Before Viewing

Revise the positive number line and demonstrate that the relative positions of two numbers determine which is greater and which is less.

Look at some simple addition and subtraction calculations and their interpretation on the number line using Lisa’s method. Start each sum at zero.

Develop the idea used in the programme of moving ‘forwards’ (in the direction of increase of positive numbers) for addition and ‘backwards’ for subtraction. Simple multiplication could also be demonstrated.

During or After Viewing

Discuss the idea of ‘greater than’ and ‘less than’ for integers on the number line. Explore questions like: Which number is bigger, 7 or —14? Students can be challenged to find numbers 1 or 10 more or less than a stated number.

Invite students to think about other real situations that can be related to the concepts of positive and negative numbers. For example:

• temperature measurement on the Celsius scale
• measurement of altitude, above and below sea level
• scoring in golf using the notion of above and below par
• the use of AD and BC for dates in the Christian calendar (careful — there is no year 0)

Discuss word pairs that imply a sense of direction or positive and negative value. For example:

• forwards — backwards
• above — below
• up — down
• before — after
• debit — credit
• hot — cold
• clockwise — anticlockwise

Discuss in detail Lisa’s method of performing addition and subtraction. Students may find it useful to recreate the method in the classroom. They could instruct a member of the group to perform calculations using a number line on the wall or floor.

Examine why it is useful to label the positive numbers with a + sign, as shown in the programme, and compare with conventions used in texts available in the classroom (and in real-life contexts such as stating temperature in weather forecasts).

Look in detail at the Wizard’s explanation of multiplication and discuss the reasonableness of the rules developed. This may be done in conjunction with completing the multiplication grid on Worksheet 1. Division can be approached as the inverse of multiplication (initial examples can be drawn from the grid), with rules consistent with those established for multiplication.

Comments

The method used in the programme for dealing with simple addition and subtraction calculations is as follows. Always start a calculation by standing on the zero on the number line, facing the positive numbers. Then walk forwards for positive numbers and backwards for negative numbers. For addition, face the positive numbers; for subtraction, face the negative numbers.

Students can make progress using this mechanism for simple addition and subtraction. As calculations become more complex, physical interpretation becomes more difficult and the need arises for students to abstract and formulate general rules.

Contexts for negative numbers are often difficult to sustain and can give rise to misconceptions. Some students may benefit from the development in depth of a single context, rather than dealing with multiple ideas.

In golf, the following terms are used to describe the difficulty of a hole:

• par: the number of strokes an expert would be expected to take to complete a hole
• bogey: a score of one over par
• birdie: a score of one under par
• eagle: a score of two under par
• albatross: a score of three under par

Distinguishing between directional concepts and positive and negative values is difficult when ideas are related to everyday language. This is reflected in the use of notation where the ‘—’ sign is used both as an operator (subtraction) and as a symbol to express a negative value or direction.

CIrcular movement illustrates the often arbitrary choice of conventions. In mathematics, anticlockwise movement is generally taken to be positive.

While the programme introduces ideas about multiplying with negative numbers, the development of rules for multiplication and division may be more easily dealt with separately. Worksheet 1 is presented in two parts to allow for this.

The Worksheets

Worksheet 1: Back and Forth

This activity aims to provide examples in the same context as the one used in the programme.

Students should progress from using the number line to calculating mentally. They may need much more practice before they can do this reliably.

In this activity the size of the numbers is limited by the length of the number line. Larger numbers can be introduced as students develop familiarity with the techniques.

Examples drawing on other contexts, such as temperature and altitude, may be helpful for some students.

Some students may enjoy the interactive game ‘Think Ahead’ at: http://gamescene.com/thinkahead.html.

(The use of the +/— button on a calculator could be introduced here and used for checking work.)

Worksheet 2: Grid of Enlightenment

This activity gives students the opportunity to look in more detail at the Wizard’s explanation of multiplication, applying the ideas to a wider range of numbers but still using quadrants in a grid.

