Identifying Factors and Multiplication: 3.75

Indicator of Progress
Teaching Strategies
Further Resources

Supporting Materials

Indicator of Progress

Students can find the factors of a given number, searching systematically if necessary. They recognise that knowing factors can help with multiplication tasks.

Illustration 1: Odd and even numbers

Students may initially think (correctly) that the even numbers are those that end in 0, 2, 4, 6 or 8 and the odd numbers are those that end in 1, 3, 5, 7 or 9.

When they achieve this indicator, they will see that the more fundamental property of even numbers is that they are divisible by 2 and odd numbers are not.

Illustration 2: Multiples of 5 and 10

Students recognise that numbers ending in 5 or 0 have 5 as a factor because such numbers are all multiples of 5. The numbers ending in 0 have 10 as a factor (and hence 2 and 5 as factors as well) because all multiples of 10 end in the digit 0.

Illustration 3: Identifying prime numbers

Students know that certain numbers (such as 5, 11, 19 and 29 for example) only have themselves and 1 as factors. These are the prime numbers. When a factor tree is completed, the numbers at the ends of the tree will be prime numbers (or 1).

Illustration 4: The language of factors and divisibility

There are many different ways of talking about factors, multiples and divisibility. Here are some illustrations of correct language use particularly focusing on factors and divisibility (rather than more general division), based on the example of 48 ÷ 6 = 8.

6 is a factor of 48 (we also note that 8 is a factor of 48)
6 goes into/can be divided into 48 exactly (or 6 goes into 48 exactly 8 times)
48 is a multiple of 6
48 divided by 6 is a whole number (or 48 divided by 6 is 8, which is a whole number)
48 is divisible by 6
6 divides 48 (this is correct but is usually reserved for later secondary school)
48 factorises as 6 × 8

Teaching Strategies

The activities begin with using concrete materials to illustrate what factors look like in arrays. Because establishing fluency in identifying factors is important to applications, a number of games are included. Other activities show how factorising is used and provide practice with mental computation strategies using factors.

Activity 1: Making rectangles is an introductory activity linking factors and arrays. Students arrange counters in rectangular arrays, allowing them to identify factors.

Activity 2: What goes in? allows students time to think of finding the factors of a given number.

Activity 3: Factor grab game allows students to practise quickly thinking of the factors of numbers.

Activity 4: What else goes in? provides teachers with a set of carefully chosen examples with which they can help students identify other factors of a number.

Activity 5: Shuffling factors provides teachers with a set of carefully chosen examples that allow students to rearrange multiplication problems using factorisation so that they will be easier to calculate.

Activity 6: Today’s number is… offers a mathematical start to any day of the school year, which can often involve factors.

Activity 1: Making rectangles

Students should already be familiar with the array model for multiplication, in which, for example, 3 × 6 = 18 can be represented as a rectangular array of 3 rows of 6 counters.

3 rows of 6 counters

This activity starts with a given number and asks the students to find as many different arrays as possible that will give that number in total. For example, 12 can be represented as 12 × 1, 1 × 12, 6 × 2, 2 × 6, 3 × 4, and 4 × 3, which is 6 arrangements. Or they can recognise that 6 × 2 and 2 × 6 are equivalent (because of commutativity), and only count this once. This would give a total of 3 array arrangements for 12.

The activity can be played as a game (with variations given below). One student gives a partner a number. The partner takes that number of counters and arranges them all in an array on the table. The partner then tries again, being awarded one point for each different array he or she can make. Then swap roles, and keep score. The winner is the one who first reaches an agreed target. Students will need to agree whether commutative variations will count (i.e. whether or not you would score 6 or 3 for making the arrays of 12 described above). The variations below are in order of complexity and students may move through them quite quickly as they get familiar with the activity.

Variation 1. Students give their partner a randomly gathered pile of counters (it should be a reasonably big handful, say 20-70 counters). The partner counts the pile and then makes as many different rectangular arrays as possible. The partner gets a point for each arrangement. Now swap roles. Play 5 rounds. The player with the most points wins. Conclude the activity by reporting to the teacher (and/or class) the strategies for factorising and for choosing numbers.

