Problem
In Garry’s Greengrocer’s shop he has those old fashioned scales where you put weights on one side and the fruit and vegetables on the other. Over the years he has lost some of the weights and he is now down to only three. But by being a little smart, he is still able to weigh exactly any whole number of kilograms from 1kg to 13kg.

What are the weights and how does he do the weighing?

What is this problem about?
The key point to this problem is to realize that weights can go on either side of the scale. After that it is a matter of doing careful and systematic addition. Some children may be able to identify the powers of 3 pattern in the weights (1, 3, 9, 27).

Achievement Objectives
Measurement (Level 4)
- read and construct a variety of scales
- carry out a measuring task involving reading of scales

Mathematical Processes
- devise and use problem solving strategies to explore situations mathematically (guess and check, be systematic, make a drawing).

Resources
Balance scales and weights (or picture of scales and two containers)
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- use balance scales to weigh objects (or describe how balance scales weigh objects)
- work systematically to solve a problem

1. Introduce the problem by weighing objects on the balance scales. Use weights in both pans. (If you don't have access to scales use 2 containers and a child acting as the balance. )
2. Pose the problem to the class.
3. Brainstorm for ways to solve the problem (guess and check, make a drawing).
4. As the children work on the problem ask questions that focus them on working systematically.
5. Encourage the children to plan ways to record their solution system.
6. Share solutions. Discuss the different ways that children have worked systematically and recorded their findings (table, organised list, systematic drawings...)

Extension to the problem
What range of weights could Garry measure with the right collection of four weights?

Solution to the problem
In order to be able to weigh a 1kg object, Garry probably is going to require a 1kg weight.

How can he measure 2kg accurately? Garry could have another 1kg weight. With the two 1kg weights he could weigh 1 kg and 2 kg objects. How could he weigh a 3kg object? He would need another 1 kg weight. But now he is never going to be able to weigh anything more than 3kg!

How can Garry be ‘a little smart’?

Let’s start again. Garry certainly seems to need a 1kg weight but how else could he weigh a 2kg object? What if he had a 3kg weight? How could he use that? He could certainly use it to weigh a 3kg pumpkin. Could he use it to weigh a 2kg lot of potatoes? No, not if he uses the weights in the normal way. Can he use them in an abnormal way then? What abnormal ways are there?

What if he puts the weights on the same side as the potatoes! The 1kg weight plus the potatoes could then be weighed against the 3kg weight. If the scales balanced, then the potatoes would weigh 2kg!

But that means that Garry can now weigh a 1kg object, a 2kg object and a 3kg object. And, of course he can weigh a 4kg object by putting the 1kg and 3kg weights on the same side of the scales.

The next challenge is to weigh 5kg. Or should Garry think about 13kg first? To get 13kg he would need to have a 9kg weight (1 + 3 + 9 = 13). Can he weigh 5 kg with these three weights? Yes, and he can also weigh all other amounts from 5kg to 13kg. We show you how below.

Extension: Garry would need 1kg, 3kg, 9kg and 27kg weights to be able to measure anything from 1 kg to 40kg. This sounds a lot but we show below how it can be done. Amounts of 1kg to 13kg have already been tackled so we start at 14kg.