Presenting multiplication as an array of results on the grid reveals the inherent patterns and symmetries.

Having filled in the first quadrant (+ x +), students should readily be able to continue rows and columns for (+ x —) and (— x +). You could refer here to the Wizard’s examples of (+3 x —5) and (+5 x —3).

The most conceptually difficult quadrant (— x —), is filled last. Students should be encouraged to recognise that the numbers increase leftwards in each row and downwards in each column.

The grid can be used to develop ideas about division as the inverse of multiplication. This leads naturally to the establishment of rules for division that are consistent with those for multiplication.

Worksheet 3: Backward Thinking

This is a challenging problem. Students may need some initial help in understanding the question. They need to appreciate that for five numbers there are only ten possible pairings, since addition is commutative. The activity provides practice in addition with positive and negative numbers, and develops problem-solving skills.

The numbers are: —3, —1, 2, 3, 8.

A much simpler problem can be constructed using three numbers added in pairs (giving only three sums) or four numbers added in pairs (giving six sums).

The idea could be extended to finding the initial numbers from products of pairs.

Worksheet 4: Backhand Games

A sheet of paper with the card values placed in 25 grid spaces would be useful for teachers to photocopy and give out to students to cut up.

This activity is designed to provide entertaining mental practice of the arithmetic, with the element of competiion motivating students to be accurate and to check each other’s totals.

Encourage students to write down the entire sum, not just the answer: this makes checking easier and develops familiarity with the notation. Alternatively a + and a — card could be given to each player to use.

The concept of ‘greater than’ and ‘less than’ is reinforced.

The number of cards (25) is probably sufficient for groups of six or fewer, but can be adapted for different numbers of players or different variations of the game. Variations involving products could be treated as a separate activity.

Further Ideas

An alternative to using the number line to introduce addition and subtraction is suggested in the Mathematical Association’s publication Getting Started.

Counters or cubes of two different colours are used to represent matter (+) and antimatter (—). The method is based on the idea that an amount of matter plus the same amount of antimatter equals zero. So if red is positive (matter) and blue is negative (antimatter), each pair of one red and one blue cube represents zero.

Addition is the act of combining two sets of cubes. So the combination of 4 red and 3 blue cubes represents the sum +4 + —3. Pairing off the red with the blue cubes leaves one red, so the result is +1.

Subtraction is represented as the act of removing cubes from a set. So 6 red minus 4 red leaves 2 red: +6 — +4 = +2. And 8 blue minus 3 blue leaves 5 blue: —8 — —3 = —5

To deal with subtractions containing both positive and negative numbers, it is necessary to add in pairs of red and blue, remembering that +1 + —1 = 0. So to work out +4 — —2, we need to take 2 blue away from 4 red. This can be done by starting with 4 red, putting in 2 pairs of red and blue (= 0), and then removing 2 blue. The result is 6 red; so +4 — —2 = +6.

Negative numbers also arise when using lifts and thermometers. In the context of a lift, the ground floor can be regarded as floor 0 with the floors above as positive and the floors below (‘lower ground floor’ and ‘basement’) as negative. When temperatures fall below zero they are represented on a thermometer as negative numbers (commonly called ‘minus’ numbers).

Hot air balloons provide another context for addition and subtraction of directed numbers (see the article ‘Hot Air’ in Mathematics in School, January 1998). Hot air ballooning uses the idea of a balloon with sandbags (‘negative’) and bursts of hot air (‘positive’). Loading and unloading sandbags and putting in and letting out hot air represent the four combinations resulting from performing the operations of addition and subtraction on directed numbers, addition being modelled by injecting or loading and subtraction by letting out or unloading. Injecting bursts of hot air causes the balloon to rise; and letting out hot air causes the balloon to fall. Loading sandbags onto the balloon causes the balloon to fall; and unloading sandbags causes the balloon to rise.