Patterns and learning to observe patterns is one of the most joyful parts of studying mathematics. In Algebra this past year, we discovered the pattern that exists when we try to simplify the value of the difference of two squares. In this middle of observing the values, we see the pattern and suddenly we're starting to see the magic of relationships and we're excited. Try to find this pattern yourself:
- Write out the values of the following equations: five squared - four squared ; four squared - three squared ; three squared - two squared
- What do you see?
- Can you see a relationship between the differences and the original numbers? For example, the difference between five squared and four squared is what number? How is that related to four and five?
- Is that relationship unique?
- What pattern seems to be formed if the two starting numbers are consecutive?
- What if we used six and four instead of five and four?
- What pattern seems to be formed if the two numbers are two places away from each other? (mathematically, they have a difference of two)
- Can you generate a pattern that would work for both scenarios? (if a squared number was a square, how would these differences look geometrically?)
- How can we generalize the difference of two squares in words?
- Can you convert this pattern into an algebraic formula?
Working through these ten questions takes time and patience. When we learn to be patient, rest in the discovery of this truth, create as many examples that we need to see a pattern, and take the time to think about the material, we are rewarded with discovering the wonder of the world of mathematics!