Mental Strategies for Multiplication: 3.5

Indicator of Progress
Teaching Strategies
Further Resources

Supporting materials

Indicator of Progress

At this level students can mentally carry out a wide range of multiplication calculations and estimations involving large numbers. Success depends on the use of strategies based on their understandings of multiplication, knowledge of base ten properties and multiplication basic facts.

Before achieving this, students will be able to multiply mentally with single digit numbers (and 11 and 12) using fewer strategies.

Good mental calculation is characterised by having many strategies, which can be applied flexibly to meet the task at hand. Sometimes, particular numbers suit particular strategies. Sometimes strategy selection is guided by personal preferences, or special number combinations that a student happens to spot. Mental strategies are not carried out like written algorithms, in a standard way. Instead, students need to choose them and adjust them to suit the calculation. Written algorithms have been designed for efficient calculation using pencil-and-paper technology, and are not so suitable for mental computation.

Mental calculation and estimation is very important for everyday life. In fact, the combination of good mental calculation and estimation plus good calculator skills is much more valuable to citizens than good written computation. Mental calculation and estimation are also important as a learning device because they allow students to practise and discuss fundamental number properties and build number sense.

Go to More about Properties of Operations - Structure: 2.75 for information on the properties of numbers on which mental calculation strategies depend.

Illustration 1: Multiplying by single digits

Previously students will be able to use a variety of strategies to multiply small numbers mentally by single digit numbers. For example, they know multiplication by 3 can be done by doubling and adding another one e.g. to calculate 14 × 3:

first double 14 getting 28, then add another 14, to get 3 times 14 which is 42, or
first double 14 getting 28, then add 10, giving 38 and then adding 4 more, to get 42

Now they will be able to use this strategy to multiply larger numbers (e.g. 140 × 3).

Illustration 2: Multiplying with powers of ten

Previously, students use their known multiplication basic facts in mental computation. For example, they can calculate 15 × 6 by adding 5 × 6 (known to be 30) to 10 × 6 (known to be 60). Now they will combine this knowledge with knowledge of place value to undertake mental calculations such as 20 × 30, 150 × 6, and 20 × 410.

Illustration 3: Visualising a written algorithm is generally a poor mental strategy

Some students do mental calculation by trying to visualise the written algorithm. For example, asked to multiply 13 by 25, they may say to themselves: “5 by 3 is 15, put down 5, carry 1, 2 by 3 is 6 and add the 1, gives 7, …………”. It is very hard to carry this calculation out correctly. Written algorithms have been honed for writing, not for mental calculation. These students need to learn about more appropriate mental strategies.

Illustration 4: Estimation is also important

It is possibly even more important that students learn to estimate mentally than to calculate mentally. For example, they should be able to estimate the area of a playground about 15m by 25m. Estimation requires at least: rounding to convenient numbers, mental calculation, and knowing if the estimate is likely to be too big or too small.

Illustration 5: Links to the Mathematics Online Interview

Examples of the types of tasks that would be illustrative of the prior knowledge for this indicator from the Mathematics Online Interview:

Question 32 a) – f) – Multiplication problems
Question 36 – In your head
Question 37 – Missing Number

Teaching Strategies

Students’ mental calculation and estimation builds on and in turn builds up number sense and the understanding of the concept and properties of multiplication.

Students need regular opportunities to practise mental computation and to discuss the mental strategies that they use. Class discussion of mental calculations plays a central role in establishing new strategies. Focussed teaching of mental strategies is worthwhile, even though many students will discover them for themselves because they intuitively understand how numbers and operations work. Other students need to be explicitly shown.

It is good to promote the habit of first mentally estimating an answer to any computation.

There are many opportunities in the normal routine of classrooms and at home for students to practise mental computation. Be sure to encourage parents to watch out for these situations and involve students in mental maths. These opportunities are not listed amongst the formal activities below, but constitute an especially important part of learning.

Activity 1: Using single digit strategies brings together students’ knowledge of the mathematical properties that they used to assist them in learning their multiplication basic facts, and extends these to multiplying larger numbers.

Activity 2: Extending single digit strategies based on distributive law applies the principles used in Activity 1 to multiplying by numbers close to a round number such as 99, 301 using the distributive law.

Activity 3: Extending single digit strategies using factors applies the principles used in Activity 1 to multiplying by numbers with known factors in a ‘broken calculator’ activity.

Activity 4: Keeping track of zeros uses number slides to visualise multiplying by powers of ten and their multiples.

Activity 5: Special numbers draws attention to large numbers with easy multiplication properties.

Activity 6: Calculator footy is a whole class game, used for practising mental computation of any type, building meta-cognitive appreciation of when it is best to reach for a calculator, and helping students understand different types of calculations by involving them in posing questions for others.

