Problem 21 - Prealgebra

Question

Name the property shown by each statement. $$(4+5)+15=4+(5+15)$$

Step-by-Step Solution

Answer

Associative Property of Addition.

Step 1: Identify the Operation

Observe the expression given. The operation applied is addition: \((4+5)+15\) and \(4+(5+15)\). Look at how the numbers are grouped and note that the operation involved is associative since grouping doesn't affect the sum.

Step 2: Recall the Associative Property

The associative property states that how we group numbers when adding or multiplying does not change the sum or product. For addition, this can be represented as \((a+b)+c = a+(b+c)\). Compare this to the given expression to confirm.

Step 3: Verify the Property

Check that both expressions yield the same result: \((4+5)+15 = 9+15 = 24\) and \(4+(5+15) = 4+20 = 24\). Since the sums are equal regardless of how the numbers are grouped, it confirms the associative property is applied here.

Key Concepts

Addition

Addition is a fundamental arithmetic operation where two or more numbers are combined to form a new total, called the sum. It is symbolized by the plus sign "+". Understanding addition is crucial because it forms the basis for many mathematical concepts and real-world applications.
In arithmetic, addition is an operation that deals with combining quantities in a straightforward manner:

• Adding positive numbers increases the total.
• Adding zero to a number leaves it unchanged.
• Additive inverses, or negative numbers, cancel each other out when summed.

In the context of our exercise, the operation is straightforward since it involves simple numbers:

• Start with one pair of numbers, such as 4 and 5, add them together to get 9.
• Then add a third number, like 15, to get the final sum for each grouping.

These basics form the foundation for tackling more complex problems like those involving the associative property.

Number Grouping

Number grouping in mathematics is the concept of organizing numbers in different clusters or parentheses during operations like addition or multiplication. It might seem like a minor change, but grouping can significantly impact the process and understanding of calculations, though it doesn't affect the sum or product when using the associative property.
In our exercise,

• Initial group:
  • \((4+5)+15\)
• Re-grouped:
  • \(4+(5+15)\)

Observing how numbers are grouped can help in streamlining calculations, especially in complex arithmetic or algebraic expressions. Proper grouping can lead to easier and quicker calculations as it might allow interim simplifications.
Ultimately, by practicing number grouping, students develop an intuitive sense of number manipulation, which is crucial for algebra and beyond.

Mathematical Properties

Mathematical properties are rules that describe the operations and their results. They are the tools that help simplify and solve equations. Among these is the associative property, which is highlighted in this exercise. This property applies to both addition and multiplication.

• Associative Property of Addition:
  • States that changing the grouping of addends does not change the sum. For instance, \((a+b)+c = a+(b+c)\).
• Associative Property of Multiplication:
  • Similarly, changing the grouping of factors does not change the product. E.g., \((a \times b) \times c = a \times (b \times c)\).

Recognizing these properties is crucial because they allow more flexibility in computation and can simplify problem-solving.Most importantly, these properties ensure consistency and predictability in mathematics, which is essential for developing more advanced mathematical understanding.

Elementary Mathematics

Elementary mathematics covers the basic concepts and operations that form the foundation for all future mathematical learning. This includes addition, subtraction, multiplication, division, and the basic properties that govern these operations. Each of these components is essential for building a solid math foundation.

• Numerical Operations: Basic operations such as addition are introduced at an early stage.
• Properties of Operations: Children begin learning properties like associative, commutative, and distributive properties to better understand how numbers work.
• Application: Real-life applications help cement these concepts, such as using addition in counting money or objects.

In our exercise, the concept of the associative property is part of elementary mathematics. Starting with simple, concrete examples helps students grasp abstract ideas more effectively. This forms the basis for all future studies in numbers and operations in mathematics.