Making pathways
Problem
We have some blue, green and red paving slabs - twelve paving slabs of each colour.
There are three different sizes of slabs for each colour. All paving slabs are the same width.
The blue paving slabs are: four of length 1, four of length 9 and four of length 24.
The green paving slabs are: four of length 1, four of length 7 and four of length 25.
The red paving slabs are: four of length 1, four of length 5 and four of length 29.
The paving slabs cannot be broken into smaller pieces. Every path in this challenge must be made out of only one colour of paving slab - we cannot have a mixture of colours in a path.
Challenge 1
If we want to make a path of length 18 in each colour, we might have:
Blue:
9+9 
Green:
7+7+1+1+1+1 
(which we count as the same solution as e.g. 7+1+1+7+1+1 
Red:
5+5+5+1+1+1 
(which we count as the same solution as e.g. 5+1+5+1+5+1 
Can you make three paths of length 22, one in each of the three colours?
Next, try making three paths of length 40, one in each of the three colours.
Lastly, try making three paths of length 64, one in each of the three colours.
Challenge 2
In how many different ways can you make paths of length 75, using only one colour in each path?
Challenge 3
We would like to find eight consecutive lengths of path that can be made out of the blue, green and red paving slabs, where each of the eight lengths can be made out of each colour.
It would look something like this picture for the consecutive lengths 59, 60, 61, 62, 63, 64, 65 and 66:
Unfortunately this is not a solution as it cannot be made without breaking some paving slabs up into smaller pieces.
Your challenge is to find eight consecutive lengths of path that can be made out of each colour separately.
Student Solutions
Well done to everybody who had a go at this activity. We received lots of solutions from the children at Ganit Kreeda in Vicharvatika, India - thank you all for sending us your ideas.
Ishaan and Samaira sent in these solutions for Challenge 1:
Well done for working these out!
Hiren, Vansh, Eshaan, Rishaan, Samaira, Renah, Udit, Uday, Viha, Gowri, Arya, Vishnu, Krishna, Rivaan, Miraya and Vraj all worked on Challenge 2, and came up with the following solution:
We tried systematically starting with the smallest length size and if doesn’t work we used a bigger length. Here are our solutions. Together we got 2 for blue, 2 with green and 1 with red.
Blue 24+24+9+9+9, 24+24+24+1+1+1
Green 25+25+7+7+7+1+1+1+1, 25+25+25
Red 29+29+1+1+5+5+5
Well done for working systematically to find all of the possible solutions.
For Challenge 3, the children sent in these solutions of eight consecutive numbers that could be made with each colour:
Blue = 72, 73, 74, 75, 76, 77, 78, 79.
Red = 29, 30, 31, 32, 33, 34, 35, 36.
Green = 21, 22, 23, 24, 25, 26, 27, 28.
I agree that these numbers can all be made with these colours of paving slab, but what we'd really like to find is eight consecutive numbers that can be made with blue paving slabs, red paving slabs and green paving slabs (separately!). If you think you've found eight consecutive numbers that can all be made with each colour, please email us with your ideas.
Teachers' Resources
Making Pathways
We have some blue, green and red paving slabs - twelve paving slabs of each colour.
There are three different sizes of slabs for each colour. All paving slabs are the same width.
The blue paving slabs are: four of length 1, four of length 9 and four of length 24.
The green paving slabs are: four of length 1, four of length 7 and four of length 25.
The red paving slabs are: four of length 1, four of length 5 and four of length 29.
The paving slabs cannot be broken into smaller pieces. Every path in this challenge must be made out of only one colour of paving slab - we cannot have a mixture of colours in a path.
Challenge 1
If we want to make a path of length 18 in each colour, we might have:
Blue:
9+9
Green:
7+7+1+1+1+1
(which we count as the same solution as e.g. 7+1+1+7+1+1
Red:
5+5+5+1+1+1
(which we count as the same solution as e.g. 5+1+5+1+5+1
Can you make three paths of length 22, one in each of the three colours?
Next, try making three paths of length 40, one in each of the three colours.
Lastly, try making three paths of length 64, one in each of the three colours.
Challenge 2
In how many different ways can you make paths of length 75, using only one colour in each path?
Challenge 3
We would like to find eight consecutive lengths of path that can be made out of the blue, green and red paving slabs, where each of the eight lengths can be made out of each colour.
It would look something like this picture for the consecutive lengths 59, 60, 61, 62, 63, 64, 65 and 66:
Unfortunately this is not a solution as it cannot be made without breaking some paving slabs up into smaller pieces.
Your challenge is to find eight consecutive lengths of path that can be made out of each colour separately.
Why do this problem?
This activity challenges pupils to use all four operations and their knowledge of number facts. It is a great opportunity to develop problem-solving skills as the calculations on their own are not enough.
Possible approach
The introduction needs to emphasise the rules of the situation, clearly stating what is allowed. Introduce the activity by working together through the example shown at the start of challenge 1.
If other examples are needed before learners set off on their own, then totals of 30, 31, 33, 34, 35, 42, 43, 46, 48, 49, 50, 51, 52, 53, 58, 59, 60 and 61 could be used as well.
Key questions
Tell me about how you are doing this.
Can you tell me why this is correct?
How are you making sure you obey the rules?
Possible extension
The pupils could set up their own challenges for different slab lengths.
Possible support
Some children may need some connecting cubes to represent the long slabs.