Suryamg Master

Chapter 6: Algebra Play

Simulation Topics

Simulation 1: 'Think of a Number' Trick Simulator

Learning Objective: Test algebraic tricks step by step and see why the final result is always constant.

Step 1 (Think): 7 → Step 2 (Double): 14 → Step 3 (Add 4): 18 → Step 4 (Divide 2): 9 → Step 5 (Subtract x): 2

Simulation 2: Number Pyramid Solver

Learning Objective: Build and solve a 3-row number pyramid where each cell is the sum of the two directly below it.

Middle Row: [32, 28] | Top Apex = 60

Simulation 3: Largest Product Finder

Learning Objective: Find how to arrange three digits into a 2-digit number and 1-digit multiplier to maximize the product.

Optimal Arrangement: 52 × 3 = 156

Chapter Sections

6.1 & 6.2 'Think of a Number' Tricks

Algebra allows us to model numerical puzzles, investigate magic tricks, and explain why they always work regardless of the starting number.

Date Guessing Trick Example - Choose a date (Month M, Day D).
- Multiply M by 5, add 6, multiply by 4, add 9, multiply by 5, add D.
- Formula resulting: 100M + 165 + D. Subtracting 165 leaves 100M + D, where M forms the first digits and D forms the last two digits.

6.3 Number Pyramids

In a number pyramid, each number is the sum of the two numbers directly below it. Using letter-numbers (like a, b, c) allows us to solve unknown cells and express top apex values algebraically.

6.4 Fun with Grids & Calendar Magic

Selecting 2x2 grids or cross shapes on monthly calendars yields predictable algebraic sums related to the top-left or center date.

6.5 The Largest Product

To maximize the product of a 2-digit number and a 1-digit multiplier formed from three digits (p < q < r), the largest digit (r) must be the multiplier, and the other two digits must be arranged in decreasing order to form the multiplicand (qp × r).

Complete Figure It Out Solutions

Figure It Out: Section 6.5 (Page 144)

1. Fill the digits 1, 3, and 7 to make the largest product possible:
- Options: 71 × 3 = 213, 73 × 1 = 73, 31 × 7 = 217.
- Answer: 71 × 3 = 213 (or check 71 × 3 vs 31 × 7 vs 17 × 3 -> 71 × 3 = 213).

2. Fill the digits 3, 5, and 9 to make the largest product possible:
- Options: 93 × 5 = 465, 95 × 3 = 285.
- Answer: 93 × 5 = 465.

6.6 Decoding Divisibility Tricks & Word Problems

Divisibility rules and number properties can be proved using polynomial expansions in base 10 (e.g., a two-digit number ab equals 10a + b).

Complete Figure It Out Solutions

Figure It Out: Section 6.6 (Page 145-147)

1. 2-digit reversal difference: Quotient when dividing difference by 9 is b - a (the difference of the digits).

2. Sum of 2-digit number and its reversal: Sum of ab and ba is 11(a + b), which is always divisible by 11.

3. Cyclic 3-digit sum (abc + bca + cab): Sum equals 111(a + b + c) = 37 × 3(a + b + c), hence always divisible by 37 and 3.

6. Horses and hens: 55 heads, 150 legs. How many of each?
- Step 1: If all 55 were hens, legs = 55 × 2 = 110 legs.
- Step 2: Extra legs = 150 - 110 = 40 legs.
- Step 3: Each horse has 2 extra legs compared to a hen (4 - 2 = 2). Number of horses = 40 / 2 = 20.
- Step 4: Number of hens = 55 - 20 = 35.

7. Mother is 5 times daughter's age. In 6 years, 3 times. Daughter's age now?
- Let daughter be x, mother be 5x.
- In 6 years: 5x + 6 = 3(x + 6) → 5x + 6 = 3x + 18 → 2x = 12 → x = 6 years old.

8. Gauri and Naina cows: Naina has twice as many as Gauri (N = 2G). If Naina gives 3 to Gauri, they have equal (N - 3 = G + 3):
- 2G - 3 = G + 3 → G = 6. Naina = 12 cows.