Sort the Street
Sort the houses in my street into different groups. Can you do it in any other ways?
Problem
Here is a picture of nine of the houses in my street:
Find as many different ways to sort them into groups as you can.
You may like to use this interactivity to drag the houses into groups.
Getting Started
Student Solutions
Thank you to everybody who sent in their thoughts about these houses.
Lauren and Toby from Llandovery Prep School in Wales each sent in a solution like this:
There are 3 houses with green doors.
There are 3 houses with blue doors.
There are 3 houses with black doors.
There are 3 houses with 3 storeys.
There are 4 houses with grey rooves.
There are 5 houses with red rooves.
There are 3 houses with small chimneys.
There are 6 houses with big chimneys.
There are 3 houses with 3 windows.
There are 6 houses with 5 windows.
Well spotted! I wonder how many different ways of sorting the houses that gives us?
Elijah from the Independent Jewish Day School wrote in with these solutions:
There are 3 houses with blue doors
There are 3 houses with green doors
There are 3 houses with black doors
There are 3 houses with 3 small windows
There are 3 houses with 5 small windows
There are 3 houses with 4 small windows and one big window
There are 3 houses big houses
There are 3 houses medium houses
There are 3 houses small houses
There are 6 houses with a door on the right
There are 3 houses with a door in the middle
There are 4 houses with a grey roof
There are 5 houses with a red roof
Well done for spotting all of these differences, Elijah. How many ways of sorting the houses does that give us?
Noah from Goodwyn School came up with these eight solutions:
Door colour - blue, green or black
Number of windows - 3 or 5
Roof colour - grey or brown
Amount of floors - 2 or 3
Position of door - right or middle
Number of sections in window - 4 or 6
Positioning of windows
Height of roof - small or big
Thank you for sending us these ideas, Noah - that's eight different ways we could sort the houses.
If you would like to get in touch with us with some different ways of sorting these houses, please email us with your ideas.
Teachers' Resources
Sort the Street
Here is a picture of nine of the houses in my street:
Find as many different ways to sort them into groups as you can.
You may like to use this interactivity to drag the houses into groups.
Why do this problem?
This low threshold high ceiling problem is ideal for children to work on in pairs or small groups as this will encourage them to talk about what they are doing, which is a great assessment opportunity for you. Its open-ended nature is a reason in itself to try it in the classroom. Some pupils may surprise you with their inventiveness! The task encourages learners to identify what is the same and what is different as they compare the images. Being able to articulate sameness and difference is a key mathematical skill.
Possible approach
This problem featured in the NRICH Primary and Secondary webinar in September 2022.
To introduce the problem, invite the children to chat in pairs about what they notice about the houses. Then, sort the houses according to a particular criterion using the interactivity and invite the class to work out how they have been sorted. This will help them to grasp the focus.
Key questions
What is the same about these houses? Are there any others like that?
Why have you grouped the houses in this way?
Possible extension
Some children will enjoy the challenge of finding two or more criteria which fit some houses, perhaps also extending to more than two groups. For example, rather than just "these houses have five windows" and "these houses don't have five windows", some might say "these houses have five windows and are tall"; "these houses have five wondows but aren't tall/are short" and "these houses don't have five windows".
The problem Sets of Numbers offers an opportunity for more experienced learners to apply the skill of identifying what is the same and what is different to numbers. (The numbers themselves could be changed to suit the class.)
Possible support
You could model one way to approach this activity by taking two houses and asking what the same is about them, as suggested in the key questions. Children can then add more houses to the pair by looking for those that fit this criterion.
If you are looking for copies of the old style houses you can print them here.
Two Dice
Problem
Printable NRICH Roadshow resource.
What other totals could you get if you roll the dice again?
You will need two dice to play this game. The children can count the total number of spots on the dice or add them together using number facts they already know.
Record the results and explore the different totals that you can get.
Help them to find all the possible combinations.
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
Two Dice
Printable NRICH Roadshow resource.
What other totals could you get if you roll the dice again?
You will need two dice to play this game. The children can count the total number of spots on the dice or add them together using number facts they already know.
Record the results and explore the different totals that you can get.
Help them to find all the possible combinations.
Why do this problem?
This activity provides a valuable experience for younger pupils to explore some simple additions while finding all possibilities.
What children need to know to play this game
The children need to be able to roll two dice and identify their score.
Possible approach
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.
What is the lowest total you could make?
If one dice shows $6$, what could the other dice be showing?
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.
Three Ball Line Up
Problem
Two children are playing with three balls, one blue, one red and one green.
