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Problem-solving Pack
MathematicsYear 7 · 7.3 Introduction to Algebra
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
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.
What is the number? Show your working using algebra.
Working space
2Problem 2 of 12
Perimeter puzzle. A rectangle has length cm and width cm. Its perimeter equals 48 cm.
Find , then calculate the area of the rectangle.
Find , then calculate the area of the rectangle.
Working space
3Problem 3 of 12
Age problem. Priya is 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?
The sum of all three ages is 60. How old is each person?
Working space
4Problem 4 of 12
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 .
(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 .
Working space
5Problem 5 of 12
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.
Write and solve an equation to find the cost of the meal before the service charge.
Working space
6Problem 6 of 12
Directed numbers + algebra. I think of a number. I subtract 8 from it, then multiply the result by 3. The answer is .
Form and solve an equation to find the number.
Form and solve an equation to find the number.
Working space
7Problem 7 of 12
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?
(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?
Working space
8Problem 8 of 12
Temperature puzzle. The temperature at midnight is °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 , the midnight temperature.
Find , the midnight temperature.
Working space
9Problem 9 of 12
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 .
(b) Which pattern uses exactly 41 tiles?
(c) Is there a pattern that uses exactly 100 tiles? Explain.
- Pattern 1: 5 tiles
- Pattern 2: 9 tiles
- Pattern 3: 13 tiles
(a) Write an expression for the number of tiles in pattern .
(b) Which pattern uses exactly 41 tiles?
(c) Is there a pattern that uses exactly 100 tiles? Explain.
Working space
10Problem 10 of 12
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, .
(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 consecutive integers starting from is .
(d) Which powers of 2 are staircase numbers? Make a conjecture.
(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 consecutive integers starting from is .
(d) Which powers of 2 are staircase numbers? Make a conjecture.
Working space
11Problem 11 of 12
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?
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?
Working space
12Problem 12 of 12
Extended reasoning. Jamie says: "If I square any integer and subtract the integer, the result is always even."
(a) Test Jamie's claim for .
(b) Prove algebraically that is always even for any integer .
(c) If , find all integer solutions for .
(a) Test Jamie's claim for .
(b) Prove algebraically that is always even for any integer .
(c) If , find all integer solutions for .
Working space