Old scales

Rawhiri has found some old-fashioned scales where you put metal weights on one side and the fruit or vegetables on the other. Unfortunately, he has found only three weights. But he is still able to weigh exactly any whole number of kilogrammes, from 1kg to 13kg.

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

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 student acting as the balance.)

2.

Pose the problem to the class.

3.

Brainstorm ways to solve the problem (guess and check, make a drawing).

4.

As the students work on the problem, ask questions that focus them on working systematically.

• Could Rawhiri measure using a 1kg, 2kg, or 3kg weight?
• How could Rawhiri combine or separate different weights to use the scales in an 'abnormal' way?
• How can you keep track of what you have measured so far?

5.

Encourage the students to plan ways to record their solution system.

6.

Share solutions. Discuss the different ways that students have worked systematically and recorded their findings (for example, table, organised list, systematic drawings, etc.).

Extension

What range of weights could Rawhiri measure with the right collection of four weights?

Rawhiri certainly seems to need a 1 kg weight, but how could he weigh a 2 kg object? What if he had a 3 kg weight? How could he use that? He could certainly use it to weigh a 3 kg pumpkin. Could he use it to weigh a 2 kg 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 1 kg weight plus the potatoes could then be weighed against the 3 kg weight. If the scales were balanced, then the potatoes would weigh 2 kg!

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

The next challenge is to weigh 5 kg. Rawhiri thinks about 13 kg first. To get 13 kg he would need to have a 9 kg weight (1 + 3 + 9 = 13). Can he weigh 5 kg with these three weights? Yes, and he can also weigh all other amounts from 5 kg to 13 kg. Here's how.

It is worth noting that Rawhiri can’t do the weighing if he uses 1 kg, 4 kg, or some other weight. With 1 kg, 4 kg and something heavier, he can’t make 2 kg.

Solution to the extension

Rawhiri would need 1 kg, 3 kg, 9 kg and 27 kg weights to be able to measure anything from 1 kg to 40 kg. One solution is to add the weights we have already (13 kg) to the next weight we want to be able to measure (14 kg) to give a total of 27 kg. Rawhiri could use the 27 weight to measure 14 kg and the largest weight he could measure would be 13 kg and 27 kg to give 40 kg. We show below how it can be done. Amounts of 1 kg to 13 kg have already been shown.

It is interesting to note that you can weigh so many things with so few weights.

You might like to think about how many different amounts could be weighed using 5 weights. Then spot the pattern and continue indefinitely.