1. Consider the following collection (in base 4):

a) If one box were to be unpacked, how would the inventory indicate the number of boxes, rolls, and pieces?

b) Enter this configuration in the computer. Unpack a box and check your answer. Were you correct? If not, what surprised you about the answer?

c) Now pack enough rolls to make a box. How will the inventory form indicate this transformation?

d) How many total candies (in base 10) are shown in the configuration above?

e) Check your answer in part (d) by accessing the graphics menu and choosing "Show total (base 10)" Were you correct? If not, what surprised you?

2. Start a new picture. (From the "File" menu, choose "New". Enter a packing rule of 5. Enter 7 boxes, 6 rolls, and 11 pieces. How would you go about figuring out how many candies are shown in base 5?

a) Express the number of candies in base 5:

b) If you were to pack up all of the candies into the canonical form, what would the resulting inventory indicate?

c) Express the number of candies in base 10:

d) Check your work. Were you correct?

e) If you were to enter the same number of boxes, rolls and pieces (in base 10), but changed the packing rule to base 8, would the total quantity of candies change?

f) Suppose you enter the same number of boxes, rolls and pieces (in base 10), but changed the packing rule to base 8. How would the inventory form change? Would the number be higher or lower than the number as written in base 5?

g) In general, if you want to indicate a given quantity in base 8 and base 4, which expression would appear (from a base 10 perspective) to be larger?

3. Consider 1 box in the candy factory in base 10.

a) How many candies are contained in this box (in base 10)? How could you prove your answer?

b) Consider this same representation (one box) in base 8. How many candies are contained in this box (in base 10)? How could you prove your answer?

c) Consider this same box in base 4. How many candies are contained in the box (in base 10) now? How could you prove your answer?

d) In general, how can one drawing represent different quantities?

e) Consider the inventory form shown to the left:

How many candies wouldbe contained in the collection if the packing rule was: 1) Base 3? 2) Base 8? 3) Base 12? 4) Base n?

f) Reflect on your answers to parts 2(g) and 3(e). What do these two tell you about place value?