Function Machines
2 out of 3
If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?
Problem
This one is a bit more of a puzzle.
Can you work out what happens in the three circles?
Getting Started
You might like to start with $5$. Follow the operations in order - add $3$, multiply by $5$, take away $2$. What comes out?
How could you get from $2$ to $22$ ? Will those same operations turn $3$ into $28$?
Student Solutions
Well done to all of you who gave a correct solution to this problem. The correct answer to the first part of the question is $33, 48, 38$.
There are several correct answers to the second part of the problem including:
Richard from Burlingame School used a systematic approach to help him find the answer to the second part of the problem:
What I did was multiply each number by $2$, then $3$, then $4$, etc till I found a pattern. If you multiply the number by $6$, you come out with a number $10$ less than the answer. I simply added ten, then subtracted $0$ to give me the answer.
Quite a few of you used a really useful technique to help find the solution more efficiently including Marley and Jake from Swarcliffe Primary and Shiv from Mayplace Primary School. Ester sent in a nice solution:
The numbers are CONSECUTIVE going in and HAVE A DIFFERENCE OF $6$ coming out. This tells us that there must be a multiplication by $6$ somewhere.
She also spotted that there was more than one solution:
If $\times 6$ is in the middle circle some solutions could be
$+2, \times 6, -2$
$+1, \times 6, +4$
$-1, \times 6, +16$
If you multiply by six first you have to add or subtract in the last two circles so that the result is $+10$
$\times ,+6,+4$
$\times 6,+11,-1$
You cannot make the $\times 6$ the last thing you do because the numbers coming out are not in the six times table.
As you can see, there are lots of possible solutions using multiply by $6$ as one of the functions, but you can also split multiplying by $6$ into multiplying by $3$ and multiplying by $2$, or as several pupils from the Mount School in York observed, multiplying by $12$ and dividing by $2$.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem gives a good deal of practice in addition, subtraction, multiplication and division in an interesting and unusual format. The second part of the problem involves much reasoning about numbers.
Key questions
Possible extension
Learners could create their own function machines for friends to complete. Asking them to make examples like the second part of the problem will involve a great deal of mental calculation and reasoning about numbers.
Possible support
Suggest trying just the first part of the problem which is relatively easy. They could then go on to making their own 'function machine' [as in the second part of the problem] with two, or even just one, circle.