Patterns and relationships with algebra

Patterns, then expressions

Algebraic expressions can do more than record a rule: they can show why a pattern must always hold. The approach is always the same. Describe the relationship in simple words, or draw it, then replace the changing quantity by a letter-number and write the expression.

Formula Detective

A number machine takes two numbers at the top of a Y and gives one at the bottom, doing the same thing every time. One machine turns 5 and 2 into 8, 8 and 1 into 15, 9 and 11 into 7, and 10 and 10 into 10. Its rule: double the first number, then subtract the second, 2a − b, so 6 and 4 give 2 × 6 − 4 = 8. Another turns 5 and 2 into 5, and 9 and 11 into 18: it adds the numbers and takes away 2, a + b − 2. A third turns 4 and 1 into 5 and 6 and 0 into 1: it multiplies and adds 1, ab + 1, so 10 and 3 give 31.

Repeating designs

Somjit sees designs A, B and C repeating along a saree border. Design C is at positions 3, 6, 9, …, the multiples of 3, so its nth appearance is at 3n. B is always one place earlier, at 3n − 1, and A two places earlier, at 3n − 2. To go the other way, divide the position by 3: a remainder of 0 means C, 2 means B and 1 means A. So position 99 has C, 122 has B and 148 has A.

Patterns in a calendar

In the November 2024 calendar, the 2 × 2 square with 12, 13, 19 and 20 has equal diagonal sums: 12 + 20 = 32 and 13 + 19 = 32. Is that true for every 2 × 2 square, even in a calendar continued without end? Checking them all is impossible, but algebra settles it. Call the top-left number a. The number to its right is a + 1, the one below is a + 7, and the one diagonally below is a + 8. The diagonal sums are a + (a + 8) = 2a + 8 and (a + 1) + (a + 7) = 2a + 8: equal for every a, so for every square.

The same idea works for other shapes. In a plus shape with centre a, the numbers are a − 7, a − 1, a, a + 1 and a + 7, and they add to 5a, five times the centre.

Matchstick patterns

A row of triangles made of matchsticks needs 3 sticks for step 1, 5 for step 2, 7 for step 3 and 9 for step 4: each step adds 2. Step y adds 2 to the first 3 sticks y − 1 times, so it needs 3 + 2(y − 1) sticks, which simplifies to 2y + 1. Seeing that the first 3 is 1 + 2 gives 2y + 1 directly. Counting another way, there are y flat sticks along the top and bottom and y + 1 slanting ones, which again add up to 2y + 1. So step 33 needs 67 sticks, step 84 needs 169 and step 108 needs 217.

Why it matters

Mathematicians do not trust a pattern until they can explain it, and algebra is how they do it. The basics of writing expressions are in letter-numbers and algebraic expressions, and the simplifying used in these proofs in simplifying algebraic expressions. More patterns, such as traffic signals, ropes and number grids, are solved in the NCERT solutions for Chapter 4; see all of the Chapter 4 notes.