- Start with the given equation \(56 = 8 \times 7\).
- Break apart the factor \(8\) into the product \(4 \times 2\), resulting in \((4 \times 2) \times 7\).
- Apply the associative property of multiplication to regroup the factors as \(4 \times (2 \times 7)\).
- Identify that the missing value is \(\boxed{2}\).
Explanation
- Analysis of the Problem and Mathematical Properties
We are given the equation \(56 = 8 \times 7\). The task is to express \(56\) as \(4 \times (? \times 7)\) by breaking apart one of the factors and regrouping them using the associative property of multiplication. The associative property of multiplication states that for any three numbers \(a, b,\) and \(c\), the product remains the same regardless of how the factors are grouped: \((a \times b) \times c = a \times (b \times c)\).
- Breaking Down the Factor 8
We start with the original factor pair \(8 \times 7 = 56\). We know that \(8\) can be broken into the product of two factors: \(4 \times 2\). Substituting this into the equation, we get \(56 = (4 \times 2) \times 7\).
- Regrouping Factors and Identifying the Missing Number
Now, we apply the associative property of multiplication. By changing the grouping of the factors from \((4 \times 2) \times 7\) to \(4 \times (2 \times 7)\), we reach the desired form. By comparing this to the target equation \(4 \times (? \times 7)\), we can see that the missing factor represented by the question mark must be \(2\). Thus, \(4 \times (2 \times 7) = 4 \times 14 = 56\).
- Final Answer
The missing factor in the expression \(4 \times (? \times 7)\) is \(2\). Therefore, the completed equation is \(56 = 4 \times (2 \times 7)\), and the missing value is \(\boxed{2}\).
Examples
Understanding the associative property is very useful in everyday life, especially when calculating totals for large groups. Imagine you are buying party supplies: if you need to buy 8 packs of juice, and each pack contains 7 juice boxes, you can calculate the total as \(8 \times 7 = 56\) boxes. If you decide to buy the juice in a different way, such as 4 large bundles where each bundle contains 2 packs, and each pack has 7 boxes, you can use the associative property to calculate \(4 \times (2 \times 7) = 4 \times 14 = 56\). This helps you simplify complex multiplication problems by breaking them into smaller, more manageable parts!