Making conjectures about patterns and relationships | Maths

Starter quiz

Which of these numbers are prime?

1

2 ✓

3 ✓

27

41 ✓
Which of these are square numbers?

1 ✓

10

22

68

100 ✓
Which of these are triangular numbers?

5

9

10 ✓

15 ✓

20
The number 100 has exactly ______ factors.

'9' ✓
Which of these expressions must represent a number 'one more than a multiple of 5' when $n$ is an integer?

$n+1$

$2n+3$

$5n+1$ ✓

$5n-4$ ✓

$10n+2$
What is the value of $3^n$ when $n=4$ ?

'81' ✓

Exit quiz

What sort of numbers always have an odd number of factors?

even numbers

odd numbers

prime numbers

square numbers ✓

triangular numbers
Give an example of a number less than 100 which is both square and triangular.

'36' ✓
A conjecture is a (mathematical) statement which __________.

is always true.

cannot be proven.

is impossible.

is not yet proven true. ✓
Lucas writes the conjecture "All numbers of the form $2^{n}-1$ (where $n$ is an integer greater than 1) are prime". Which of the following are true?

This does not hold when $n=2$

This does not hold when $n=3$

This does not hold when $n=4$ ✓

This does not hold when $n=7$

This does not hold when $n=9$ ✓
Which of these shows the first 4 numbers of the form $7^n$ where $n$ is a positive integer?

7, 14, 21, 28

1, 7, 49, 343

1, 128, 2187, 16384

7, 49, 343, 2401 ✓

7, 77, 777, 7777
Jacob has written out the first 4 numbers of the form $7^n$ where $n$ is a positive integer. Which of these conjectures hold true for these 4 values?

Numbers of this form always end in either 1, 3, or 9

Numbers of this form cannot be multiples of 10 ✓

Numbers of the form $7^{2n}$ where $n$ is a positive integer end in 9

Numbers of the form $7^{4n}$ where $n$ is a positive integer end in 1 ✓

Worksheet

Loading worksheet ...

Presentation

Loading presentation ...

Video

Lesson Details

Key learning points

A conjecture can be made about underlying patterns.
The conjecture can be tested to see if it holds.

Common misconception

Conjectures have to be correct so it's really hard to make a conjecture as you don't want to be wrong.

Conjectures are thought to be true but they may not be. This is why we test and see ways to prove whether a conjecture is true or not. It is perfectly acceptable for a conjecture to be proved wrong.

Keywords

Prime number - A prime number is an integer greater than one with exactly 2 factors.
Triangular number - A triangular number is a number that can be represented by a pattern of dots arranged into an equilateral triangle. The term number is the number of dots in a side of the triangle.
Conjecture - A conjecture is a (mathematical) statement that is thought to be true but has not been proved yet.