# Reasoned Rounding

## Problem

*Reasoned Rounding printable sheet*

This is a rounding game for two players.

You will need a recording sheet shared between the two players. One sheet is enough for three games.

The first player rolls a 0-9 dice twice and chooses which two-digit number they would like to make from the numbers rolled.

They then round this number to the nearest 10, find the matching circle on the recording sheet and write their two digit number in one of the spaces in that circle.

The second player rolls the dice twice to take their turn in a similar way.

A circle is complete when it has two numbers in it and this scores a point for the person who wrote the **second number** in. (Even if the first number was written in by the other player.)

The game is over when all the circles are full and then the points are counted up to find the winner.

## Student Solutions

We received some solutions from children who wanted to share their ideas about this game with us. Sky from Cubitt Town Junior School in the UK explained their strategy:

When I rolled the dice, I looked at the sheet to see what spaces were available. I looked to see if I could block my partner or complete my own circles. When we played a round where we could choose the order of the digits, I switched them around to help me. If I was blocked with one way, I could switch the digits and round the other way.

Georgina from Wavell Junior in the UK thought that going second gives a better chance of winning:

If you want to win the game you would need to think carefully about the choices you have. If your opponent had got half the circle of 70 and you rolled 3 and 7 you could make 73 and you would get the point. It is best not to roll first because it is less likely for you to get a point. If you want to get a point you should not start a circle. It also is a bit of luck for the numbers you roll.

Rosie from King's College Madrid in Spain agreed with Georgina, and also spotted some circles that are more difficult to fill than the others:

In Maths class we decided that the 0 and 100 circles were the hardest to fill as only 5 numbers rounded to each circle whereas 10 numbers round to the other circles. We tried to fill the 0 and 100 circles as soon as we could. We also found that out of 14 pairs the player who went second won 12/14 times. We decided that because each circle needs two numbers it was better to go second.

The children from Central CofE Junior School in the UK noticed lots of things about this game:

We really enjoyed playing Reasoned Rounding and noticed lots of things to help us win the game and develop our reasoning.

- The first and last circles are the hardest to win as there are fewer chances of rolling a number that rounds to it. E.g. to win 0 you have to roll 00, 01, 02, 03 or 04 and for 100 you would need 95, 96, 97, 98 or 99 so there are only 5 chances but for the other circles there are 10 chances - from the 5 before to the 4 after. So, if you rolled a 9 and 6 it would be better to make 96 to round to 100.
- Also, if you go second you have more chance of winning as your partner will already have filled in one number of the circle so you can get the other one.
- One pair got down to the last circle which was 0. They knew that if they rolled a number over 5 with their first throw then there was no point rolling again as they couldn't win the circle.
- As you get further into the game it is harder to find a place to put your number as the circles are full.
- To vary the game, you could roll 3 times to create a number with 2 decimal places and round to the nearest tenth.

These are all great ideas! I like your idea for varying the game - we've made some similar extensions to this game on the Teachers' Resources page for this activity.

## Teachers' Resources

**Why do this problem?**

The game offers a competitive context which provides pupils with lots of practice in rounding numbers and helps develop their reasoning by encouraging them to talk about their strategies for winning points whilst playing.**Possible approach**

Play as a class first by projecting the recording sheet onto the board and asking pupils to roll dice, suggest what numbers could be made and explain to the class where they would want to put them and why.**Key questions**

What two-digit numbers can you make from the digits you rolled?

Where might be a good place to put your number?

What numbers are you hoping might come up on your next turn?

Can you explain why you chose that circle?**Possible extension**

Some other versions of the same game are available here:*Reasoned Rounding to the nearest hundred recording sheet*

This time the players roll the dice three times each per turn and can be encouraged to record the different 3-digit numbers possible before making strategic decisions as to where they could best place their chosen number.*Reasoned Rounding to the nearest whole number recording sheet*

Players roll the dice twice and then place one number on each side of a decimal point. They round their chosen number to the nearest whole number and then place it in the appropriate circle as before.*Reasoned Rounding to the nearest tenth recording sheet*

Players still roll the dice twice, but then decide in which order to place them *after* a decimal point. Having created a number which has two decimal places, they must then round it to the nearest tenth and place it in the appropriate space on the recording sheet.

What other variations can pupils come up with for themselves?

What if they were to use different dice? What recording sheets would be needed now?**Possible support**

Ask pupils to play in groups of four, one pair against another, in order to support their discussion of options whilst carrying out the rounding.