7.3 Introduction to Algebra

**Think-of-a-number chain.** I think of a number. I triple it, then add 5. Then I subtract twice the original number. The result is 11. What is the number? Show your working using algebra.

**Perimeter puzzle.** A rectangle has length $(3x + 1)$ cm and width $(x + 3)$ cm. Its perimeter equals 48 cm. Find $x$, then calculate the area of the rectangle.

**Age problem.** Priya is $x$ years old. Her brother is 5 years older. Their mother is three times Priya's age. The sum of all three ages is 60. How old is each person?

**Consecutive integer magic.** Alex claims: "Pick any three consecutive integers. Multiply the outer two together, then subtract the square of the middle one. You always get −1." (a) Test this with the consecutive integers 5, 6, 7. (b) Test it with −3, −2, −1. (c) Prove algebraically that it always works. Let the middle integer be $n$.

**Equation from context.** Three friends share the cost of a meal equally. A service charge of £6 is added to the total before splitting. Each person ends up paying £14. Write and solve an equation to find the cost of the meal before the service charge.

**Directed numbers + algebra.** I think of a number. I subtract 8 from it, then multiply the result by 3. The answer is $-6$. Form and solve an equation to find the number.

**The number machine — fixed points.** A number machine applies the rule: "Multiply by 3, then subtract 10." (a) What output does an input of 7 produce? (b) What input gives an output of −1? (c) Find the "fixed point" — the input that gives the same value as the output. Form and solve an equation. (d) What happens to inputs above the fixed point after many repeated applications? What about inputs below it?

**Temperature puzzle.** The temperature at midnight is $t$ °C. By 6 am it has risen by 7 °C. By noon it has doubled from the 6 am temperature. The noon temperature is 8 °C. Find $t$, the midnight temperature.

**Pattern and algebra.** A sequence of patterns is made from square tiles: - Pattern 1: 5 tiles - Pattern 2: 9 tiles - Pattern 3: 13 tiles (a) Write an expression for the number of tiles in pattern $n$. (b) Which pattern uses exactly 41 tiles? (c) Is there a pattern that uses exactly 100 tiles? Explain.

**Staircase numbers.** A "staircase number" is any positive integer that can be written as the sum of two or more consecutive positive integers. For example, $9 = 4 + 5 = 2 + 3 + 4$. (a) Show that 15 is a staircase number in at least **three** different ways. (b) Test whether 16 is a staircase number. (Try all possible staircases with 2, 3, 4, or 5 consecutive integers.) (c) Show algebraically that the sum of $k$ consecutive integers starting from $n$ is $kn + \dfrac{k(k-1)}{2}$. (d) Which powers of 2 are staircase numbers? Make a conjecture.

**Balance puzzle.** On one side of a balance: 3 identical boxes and a 5 kg weight. On the other side: 7 identical boxes. All boxes are the same mass. The scales are balanced. Find the mass of one box. What total mass is on each side?

**Extended reasoning.** Jamie says: "If I square any integer and subtract the integer, the result is always even." (a) Test Jamie's claim for $n = -3$. (b) Prove algebraically that $n^2 - n$ is always even for any integer $n$. (c) If $n^2 - n = 30$, find all integer solutions for $n$.