Question

LOGICAL REASONING Your friend does not understand how the product of \(a\) and \((b+c)\) is given by both \(a b+a c\) and \(b a+c a\) Use the properties of multiplication to write a convincing argument to show your friend that both statements are correct.

Short answer

The expressions \(a \cdot b + a \cdot c\) and \(b \cdot a + c \cdot a\) are the same since the distributive property allows us to expand \(a \cdot (b + c)\) into \(a \cdot b + a \cdot c\), and the commutative property lets us swap the order of multiplication, yielding \(b \cdot a + c \cdot a\).

Step-by-step solution

Apply the Distributive Property

The distributive property of multiplication over addition states that if you have a number multiplied by the sum of two other numbers, you can distribute the multiplication over the addition. Thus, \(a \cdot (b + c)\) equals \(a \cdot b + a \cdot c\).

Apply the Commutative Property

In order to compare the first part of this expression with the term \(b \cdot a + c \cdot a\), the commutative property of multiplication must be applied. The commutative property states that the order in which two numbers are multiplied does not change their product, hence \(a \cdot b = b \cdot a\) and \(a \cdot c = c \cdot a\).

Compare the Results

By applying the commutative property to \(a \cdot b + a \cdot c\), it becomes \(b \cdot a + c \cdot a\). Thus, it can be seen that the terms \(a \cdot b + a \cdot c\) and \(b \cdot a + c \cdot a\) are equivalent.

Key concepts

Logical Reasoning and the Distributive Property

Understanding the concept of the distributive property involves a bit of logical reasoning, a crucial skill in mathematics. Logical reasoning helps you derive conclusions based on given facts or properties. To showcase the distributive property in action, let's start with the equation \(a \cdot (b + c)\). Here, we have a number \(a\) being multiplied by the sum of two other numbers, \(b + c\). According to the distributive property, you can multiply \(a\) by each value inside the parentheses and then add the products. So, \(a \cdot b + a \cdot c\) becomes the expanded form.

Improving the exercise understanding, let's break it down further. If we take \(a\) and multiply it by both \(b\) and \(c\) separately, we get two products, \(a \cdot b\) and \(a \cdot c\), which we then sum up. This method of distributing the multiplication over addition allows students to work with each number individually, simplifying the process and reinforcing the understanding of multiplication as an additive concept.

Commutative Property - A Simple Yet Powerful Rule

The commutative property is like a friendly handshake in mathematics; it doesn't matter which hand you extend first, the gesture remains the same. Translating this into numbers, if you multiply two numbers, the order in which you multiply them doesn't affect the product. Mathematically, this means \(a \cdot b = b \cdot a\) and similarly \(a \cdot c = c \cdot a\).

Let's apply this to our exercise. We have established that \(a \cdot (b + c)\) becomes \(a \cdot b + a \cdot c\). Now, when we switch the order of multiplication using the commutative property, \(a \cdot b\) becomes \(b \cdot a\) and \(a \cdot c\) becomes \(c \cdot a\). Thereby, \(a \cdot b + a \cdot c\) effortlessly transforms into \(b \cdot a + c \cdot a\), illustrating that the outcome of multiplication remains constant, irrespective of the order of the factors. This rule not only simplifies calculations but also helps in understanding that multiplication is flexible and can be approached from different angles.

Properties of Multiplication - Ground Rules for Solving Problems

To fully appreciate multiplication's behavior, it's essential to grasp its fundamental properties, which are rules that define how multiplication operates within the realm of numbers. These are not mere suggestions; they're the nuts and bolts of solving multiplication problems.

Let's highlight a few key properties:

• Associative Property: This property tells us that when multiplying three or more numbers together, the grouping doesn't matter (e.g., \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)).
• Identity Property: This states that any number multiplied by one remains unchanged (e.g., \(a \cdot 1 = a\)).
• Zero Property: When any number is multiplied by zero, the result is always zero (e.g., \(a \cdot 0 = 0\)).

Embracing these properties allows students to approach problems systematically and with confidence, ensuring they can apply these rules to simplify complex multiplication tasks. By connecting the dots between these properties, students develop a deeper understanding and become equipped to tackle a myriad of mathematical challenges.