Activity 1: Using single digit number strategies

Make a class list of the various strategies that can be used for mental multiplication by single digit numbers. The main possibilities are listed below. As the list is made, encourage students to explain why the strategy works and to give examples. Then practise applying these strategies to multiply large numbers. Note, and discuss with students, that if you know your multiplication basic facts, it is often easier to multiply directly than to use the special strategy (e.g. 71 × 6 is easier than the method shown below).

To multiply by:	Strategy	Example	Reason
2	Double or add number to itself	2 × 71 is 71 + 71 = 142	A + A = 2 × A
3	Double, then add the number	3 × 71 is 142 + 71 = 213	Distributive law
4	Double, then double again	double 71 = 142, double again = 284	Multiplying and dividing with factors
5	Multiply by 10, then halve	10 × 71 = 710, halved = 355	Multiplying and dividing with factors
6	(a) multiply by 3, then double (b) Multiply by 5 then add number	(a) 3 × 71 = 213, double 213 = 426 (b) 5 × 71 = 355, 355 + 71 = 426	(a) Multiplying and dividing with factors (b) Distributive law
7	Just do it! No easy way.
8	Double, double and double again	double 71 = 142, double again = 284, double again = 568	Multiplying and dividing with factors
9	Multiply by 10 and subtract the number	9 × 70 = 700 - 70	Distributive law
10	Append zero (but be careful with decimals)	10 × 70 = 700	Place value shifts

These strategies depend on two main principles. Students can explain the distributive law using groups of counters or arrays. See, for example, Properties of Operations: Spin, shuffle and split: 2.75.

It is harder for students to explain why ‘multiplying and dividing with factors’ works, although many students will intuitively feel that it is correct. Also see Properties of Operations: Spin, shuffle and split: 2.75.The most important thing here is for students to be able to explain how the factors are involved in the strategies. For example, they should link the fact that 8 = 2 × 2 × 2 with the strategy above of ‘double, then double and double again’, and link the fact that 5 = 10 ÷ 2 to the strategy above for multiplying by 5.

Activity 2: Extending single digit strategies based on distributive law

The principles behind the single digit strategies in Activity 1 can be extended to larger numbers. This activity focuses on extending with the distributive law to multiply by numbers close to 100. It can be extended to numbers close to 1000 etc.

Students write down their answers to 100 × 1, 100 × 2, 100 × 3 … 100 × 10, and noting that this is easy. Then they write down their answers to the rows involving 99, 101, 98 and 102 using a calculator to help. Many patterns are evident. Students should work in groups to write some of these down. Emphasise those patterns that assist with mental multiplication:

To multiply by 101, multiply by 100 and add the number
To multiply by 102, multiply by 100 and add twice the number
To multiply by 99, multiply by 100 and take away the number
To multiply by 98, multiply by 100 and take away twice the number

×	1	2	3	4	5	6	7	8	9	10
98	98	196	294	392	490	588	686	784	882	980
99	99	198	297	396	495	594	693	792	891	990
100	100	200	300	400	500	600	700	800	900	1000
101	101	202	303	404	505	606	707	808	909	1010
102	102	204	306	408	510	612	714	816	918	1020

Follow with class discussion about other places where this principle can be used e.g. multiplying by 21, or by 301. The principle involved here is the distributive law – it can be illustrated as shown in Properties of Operations: Spin, shuffle and split: 2.75.

Activity 3: Extending single digit strategies using factors

This activity focuses on extending the use of factors to multiply, using a ‘broken calculator’.

Each student has a calculator, with which to work out the calculations below. There is only one obstacle. They must pretend that the entry key for digit 1 is broken along with the addition and subtraction keys and cannot be used. The digit 1 can appear in the display.

Use the first three examples in the table below as a class discussion. Teachers should extend the table with questions to suit the class number skills. It is a good idea to ask for two alternative calculations for each, as this enhances flexibility and creativity.

This activity strengthens mental computation in two ways: firstly by focussing on number principles that provide sound mental strategies, and secondly, by giving students practise in manipulating numbers mentally to find the alternatives. It should be followed by practice implementing the discovered strategies.

Sample ‘broken calculator’ calculations

Broken Keys	Calculation	Answer	Alternative Calculation	Alternative Calculation
1, +, -	12 × 3	36	4 × 3 × 3	6 × 2 × 3
1, +, -	12 × 12	144	4 × 3 × 4 × 3	2 × 6 × 3 × 4
1, +, -	16 × 32	512	2 × 8 × 32	8 × 64
1, +, -	16 × 22
1, +, -	18 × 223
1, +, -	16 × 18
1, +, -	120 × 81
1, +, -	111 × 101		(3 × 37) × (5 × 20.2)
Challenge 1, +, -	17 × 23		8.5 × 2 × 23

Extensions: Increase the number of broken digit keys and use more difficult numbers.

