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). Students need to investigate these relationships many times. Until students believe that 2 and 3 is always 5, they see no reason to remember it.
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?
- Read the first part of the problem to the class to ensure that they understand that they are working with 5 cars.
- Brainstorm for ways to solve the problem.
- Having students tell how they know the number of cars in the garage is the most important part of this problem. Allow the students 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?
- Get the students to plan ways to record their solution.
- Read the second part of the problem and have students 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). Support the students as they problem solve with questions such as:
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?
- Share and discuss solutions.
Other Contexts for the Problem
Trains in tunnels
Cups in the cupboard
Shells in a bucket
Frogs in a pond or on lily pads
Because 2 + 3 = 5 , if there are 2 cars inside the garage there must be 3 outside.
6 possibilities: (0,5) (1,4) (2,3) (3,2) (4,1) (5, 0)