Going bananas

This activity leads students through a process that involves dividing composite (non-prime) numbers by a prime factor and follows a set procedure. You can help your students to understand this process by modelling the distribution of bananas in the flow chart on the students’ page using counters or cubes. (The flow chart uses 18 bananas, but it is set out in a way that covers any number of bananas.) In pairs, the students could then use counters or cubes to model the distribution of the 30 bananas in questions 1–4.

The students need to realise that the total pool of bananas is not re-formed before each division calculation and that the bananas that are not going into a new pile are kept by the gorilla who is doing the dividing.

For example, in the first distribution of the 30 bananas shared out in questions 1–4, the bananas are divided into groups of two bananas: 30 ÷ 2 = 15. One banana from each group goes into the new pile, which means there are 15 bananas in the second round. Father can’t use his prime number on this new pile, so he has to be satisfied with the 15 bananas that are not in this pile.

Mother divides the 15 bananas in the new pile into five groups of three (her prime number): 15 ÷ 3 = 5. Again, students need to remember that one banana from each group goes into a new pile. Mother can’t use her prime number on this new pile of five bananas, so she gets only the 10 bananas that don’t go into this pile.

On Gus’s turn, he has to divide the new pile created by Mother into groups of five, his prime number. There are only five bananas in the pile, so that is only one group: 5 ÷ 5 = 1. One banana from that pile goes into a new pile (that’s what Junior gets), so Gus will get the four bananas that don’t go into that pile.

In this process, the students need to think of division as grouping (that is, the number of groups of twos in 30, threes in 15, and fives in 5), not as partition (sharing) into equal parts.

Flow charts for 20 and 21 could look like this:

The more confident students could try keeping track of developments in a table or using a table as their overall strategy. For example, for question 5b, with 18 in a bunch, a table could look like this:

There are no third-turn splits for 17, so a column for this is not necessary at this stage. You could extend the activity by getting the students to try other numbers to see if they can obtain more equitable shares for the gorillas or by suggesting to the students that they use their knowledge of number to create an equally interesting but perhaps fairer system for sharing the bananas.

1.

a. 15. (30 divides into 15 groups of 2. One from each group goes into the new pile.)

b. No, 15 does not divide by 2 without a remainder.

c. Any odd number.

2.

a. 10. (15 divides into 3 groups of 5. One from each group goes into the new pile. Mother gets the 10 left over.)

b. A bunch that is not a multiple of 3 or a multiple of 3 after it has been halved by Father. (For example, he could pick prime numbers greater than 3 or doubles of primes.)

3.

To get some bananas, Gus should get multiples of 5, for example, 5, 10, 15, 20. With powers of 5, such as 25, he gets first pick of the bunch.

4.

Junior should choose prime numbers higher than 5 (for example, 7, 11, 13).

5.

a. 14: Father 7, Mother 0, Gus 0, Junior 7

b. 18: Father 9, Mother 8 (6 in first split and then 2 out of the 3 left), Gus 0, Junior 1

c. 17: Father 0, Mother 0, Gus 0, Junior 17

d. 6: Father 3, Mother 2, Gus 0, Junior 1