Variation 2. As above, but the students select the number of counters for their partner to be given. This allows an element of strategy, when students aim to select numbers with few factors (e.g. prime numbers) so that the partner does not score many points. Let students discover this strategy for themselves and then discuss it in class.

Variation 3. As variation 2, but not permitting prime numbers.

During variation 2 and 3, students can gradually move to more mental work. Challenges can be settled by trying to make the array of counters, or by calculator. Conclude the activity by reporting to the teacher (and/or group), the strategies for factorising and for choosing numbers.

One systematic way of factorising is to try numbers in order, until the factors begin to repeat.

24
1	×	24
2	×	12
3	×	8
4	×	6
5	×	doesn’t work
6	×	I can stop here, because I already had a 6 as factor in 4 ×6 = 24

35
1	×	35
2	×	doesn’t work
3	×	doesn’t work
4	×	doesn’t work
5	×	7
6	×	doesn’t work
7	×	I can stop here, because I already had a 7 as factor in 5 × 7 = 35

Activity 2: What goes in?

This game can be played with a small group. Players need to be seated in a circle. They each have a pencil and some scrap paper to tally their own score and to record any numbers that will help them keep track of the factors. One player is selected to go first, and picks a number between 1 and 100. The player on the right then has to name one factor of that number (a number that ‘goes in’), and gets a point for doing so. The next player has to name a different factor, and gets a point if correct; and so on, around the circle. If a player says “There are no more factors” and is correct, then he/she gets a point and then chooses a new number to start the next round. If a player says “There are no more factors” and the next player can think of a factor that hasn’t been said yet, then that next player gets two points, and can start the next round.

Students can use a calculator for checking the factors if this seems appropriate for the ability of the students. You may also decide whether or not to allow 1 and the chosen number itself to be listed as factors. They certainly are factors, and so if you allow them to be counted there are always two easy correct answers, which is itself a valuable lesson to learn.

Activity 3: Factor grab game

This is a fast competitive game for two players, which allows further practice in determining factors. Ideally the two players should be of reasonably similar ability and speed. See: Factor Cards (PDF - 27Kb).

Players use the accompanying game cards which need to be printed off and laminated as follows:

The standard numbers should be printed on one colour of card. These are the Factor Cards

The bold numbers should be printed on a different colour. These are the Find My Factors Cards.

The Factor Cards are spread out face up between the two players in any order, and the Find My Factors Cards are shuffled and kept in a pile, face down. The top card in the pile is turned over for both players to see. The two players then grab any of the cards that are spread out on the table, that are factors of the nominated number.

Players can only take a factor once in the round. So, for example, if the Find My Factors Card is 28, a player can grab one (only) of each of the following Factor Cards if they are available: 1, 2, 4, 7, 14 and 28. The player must not take more than one of each of these.

Each player then checks the other’s chosen Factor Cards and challenges any incorrect selections. If the player is right about the opponent’s wrong selection, then that player can take any two of the opponent’s cards (it doesn’t matter which cards are taken).

Once the challenge period is over the players put their grabbed cards aside in a pile, as these are used to keep score.

The next Find My Factors Card is turned over, and players again grab Factor Cards from the pile still remaining between them. As the game progresses some factors will become unavailable, making knowledge of other possibilities and speed desirable. Once all the Factor Cards have been grabbed, players count the number of cards they have collected to determine the winner. (A calculator may be used to resolve disputes).

Note: The game can also be made less speed oriented by having students take it in turns to select single cards from the pile of Factor Cards, continuing to do so until one player can’t find any more, at which point the other player can grab any others that have been overlooked.

Activity 4: What else goes in?

Teachers should model how knowing pairs of factors of a number can help us find other factors of a number. For example, if we know that 28 is 4 × 7 we also know that 28 = 2 × 14 by breaking up the 4 as 2 × 2. This means that we know that 2 and 14 are factors of 28, as well as 4 and 7. This is a consequence of the associative property of multiplication, which allows us to ‘shuffle’ factors of numbers.

For more information see: More About Properties of Operations. (Level 2.75) which addresses the Commutative, associative and distributive properties.

To help students appreciate this way of using factors to find other factors, teachers could try asking questions like “If 36 = 4 × 9, what else goes into 36?” and get students to explain why. Some good answers would be “I see that 3 must be a factor of 36, because it is a factor of 9” and “I see that 6 is a factor of 36, because 2 is a factor of 4 and 3 is a factor of 9”.

