4. In the Head Calculations

4.1 Introduction

Adult students come to numeracy class expecting to relearn all of the formal, pen and paper methods of calculation from their school days. However, in these days of calculators, spread sheets and computerised cash registers, it is just as important to be able to use 'in the head' or informal methods, and to be able to estimate and use a calculator efficiently. For these informal methods, what is written on paper will not a neat and orderly 'sum', but a few 'jottings' to help the shortcut or 'back-of-envelope' calculations.

These informal or 'in-the-head' approaches are powerful and important in the adult world. In industry and workplace settings estimation and quick checks of calculator and spread sheet results are essential to ensure that you don't get nonsense from the technology. Mistakes leading to wasted materials and labour are expensive in a work situation.

Informal approaches are not only important in the workplace. Many adults use them without thinking for rapid figuring of their capacity to save money from their annual salary or to pay off loans and bills.

4.2 Useful Number Pairs

This activity presents a number of strategies which will help students recognise and recall useful number pairs fundamental to 'in the head' addition and subtraction. These include pairs which total 10, such as, 2 & 8, 3 & 7 and pairs which total 100, such as, 20 & 80, 30 & 70, 75 & 25 or 35 & 65.

4.3 Calculating Change

This activity demonstrates how to use the number pairs which total 10 and 100 to perform simple, in the head change calculations. It could be used in conjunction with Useful Number Pairs or as a separate follow up activity, depending on students' prior skills and knowledge.It also describes the more general 'counting on' method for calculating change, which can be applied to a variety of situations that would otherwise use subtraction.

4.4 Subtraction Using Counting On

This activity extends the 'Counting On' method of calculating change using 'Useful Number Pairs' to apply to any subtraction, including those involving time. Ideally it should follow the activity Calculating Change.

4.5 Addition with Useful Number Pairs

This activity uses the knowledge developed in the Useful Number Pairs Activity to demonstrate a quick technique that can be helpful when adding a collection of numbers. The activity should be done after Useful Number Pairs.

4.6 Doubling Up

Doubling is a straightforward process that is useful for a range of in the head calculations involving multiplication by 2, 4 and 8. Because it relies on remembering only a few simple number facts, it can act provide a remarkable boost to students' numeracy confidence.

4.7 The Power Of Halving

The skill of halving numbers is very useful for a range of in the head calculations, including division by 2, 4, and 8. Since many students find division one of the most difficult skills to master, this can be extremely helpful to them. Like doubling, halving relies on only a few number facts, so these strategies can make a powerful contribution to building students' confidence at in the head calculations.

4.8 Splitting Numbers for Addition and Multiplication

This activity explores an alternative, common sense method for in the head addition which relies on the technique of 'splitting numbers' into tens and units, used in Doubling Up, and The Power of Halving. This method can also be extended to make sense of multiplication of larger numbers.

4.9 Multiplying By Tens

The shortcuts of adding or removing zeros when multiplying and dividing by 10, 100 and 1,000 are extremely important in a society that uses decimal (10-based) systems of currency and measurement. They are also very useful for approximation of calculations, whether for making predictions or estimates, or for checking results given by spread sheets and calculators. These are skills that are often taken for granted in our society but which may need specific attention for adult numeracy students.This activity introduces these important skills for multiplication in gradual steps using an approach that allows adult students to recognise the patterns then apply the shortcuts to a range of whole number calculations.

4.10 Division By Tens

The shortcut of removing zeros when dividing by 10, 100 and 1,000 is extremely important in a society that uses decimal (10-based) systems of currency and measurement. It is also very useful for approximation of calculations, and for in the head methods of calculating 10%. This is a skill which may need specific attention for adult numeracy students. This activity introduces these important skills for division in gradual steps using an approach that allows adult students to recognise the patterns then apply the shortcuts to a range of whole number calculations.