Reflections on a problem

Our previous blog set a problem in which the goal was to make two numbers whose total is 9 and which use the digits 1, 2, 3, 4, 5, 6, 7 and 8 once only. We have been working with groups of teachers for the last two weeks and have found that, even for adults, the problem generates a lot of useful discussion at the S stage of the STAR model (Sort it out). During the same period, we read about the Manu Kapur concept of ‘productive failure’. The experiences of the teachers was of the ‘productive failure’ type, because as they struggled with sorting out what the question meant, they had to cycle through a number of concepts and clarify what was involved in this problem.

We heard questions such as:

“What is a digit?”

“How is a digit different from a number?”

“What does total actually mean? Can it be the result of a subtraction or even a multiplication?”

These questions were answered through discussion and discarding ideas that did not fit the consensus views. For example, a digit isn’t a number, it is one of the symbols that we use to make a number. That is, 5 and 7 are digits but 75 and 557 are numbers that can be made with those digits. A total is made when two or more numbers are added and does not refer to the outcome of a subtraction or multiplication. 

The initial failure to find the right meaning for these terms eventually gave way to exploring how two numbers could be made with the digits 1 – 8 to reach a total of 9.

“We’ll have to use fractions. Otherwise the total is going to be much more than 9.”

“Could we use decimals? That could use all the digits but the two numbers could be less than 10.”

These ideas were the key that unlocked the solutions. Soon it was realised that two 4-digit numbers were needed and for their total to be equal to 9 the decimal fraction parts would have to reduce to zero. For this to happen, the thousandths have to add to 10. The tenths and hundredths have to add to 9 and the units have to add to 8 as shown below.

It was full steam ahead from here on. For example, to make 8 in the ones position, we could try 1 and 7. This means that 8 and 2 cannot be used in the tenths and hundredths positions, but 8 + 2 = 10, which is what we need for the thousandths position. This reasoning leads to the answer:

Can 9 be made in any other way with these digits? It certainly can, as there are other ways in which the 8 in the ones position can be made. By our reckoning, there are 96 different ways in which the digits can be arranged to make a total of 9 but you don’t have to find all of them!