Hot Air: Making Sense of Minus and MinusAuthor(s): Val Butterfield-Wallbank and Susan MartinSource: Mathematics in School, Vol. 27, No. 1 (Jan., 1998), pp. 23-25Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211841 .

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Hot

Air

Making sense

of minus

and minus

Val Butterfield-Wallbank and Susan Martin

Hot air balloons might seem to have more in common with Bath than with making sense of some aspect of mathematics. This article outlines and discusses the potential of a tool based on hot air balloons to help students handle the often problematic concept of using the number operations of ad- dition and subtraction with directed numbers.

The article is based on an idea generated for a coursework assignment by a novice teacher on a one year PGCE (Secon- dary) course. The main aim of the assignment is to give novice teachers an opportunity to devise a resource to meet a par- ticular learning need that they have identified as a result of their teaching placement(s) using an appropriate and creative approach to stimulate and enhance pupil learning.

It is not uncommon for novice teachers to come across difficulties encountered by different year groups with operat- ing on directed numbers, and to experience that addressing these is not simple. Val's experience was no different. She decided to focus on this area for her project and to produce a resource for introducing the concept to lower secondary year groups in a tangible way, one of the problems that she identified, both from her experience and from looking at text books, being that students often had nothing to relate to, apart from a set of rules governing the combination of nega- tive and positive numbers for the operations of addition and subtraction. No reasoning can be used to find the answers to all the combinations in a logical way (logical in the sense that they could relate it to a physical phenomenon that in turn would enable students to reason out the answer for them- selves). Text books often use a rule table to establish the result of combining signs and operations but this tends to assume

Mathematics in School, January 1998

the answers as prerequisite knowledge. Val wanted students to establish the rules for themselves and to have some own- ership of them rather than relying on remembering them, that is, her aim was to teach for understanding rather than teach rules (Skemp, 1976). Given that the rules are too obscure for students to understand fully, a plausible deduction was sought. The aim was to produce a resource which enabled students to carry out simple calculations, using both negative and positive numbers, with ease.

Hot Air Ballooning uses the idea of a balloon with sand- bags and bursts of hot air. Sandbags are negative and bursts of hot air are positive. Loading and unloading sandbags and putting in and letting out hot air represent the four combina- tions resulting from performing the operations of addition and subtraction on directed numbers; addition being mod- elled by injecting and loading and subtraction by letting out and unloading. Injecting bursts of hot air result in the balloon rising and the opposite action, letting out hot air, results in the balloon falling (see Figure 1 for further details). If we consider loading a sandbag into a balloon it will go down; if several sandbags are loaded into the balloon it will go down further. Thus, loading four sandbags and then another four will mean the balloon drops eight units. If sandbags are unloaded, the balloon will rise conveying the idea that remov- ing or taking away a negative entity will result in an upwards or positive movement. Thus, adding - 8 (loading eight sandbags) is equivalent to subtracting + 8 (letting out eight units of hot air) in that both result in a downward movement of eight units. Similarly, adding +8 (injecting eight bursts of hot air) is equivalent to subtracting -8 (unloading eight sandbags), both resulting in an upward movement of eight units.

The resource, a game, has two vertical scales each running from +20 to -20, each with a hot air balloon attached. Cards of various difficulty are used from those dealing with straight loading or unloading to those requiring combinations of actions or negative and positive numbers, for example, 'load a sandbag and let out 3 bursts of hot air'. Each player turns over a card and moves their balloon to correspond with the instruction (see also Figures 1 and 2). There is no skill in terms of strategies used, it is simply a test of operating on directed numbers; the game is finished when someone floats away.

A worksheet was also part of the resource for use after the game. This sheet dealt with the same concepts, each section moving away from the hot air balloon principle and ended with straight calculations involving negative numbers ie with no mention of sandbags or hot air.

Experience of using the resource and of teaching directed numbers in subsequent lessons without the aid of hot air balloons suggested that the work needed some further con- solidation, the main problem being that students needed to be reminded of the hot air balloon 'theory'. A further caveat of the resource is that it uses a vertical number line which may or may not fit with students' prior conceptions and may, for those who have a well established horizontal number line, make the transition to de-contextualised problems with di- rected numbers more difficult.

In addition to the varied success Val had using it with various Y7 and Y8 groups, fellow novice teachers were im- pressed with the resource as a very appropriate approach to promote a better grasp of the concept by students. M

Reference Skemp, R. (1976) Relational understanding and instrumental under-

standing, Mathematical Teaching, Bulletin of ATM No. 77.

Authors Val Butterfield-Wallbank, Hayesfield School, Bath and Susan Martin, University of Bath.

23

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HOT AIR BALLOONING

A hot air balloon starts at ground level, zero, beside a cliff. If it wants to rise it needs a burst of hot air. If it wants to go slower it needs to load sand bags from the nearby helicopter.

Hot Air Bursts are POSITIVE

Sand Bags are NEGATIVE

+ (+ + etc.

4 -3 -4 etc.

Sand Bags

If we load a sand bag of value -1 the balloon will sink by one unit.

If we then take the sand bag off, the balloon will rise by one unit.

The same for two sandbags, if we load two sandbags the balloon will sink by two units.

Hot Air Bursts

If we give 1 burst of hot air + I the balloon will rise by one unit.

If we take away or let out a unit of hot air, the balloon will sink by one unit

If you get to positive 20, you'll float away and you lose!!!

If you get to negative 20, you'll sink into the sea and you lose!!!

GOOD LUCK!!

Fig. 1

24 Mathematics in School, January 1998

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HOT AIR BALLOONING

20

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18 17

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10 9

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2

1

0

-1

-2

-3

-4

-5

-6

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-20

FLOAT AWAY!!! FLOAT AWAY!!!

S20 19

18 17

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S-20

SINK INTO THE SEA!!!

SINK INTO THE SEA!!!

. . .. . .. .

Fig. 2

Mathematics in School, January 1998 25

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