Hinglish

Explorations with Integers

Class 6 Maths · Chapter 10: The Other Side of Zero · NCERT Ganita Prakash section 10.4, 10.5

Learn

Border puzzle. Here is a square of 8 boxes with an empty middle. Along each edge, the three numbers add up to the same total, the border sum. Check: top 5 + 1 + (−2) = 4, right (−2) + 2 + 4 = 4, bottom (−6) + 6 + 4 = 4, left 5 + 5 + (−6) = 4.

border sum = 4

To fix a side, find what is missing: if a side must add to 4 and you know 5 and −6, the missing number is 4 − 5 − (−6) = +5. Every empty box can be found like this.

4 − 5 − (−6) = 5

Patterns. Count backwards in 3s from 7: 7, 4, 1, −2, −5, −8 … The jump is −3 each time, and the pattern carries on smoothly past zero.

From history: negative numbers took a long time to be accepted. Over 2,000 years ago, Chinese mathematicians used red and black counting rods for gains and debts, just like our green and coral tokens. In 628 CE, the Indian mathematician Brahmagupta wrote clear rules for adding and subtracting "fortunes" (positive numbers), "debts" (negative numbers) and zero. Many European mathematicians still called negative numbers "absurd" more than a thousand years later!

Explore: solve the border puzzles.

  1. Make every side add up to 4.
  2. Now every side must add up to −2.
  3. Only the corners are given. Border sum 0.

Remember

  • A missing number = the total − the numbers you know.
  • Patterns of integers carry on smoothly through zero.

Quick check

What comes next? 6, 2, −2, −6, ?

  1. −10
  2. −8
  3. −2
  4. 10
Answer

−10. The jump is −4 each time: −6 − 4 = −10.

Think about it

In a border puzzle with border sum 0, all four corners are 0. What must the middle boxes be?

One answer

Each side is corner + middle + corner = 0 + middle + 0, so every middle box must be 0 too.

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.