- Understand that the expression \(-5(-2)\) represents subtracting 5 groups of \(-2\).
- Recognize the need for "zero pairs" to provide the negative tiles required for subtraction without changing the total value.
- Identify the steps: first, add zero pairs to the window, then remove the 5 groups of \(-2\) tiles to leave the remaining positive tiles.
- The correct options are: "Drag zero pairs to the window" and "Remove the 5 groups of -2 tiles". The final value is \(\boxed{10}\).
Explanation
- Analyzing the Mathematical Expression and Tool Model
The problem asks to evaluate the expression \(-5(-2)\) using an integer tiles model. Integer tiles are a common pedagogical tool used to represent numbers visually, where positive integers are represented by one color (e.g., yellow) and negative integers by another color (e.g., red). A "zero pair" consists of one positive tile and one negative tile, which together represent the sum of zero. The expression \(-5(-2)\) can be interpreted as "subtracting 5 groups of \(-2\)".
- Adding Zero Pairs to the Workspace
To perform subtraction with tiles, we need the tiles we are subtracting to be physically present in the workspace. Since we want to subtract 5 groups of \(-2\), we need to have these tiles available. However, our workspace starts with a net value of zero. Therefore, to make these tiles available without changing the total value of our work, we must introduce "zero pairs". By adding 5 groups of \((-2, +2)\) pairs, we maintain a total value of zero while providing the 5 groups of \(-2\) tiles that need to be removed. Thus, the first necessary step is to drag these zero pairs into the window.
- Performing the Operation by Removing Tiles
Once the zero pairs are in the window, we have the necessary 5 groups of \(-2\) tiles ready for subtraction. The operation \(-5 \times (-2)\) means we must remove 5 groups of \(-2\). By removing these negative tiles, we are left with the 5 groups of \(+2\) tiles that were previously paired with them. Removing 5 groups of \(-2\) leaves us with a final total of \(+10\). Therefore, the second necessary step is to remove the 5 groups of \(-2\) tiles.
- Conclusion
Based on the analysis of the integer tile model, the two required steps to evaluate the expression \(-5(-2)\) are:
- Drag zero pairs to the window.
- Remove the 5 groups of \(-2\) tiles.
These actions correctly represent the concept that multiplying by a negative number is equivalent to removing negative values from a neutral starting state, resulting in a positive outcome. The final result of \(-5(-2)\) is \(\boxed{10}\).
Examples
The concept of using "zero pairs" in algebra is highly useful for understanding financial transactions and inventory management. For instance, if a company has a debt represented as a negative value, "subtracting" that debt is equivalent to an increase in their actual assets. If an accountant removes 5 entries of \(\$2\) worth of debt, they are essentially adding \(\$10\) back into the company's net value. Mastering this logical framework helps in grasping complex accounting and resource allocation processes.