Algebra Play

Algebra Play — NCERT Class 8 Mathematics, Ganita Prakash Part II

Direct answer: Algebra Play (Part II Chapter 6; overall Class 8 Chapter 13) explores number tricks, number pyramids, calendar and shape grids, maximum products, divisibility and word puzzles. Algebra explains why patterns work for every allowed number, not just a few examples.

Can a friend predict your chosen date from a single number? Can you find the top of a pyramid without calculating each row? These puzzles turn algebra into a practical tool for discovering and proving hidden rules.

6.1 Algebra Play: mathematical modelling

Represent an unknown by x, translate operations into expressions, simplify, and check the answer in the original problem. Equal quantities remain equal when the same operation is applied to both sides.

6.2 Thinking about Think of a Number Tricks

Choose x, double to 2x, add 4 to obtain 2x+4, halve to x+2, subtract x to obtain 2. The starting number cancels. To force answer k, add 2k before halving. In the calendar-date trick, month M and day D lead to 100M+165+D. Subtract 165 and read the month and two-digit day. For final answers 1269, 394 and 296, subtract 165 to obtain 1104,229 and 131: these represent 4 November, 29 February (only a valid date in leap years), and 31 January, respectively. For the stated August 2025 calendar, the February date is not an August date; the trick can use any valid date.

6.3 Number Pyramids

Each upper cell is the sum of the two cells directly below. For bottom a,b,c, top=a+2b+c; for bottom a,b,c,d, top=a+3b+3c+d. Coefficients follow Pascal-triangle rows. If the first n Virahāṅka–Fibonacci terms 1,2,3,5,… form the base, the top is term F(2n−1), with F1=1 and F2=2.

6.4 Fun with Grids

In a calendar 2×2 block with top-left a, entries are a,a+1,a+7,a+8; sum=4a+16. A sum of 36 gives a=5, hence 5,6,12,13. Check that the selected cells exist within the calendar month and week. Shape grids translate repeated icons into simultaneous equations; preserve the printed row and column positions before solving.

6.5 The Largest Product

For three distinct positive digits p<q<r, the maximum two-digit-by-one-digit product is (10q+p)×r. Example: 2,3,5 gives 32×5=160, greater than 52×3=156. Comparing algebraic expansions proves the rule.

6.6 Decoding Divisibility Tricks

A two-digit number minus its reversal is a multiple of 9; their sum is a multiple of 11. Three cyclic arrangements of a three-digit number sum to 111 times its digit sum and are divisible by 37 and 3. Repeating a three-digit number yields 1001 times the original; 1001=7×11×13.

Essential formula table

Activities, errors and revision

Try the date trick with three valid dates and verify by reversing the arithmetic. Build a number pyramid on squared paper. Use a 2×2 calendar block to check its sum. For a shape grid, write one equation per complete row before solving. Typical errors include mixing up digit place values, overlooking that the top of a pyramid weights central base cells more heavily, ignoring leap years, and treating an equation with infinitely many scaled solutions as having a unique answer.

Textbook reference and scope

Source: official NCERT Ganita Prakash Part II Chapter 6 (pages 135–147). All six chapter sections are covered. The chapter has three numbered Figure it Out sets with 6, 2 and 11 main questions. The 26 worked items here include separately numbered subparts and extended applications. Some picture-based number-pyramid and shape-grid prompts require the printed diagrams for precise cell values and are not represented as verified complete solutions.