Cars in Garages                                               Number, Level 1

Problem
I own 5 cars and a very large garage.
If I can see 2 cars parked outside the garage, how many are inside?

How many different ways can I park my cars inside and outside the garage?

What is this problem about?
This problem is all about how numbers are made up of other, smaller numbers, an essential concept underlying addition and subtraction. The problem helps develop two ideas: first, that there is a finite set of whole number pairs for a given number (for example, 5 can be thought of as 0 and 5, 1 and 4 , 2 and 3, and no other pairs can be found); second, that numbers are uniquely paired (if 2 is one of the parts of 5, the other part must be 3). Children need to investigate these relationships many times. Until children believe that 2 and 3 is always 5, they see no reason to remember it.

Achievement Objectives
Number (Level 1)
- make-up, tell, and record number stories, up to 9, about given objects and sequence pictures.

Mathematical Processes
- devise and use problem solving strategies (act it out, draw a picture)
- use equipment appropriately

Resources required
- toy cars (5 identical cars for each pair of children)
- shoe boxes or similar
- half sheets of paper
- Blackline master of the problem (English)
- Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- Give many names for the same number by drawing a picture, using equipment or acting it out.
- Use a mental image of 5 to work out the missing number in the number pair.

Teaching Sequence

1. Read the first part of the problem to the class to ensure that they understand that they are working with 5 cars.
2. Brainstorm for ways to solve the problem.
3. You could suggest that they use equipment to solve the problem or it could be play acted with 5 children pretending to be cars. A table covered with a cloth could be the garage..
4. Having children tell how they know the number of cars in the garage is the most important part of this problem. Allow the children to describe their ideas. Encourage explanations. 
   How did you know how many cars were hidden?
   Tell us about your thinking?
   Could there be any other number of cars in the garage when 2 are parked outside? How do you know?
5. Get the children to plan ways to record their solution. A more structured approach would be to give them a flip-fold page to record the activity.
6. Read the second part of the problem and let the children try to solve this in pairs or on their own.  You need to use identical cars or there are multiple solutions for each pairing (for example:  there would be 5 ways to complete the 1-4 pairing if all the cars were different).  Let the children play around with the pairings of digits. You may like to use the following questions to support their problem solving.
   How do you know how many cars are parked inside?
   Does there always have to be a car in the garage? or parked outside?
   How do you know that you have found all the ways that the cars can be parked?
   How are you keeping track of the ways that you find?
7. At the end of the session read the flip-fold books with the class, opening the garages to reveal the hidden cars. Discuss with the class ways they used for thinking about the missing cars.

Other Contexts for the Problem
Trains in tunnels
Cups in the cupboard
Shells in a bucket
Frogs in a pond or on lily pads