This is the third section of Chapter 1: Number Relationships. The learning goal for this section was for students to understand the different between common factors and common multiples. Furthermore, we wanted to find the best ways to determine common factors and common multiples of big numbers.

## Introduction to Common Factors and Common Multiples

The first thing we did was define *Greatest Common Factor (GCF)* and *Least Common Multiple (LCM)*. We actually started by playing with linking cubes, using them to represent different factorizations of numbers, in order to convince ourselves of what factors and multiples really are.

**Example 1** is a visual representation of how we compared the common factors of 15 and 36, using linking cubes.

**Example 2** is a technique that one student developed to find the Greatest Common Factor of two numbers. What he did was to find all the factors of the smaller number, and test them from largest factor to smallest factor to see if any were also factors of the bigger number.

## Generalizing Our Ideas

Next, we tried to figure out an easier way to find out the Greatest Common Factor of two numbers. The problem we were having was that it's easy to find just *any* common factor. It's hard to figure out if it's the *greatest* common factor. So, we used the scientific method to try to work out a solution using bigger numbers. By using *factor trees* and a *Venn diagram*, we were able to figure out a good method.

## Support Questions

Finally, we chose some support questions to help us ensure that we understood. Please do any five of the following: p. 14, #4, 5, 6, 7, 9, 10, 12, 18.