Round the Three Dice
What happens when you round these three-digit numbers to the nearest 100?
Problem
Round the Three Dice printable sheet
There are three dice, each of them with faces labelled from 1 to 6.
When the dice are rolled they can be combined in six different ways to make a three-digit number.
For example, if I roll a 2 and a 4 and a 5, I can combine them to make 245, 254, 425, 452, 524 or 542.
Now round each of these numbers to the nearest 100:
245 rounds to 200, 254 rounds to 300, 425 rounds to 400, 452 rounds to 500, 524 rounds to 500 and 542 rounds to 500.
Repeat for other rolls of the dice.
Can each of the six numbers round to the same multiple of 100?
Can each of the six numbers round to a different multiple of 100?
There are some interactive dice that you can use for this problem.
Getting Started
Begin by rolling the dice and making the different three-digit numbers. Then round each of them to the nearest 100. Remember that sometimes you round up and sometimes you round down.
You could record your results in a table like this one:
Student Solutions
Thank you to everybody who sent in their ideas about this activity. In this solution page, we're going to focus on solutions which included some explanation.
Maddy, Chloe and Georgia from Clifton Hill Primary School in Australia tried this activity with a few different sets of numbers:
4, 1, 3 you can make 413, 134, 143, 341, 314, 431, round all the numbers to the nearest 100
413 = 400
134 = 100
143 = 100
341 = 300
314 = 300
431 = 400
6, 3, 3 you can make 633 336 363
633 = 600
336 = 300
363 = 400
6, 1, 2 you can make 612 621 126 162 216 261
612 = 600
621 = 600
126 = 100
162 = 200
216 = 200
261 = 300
We didn't find any with all the same rounded multiples but I'm sure there are many more solutions.
You're right, there are many more solutions! I wonder if any will give six of the same rounded multiple, or six different rounded multiples? I can see that one of your sets of numbers gives all different rounded multiples, but only three possibilities rather than six.
Nina from Clifton Hill Primary School thought that it wouldn't be possible to have all six numbers rounding to the same multiple of 100:
1) round these numbers to the nearest 100.
245, 254, 425, 452, 524, 542
200, 300, 400, 500, 500, 500
2) round some more numbers:
256, 265, 526, 562, 625, 652
300, 300, 500, 600, 600, 700
Can each six numbers round to the same multiple of 100: No
Because: each set of six numbers has to at least have a 2 digit difference which means that there will always be a different multiple:
e.g. 123, 132, 213, 231, 312, 321
100, 100, 200, 200, 300, 300
Six numbers can't be rounded to make the same multiple of 100
I agree that if all the numbers rolled are different, they can't all round to the same multiple of 100. Your example uses numbers which are as close to each other as possible, without being the same, which is quite convincing.
Daniel from Holy Rosary Catholic School in Canada also rolled some numbers which were very close to each other:
423 234 243 324 342 432
400 200 200 300 300 400
As you can see, 2 round to 400, 2 round to 300, and 2 round to 200. So therefore the numbers cannot round to the same multiple of 100, and they round to different multiples of 100.
I wonder if this would be true for any other sets of numbers as well?
Lucy from Clifton Hill Primary School noticed that if the same number was rolled three times, all of the rounded numbers would be the same:
I did not get both answers but I got one. I got lucky and I rolled 2,2 and 2 so for all the rounded answers were 200. For the other answer required they have to be far apart and some low and some high so they can not be similar. It was a hard problem but it was also very fun:).
That's true - when all the dice are the same, the rounded numbers will all be the same as well. I agree that for all the rounded numbers to be different, the numbers rolled would have to be far apart from each other. I wonder if anybody can think of numbers that would give six different rounded multiples of 100?
Lila from Holy Rosary Catholic School in Canada had an idea for how to make the rounded multiples all the same or all different:
In my case the six numbers I made were not all different multiples and not all the same multiples. But with other numbers like 111 or 222 because the only way to get the same multiple of 100 is to have all the same digits. To have all different multiples of 100 you would need three numbers like 1, 5, 8, which can round to different multiples.
1, 5 and 8 certainly seem to be far apart from each other. What multiples of 100 would those give?
If anybody does find a set of three numbers that would give six different rounded multiples of 100, or thinks of a reason why this wouldn't be possible, please email us with your thoughts.
Teachers' Resources
Round the Three Dice
Round the Three Dice printable sheet
There are three dice, each of them with faces labelled from 1 to 6.
When the dice are rolled they can be combined in six different ways to make a three-digit number.
For example, if I roll a 2 and a 4 and a 5, I can combine them to make 245, 254, 425, 452, 524 or 542.
Now round each of these numbers to the nearest 100:
245 rounds to 200, 254 rounds to 300, 425 rounds to 400, 452 rounds to 500, 524 rounds to 500 and 542 rounds to 500.
Repeat for other rolls of the dice.
Can each of the six numbers round to the same multiple of 100?
Can each of the six numbers round to a different multiple of 100?
There are some interactive dice that you can use for this problem.
Why do this activity?
This activity provides a meaningful task for practising rounding three-digit numbers to the nearest multiple of 100. It encourages children to record their results, notice patterns and make predictions.
Possible approach
A possible starting point is using a number line to remind the class what is meant by rounding.
The interactive dice can be used to model the activity on an interactive whiteboard or can be used by the children as they engage with the task. You may also find that individual dice are useful.
As the task is being modelled, the results should be recorded in a table, perhaps like the one below:
This organisation of results will support the children to notice patterns and conjecture about when numbers will round to different multiples of 100.
Some children may move onto the extension tasks (below).
Key questions
Which numbers can we make?
What will they round to? Will they round up or down? Why?
Do they round to the same multiple of 100? Why? When will this happen?
Does it make a difference if the digits rolled are unique?
Possible extensions
Extension 1: Did the class find examples where some of the three-digit numbers round to the same multiple of 100? Can they then come up with a rule about when this will happen? When will two round to the same? Or three? Or four? Or five? Or all six?
Extension 2: Did the class find examples where each of the three-digit numbers round to a different multiple of 100? Why? Why not?
Extension 3: Having completed the original task, ask the children to add a column to the right hand side of their table to note when numbers round up or down. Can they predict from the initial dice roll, how many of the three-digit numbers made will round up or down?
Extension 4: What if you change the numbers on the faces of the dice? Can you pick numbers so that all of the six three-digit numbers round to the same multiple of 100? Can you pick numbers so that all of the six three-digit numbers round to a different multiple of 100?
Extension 5: What happens if you round to the nearest multiple of 10 instead of the nearest multiple of 100?
Extension 6: Have a go at the activity Round the Four Dice and practise rounding numbers to the nearest multiple of 1000.
Possible support
Copies of the table for children to write the numbers into would be useful.
A number line might also be useful to support children in reasoning about which multiple of 100 is closest to a given number.
Children may like to try Round the Two Dice before tackling this task.