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Operations & Algebraic Thinking

Represent and solve problems involving multiplication and division


Understand properties of multiplication and the relationship between multiplication and division


Multiply and divide within 100


Solve problems involving the four operations, and identify and explain patterns in arithmetic


The Operations and Algebraic Thinking Standards emphasizes relationships among quantities, including functions, patterns, ways of representing mathematical relationships, and the analysis of change. Algebra is best learned as a set of concepts and techniques tied to the representation of quantitative relations. It is a style of mathematical thinking for formalizing patterns, functions, and generalizations. Although many adults think that algebra is an area of mathematics more suited to middle school or high school students, even young children can be encouraged to use algebraic reasoning as they study numbers and operations and as they investigate patterns and relations among sets of numbers. In these standards, the connections of algebra to numbers and everyday situations are extended in the later grade bands to include geometric ideas.

In order to “fluently” multiply and divide within 100 and add and subtract within 1000, third grade students must begin work on multiplication and division very early in the year and continue to work with these concepts throughout the year. Multiplication is more than memorizing facts. Students need to recognize that multiplication is finding an unknown product and division is finding an unknown factor. Experiencing different strategies for finding solutions to problems involving multiplication and division allows students to recognize that multiplication and division are inverse operations.

Students recognize that multiplication is a more efficient way to perform repeated addition, as division is for repeated subtraction. Students can identify and build upon patterns that are found when multiplying and dividing. They learn to look for relationships between quantities and make generalizations about them. These strategies help students not only develop fluency with these operations, but to build upon this fluency as they encounter increasingly more complex mathematical situations in future grades. Center activities are a good place for students to gain the practice, and experience, they need to attain the fluency that is required to solve problems involving these two operations. Tasks that involve real-life experiences are crucial for students to gain the understanding of when and how to use each operation to obtain a solution.

This continuum contains Counting and Cardinality standards for K, Number and Operations and Algebraic Thinking standards K-3, Number and Operations in Base Ten standards K-3, Number and Operations – Fractions standards for grade 3.