Quandaries
and Queries 

Person asking: Home schooling parent While teaching the multiplication table to my daughter I noticed an interesting pattern. It goes something like ...add two to it ...multiply the two numbers together ...add one to the resulting number ...it will now be equal to the original number plus one, squared. 

Hi Melissa, You can "picture" the equation [X(X+2)]+1=(X+1)(X+1)
using the rows of pennies: then you can move the last column to make it a row of 6 pennies on top: now you only need to put the single penny in the northeast corner to complete a 7 by 7 square of pennies. Later on in school, your daughter will learn algebra so she will learn to expand expressions such as X(X+2), to get X^{2} + 2X, and also to expand (X+1)(X+1) to get X^{2} + 2X + 1 (using the identity (A+B)^{2} = A^{2} + 2AB + B^{2}). Therefore she will have an algebraic justification of the identity [X(X+2)]+1=(X+1)(X+1): 

