Discovering Patterns

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: 

1. Write out the val­ues of the fol­low­ing equa­tions: five squared — four squared ; four squared — three squared ; three squared — two squared
2. What do you see? 
3. 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? 
4. Is that rela­tion­ship unique? 
5. What pat­tern seems to be formed if the two start­ing num­bers are consecutive?
6. What if we used six and four instead of five and four? 
7. What pat­tern seems to be formed if the two num­bers are two places away from each oth­er? (math­e­mat­i­cal­ly, they have a dif­fer­ence of two)
8. Can you gen­er­ate a pat­tern that would work for both sce­nar­ios? (if a squared num­ber was a square, how would these dif­fer­ences look geometrically?)
9. How can we gen­er­al­ize the dif­fer­ence of two squares in words?
10. 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!