Two Dice
1 out of 3
Find all the numbers that can be made by adding the dots on two dice.
Problem
Printable NRICH Roadshow resource.
Here are two dice.
If you add up the dots on the top you'll get 7.
Find two dice to roll yourself. Add the numbers that are on the top.
What other totals could you get if you roll the dice again?
Can you find all the possible totals that you could get? How do you know that you've found them all?
Getting Started
If one dice shows $6$, what could the other dice be showing?
What totals would they give?
What could the other dice be if one is a $5$? A $4$?
How do you know you've found all the totals?
Student Solutions
It seems that quite a few of you enjoyed the dice rolling needed here.
Joe from Clutton Primary had this work of his commented upon by an adult.
Joe worked systematically to find all the possible combinations for throwing two dice. He identified the lowest possible score of 2 and the highest score of 12. At first he thought that 4 was the lowest score and was then able to correct himself.
There are eleven possible solutions to 'Two Dice'.
Let's say Dice One is 'A' and Dice Two is 'B'.
They are as follows :
$2$ (A=$1$ and B=$1$)
$3$ (A=$1$ and B=$2$; A=$2$ and B=$1$)
$4$ (A=$1$ and B=$3$; A=$2$ and B=$2$; A=$3$ and B=$1$)
$5$ (A=$1$ and B=$4$; A=$2$ and B=$3$; A=$3$ and B=$2$; A=$4$ and B=$1$)
$6$ (A=$1$ and B=$5$; A=$2$ and B=$4$; A=$3$ and B=$3$; A=$4$ and B=$2$; A=$5$ and B=$1$)
$7$ (A=$1$ and B=$6$; A=$2$ and B=$5$; A=$3$ and B=$4$; A=$4$ and B=$3$; A=$5$ and B=$2$; A=$6$ and B=$1$)
$8$ (A=$2$ and B=$6$; A=$3$ and B=$5$; A=$4$ and B=$4$; A=$5$ and B=$3$; A=$6$ and B=$2$)
$9$ (A=$3$ and B=$6$; A=$4$ and B=$5$; A=$5$ and B=$4$; A=$6$ and B=$3$)
$10$ (A=$4$ and B=$6$; A=$5$ and B=$5$; A=$6$ and B=$4$)
$11$ (A=$5$ and B=$6$; A=$6$ and B=$5$)
$12$ (A=$6$ and B=$6$)
Two $1-6$ dice will not add up to any other number. This solution shows the
amount of numbers possible to make. However there are 36 possible
Pupils from David Hoy Elementary School worked on the challenge and this was the result;
Step 1: Explore
In groups of three we took turns rolling the dice. Then we recorded our answers, as this was going to help us remember what numbers we found. In some groups everyone wrote down the number sentences, in other groups, each person wrote one of the addends, and the third person wrote the solution.
(Ex: 4+3=7)
Step 2: Making a class list of all the numbers we found and how
We read the numbers we found off of our sheets checking to see if we had rolled the same as other groups, and then looking for other numbers we missed.
Step 3: To make sure we had found all the numbers, the teacher asked what the largest number was and why
"12, it can't be bigger than 12"
"It can't be a very big number cause there is only like 4 maybe 5 sides on the dice, so not many numbers."
"Six was the biggest number on the dice."
"Double six is 12, so that is the biggest number."
"Then the smallest has to be 2, cause double 1 is 2, and 1 is the smallest number."
Teacher: Can we get all the numbers in between?
"Look at the board, and put the solutions on a number line."
Which is what we did, and we discovered that there were multiple ways to get some of the numbers, but only one way to get 2 and 6.... Next I think we will try and figure out how many ways we can get the numbers 3-11 using two dice.
Their teacher said:
What a nice problem! My class loved it, it was a great way to practise counting, counting on, and writing number sentences.
Well done, thank you so much for all the contributions. Submissions that include some reflections of what happened are always good to share.
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 provides a valuable experience for younger pupils to explore some simple additions while finding all possibilities.
Possible approach
Using a dice with dots on, encourage discussion as to what numbers are represented by the faces of the dice before introducing the challenge itself.
You could support the children to collect their totals on the board. Ask them how they should be arranged and see if they can suggest a systematic way of recording their results. For example, they might start with all the totals that use a 1. In this way, you can ask the class to talk about the patterns they notice and this will help to reveal any combinations that are missing.
Key questions
These questions have been phrased in ways that will help you to identify the children's prior knowledge about both the number concepts involved and the strategies and mathematical thinking needed to solve the problem.
Can you make a bigger/smaller total?
What is the highest total you could make?
What is the lowest total you could make?
If one dice shows 6, what could the other dice be showing?
How will you know when you've found all the totals?
Possible extension
You could make use of more dice and/or dice with different numbers of faces. Alternatively, consider finding the difference between the two numbers or the product of the two numbers.
Possible support
Children who struggle with addition may count the dots to help them but encourage them to articulate the number sentence once they have done so. This will help them to build the visualisations of the numbers as dotty dice patterns which will support their learning of number bonds.