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Quick Check
Find the value of the following using the token/counter method.

1. 5 + 8 2. 8 – 11 3. 13 – 17 4. –5 – 6
5. 12 – 7 6. –8 + 9 7. –3 – (–2) 8. 5 – 6

Explorations with Integers

Hollow Integer Grid

Look at the integer grids.

What did you notice? 4 –1 –3 5 –3 –5
In the first grid, –3 1 0 –5

Sum of 1st row = 4 + (–1) + (–3) = 0 –1 –1 2 –8 –2 7
Sum of Last row = (–1) + (–1) + 2 = 0

Sum of 1st column = 4 + (–3) + (–1) = 0
Sum of Last (3rd) column = (–3) + 1 + 2 = 0

In the second grid,
Think and Answer
Sum of 1st row = 5 + (–3) + (–5) = –3
Sum of Last row = (–8) + (–2) + 7 = –3 Fill in the hollow grid given below with integers
–1 to –8 so that the border sum is –12.
Sum of 1st column = 5 + 0 + (–8) = –3
Sum of Last (3rd) column = (–5) + (–5) + 7 = –3

You can make a similar integer grid where the
middle box is empty and the sum of the first and
last rows and left and right columns are the same.
This is also called border sum.

Integer Magic Square

Like the magic square we have discussed in the book earlier, you can create a magic square of 3
by 3 or 4 by 4 including the positive and negative integers, where the sum of rows, columns and
diagonals are equal.
Example 15: Complete the magic square given below.

–4 0

3 –1

–2 2

Solution: Here, sum of 1st column = (–4) + 3 + (–2) = –3

And, the diagonal = (–4) + (–1) + 2 = –3
Hence, the magic constant is –3.

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