1.Find the sum 1+2+3...+9+10 through a conceptual shortcut. Demonstrate on a picture, perhaps using a spreadsheet how to use manipulatives in support of the conceptual shortcut. Connect this summation problem to the history of mathematics.

2.Represent the sum 1+5+9+13+17 as a product of two integers greater than one ( do not consider different orders of factors.) Can this be done i ore than one way? why or why not?

3.Using pictures, solve the following problem. If 2 twiggles and 5 gloves have 18 legs, how many legs does each creature have? Alternatively, if one spent $18 on buying notebooks and pens of which costs $2 and $5, respectively, how many notebooks and how many pens did one buy? Now. solve the equations 2x+5y= 18 in whole numbers using a conceptual shortcut. Here the variables x and y stand for the number of legs each creature has; alternatively, x and y are the prices of a notebook and pen. What if we replace 18 by 24? Does the corresponding equation (with 24 in the right hand side) have only one solution? why or why not?