Mental Strategies for Division: 3.5

Indicator of Progress
Teaching Strategies
Further Resources

Supporting materials

Inidicator of Progress

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

Before achieving this, students will be able to divide by single digit numbers (and 11 and 12) using written algorithms. With mental calculation, they will have good fluency with division number facts. They will use fewer strategies.

Good mental calculation is characterised by having multiple 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. The goal is efficiency and an appropriate level of accuracy.

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.

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, so this is a major part of the teaching. It is generally also the case that written algorithms, which have been honed for a pencil and paper environment, are not well suited to being carried out mentally. Students who calculate mentally by visualising the written algorithm will be more limited in their mental calculation than if they utilise other strategies.

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

Illustration 1: Moving from single digits to multiples of 10

Previously students will be able to demonstrate fluency with multiplication basic facts, know the inverse relationship between multiplication and division and the fact families that this produces (e.g. 2 × 7 = 14, 14 ÷ 2 = 7 etc). Now students combine this knowledge to be able to solve problems such as 140 ÷ 20 = 7 (14 tens, how many 2 tens?) and 1400 ÷ 20 = 70 (140 tens, how many 2 tens?).

Illustration 2: Estimation is important

It is equally important that students learn to estimate mentally as well as to calculate mentally. For example, they should be able to estimate the number of litres of petrol at $1.23 per litre that can be bought for $50 as being less than 50 litres but more than 25 litres. Estimation requires at least: using convenient numbers, mental calculation, and knowing if the estimate is likely to be too big or too small.

Illustration 3: 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 33 a) – f) – Division problems
Question 35 – Sharing our money

Teaching Strategies

Students need regular practice with mental computation. Students’ mental calculation and estimation builds on and in turn builds up number sense and the understanding of the concept and properties of multiplication. Teaching specific strategies for mental computation is worthwhile. Many students will discover them for themselves, because they intuitively understand how numbers and operations work. However, other students need to be explicitly shown. It is good for students to form the habit of first mentally estimating the answer to any computation; this is a powerful checking device

Mental calculation is useful in everyday life, so there will be many situations that arise in the classroom where students can use their skills. Capitalise on these opportunities. Parents can also help by involving their children in mental computation and estimation in real life settings.

Activity 1: Dividing using the distributive law applies the principle of the distributive law in mental computation.

Activity 2: Estimation of division using close numbers uses the principle of the distributive law (Activity 1) to obtain quick estimates.

Activity 3: Dividing using factors uses a ‘broken calculator’ activity to focus on another principle of division that is useful for mental computation.

Activity 4: Keeping track of zeros uses number slides and number expanders to explain division of special numbers.

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

Activity 1: Dividing using the distributive law

This activity focuses on mental division with the distributive law. The strategy is to split the dividend into parts that are easy to divide. The lesson would start with cases where the natural place value splits are easy (e.g. 69 ÷ 3) and then progresses to examples where more creative splits are required (e.g. 42 ÷ 3).

Begin by illustrating the distributive law, for example, by using arrays. Ask students to explain all of these images with appropriate mathematical language, drawing on the partition meaning of division. For example, to divide 69 counters into 3 equal groups, first divide 60 counters into 3 equal groups (with 20 in each group) and then divide the remaining 9 into 3 groups. Altogether there are 23 in each group.

Continue by applying the distributive law to mental calculation. The examples below are written in formal mathematical notation, but students do not need to read or write this notation to use this strategy. They may say “42 is 30 + 12 and both of these are divisible by 3. 30 divided by 3 is 10 and the 12 left over when divided by 3 is 4. So the answer is 14.” There are many correct ways of doing this division.

A variety of examples (choose size of numbers to suit group) are shown below. In each row, there is a division calculation, a way of splitting the dividend to make the mental calculation easier, the mental calculation and answer, and possibly a note on the calculation.

Division	Possible split	Calculation	Answer	Note
69 ÷ 3	69 = 60 + 9	69 ÷ 3 = (60 ÷ 3) + (9 ÷ 3)	20 + 3 = 23
391 ÷ 3	391 = 390 + 1	391 ÷ 3 = (390 ÷ 3) + (1 ÷ 3)	130 remainder 1	It is easy to divide 390 by 3.
42 ÷ 3	42 = 30 + 12	42 ÷ 3 = (30 ÷ 3) + (12 ÷ 3)	10 + 4 = 14
100 ÷ 3	100 = 99 + 1	100 ÷ 3 = (99 ÷ 3) + (1 ÷ 3)	33 remainder 1	There are several ways to deal with remainders.
670 ÷ 6	670 = 660 + 10	675 ÷ 6 = (660 ÷ 6) + (10 ÷ 6)	110 rem 10	There are several ways to deal with remainders.
385 ÷ 4	385 = 400 - 15	385 ÷ 4 = (400 ÷ 4) - (15 ÷ 4)	100 – 3.75 = 96.25	It can be tricky to deal with remainder when subtracting.

Follow up with some focussed practice on this strategy and then mixed practice which includes calculations where this strategy is not very helpful. The principle involved here is the distributive law – it can be illustrated as shown in Indicator of Progress: Properties of Operations: Spin, shuffle and split: 2.75.

