# 3.3 - Single-step and Multistep Problems

The student will

a)  estimate  and determine the sum or difference of two whole numbers; and

b)  create and solve single-step and multistep practical problems involving sums or differences of two whole numbers, each 9,999 or less.

### BIG IDEAS

• So that I can understand that numerical quantities can be estimated by using numbers that are close to the actual values

• So that I can understand that estimation skills are valuable, time-saving tools, particularly in practical situations when exact answers are not needed

• So that I can find the sum or difference when shopping or counting change

### UNDERSTANDING THE STANDARD

• Flexible methods of adding whole numbers by combining numbers in a variety of ways, most depending on place values, are useful.
• Grade three students should explore and apply the properties of addition as strategies for solving addition and subtraction problems using a variety of representations (e.g., manipulatives, diagrams, symbols, etc.).
• 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 of addition are most appropriate for exploration at this level:

• The commutative property of addition states that changing the order of the addends does not affect the sum (e.g., 4 + 3 = 3 + 4).

• 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 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).

• Using concrete materials (e.g., base-ten blocks, connecting cubes, beans and cups, etc.) to explore, model and stimulate discussion about a variety of problem situations which helps students understand regrouping and enables them to move from the concrete to the abstract. Regrouping is used in addition and subtraction algorithms.
• Exploring concepts through concrete experiences develops conceptual understanding. Next, the children must make connections that serve as a bridge to the symbolic. Student-created representations, such as drawings, diagrams, tally marks, graphs, or written comments are strategies that help students make these connections.
• 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 addition and subtraction.  Examples are included in the following chart: • Combination problem types are introduced in grade five (e.g., How many different outfits can be made given 3 shirts and two pants?).
• In problem solving, emphasis should be placed on thinking and reasoning rather than on key words.  Focusing on key words 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 key-word 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.
• Addition is the combining of quantities; it uses the following terms:

sum:         669
• Subtraction is the inverse of addition; it yields the difference between two numbers and uses the following terms:

minuend:          7,698

subtrahend:   – 5,341

difference:        2,357

• An algorithm is a step-by-step method for computing.
• The least number of steps necessary to solve a single-step problem is one.
• Estimation skills are valuable, time-saving tools particularly in practical situations when exact answers are not required or needed.
• Estimation skills are also valuable in determining the reasonableness of the sum or difference when solving for the exact answer.
• When an exact answer is required, opportunities to explore whether the answer can be determined mentally or must involve paper and pencil or calculators help students select the most efficient approach.
• Determining whether an estimate is appropriate and using a variety of strategies to estimate requires experiences with problem situations involving estimation.
• There are a variety of mental mathematics strategies for each basic operation, and opportunities to practice these strategies give students the tools to use them at appropriate times. For example, with addition, mental mathematics strategies include:
• using addition by counting up for solving subtraction problems;
• making 10: for column addition, looking for numbers that can be grouped together to make 10.

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

• Determine whether an estimate or an exact answer is an appropriate solution for practical addition and subtraction problems involving single-step and multistep problems. (a, b)
• Estimate the sum of two whole numbers with sums to 9,999. (a)
• Estimate the difference of two whole numbers, each 9,999 or less. (a)
• Apply strategies, including place value and the properties of addition, to add two whole numbers with sums to 9,999. (a, b)
• Apply strategies, including place value and the properties of addition, to subtract two whole numbers, each 9,999 or less. (a, b)
• Use inverse relationships between addition and subtraction facts to solve practical problems. (b)
• Create and solve single-step and multistep practical problems involving the sum or difference of two whole numbers, each 9,999 or less. (b)

### KEY VOCABULARY

Updated: Aug 22, 2018