Content

The problem-solving process is enhanced when students create and solve their own practical problems and model problems using manipulatives and drawings.

In problem-solving, emphasis should be placed on thinking and reasoning rather than on keywords. Focusing on keywords such as in all, altogether, difference, etc., encourages students to perform a particular operation rather than make sense of the context of the problem. A keyword focus prepares students to solve a limited set of problems and often leads to incorrect solutions as well as challenges in upcoming grades and courses.

Estimation can be used to determine a reasonable range for the answer to computation and to verify the reasonableness of sums, differences, products, and quotients of whole numbers.

The least number of steps necessary to solve a single-step problem is one. A multistep problem incorporates two or more operational steps (operations can be the same or different).

Extensive research has been undertaken over the last several decades regarding different problem types. Many of these studies have been published in professional mathematics education publications using different labels and terminology to describe the varied problem types.

Students should experience a variety of problem types related to multiplication and division. 

Investigating arithmetic operations with whole numbers helps students learn about several different properties of arithmetic relationships. These relationships remain true regardless of the numbers.

Grade five students should explore and apply the properties of addition and multiplication as strategies for solving addition, subtraction, multiplication, and division problems using a variety of representations (e.g., manipulatives, diagrams, and symbols). 

The properties of the operations are “rules” about how numbers work and how they relate to one another. Students at this level do not need to use the formal terms for these properties but should utilize these properties to further develop flexibility and fluency in solving problems. The following properties are most appropriate for exploration at this level:

1. The commutative property of addition states that changing the order of the addends does not affect the sum (e.g., 4 + 3 = 3 + 4). Similarly, the commutative property of multiplication states that changing the order of the factors does not affect the product (e.g., 2 * 3 = 3 * 2).

2. The identity property of addition states that if zero is added to a given number, the sum is the same as the given number. The identity property of multiplication states that if a given number is multiplied by one, the product is the same as the given number.

3. The associative property of addition states that the sum stays the same when the grouping of addends is changed (e.g., 15 + (35 + 16) = (15 + 35) + 16).  The associative property of multiplication states that the product stays the same when the grouping of factors is changed [e.g., 6(3 * 5) = (6 * 3)5].

4. The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products.