Over a barrel

Problem 1

Students may solve the problem by trial and improvement. This will involve trying numbers for the mass of the barrel and seeing if the resulting masses for wheat and golden syrup work.

Another way to solve the problem is to use logical reasoning. If golden syrup is twice as heavy as wheat, this collection of barrels would have the same mass:

A barrel of golden syrup has a mass of 50 kilograms, and a barrel of wheat has a mass of 30 kilograms, so this can be represented as:

•  + 50 = 30 + 30 where  is the mass of an empty barrel.
• So  + 50 = 60
• So  = 10

Problem 2

Students should realise that because each equation involves multiplication of a number by itself and the answers are less than 100,

must be less than 10.

They can then solve each equation by trial and improvement or by eliminating all the possibilities:

The numbers in the right-hand column show a pattern of differences that provides an interesting extension:

Problem 3

This problem is similar to Problem 1 on this page because it involves possibilities (that is, the number of kittens) and a constraint (there are 21 more legs than tails).

As with Problem 1, students could use trial and improvement. However, a more efficient method might be to realise that each kitten has three more legs than tails (4 – 1 = 3). To get 21 more legs than tails would require seven kittens because 7 x 3 = 21.

Similarly, a kitten has two more legs than eyes, so 14 more legs than eyes means seven kittens as well because 7 x 2 = 14.

Problem 4

Combinations such as this can be solved in a variety of ways (see the notes on probability, Answers and Teachers’ Notes: Statistics, Figure It Out, Level 3). These include:

• An organised list:

All possibilites with ham:

• ham – salami – olives
• ham – salami – peppers
• ham – salami – pineapple
• ham – olives – peppers
• ham – olives – pineapple
• ham – peppers – pineapple

All possibilities with salami but no ham:

• salami – olives – peppers
• salami – olives – pineapple
• salami – peppers – pineapple

Remaining possibility

• olives – peppers – pineapple

A tree diagram:

• Tables:

Ham with …

Salami with ...

Olives with ...

1.

10 kg

2.

The circle represents 4, the triangle represents 5, and the square represents 9.

3.

a. 7

b. 7

4.

There are 10 different pizzas that could be made:

• ham – salami – olives
• ham – salami – peppers
• ham – salami – pineapple
• ham – olives – peppers
• ham – olives – pineapple
• ham – peppers – pineapple
• salami – olives – peppers
• salami – olives – pineapple
• salami – peppers – pineapple
• olives – peppers – pineapple