Courtney visited her grandmother. Her grandmother used to collect stamps. She had a shoebox full of 5 cent, 6 cent and 7 cent stamps. Courtney thought, I could mail a lot of different-sized letters and postcards with these stamps.

She tried to figure out the different amount of postage she could make with a combination of those stamps. What totals could she make and what totals are impossible?

Explain how you found your solution. How do you know your solution is correct?

Courtneys grandmother said, Your grandpa has different shoeboxes with other stamps. His shoebox has just 4 cent, 6 cent and 8 cent stamps.

Which totals can you make with these stamps?

Courtney said, I wonder why there is a difference between Grandmas and Grandpas shoeboxes. I like finding the largest total I cant make, but Grandpas box has an unlimited amount of totals I cant make.

I wonder why it works for some and not for others. She continues, I am going to investigate which three stamp amounts are like Grandmas shoebox and which three stamp amounts are like Grandpas shoebox.

I am going to compare three different sets. I will try 6 cents, 7 cents and 8 cents.

Then I will try 6 cents, 9 cents and 12 cents. Finally I will try 6 cents, 8 cents and 9 cents.

Explain how the three different sets are like either Grandmas shoebox or Grandpas shoebox.