# Grades 3–5, Operations and Algebraic Thinking

This module will extend participants’ knowledge and understanding of operations from addition and subtraction to multiplication and division. Models for multiplication will start with arrays and equal groups, which will provide the context in which these operations are studied. Area models follow from the array model. The compare model for multiplication will be studied contrasting it with the additive compare model. This will form the basis for eventual work with size change, fractions, and proportional reasoning, which come after fifth grade. Number theory, factors, multiples, primes, will be studied in a meaningful way, in the context of problems, activities, or explorations.

Another focus is the connection between operations, specifically multiplication and division, considering division as a missing factor problem as an example. Each session will focus on giving meaning to the operations in a context as well as to develop an understanding of the properties, which govern their use. Operations and Algebraic Thinking is truly the underpinning of algebra, the language of mathematics, and therefore deep understanding is essential. It is not enough to know how to compute, but to know how and why each operation can be used in a meaningful way to answer a question or solve a problem is the goal.

**Goals**

- Become familiar with ways to find and represent patterns, know and interpret remainders.
- Understand symbolic ways to represent mathematics and gain proficiency in using visual representations, i.e., graphs, tape diagrams, and other models.
- Become proficient with computation and to know how and why each operation can be used and how they relate to one another.
- Understand the critical nature of this foundational learning as it applies to Algebra.

# Grades 3–5, Number and Operations – Fractions

This module focuses on deepening understanding and proficiency with fractions aligned to the expectations of the Common Core State Standards for Mathematics (CCSSM). Participants study the development of fraction knowledge in Grades 3-5, drawing from the domains of Number and Operations–Fractions in Grades 3-5 and The Number System in Grade 6 with connections to the Geometry domain in Grades 1-3. Emphasis is placed on knowing the standards, understanding how they relate to the progression of mathematics learning across grades, examining visual models, and studying meaningful contexts in which to apply the standards. In addition, participants are engaged in the Standards for Mathematical Practice while immersed in studying the Standards for Mathematical Content. In particular, participants focus on four of the Standards for Mathematical Practice—(MP1) Make sense of problems and persevere in solving them, (MP2) Reason abstractly and quantitatively, (MP3) Construct viable arguments and critique the reasoning of others, and (MP6) Attend to precision.

**Goals**

- Become familiar with progression of standards for fractions across Grades 3-5.
- Understand, represent, and explain the important ideas in CCSSM for unit fractions, fraction equivalence, comparing fractions, operations with fractions, and relationships between decimals and fractions.