Before Viewing

Revise the terms ‘multiple’ and ‘factor’.

Ask students to list some sets of multiples and some sets of factors.

Remind students of the meaning of the term ‘prime number’ and ask for examples of ones they already know.

Revise the index notation for expressing powers.

Remind students of some simple sets of powers, such as powers of 2, 3 or 10. (Alternatively this could be done after viewing the programme.)

During or After Viewing

Students are offered the opportunity in Worksheets 1 and 2 to carry out the investigations that Lisa undertook on the number grid. This should allow a fuller appreciation of the concepts involved and greater opportunity to explore and discuss patterns.

Section 1: Priming the Grid

Discuss the patterns that arise on the 6-column grid as the activity is carried out by Lisa. When students try this activity they could use different colours for the different multiples. (Note that different patterns would appear on a 10x10 grid.)

Point out the links between sets of multiples. For example, the multiples of 4 are already coloured because they are all even. The multiples of 6 are all multiples of both 3 and 2.

Encourage students to formulate simple rules for divisibility by 2, 3 and 5. These will help in testing for larger prime numbers.

Lisa carried on as far as multiples of 19. Did she need to? Can students identify when they can stop the activity. Can they explain how they know they have not missed any multiples?

Invite students to make general statements about the prime numbers they have generated, such as:

• They are all odd, except 2.
• They get less frequent as the numbers get bigger.
• A lot of them end in 7.

Give examples of larger numbers that students can test to see if they are prime.

Discuss the fact that the number 1 is not prime. Why is this?

Section 2: Consecutive Sums

Lisa quickly concluded that summing in pairs would give all the odd numbers. Invite students to check some larger odd numbers to find the pairs of consecutive numbers that generated them. Can students justify her conclusion?

Lisa found that adding three consecutive numbers produced the multiples of 3. Do students agree, and can they explain it?

Can students describe the patterns for adding four, five and six consecutive numbers? How do they decide when to stop calculating? Why?

The unshaded numbers are 2, 4, 8, 16, 32, 64. Discuss writing these as powers of 2. Can students continue the sequence?

Can they give sequences for powers of 3, 4, 5, 10 and so on? How far can they get without a calculator? How are the powers of 2 and of 4 related? What other sets of powers are related in this way?

Comments

Section 1: Priming the Grid

The use of a 6-column grid makes the early stages of the prime number investigation easier and more accessible. The multiples of 2 fall in the second, fourth and sixth columns, those of 3 in the third. These patterns make mistakes in recording less likely. The multiples of 5 and 7 fall diagonally on this grid.

Point out that this ancient method of finding primes (often called the Sieve of Eratosthenes) is still used by powerful computers testing large numbers for primality.

Students should appreciate that a systematic approach, testing the smallest candidates first, is helpful in finding factors. To establish if a number is prime, only prime factors need be sought, since all other potential factors are multiples of previous primes.

Students could explore common multiples through examination of squares containing more than one colour of shading.

Consideration could be given to the usefulness of tests for divisibility by different numbers. Some are very simple while others are quite complex.

Section 2: Consecutive Sums

Here it is assumed that students will work systematically, summing pairs, then three, four, five and six consecutive numbers, and looking for patterns each time.

Most students should be able to reason that ‘odd + even = odd’. Many students should be able to follow a generalisation that if the first number is n, the second is n + 1, so their sum is n + n + 1 = 2n + 1. Since 2n is even, 2n + 1 must be odd.

Sums of 3 consecutive numbers are all multiples of 3; sums of 5 consecutive numbers are all multiples of 5; sums of 7 consecutive numbers are all multiples of 7; and so on.

Sums of 4 consecutive numbers are all even; and sums of 6 consecutive numbers are all multiples of 3

Students can find the following expressions for the general sums by considering

n + (n + 1) + (n + 2) + ... + (n + k — 1)

where n is the first number and k is the number of terms.

For 2 consecutive numbers, the sum is 2n + 1.
For 3 consecutive numbers, the sum is 3n + 3.
For 4 consecutive numbers, the sum is 4n + 6.
For 5 consecutive numbers, the sum is 5n + 10.
For 6 consecutive numbers, the sum is 6n + 15.

and so on.

Able students could be asked to find an expression for the sum of k consecutive numbers.

There is a proof of why a power of two cannot be expressed as a sum of consecutive integers at http://www.nrich.maths.org.uk/mathsf/journalf/may97/prob4.html, though most students would probably find it too difficult at this level.

The work could be extended to expressing numbers in terms of their prime factors.

The Worksheets

Worksheets 1 and 2

For more information, see the Programme Overview above.

Worksheet 3: Consecutive Products

This structured investigative activity builds on the understanding that students have gained of consecutive numbers and provides an opportunity to look for factors and highest common factors.

The product of two consecutive numbers is always even. (In each pair one number will be odd and the other even.)

For three consecutive numbers, at least one must be even, and one must be a multiple of 3. So the product must be a multiple of 6.

The product of four consecutive numbers is always divisible by 24. Of the four numbers, one must be a multiple of 4, at least one must be a multiple of 3, and one (apart from the multiple of 4) must be a multiple of 2: 4 x 3 x 2 = 24.

Worksheet 4: Generation Games

This open-ended activity provides practice in substitution, further work on powers of 2, and opportunities to test for prime numbers.

(a) Students should obtain the results: 1, 3, 7, 15, 31, 63, 127 (prime), 255, 511 (7 x 73), 1023, 2047 (23 x 89), 4095.

If n is a multiple of 4, the Mersenne number M_n always ends in 5.

Students should be aware that if a number factorises into a product of two smaller numbers, the smaller factor must be no greater than than the square root of the number.

Mersenne (in the seventeenth century) found a number with 69 digits that he thought was prime. Three centuries later, in 1984, a team of mathematicians using a computer finally managed to find three factors of this Mersenne number.

(b) Students could work in groups here to explore substitution and testing small sets of values.

The first formula works for x < 16.

The second formula generates primes for all other numbers from 0 to 80 except for x = 40, 41, 44, 49, 56, 65 and 76.

Other interesting formulae are:

2x² + 29 (for x < 29)
x² — 79x +1601 (for x < 80)

Some students may appreciate the following proof that there is no greatest prime number, which was discovered by the Greeks:

Imagine there is a greatest prime, p. Multiplying all the primes up to and including p and adding 1 gives a number that is not divisible by any of the known primes. This number must either be prime or have a prime factor greater than p. Hence p cannot be the greatest prime.

Further Ideas

Factors for Numbers 1 to 40

Build up a table of factors for the numbers 1 to 40.

Recognise which numbers have two, three, four factors, and so on.

By consideration of the number of factors, students can begin to classify the numbers as follows:

The only number with one factor is 1.
The numbers with two factors are the prime numbers.
The numbers with three factors are all square.
Square numbers all have an odd number of factors (for example, 16 has 5 factors: 1, 2, 4, 8 and 16).
Rectangular numbers that are not square all have an even number of factors.
Some numbers with 4 factors are perfect cubes (such as 8 and 27).

Some students may find it helpful to approach this activity in a more visual way, by finding how many different ways there are to represent each number with an array of dots.

Summing the First n Numbers

A method for finding the sum of the numbers form 1 to n may appeal to potential wizards!

For example, to find the sum of the numbers

1 + 2 + 3 + 4 + ... + 12

Add the numbers in reverse:

12 + 11 + 10 + 9 + ... + 1

Adding these together, we see that twice the required sum is:

13 + 13 + 13 +13 + ... + 13 = 12 x 13 = 156

So the sum is 156 ÷ 2 = 78.

What is the sum of the first 100 whole numbers? The first 1000?