A list of suitable factorisation equations for such an activity is given below:

36 = 4 × 9	32 = 4 × 8	60 = 6 × 10
80 = 8 × 10	72 = 8 × 9	42 = 7 × 6
56 = 8 × 7	64 = 8 × 8	40 = 5 × 8
100 = 10 × 10	96 = 12 × 8	108 = 12 × 9

Teachers should also pose the following challenging exercises, because they highlight how much we can learn about a number by knowing some of its factors. Later this leads to prime factorisation.

144 = 12 × 12	256 = 64 × 4	1000 = 100 × 10
216 = 6 × 36	280 = 4 × 70	600 = 60 × 10

Example:

If we know that 280 = 4 × 70, then we also know that:

280 = 4 × 70 so 280 =
2	×	140	shuffle the factor of 2 from the 4 to the 70
40	×	7	shuffle the factor of 10 from the 70 to the 4
28	×	10	shuffle the factor of 7 from the 70 to the 4
20	×	14	shuffle the factor of 5 from the 70 to the 4
2 × 2	×	7 × 10	(this way of writing the factors begins to reveal the factor structure and the importance of prime numbers)
2 × 2	×	2 × 5 × 7

Activity 5: Shuffling factors

Teachers should model how to use factorising and ‘shuffling’ numbers around to make mental computation easier. For example, to multiply by 200, it is easier to multiply by 2 (or double) and then multiply by 100 than to multiply by 200 itself. For example, 17 × 200 = 34 × 100 = 3400.

A different example is 35 × 6, which might be calculated by breaking the 35 back into 5 × 7, so that we have 35 × 6 = 5 × 7 × 6. We can then work out the 7 × 6 first (which is 42) and then multiply by the 5 (which is easily done by multiplying by 10 and then halving), and this gives 210.

A final common example, is the halve/double method, which is useful when multiplying by powers of 2 (2, 4, 8, 16, 32 etc). Here, something like 43 × 32 is done by halving the 32 part repeatedly (since it is a power of 2) and doubling the 43 part correspondingly. So, we have 43 × 32 = 86 × 16 = 172 × 8 = 344 × 4 = 688 × 2 = 1376.

Teachers should model and explain these strategies, and provide students with opportunities to practise them. Some suitable examples are given below. Remind students that it may help to look at the commuted (spun around) equivalent expression, e.g. look at 43 × 32 instead of 32 × 43. These relationships come from the properties of multiplication that are part of Structure dimension.

For more information see: More About Properties of Operations. (Level 2.75) which addresses the Commutative, associative and distributive properties.

Multiples of 10 and 5

4 × 90	14 × 20	6 × 300
8 × 2000	7 × 900	3 × 4000
8 × 70	800 × 40	500 × 8000
35 × 8	25 × 6	7 × 45
250 × 7	40 × 450	70 × 150

Powers of 2

4 × 72	34 × 8	64 × 12
8 × 300	3000 × 16	70 × 64
32 × 71	80 × 85	15 × 160

Mixed (to encourage students to make their own decision about the best strategy)

6 × 32	140 × 8	70 × 45
16 × 53	32 × 50	600 × 40
800 × 25	150 × 4	35 × 40
250 × 6000	120 × 8	2000 × 64

Activity 6: Today’s Number Is…

Each day of the school year can be given a number. Start the day by looking at the properties of this number. A good question to ask on any day is “What are the factors of this number?” and to discuss quick ways of finding the factors (e.g. number is even so 2 is a factor, number ends in 5 or 0, sum of digits is multiple of 3 so number is multiple of 3 etc).

Further Resources

The following resource contains sections that may be useful when designing learning experiences:

Digilearn object *

Arrays: explore factors – students explore how numbers can be broken up with factors. For example, the number 9 can be expressed as 9x1 or 3x3. Students predict the factors of a number in the range 1 to 50. Students make an array of equal rows and columns with the number to check its factors. Students choose a statement to describe how many factors the number has.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4492)

* Note that Digilearn is a secure site; SMART::tests login required.

Mathematics Developmental Continuum P-10