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 kilograms from 1kg to 13kg.

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

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

Copymaster of the problem (Māori)

Balance scales and weights (or picture of scales and two containers)

### Problem

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 kilograms from 1kg to 13kg.

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

### Teaching Sequence

- 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. )
- Pose the problem to the class.
- Brainstorm for ways to solve the problem (guess and check, make a drawing).
- As the students work on the problem ask questions that focus them on working systematically.
- Encourage the students to plan ways to record their solution system.
- Share solutions. Discuss the different ways that students have worked systematically and recorded their findings (table, organised list, systematic drawings...)

#### Extension to the problem

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

#### Solution to the problem

Rawhiri certainly seems to need a 1kg weight but how 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!

Rawhiri 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. Rawhiri thinks 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. Here's how.

5: | 9 | v | 1 + 3 + ? |

6: | 9 | v | 3 + ? |

7: | 9 + 1 | v | 3 + ? |

8: | 9 | v | 1 + ? |

10: | 9 + 1 | v | ? |

11: | 9 + 3 | v | 1 + ? |

12: | 9 + 3 | v | ? |

13: | 9 + 3 + 1 | v | ? |

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

Extension:

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

14 | 27 | v | 9 + 3 +1 + ? |

15: | 27 | v | 9 + 3 + ? |

16: | 27 + 1 | v | 9 + 3 + ? |

17: | 27 | v | 9 + 1 + ? |

18: | 27 | v | 9 + ? |

19: | 27 + 1 | v | 9 + ? |

20: | 27 + 3 | v | 9 + 1+ ? |

21: | 27 + 3 | v | 9 + ? |

22: | 27 + 3 + 1 | v | 9 + ? |

23: | 27 | v | 3 + 1 + ? |

24: | 27 | v | 3 + ? |

25: | 27 + 1 | v | 3 + ? |

26: | 27 | v | 1 + ? |

27: | 27 | v | ? |

28: | 27 + 1 | v | ? |

29 | 27 + 3 | v | 1 + ? |

30: | 27 + 3 | v | ? |

31: | 27 + 3 + 1 | v | ? |

32: | 27 + 9 | v | 3 + 1 + ? |

33: | 27 + 9 | v | 3 + ? |

34: | 27 + 9 + 1 | v | 3 + ? |

35: | 27 + 9 | v | 1 + ? |

36: | 27 + 9 | v | ? |

37: | 27 + 9 + 1 | v | ? |

38: | 27 + 9 + 3 | v | 1 + ? |

39: | 27 + 9 + 3 | v | ? |

40: | 27 + 9 + 3 + 1 | v | ? |

It always seems surprising to us that you can get so many things weighed with so few weights.

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