use evidence to support your thoughts.
When you provide students with the opportunity to form conjectures and prove/disprove these conjectures, what effect does this have on their identities as math learners? Use evidence to support your thoughts.
Looking ahead, what changes could you make to your practice to help students become proficient in your standards focused on algebraic properties and conjectures.
Review Criteria
Criterion 1: The reflection indicates that the educator understands the importance of students forming and proving conjectures and linking this process to existing rules and properties and the impact this practice would have on student learning.
Criterion 2: The reflection discusses changes the teacher could make to their practice to help students become proficient in standards focused on algebraic properties and conjectures.
https://www.schools.utah.gov/file/f0e86540-5617-4166-a701-fea403f2f848
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http://ime.math.arizona.edu/progressions/
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