The first 3 digits after the decimal for the decimal expansion of have been calculated. Find the next 4 digits.

Long division calculations for decimal expansion, showing place value. The first line indicates the given place values of the quotient, 0 point 4 2 8. The second line indicates the division sentence 3 divided by 7; the number 7 is the left most number, followed by the long division symbol, and the number 3 inside; the 3 lines up vertically with the 0 above. On the third line reads as "minus twenty eight," with the 2 directly below the 3 from above. The fourth line reads "twenty," with the 2 directly below the 8 in 28. The fifth line reads "minus fourteen," with the 1 directly below the 2 in 20 and the 4 directly below the 0 in 20. The sixth line reads "sixty," with the 6 directly below the 4 in 14. The seventh line reads "minus fifty six" with the 5 directly below the 6 in 60, and the 6 directly below the 0 in 60. The eight line reads "4" with the 4 directly below the 6 in 56. A vertical line is drawn through all of the lines, falling between the 0 and the 4 in "0 point four two eight" and the 2 and 8 in twenty eight.

Your teacher will give your group a set of cards. Each card will have a calculations side and an explanation side.

1. The cards show Noah’s work calculating the fraction representation of . Arrange these in order to see how he figured out that without needing a calculator.

2. Use Noah’s method to calculate the fraction representation of:

1. 1. Why is between 1 and 2 on the number line?
   2. Why is between 1.4 and 1.5 on the number line?
   3. How can you figure out an approximation for accurate to 3 decimal places?

   4. Label all of the tick marks. Plot on all three number lines. Make sure to add arrows from the second to the third number line.

      

2. 1. Elena notices that a beaker in science class says it has a diameter of 9 cm and measures its circumference to be 28.3 cm. Using these values and the equation for circumference, , what value do you get for ?
   2. Diego learns that one of the space shuttle fuel tanks has a diameter of 840 cm and a circumference of 2,639 cm. Using these values and the equation for circumference, , what value do you get for ?

   3. Label all of the tick marks on the number lines. Use a calculator to get a very accurate approximation of and plot that number on all three number lines.

      

      
       

   4. How can you explain the differences between these calculations of ?