- For equation (a), grouping changes from \(3 \times (5 \times 9)\) to \((3 \times 5) \times 9\), identifying the Associative Property.
- For equation (b), factor order changes from \(4 \times 5\) to \(5 \times 4\), identifying the Commutative Property.
- For equation (c), multiplying by \(0\) results in \(0\), identifying the Zero Property of Multiplication.
- For equation (d), multiplying by \(1\) leaves the number unchanged, identifying the Identity Property of Multiplication.
- The completed table is:
A\(3 \times (5 \times 9) = (3 \times 5) \times \boxed{9}\) | Associative Property
B\(4 \times \boxed{5} = 5 \times 4\) | Commutative Property
C\(\boxed{0} \times 5 = 0\) | Zero Property of Multiplication
D\(3 \times \boxed{1} = 3\) | Identity Property of Multiplication
Explanation
- Analysis of the problem
We are given four equations that illustrate fundamental properties of multiplication. Each equation contains a missing value, and we are tasked with identifying both the missing value and the property of multiplication being demonstrated in each case. The four fundamental properties we are dealing with are the Associative Property, the Commutative Property, the Zero Property of Multiplication, and the Identity Property of Multiplication.
- Solving Equation (a)
Equation (a) is \(3 \times (5 \times 9) = (3 \times 5) \times \square\).
The grouping of factors has changed from \((5 \times 9)\) to \((3 \times 5)\). This is the hallmark of the Associative Property, which states that \(a \times (b \times c) = (a \times b) \times c\).
Comparing \(3 \times (5 \times 9)\) to \((3 \times 5) \times \square\), we see that the missing factor must be \(9\).
Thus, the property used is the Associative Property and the missing value is \(9\).
- Solving Equation (b)
Equation (b) is \(4 \times \square = 5 \times 4\).
The order of the factors has changed from \(4 \times \square\) to \(5 \times 4\). This is the Commutative Property, which states that \(a \times b = b \times a\).
Comparing \(4 \times \square\) to \(5 \times 4\), we see that the missing factor must be \(5\).
Thus, the property used is the Commutative Property and the missing value is \(5\).
- Solving Equation (c)
Equation (c) is \(\square \times 5 = 0\).
We know that any number multiplied by \(0\) equals \(0\), which is known as the Zero Property of Multiplication.
Therefore, for the product to be \(0\), the missing factor must be \(0\).
Thus, the property used is the Zero Property of Multiplication and the missing value is \(0\).
- Solving Equation (d)
Equation (d) is \(3 \times \square = 3\).
We know that any number multiplied by \(1\) remains unchanged, which is the Identity Property of Multiplication.
Since \(3 \times \square = 3\), the missing factor must be \(1\).
Thus, the property used is the Identity Property of Multiplication and the missing value is \(1\).
Examples
Understanding these properties of multiplication helps in mental math and simplifying algebraic expressions. For instance, if you have to calculate \(17 \times 5 \times 2\), the associative property allows you to group it as \(17 \times (5 \times 2) = 17 \times 10 = 170\), making the calculation much easier. Recognizing these patterns is a foundation for higher-level mathematics.