Activity 4: Keeping track of zeros

Multiplying by numbers ending in zeros is easy, provided students can keep track of the zeros. Rules such as ‘counting up the number of zeros’ are useful AFTER students understand what they are doing. Without understanding, they will muddle the rules or use them inappropriately (e.g. with decimals).

Examples:

Calculate 30 × 40. This can be thought of in several ways. For example 30 groups of 40 is 10 groups of 3 groups of 40, which is 10 groups of 120, which is 1200.

Number slides provide a visual image of the effects of multiplication by ten. Develop these skills in stages – first mental multiplication by ten and powers of ten (e.g. 100, 1000), then by multiples such as 30 and 300. The number slide below goes down to ten-thousandths, but slides with the ones column on the right-most edge can be easily constructed, if desired.

Activity 5: Special numbers

This activity shows how it is easy to multiply by factors of 100, 1000, etc.

First students copy the table structure below (first column and first row) and then ask them to complete, possibly using calculators. Then they work in groups to write down the patterns that they see. (If this will take too long, have some students work on the top 3 rows and others on the bottom 4 rows). In the ensuing class discussion, note the many patterns, with special emphasis on those that help multiplication:

To multiply by 50, multiply by 100 and halve.
To multiply by 25, multiply by 100 and divide by 4.
To multiply by 500, multiply by 1000 and halve.
To multiply by 250, multiply by 100 and divide by 4.
To multiply by 125, multiply by 100 and divide by 8.

×	1	2	3	4	5	6	7	8	9	10
100	100	200	300	400	500	600	700	800	900	1000
50	50	100	150	200	250	300	350	400	450	500
25	25	50	75	100	125	150	175	200	225	250
1000	1000	2000	3000	4000	5000	6000	7000	8000	9000	10000
500	500	1000	1500	2000	2500	3000	3500	4000	4500	5000
250	250	500	750	1000	1250	1500	1750	2000	2250	2500
125	125	250	375	500	625	750	875	1000	1125	1250

Activity 6: Calculator Footy

Calculator footy is a whole class game for all types of mental calculation. It is very flexible, because it can be used for mixed or focussed practice and it develops students’ ability to select mental or calculator methods, depending on the difficulty of the calculation. There are possibilities to use students as question designers, so that they begin to think strategically about the difficulty of questions. Keep the pace reasonably fast by using questions that are not too difficult and imposing time limits if required: a fast game is more fun.

Teacher selects two teams of 5 students who line up across the front of the room. Team A has a goal on the far left, and team B has a goal on the far right. Team A player 1 and team B player 1 stand together in the middle, with the other players lined up towards the goals, which are near player 5 of each team. A calculator is positioned on each side of the room, beyond the goal. The rest of the class are divided into supporters of either team A or team B and each child has a calculator. One supporter of team A and one supporter of team B work together as the scorers. Another child is the timekeeper.

Bounce the ball to decide which team starts first. Say team A wins the bounce. Play begins with Team A player 1 holding the football. The teacher, or any supporter chosen by the teacher, poses a calculation (e.g. 20 × 30) to be answered by the player holding the ball and their opponent with the same number on the other team. These two players can answer immediately by mental calculation, or run to their team’s calculator, do the calculation, return to their place and then answer the question. The first player to answer correctly scores 1 point, gets the football, and immediately passes it to the next team player (e.g. team A player 1 passes it to team A player 2). If neither student is correct, the teacher can ask any supporter for the correct answer, and if they are correct, the football passes to their team but without scoring a point. If a supporter has asked the question, then a supporter from the opposite team would be asked to answer. Play continues, and a goal is scored (6 points) when player 5 correctly answers a question first. The football then returns to the centre, held by the team who did not score the goal, and play begins again. When the game time has elapsed, the team with the largest number of points is the winner.

The teacher can select questions that are well suited to particular mental strategies, questions that are easy although they appear hard (e.g. multiplication by zero), questions that are best done with a calculator, as well as arranging students so they can be given questions of appropriate level of difficulty. Supporters are kept engaged by the possibility of being called on to supply an answer. They can also be instructed to design questions of a particular type.

Further Resources

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

McIntosh, A. (2005). Mental Computation: A Strategies Approach. (Module 4: Two-digit whole numbers). Department of Education, Hobart.

Digilearn object *

Arrays: word problems with products from 35 to 64 – Students apply knowledge of factors of numbers to solve problems with products up to 64. Students apply the commutative property of multiplication. (https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4500)

* Note that Digilearn is a secure site;DEECD login required.

Mathematics Developmental Continuum P-10