CAUTION: Some students who learn this strategy without understanding may try to divide by splitting the divisor instead of the dividend. For example, they might want to divide by 7 by dividing by 5 and then dividing by 2 because 7 = 5 + 2. Illustrating division by partitioning groups of counters will show that this process does not produce 7 equal groups.

Activity 2: Estimation of division using close numbers

The process used in Activity 1 can also be used to find quick estimates to division by using convenient close numbers. For example, to estimate 670 ÷ 6, note that it is near 666, which is easy to divide by 6. To estimate 1000 ÷ 12, note that 1000 is near 960, which is 80 × 12. (I selected 960 because I know that 8 × 12 = 96).

Activity 3: Dividing using factors

This activity focuses on the principle that if I divide by two numbers successively, I get the same answer as dividing by their product. For example, (48 ÷ 3) ÷ 2 is the same as 48 ÷ 6. At the first stage, the 48 objects are divided into 3 equal groups, and then each of these is divided into 2 equal groups, so there are 6 equal groups altogether. Symbolically (A ÷ n) ÷ m = A ÷ (n × m).

Begin by explaining this property using a partition model, before starting the broken calculator activity.

Broken Calculator

This activity uses the idea of the broken calculator. Many variations on the ‘broken calculator’ theme can be created. Adapt the rules to teach different properties.

Each student has a calculator, with which to work out the answers to the calculations below. There is only one obstacle. They must pretend that the entry key for digit 1 and the addition and subtraction keys are broken and cannot be used. The digit 1 can appear in the display, but the input key cannot be used. Students will use mental calculation to find the alternative button presses

Use the first 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 at least two alternative calculations for each, as this enhances flexibility and creativity.

Even though students have to say how to use a calculator to do each of these calculations, 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 calculations

Broken Keys	Target Calculation	Answer	Alternative calculation	Alternative calculation
1, +, -	240 ÷ 16	15	(240 ÷ 2) ÷ 8	(240 ÷ 4) ÷ 4
1, +, -	240 ÷ 16	15	[(240 ÷ 4) ÷ 2] ÷ 2	240 ÷ 2 ÷ 2 ÷ 2 ÷ 2
1, +, -	70 ÷ 14	5	(70 ÷ 2) ÷ 7	(70 ÷ 7) ÷ 2
1, +, -	342 ÷ 18	19	(342 ÷ 2) ÷ 9	[(342 ÷ 3) ÷ 2] ÷ 3
1, +, -	225 ÷ 15	15	(225 ÷ 5) ÷ 3	(225 ÷ 3) ÷ 2
Challenge 1, +, -	286 ÷ 11	26	11 is prime, so factors strategy must be modified	(286 ÷ 22) × 2
Challenge 1, +, -	111 ÷ 15	7.4	(333 ÷ 45)	(222 ÷ 30)

Extensions: Increase the number of broken digit keys. Allow more strategies.

Conclude the lesson by revising the fundamental principles that allow these strategies to work.

Activity 4: Keeping track of zeros

Calculations with many zeros (e.g. 1200 ÷ 20) are easy in theory but often hard in practice. It seems hard to calculate 1200 ÷ 20 mentally, but it is not so difficult to calculate how many lots of 2 tens there are in 120 tens, because this is just 120 ÷ 2. The key to calculations with special numbers is therefore flexibility in interpreting 1200 (for example) in many ways. Using a number expander is a good way to do this - Number expander.

By appropriately folding a number expander, 1200 can be shown as:

Discuss with the students why these are all equal to 1200. Discuss how this is helpful for mental calculations. Practise these skills. This work with zeros is especially important for mental calculation with metric measurement. For example, how many lengths of 20mm ribbon can be made from 1.2m ribbon? Include mental division problems like this.

An adapted number expander for metric measurement could be used.

Students who are unsure about the effect of multiplying by ten and its powers can profitably use a number slide.

Activity 5: Special numbers

This activity shows how it is easy to divide by factors of 100, 1000, etc. (N.B. The later patterns go beyond Level 3.5).

First students calculate various divisions for select numbers, probably 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 rows and others on the bottom rows). In the ensuing class discussion, note the many patterns, with special emphasis on those that help division:

To divide by 5, divide by 10 and multiply by 2.
To divide by 50, divide by 100 and multiply by 2.
To divide by 25, divide by 100 and multiply by 4.
To divide by 500, divide by 1000 and multiply by 2.
To divide by 250, divide by 1000 and multiply by 4.
To divide by 125, divide by 1000 and multiply by 8.

Students should link the observations to the inverse relationship between multiplication and division. For example, 5 is half of 10, so dividing by 5 gives an answer twice that of dividing by 10. They should also be able to explain that if you divide a certain number into 5 equal groups, then there will be same number in each group as it you divided double the number into 10 equal groups.

Look in this table for patterns that can be used as shortcuts for division. The numbers in the first column are specially selected because of the relationships with 100 and 1000, whilst the numbers along the top row are simply selected for convenience.

Dividing by	1000	1500	2000	2500	3000	3500	4000
10	100	150	200	250	300	350	400
5	200	300	400	500	600	700	800
100	10	15	20	25	30	35	40
50	20	30	40	50	60	70	80
25	40	60	80	100	120	140	160
1000	1	1.5	2	2.5	3	3.5	4
500	2	3	4	5	6	7	8
250	4	6	8	10	12	14	16
125	8	12	16	20	24	28	32

Further Resources

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

Mathematics Developmental Continuum P-10