State whether the statements are true or false. Explain why each time. 4.5 is a multiple of 3 1002 is a multiple of 3 0 is a multiple of 3 Freyas age is a multiple of 7 Next year her age will be a multiple of 5

How old could Freya be? Freya then says that two years ago her age was a multiple of 4 How old must Freya be? If is a positive integer, state whether the following is a positive integer, state whether the following expressions are always, sometimes or never a multiple of 4 Explain your choice of answer each time. Pens come in packs of 8. Pencils come in packs of 6

Mrs Potts buys packs of both and has 120 pens and pencils in total. How many packs of each did she buy? Ravi thinks there are 4 different combinations of packs of pencils and pens that Mrs Potts could have bought. Find the four ways. Use arrays of 24 counters to find all the factors of 24 How many ways can you find to show that 3 is not a factor of

10? What diagrams can you use? What sentences can you write? Sally uses a bar modelling to represent 4 + 8 + 8 Sally notices that 2 lots of 2 + 8 + 4 is the same as 4 + 8 + 8 She concludes that both 2 and 2 + 8 + 4 are factors of 4 + 8 + 8 Draw another bar model, or an array, to represent different factors of 4 + 8 + 8

In each list, whats the same? Whats different? Find the factors of each expression e.g. so and are both factors of so . Find the factors of all the integers from 1 to 20 How many of the first 20 integers are prime? How many have an odd number of factors?

Investigate further. Explore these statements by substituting in values of is a positive integer, state whether the following into the expressions. Are Dora and Whitney correct all of the time, some of the time or never? Why? Dora That means the expression could be a

prime number. Whitney The expression is always even and so is never prime. Raffle tickets with the numbers 1 to 100 on are placed in a

bag. To play the game, you randomly select a raffle ticket. You choose the winning criteria. Which would you choose? Justify your answer. Win a prize for a multiple of 8 Win a prize for a prime number. Make the following square numbers using counters.

What do you notice about the way square numbers increase. How many counters will you need for the 10th pattern? 20th pattern? Make the following triangular numbers using counters. What do you notice about the way triangular numbers increase. How many counters will you need for the 10th pattern? 20th pattern?

Which triangle numbers can you see within this square number? Investigate with other square numbers. What relationships can you find? Mo A triangle is half of a square. This will mean a triangular

number will be half of a square number. Draw as many different rectangles as possible, with integer lengths and widths, with an areas of: 12cm2 18cm2 Write down the highest common factor of 12 and 18

Use this fact to help work out the highest common factors of these pairs of numbers. 120 and 180 6 and 9 24 and 36 18 and 27

Bob has two pieces of ribbon, one 75 cm long and one 45 cm long. He wants to cut them up into smaller pieces that are all of the same length, with no ribbon left over. What is the greatest length of ribbon that he can make from the two pieces of ribbon? Keira works out the HCF of some pairs of numbers. Pair 12 and 30

30 and 60 12 and 60 HCF 6 30 12 She says the HCF of 12, 30 and 60 is 30 because it is the highest HCF of the pairs.

Do you agree? Why or why not? Marcel uses number lines to find the LCM of 9 and 12 27 18 9 12

24 36 36 45 60 48

Explain how Marcel uses this to find the LCM of 9 and 12 Find the LCM of the following numbers. 6 and 8 6 and 15 6, 8 and 15 Marcel notices that the LCM of 6, 8 and 15 is 5 times as large as the LCM of 6 and 8. Explain why this is true.

Explain how the LCM can be used to compare the size of the following pairs of fractions: and and and What other ways could you use to compare the size of the fractions? At a bus stop, Bus A arrives every 4 minutes and bus B

arrives every 6 minutes. Bus A and B both arrive at 10am. At what time to Bus A and Bus B arrive together next? Bus C arrives every 10 minutes. How many times per hour do buses A, B and C arrive at the same time? Tom knows that 90 = 5 18 He starts his prime factor tree: 90

5 18 90 Amir knows that 90 = 6 15 He starts his prime factor tree: 6 Complete both trees.

What do you notice about your final answers? 15 Whats the same and whats different about each question? Which of the questions can you answer in more than one way? Why is it only possible to express the product of 80 in terms of its prime factors in one way only? List the factors of 80

Express 80 as a product of its prime factors. Factorise 80 Liam is investigating the prime factorisation of some numbers. He lists his results so far in a table. Copy and complete the table.

Number Prediction How do you know? 12 2 2 3

12 = 4 3 and 4 = 2 2 so 12 = 2 2 3 24 2 2 3 2 24 is double 12, so 24 = (2 2 3) 2 72 36

144 Use a prime factor trees to check your answers. The Venn Diagram shows the prime factors of 24 and 60 Why does 2 appear twice in the intersection? Why does 5 not appear in the circle representing 24? How does the intersection help us find the all the common factors, and so the HCF of 24 and 60? How can we use the diagram to find the LCM of 24 and

60? 24 2 2 3 2

60 5 Express 105 and 120 as products of their prime factors. Use a Venn diagram to find the HCF and LCM of 105 and 120. Use your answers to work out: The HCF of 105 and 60 The HCF of 1050 and 120

The LCM of 105 and 240 The LCM of 105 and 12 Mo expresses two numbers as a product of their prime factors: 30 = 2 3 5 36 = 2 2 3 3 Mo said: I can use these prime factors to calculate the lowest common multiple of 30 and 36: 2 3 5 2 2 3 3 = 1080

Mo Explain Mos mistake and find lowest common multiple of 30 and 36 Sort these conjectures into: always true, sometimes true, never true. Birds can fly. Odd + Odd = Even

1, 2, 4.. The numbers in the sequence are doubling each time. In an equilateral triangle, each angle is 60 To find the area of a shape, you

multiply length by the width. = 1, 1, 2, 3, 5 Here is a 5-term Fibonacci sequence. Add together the first and last terms in the sequence. What do you notice about the relationship between this, and the middle term of the sequence?

Repeat with several other 5-term Fibonacci sequences. Is this result always true? Can you use counters or cubes to prove it? Sarah finds the HCF of 12 and 18 is 6 She also finds the LCM of 12 and 18 is 36 She notices that She conjectures that that the product of the HCF and LCM of two numbers is always the same as the product of the numbers themselves.

Investigate Sarahs conjecture. Exemplification Exemplification Eva 42 = 16 So, it must be true that if I square a number, the result is

always greater than the number I start with. Write down a counterexample to show that this conjecture is not always true. Saffi rolls two dice. She subtracts the scores on the two dice and makes the conjecture: The difference between the scores on two dice is always even..

Do you agree, or can you find a counterexample? She conjectures again. Is this conjecture true? If not, find a counterexample. If the total of the scores on the two dice is even, then the difference between the scores on two dice is also even.. Ali works out the perimeter and area of this square. Perimeter = 24 cm Area = 36 cm2 He thinks The perimeter of a square can never be equal to

its area. Do you agree? Justify your answer. 6 cm 6 cm