Highest and Lowest
1 out of 3
Put operations signs between the numbers 3, 4, 5 and 6 to make the highest possible number and lowest possible number.
Problem
Highest and Lowest printable sheet
Put operations signs ($+$ or $-$ or $\times$ or $\div$) between the numbers 3, 4, 5, 6 to make the highest possible number and lowest possible number.
How about trying with numbers 1, 2, 3, 4, 5 and 6?
Student Solutions
Ashkan from Gorsefield Primary used the numbers 3, 4, 5 and 6, and tried not to use an operation sign more than once in each solution. They explained:
The highest I found was:
3 - 4 + (5 × 6) = 29.
I did 5 × 6 which = 30. Then I did 3 - 4 which is -1. Finally I did -1 + 30 = 29.
The lowest I found was:
3 - 4 + (5 / 6) = 0.167
I did 5 / 6 = 0.833. Then I did 3 - 4 = -1. Finally I did -1 + 0.833 = 0.167.
Kestrel class at Churchfields, the Village School in Wiltshire also used the numbers 3, 4, 5 and 6, and they experimented with using the same operations signs and different operations signs:
We decided that we had to keep the numbers in order.
At first we could use the operations as many times as we wanted.
Our highest score was 360
3 × 4 × 5 × 6 = 360
Our lowest score was -117.
3 – (4 × 5 × 6) = -117
At first we experimented with division before we realised that negative numbers were lower.
Then we decided to see what would happen if we could only use each symbol once (but we kept the numbers in order).
Our highest score was 34.5.
(3 ÷ 4 + 5) × 6 = 34.5
Our lowest score was -51.
3 – ((4 + 5) × 6) = - 51
Thank you, Kestrel class. What a good idea to decide on particular 'rules', like having the numbers in order, or using each operation once.
Brodie and Tully from St Patrick's School Macksville NSW in Australia looked at the numbers 3 to 6 and then at the numbers 1 to 6:
Highest with 3 to 6:
(4-3+5)×6=36
Lowest with 3 to 6:
6/3+4-5=1
(4-3+5)/6=1
Highest with 1 to 6:
(3+4)/(2-1)×5×6=210
Lowest with 1 to 6:
(4-3)×2/1+(5-6)=1
M, T and G from Monteney Primary School, Sheffield looked at the numbers from 1 to 6, and just used the multiplication symbol to find the highest possible number. They explained:
You need to multiply all the numbers together but not one. Multiplying all of the numbers 2, 3, 4, 5 and 6 together generate a total of 720.
2 × 3 = 6
6 × 4 = 24
24 × 5 = 120
120 × 6 = 720
If you add one to your total you get a total of 721.
720 + 1 = 721
You add one to make the biggest total, because if you multiplied your total by one then your answer of 720 wouldn't change.
The children at Holmwood House School in Colchester, Essex had a similar idea, but they did something different instead of adding the 1 at the end:
For the highest, we worked out that multiplying was the best way to enlarge the numbers, so we did 6 × 5 × 3, which came to 90. Multiplying that by 4 made 360. We then added the 2 and the 1 to make another 3, which we then multiplied by the 360 to make 1,080.
This can be written as 6 × 5 × 3 × 4 × (2 + 1) = 1080, which is the highest number anybody has found so far!
They looked for the lowest possible number by going into negative numbers:
For the lowest, we subtracted 3 from 1 to make -2, before multiplying the other remaining numbers by -2 to make an even larger negative number, using our knowledge of multiplying negatives and positives together. This ended up as -2 × 2 = -4, then -4 × 4 = -16, then -16 × 5 = -80, then finally, -80 × 6 = -480.
This can be written as (1 - 3) × 2 × 4 × 5 × 6 = -480.
Taine, Vinnie, Alex and Zac from Woodville School in New Zealand wrote to say that they had found an even lower number using all six digits:
1 - (6 x 5 x 4 x 3 x 2) = -719
Well done!
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 activity gives a good opportunity to explore using the knowledge and skills the pupils already have in a "safe" environment.
Possible approach
Start off by writing the four numbers down in order and putting the same sign between each pair. Which operation gives the highest total and which the lowest?
The children can vary the order themselves either working in pairs or individually. After a short period of independent work ask some of the children to explain their thinking to the others before continuing to see what the highest and lowest possible solutions are.
Having tried this challenge, many children will be able to explore further some of the attributes associated with the four rules of number and place value.
Key questions
Could you make this answer bigger somehow?
Could you make this answer smaller somehow?
How have you got your ideas?
How do you know that this is the biggest possible answer?
How do you know that this is the smallest possible answer?
Possible extension
Other forms of number manipulation may be applied e.g. using powers. The children could also try to make a range of target numbers using the same numbers or alternatively choose a different set of starting numbers and see what results they can make.
Possible support
Many pupils will benefit from using a calculator so that their energies can be applied to exploring ideas and reasoning rather than just be taken up with calculating. However if your focus is on gaining fluency in calculating skills then it would be better to restrict the set to just three of the numbers such as 3, 4 and 5. A further reduction in the challenge would be to take two numbers and consider all the different solutions that could be made by using any of the operations on them.