Multiples and factors

Activity 1 and Game

Discuss the definitions of multiples and factors (on the student book page) before your students begin this activity. Make sure that they understand that every number is a factor of itself, because if they divide a number by itself, there is no remainder. For example, 12 ÷ 12 = 1 without a remainder, so 12 is a factor of 12.

A prime number is a number that has only two factors, itself and 1, for example: 5, 7, 13, and 29. (Note that 1 itself is not considered to be a prime number.)

Before the students play the game, ask the following questions:

• Imagine you threw a 4 and a 6. Which squares could you choose to cover with your counter?(a number with more than two factors, a factor of 24, a multiple of 2, a multiple of 3, a multiple of 4, a multiple of 8, an even number, or a multiple of 6)
• Imagine you need a multiple of 5 to get four counters in a row. Which throws of the dice would give you a multiple of 5? (1 and 5, 2 and 5, 3 and 5, 4 and 5, 5 and 5, 6 and 5, 7 and 5, 8 and 5, or 9 and 5)

This game could be extended by asking:

• What are all the different products you could throw with the two game dice, one labelled 1–6 and the other 4–9? (4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 54)
• There are two different ways of getting a product of 12: throwing a 3 and a 4 or a 2 and a 6. Which other products can you throw more than one way using the game dice? (8: 1 x 8 or 2 x 4; 16: 2 x 8 or 4 x 4; 18: 2 x 9 or 3 x 6; 20: 4 x 5 or 5 x 4; 24: 3 x 8 or 4 x 6 or 6 x 4; 30: 5 x 6 or 6 x 5; 36: 4 x 9 or 6 x 6)
• What’s the probability of throwing a double? (There are 36 possible combinations that can be thrown with these dice, and only 3 of these are doubles: double 4, 5, or 6. So the probability of throwing a double is or .)
• Which squares in the game are easier/harder to cover? Can you use the information you have about the possible products that can be thrown to explain why? (Easier to cover: a number with more than two factors [34 out of 36 possible combinations have more than 2 factors; only 5 and 7 don’t], an even number, a multiple of 2 [27 out of 36 possible combinations are even and are therefore also multiples of 2], and a multiple of 3 [20 out of 36 possible combinations].

Harder to cover: a prime number [only 2 out of 36 combinations] and a multiple of 7 [only 6 chances out of 36].)

These problems ask students to find highest common factors and lowest common multiples. An understanding of these ideas is important for working with problems involving fractions and in algebra.

It may help the students if they make a list of all the possible products that can be thrown with the two game dice so that they can then compare this list with the factors and multiples needed in the questions.

Activity 1

1.

7, 14, 21, 28, 35, 42, 49, and so on 2. 1, 2, 3, 4, 6, 8, 12, 24

Game

A game involving factors and multiples.

Activity 2

1.

a. 8, because 8 is the highest factor that 16 and 24 have in common (highest common factor). The dice throws would be 1 and 8 or 2 and 4.

b. 4 is the highest common factor of 12, 16, and 60. (The other common factors are 1 and 2.)

2.

a. 21. (3 x 7 = 21)

b. 30. (2 x 3 x 5 = 30. 15 is a multiple of 3 and 5 but not of 2, and the only even numbers less than 30 that are multiples of 5 are 10 and 20, neither of which is a multiple of 3.)