In the table shown, the top row and leftmost column are grey, and the remaining numbers form a \(3 \times 3\) array. Each number in the array is equal to the sum of the numbers in grey in its row and column.
| \(5\) | \(2\) | \(7\) | |
|---|---|---|---|
| \(3\) | \(8\) | \(5\) | \(10\) |
| \(4\) | \(9\) | \(6\) | \(11\) |
| \(6\) | \(11\) | \(8\) | \(13\) |
Follow the steps below.
Circle any number in the array (for example, \(6\)).
Cross off the other numbers in the same row and column of the
array.
(For our example, we would then cross out \(9\), \(11\), \(5\), and \(8\).)
Circle any remaining number in the array (for example, \(13\)).
Cross off the other numbers in the same row and column of the
array.
(For our example, we would then cross out \(10\) and \(11\).)
Circle the remaining number in the array (for our example, this is \(8\)).
What is the sum of the three circled numbers?
Repeat the steps in part (a) two more times, starting with a different number each time. What do you notice about the sum of the three circled numbers?
Will your result from (b) be true if we create a \(3\times 3\) array using different initial numbers in the grey row and column? Explain why it will be true or give an example where it would not be true.
Theme: Number Sense