They toss up the balls, which run down a slope so that they land in a row of three.
In how many different ways could the balls land?
How do you know you have found them all?
You might like to use the interactivity below to explore the problem.
If you would like to try another task which involves finding all possible solutions, you could look at Inside Triangles.
You may be interested in the other problems in our Working Systematically Feature.
Getting Started
If you are not using the interactivity, perhaps you could find three differently coloured balls, counters or cubes to try out some ideas. Or you could just use coloured pencils and draw three spots.
If red landed in the middle, how could the blue and green fall?
Where else could red land if it wasn't in the middle? Can you use this idea to find all the ways?
Student Solutions
We received some very well explained solutions to this problem. Many of you described how you used ordered your solutions so you knew you had all the solutions. We can call this 'working systematically'.
Jack from Tetherdown Primary sent his solution and his teacher said:
Jack worked out that there were six ways that three different coloured balls could land. At first, he stopped at four ways but then realised that if there were two ways of a red ball landing first, then there would be two ways
of a blue ball landing first and two ways of a green ball landing first.
Here is a picture of Jack's recording:
AT from ASB in the Netherlands used the same method:
So, what I did was I first started with one colour, red and then I added blue and green. I did the same on the next one (still starting with red) but then green-blue.
So basically, I did the same for each of them and found six different outcomes...
It is easier if you just draw it on a piece of paper because then you can keep track of them easily!
Here is AT's drawing:
Pranaiv from Saint Bartholomews Primary School in Wolverhampton used a similar approach:
Thank you also to Roheena and Esther from Edgware Primary, who agreed with the six solutions.
Maryam from the International Community School explained how she worked on the task, which again, was similar:
The balls can land in 6 different ways; each coloured ball (green, blue, red) can be in three different positions - front, middle and end. So if (for example) the green ball is at the front, the red and blue need to follow it, creating one solution. Then swapping the red and blue ball (with the green ball still at the front), we created the second solution. If we do the same with a turn for each coloured ball at the front, we get six possibilities in total.
Her recording is very clear:
Mehetabel from Corpus Christi also organised her recording very clearly:
A pupil at St John the Baptist School in Wales shared this fantastic video explanation with us:
Thank you also to Charles from Clear Water Bay School in Hong Kong, Alex from Longsands School and Maria from Wimbledon High Junior School who all took the trouble to explain how they worked it out.
Aiste from VDU Rasos Gymnasium, Lithuania, wrote:
There are three balls and also three spaces for them.
So: _ _ _
We should put three different balls.
In the first place you can put all three balls. In the second place - 2 and at last in the third - 1.
So when we multiply these numbers and then will find the answer. 3x2x1 is 6.
I wonder why we multiply the numbers?
Teachers' Resources
Why do this problem?
This problem is a good context for encouraging learners to work systematically. Then they will be able to convince someone else that they have found all the possibilities.
Possible approach
You could introduce the task by showing the class the interactivity and, without saying anything at all, roll the balls three or four times. Ask children what they notice and what they wonder, and invite them to talk to a partner.
Take some noticings and questions from the whole group, but try not to respond yourself. Instead, rely on other members of the group to comment or answer questions.
You can then clarify what is happening in the simulation, and ask how many different ways the balls could land. Look back at the results you have recorded so far in the interactivity, and invite learners to explore whether there are any other combinations. As they work in pairs on this (perhaps using coloured counters to represent the balls), listen out for those who are developing a system to find all possibilities. You may like to draw attention to those who have created a good recording system too.
In the plenary, you could ask particular pairs to share what they have been doing and you could facilitate a whole class discussion so that a conclusion is reached. You may wish to ask learners to record individual solutions on individual strips of paper, then you can pin some of these up on the board. Invite the group to reorder the solutions to reflect a pattern and use this to create any solutions that are not yet displayed. Emphasise that although they have worked together using one pattern, or one particular way of working systematically, there is no 'right' way to work systematically. There will be many different systems in the room and that should be celebrated.
Key questions
If red landed in the middle, how could the blue and green fall?
Where else could red land if it wasn't in the middle?
Can you use this idea to find all the ways?
How will you remember the ways you have found so far?
Possible support
Pupils would benefit from having three differently-coloured counters to use while tackling this problem. If they are finding it difficult to work systematically, you could offer them 6 Beads first, which might be a more familiar context.
Possible extension
Encourage children to use four balls/counters. If learners would like another context in which to practise working systematically, Inside Triangles would be an appropriate task.
Finding all possibilities Lower Primary
Ordered Ways of Working Lower Primary