standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Task
Below is a table of diameters of different denominations of United States
Coins:
{"table": [{"_attributes": {"class": ["centered", "narrow"]}, "tbody": [{"tr": [{"th": [{"_value": "Denomination"}, {"_value": "Diameter"}]}, {"td": [{"_value": "Dime"}, {"_value": "$17.9$ mm"}]}, {"td": [{"_value": "Nickel"}, {"_value": "$21.2$ mm"}]}, {"td": [{"_value": "Quarter"}, {"_value": "$24.3$ mm"}]}, {"td": [{"_value": "Half Dollar"}, {"_value": "$30.6$ mm"}]}]}]}]}
If we place nickels around a central dime, as in the picture below, there is room
for five nickels with extra space but not enough room for a sixth nickel:
[IMG]
How many dimes fit around a central dime?  What about around a
central nickel, quarter, or half dollar?  Do the dimes fit snugly or is there
extra space?
How many half dollars fit around a central dime?  What about around
a central nickel, quarter, or half dollar?  Do the half dollars fit snugly or
is there extra room?
Extending the work in (a) and (b) above, for positive numbers $r$ and $s$
how many circles of radius $s$ will fit around a central circle of radius $r$?


aligns: no