Write number like this:

As in the calendar mark four numbers in a square and find the difference of diagonal products. Is it the same for all squares of four numbers?

Explain using algebra.

Write number like this:

Take a square of nine numbers and mark only the numbers at the four corners.

  What is the difference of diagonal products? Explain using algebra.

In the multiplication table below:

Take a square of nine numbers as given below, and mark the numbers at the four corners:

What is the difference of diagonal sums?

In the multiplication table made above, take a square of nine numbers, and mark the numbers at the four corners:

Explain using algebra, why this difference is the same for all such squares.

In the multiplication table above, take a square of sixteen numbers, instead of nine, and mark the numbers at the four corners:

Find the difference between diagonal sums of a square having sixteen numbers.

Look at these:

$1 \times 4 = (2 \times 3) – 2$

$2 \times 5 = (3 \times 4) – 2$  

$3 \times 6 = (4 \times 5) – 2$  

$4 \times 7 = (5 \times 6) – 2$  

Write the next two lines in this pattern.

If we take four consecutive natural numbers, what is the relation between the products of the first and the last, and the product of the middle two ?

Look at these:

$1 \times 4 = (2 \times 3) – 2$

$3 \times 6 = (4 \times 5) – 2$  

$4 \times 7 = (5 \times 6) – 2$ 

Write this as a general principle in algebra and explain it.

Shown below is a method to the product $46\times 28.$

$$\begin{gathered} \begin{array}{lrr}4\times 2=8& 8\times 100& 800\\ (4\times 8)+(6\times 2)=44& 44\times 10& 440\end{array} \\ 6\times 8 48 \\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ 46\times 28 1288 \end{gathered}$$

Check this method for some other two-digit numbers.