Paw prints

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Age

7 to 11

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

A dog is looking for a good place to bury his bone.

He goes along each and every path only once.

Where did he start in each case?

Where has he ended up?

What possible routes could he have taken?

Student Solutions

We had some good solutions sent in and here are just some of them. Many pupils from Bradon Forest in the U.K. sent in solutions. Hannah produced a Powerpoint presentation for the solutions she found. Well done Hannah - here are her slides.

Georgia, also from Bradon, sent in a Word document with pictures to show the solutions. Here is what she wrote.

In the case with the three obstacles, the dog starts at the kennel and walks on the path that leads to the signpost.

When he gets to the signpost, he goes down the path straight ahead (the one with the arrow pointing at the tree).

He then goes down the path with the arrow pointing towards the kennel, and then goes down the path pointing at the tree.

Finally, he goes down the only remaining path that he hadn't previously walked on, arriving at the signpost.

In the second case, you start at the kennel and proceed to the tree.

Next, you carry on until you reach the signpost.

Next take the path pointing at the lamppost, until you reach the lamp.

Turn and head back to the signpost.

Then, you take the path round to the kennel and, when you reach it, go down the path to the lamppost, then turn and reach your final destination, the tree.

In the third case you start at the big tree, and go to the small tree.

Then, you continue on the path to the kennel.

Once you reach the kennel, you go to the signpost.

Then, back to the big tree and take the last path going away from the tree.

You then go on the unused path to the kennel, and loop back to the small tree.

To finish off, you go from the little tree directly to the lamppost.

Lastly Krystof's dad emailed us his son's solutions to four of the problems and here's his solution for this activity.

Teachers' Resources

Paw Prints

A dog is looking for a good place to bury his bone.

He goes along each and every path only once.

Where did he start in each case?

Where has he ended up?

What possible routes could he have taken?

Why do this problem?

This problem requires pupils to use visualising, trial and improvement, and their own recording methods to get at solutions. The challenge may be unlike others they have seen in mathematics before, and this gives opportunities for all learners to shine.

Possible approach

Some pupils will enjoy and benefit from acting this out, so this might be a good place to start. You could draw the first 'map' on the playground with chalk, or use tape on the hall floor, with something being set up to represent each of the main landmarks (for example a chair or hoop). Explain the challenge to the group and ask them to talk in pairs about the dog's route. You could then challenge one pair to demonstrate the route, perhaps with one of them walking it through. Did any pair have a different way?

Pupils could then work in smaller groups to have a go at the other routes. This sheet has a picture of each route. You may wish to allow groups to physically recreate the routes in the same way as you did as a whole class, but this is certainly not necessary - many will be able to use visualisation to plan the dog's route. However they tackle the problem, encourage learners to find a way to represent their ideas so they remember their solutions.

In a plenary, you can share findings but in particular you can focus on how children approached the problem. Encourage pairs to explain how they found their solutions and draw attention to the differences and the similarities. For example, it is likely that some trial and improvement was used by more than one group, but it might have been through physically walking, through imagining what would happen when the dog took certain paths, or through jottings. You could also encourage learners to share their different recording methods.

Key questions

How have you got to your solution?

Did you all agree?

What other ideas did you have?

How have you checked that it's ok?

What do you notice about the starting point each time?

How did you know the dog started there?

What do you notice about the finishing point each time?

How did you know the dog ended there?

If you count the number of pathways to and from each landmark, what can this tell you about the dog's route?

Possible extension

Pupils could be given the opportunity to create their own questions that are of a similar style. Having completed this challenge, learners could try The Bridges of Konigsberg.

Possible support

Annotating a print-out of this sheet might help some children work on the problem. Others will find it easier to physically walk it through and this may mean they require your help creating the map on the ground.

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