According to the Common Core, third-grade students should be able to round whole numbers to the nearest 10 or 100. They should also be able to add and subtract with fluency within 1,000 using properties of place value -- for example, 885 - 200 = 685 because all that changes is the hundreds place. At this level, students should also be able to multiply multiples of 10 by single-digit numbers using place value knowledge. For example, 5 x 20 = 100 because 5 x 2 = 10.
Properties of operations that are learned in third grade include the commutative, associative and distributive properties. These become particularly useful for multiplication at this grade level. The commutative property illustrates that if 3 x 2 = 6, then 2 x 3 has the same product. The associative property is such that 3 x 5 x 1 can be found by breaking the problem into smaller pieces. If 3 x 5 = 15, then 15 x 1 = 15. Finally, using the distributive property of multiplication, knowing that 2 x 2 = 4 and 2 x 5 = 10, you can find 2 x 7 by adding the products of each (4 + 10 = 14, so 2 x 7 = 14).
Third-grade students are expected to understand that fractions are parts of a whole. They should be able to use a number line to represent fractions. For example, if zero to 1 on the number line represents a whole, students should be able to divide the whole into equal lengths, for example, one-third, two-thirds.
Students should be able to compare fractions with the same numerator or denominator by reasoning about the size of the parts. For example, 2/3 will be greater than 1/3 because one part of three equal parts is less than two parts of three equal parts.