Pick Patterns and Reveal Relationships

Class 7 Maths · Chapter 4: Expressions using Letter-Numbers · NCERT Ganita Prakash section 4.5

Learn

Formula detective: a number machine takes two numbers and gives one: (5, 2) → 8, (8, 1) → 15, (9, 11) → 7. Try "two times the first minus the second": 2 × 5 − 2 = 8 ✓, 2 × 8 − 1 = 15 ✓, 2 × 9 − 11 = 7 ✓. Its formula is 2a − b. Make your own machine and challenge a friend!

2a − b

Repeating designs: a border repeats A, B, C, A, B, C … C sits at positions 3, 6, 9 …, so its nth appearance is at 3n; B is one before, at 3n − 1; A at 3n − 2. Which design is at position 122? 122 ÷ 3 leaves remainder 2, like position 2 → B. A remainder of 0 means C, and 1 means A.

C: 3n · B: 3n − 1 · A: 3n − 2

Calendar magic: in any 2 × 2 square of dates with a at the top left, the others are a + 1, a + 7 and a + 8. The diagonal sums are a + (a + 8) = 2a + 8 and (a + 1) + (a + 7) = 2a + 8: always equal! Algebra proves it for every square at once, which checking a few could never do.

a + (a + 8) = (a + 1) + (a + 7) = 2a + 8

Matchstick triangles: a row of 1, 2, 3, 4 … triangles uses 3, 5, 7, 9 … sticks: 2 more each time. Step y has 3 plus (y − 1) twos: 3 + 2(y − 1), which simplifies to 2y + 1. So step 33 needs 67 sticks and step 108 needs 217, without drawing a single one!

3 + 2(y − 1) = 2y + 1

Explore: which design is where?

1. The border repeats A, B, C. Sort each position by its design.

Remember

• Describe a pattern in words first, then as an expression (C at 3n, B at 3n − 1, …).
• Algebra can show a pattern holds for every case: 2 × 2 calendar diagonals both equal 2a + 8.

Quick check

Using 2y + 1, how many matchsticks are in step 10 of the triangle row?

1. 21
2. 20
3. 30
4. 12

Think about it

A rope folded once and cut gives 3 pieces; folded twice, 5 pieces. What is the rule for r folds?

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.