Operations and Algebraic Thinking

In kindergarten through grade two, students focused on developing an understanding of addition and subtraction. Beginning in grade three, students focus on concepts, skills, and problem solving for multiplication and division. Students develop multiplication strategies, make a shift from additive to multiplicative reasoning, and relate division to multiplication. Third-grade students become fluent with multiplication and division within 100. This work will continue in grades four and five, preparing the way for work with ratios and proportions in grades six and seven (adapted from the University of Arizona Progressions Documents for the Common Core Math Standards [UA Progressions Documents] 2011 and PARCC 2012). Multiplication and division are new concepts in grade three, and meeting fluency is a major portion of students’ work. Reaching fluency will take much of the year for many students. These skills and the understandings that support them are crucial; students will rely on them for years to come as they learn to multiply and divide with multi-digit whole numbers and to add, subtract, multiply, and divide with rational numbers.

#### Number Sense, Classifications & Patterns

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Number and Operations in Base Ten

In grade three, students are introduced to the concept of rounding whole numbers to the nearest 10 or 100, an important prerequisite for working with estimation problems. Students can use a number line or a hundreds chart as tools to support their work with rounding. They learn when and why to round numbers and extend their understanding of place value to include whole numbers with four digits.

Third-grade students continue to add and subtract within 1000 and achieve fluency with strategies and algorithms that are based on place value, properties of operations, and/or the relationship between addition and subtraction. They use addition and subtraction methods developed in grade two, where they began to add and subtract within 1000 without the expectation of full fluency and used at least one method that generalizes readily to larger numbers—so this is a relatively small and incremental expectation for third-graders. Such methods continue to be the focus in grade three, and thus the extension at grade four to generalize these methods to larger numbers (up to 1,000,000) should also be relatively easy and rapid.

Students in grade three also multiply one-digit whole numbers by multiples of 10 (3.NBT.3) in the range 10–90, using strategies based on place value and properties of operations (e.g., “I know 5 x 90 = 450 because 5 x 9 = 45 , and so 5 x 90 should be 10 times as much”). Students also interpret 2 x 40 as 2 groups of 4 tens or 8 groups of ten. They understand that 5 x 60 is 5 groups of 6 tens or 30 tens, and they know 30 tens are 300. After developing this understanding, students begin to recognize the patterns in multiplying by multiples of 10 (ADE 2010). The ability to multiply one-digit numbers by multiples of 10 can support later student learning of standard algorithms for multiplication of multi-digit numbers.

Number and Operations—Fractions

In grade three, students develop an understanding of fractions as numbers. They begin with unit fractions by building on the idea of partitioning a whole into equal parts. Student proficiency with fractions is essential for success in more advanced mathematics such as percentages, ratios and proportions, and algebra.

In grades one and two, students partitioned circles and rectangles into two, three, and four equal shares and used fraction language (e.g., halves, thirds, half of, a third of). In grade three, students begin to enlarge their concept of number by developing an understanding of fractions as numbers (adapted from PARCC 2012).

Grade-three students understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts and the fraction a/b as the quantity formed by a parts of size 1/b.