# Cars In A Garage Years 2 - 10

### Preparation

• Collect five (5) or more toy cars like the ones in the picture.
Each one has to be a different colour.
If you don't have cars, blocks of different colours can be pretend cars.
• Print this sheet of garages.
Your cars have to be able to park in the garages.
• Write the title of this challenge and today's date on a fresh page in your maths journal.

### Getting Started

• Use three (3) cars and three garages.
• Cover the garages you aren't using with paper.

In the photo the cars are parked in the order ... Green, Purple, White
They could have been parked in a different order, perhaps ... White, Purple, Green

• How many different ways can your three cars be parked in your three garages?
• How do you know when you have found them all?
• Explain your thinking in words and pictures.

 After you finish your journal entry, you can watch these two (2) videos. They are Year 4 children explaining the answer. They are short. You may have to listen more than once to see and hear everything. Video A: William (37 seconds) Video B: Jessica & Madeleine (17 seconds) Pretend you are the Year 4 teacher and the children gave you these videos as their homework. Write a comment for them that will encourage them and help them learn a little more.

### The Next Challenge

• Open this Cars In A Garage Starter. You can read it on screen or print it.
• You have already done Question 1.
Your challenge now is to try Question 2.
• Remember to keep good notes and diagrams in your journal.

Have fun exploring Cars In A Garage.

When you have written your own journal you can look at some journals from the Year 4 class that made the videos.
Go to the Answers & Discussion further down the page.

Note: If your school is a member of Maths300, your teacher can provide software that will help you explore Cars In A Garage.

You can stop the activity here if you want to ... or you can take yourself to the next level.

### The Big Challenge

The challenge on the Starter has two parts.
The first part is to try five cars in five garages.
• Find a method to calculate the number of ways without having to draw them all?
• Explain your calculation to someone else.
The second part is the one hundred (100) car challenge. 100 cars in 100 garages.
• It might be too hard to find the exact number. It's pretty big.
• What the challenge really wants you to do is explain how to do it.

### Digging Deeper

Calculating the 100 Challenge
• There are 100 cars that could go in the first garage.
• Choose 1, then there are 99 left that could go in the next garage.
• Choose 1, then there are 98 left that could go in the next garage.
• And so one until there is only one car left and that has to go in the last garage.
But the very first car could be different, so you have to go back and start again.
• In fact the first car could be different 100 times.
• Which means the second car could be different 99 times.
• And the third car could be different 98 times.
• And so on...
That's a lot of times-ing:
100 x 99 x 98 x 97     ...     4 x 3 x 2 x 1 = ???
Try it on your ordinary calculator. Can you get all the way?
• Describe what happens.
Try it with a scientific calculator. Can you get all the way?
• Describe what happens.
What happens if ...?

A mathematician is never finished with a problem.
They are only finished until someone asks the next interesting question.
Here are a few of questions that might interest you.

• What happens if there are 3 cars and 4 garages? How many ways to park?
• What happens if there are 4 cars and 4 garages, but the green car must always be in the left garage?
• What happens if there are 4 cars and 4 garages, but the red and the blue must not be beside each other?
You don't have to do any of them. We just want you to know that asking interesting questions is where a mathematician's work starts.

### Just Before You Finish

• In your journal explain what parts of this activity you liked and why you liked those parts.
• Does this activity make you think of something you want to learn better, or perhaps something new you want to know?