10th Test - Numbers

1) Write an integer that is not also a whole number? ___-3______

    Any negative whole number is acceptable. Only 2 or 3 students answered correctly.
    The set of integers is {  -Infinity, ...... -3, -2, -1, 0, 1, 2, 3, .......... Infinity }

2) What kind of number is the ratio of the circumference of a circle to its diameter? ___irrational__
What is this ratio called? ___Pi___     What is its approximate value? __3.14159...._____ 

    Pi is a very important irrational number. Its decimal expansion is infinite.
    I would have accepted any number between 2.9 and 3.4 as an approximation.

3) What is the square root of 36? ___6_____

   Most got this.    6 times 6 = 36

4) What are the factors of 18? ____1, 2, 3, 6, 9, 18_________

   Most got this. I would have accepted 2, 3, 6, 9 (1 and 18 go without saying)

5) What are the prime factors of 420? ___2, 3, 5, 7________
   
    These are the prime factors. You can determine them by creating a factor tree.
     I was very pleased to see that most knew how to create a prime factor tree.
   
6) Write the prime factorization of 420 ___2*2*3*5*7________

   The prime factorization is the unique product of primes for the specified whole number.
    Recall that the fundamental theorem of arithmetic says that every integer greater than 1
   can be written as a unique product of primes.
 
7) Write an example of a rational number that is not a whole number. ____2/3___________

    Any fraction (a/b) that does not reduce to something in the set of whole numbers { 0, 1, 2, .......}

8) Write a fraction that is bigger than the square root of 2 and smaller than ? (Pi) ____2/1_______

     You needed to write a fraction that was greater than 1.414.. and less than 3.1415926
     Such as  3/2, 4/4, etc.
   
9) What is 2^6? __64_____

    2^2 = 4;  2^3 = 8;  2^4 = 16, 2^5 = 32;  2^6 = 2*2*2*2*2*2 = 64

10) What is 7^0?  ___1____

     x^0  = 1 for all values of x where x is not equal to 0.

11) What is 9^-1?  ___1/9____

   When there is a negative in the exponent, you must take the reciprocal.
   So 9^-1 = 1/ 9^1.   9^-2 = 1/81.
   We discussed this a bit, but it looks like most did not remember.

12)  Write the largest prime number less than 100 ____97_______

    97 has only 1 and 97 as factors. 99 has 9 and 11 as factors.

13)  Write your best estimate for the square root of 5 in decimal form ____2.236....________

     I gave only have credit for 2.5.   A lot of people wrote 2.5, but that is half of 5, not the square root.
     Some people spent a whole lot of effort trying to iterate and find it to many decimal places.
     That was not necessary.

14)  What is the volume of a rectangle that is 3 inches by 4 inches by 5 inches? __60 cubic inches____

       I gave half credit if you did not get the units right. Volume always has cubic units
      (or volume units like liters or quarts).

15) What are all values for x that work for the equation x^2 = 4?  ___2, -2___

  Almost everyone understood that 2 was a solution, but only one person found both 2 and -2.
  It is important to realize that -2 * -2  = 4

16) What is 2/3 - 2/5?   _____4/15________

     10/15 - 6/15 = 4/15
   You must find the new denomicator by finding the LCM of 3 and 5.
   There are still a few that do not know how to add fraction with different denomators.
  
17) Write a decimal that is greater than 4/7, but less than 5/8 ___0.572_________

     This was a little tricky both 4/7 and 5/8 are a little bigger than 1/2 and they are very close together.
     One way to approach the problem is to divie 4 into 7 using long division (0.571), then
     use a number that is a little bigger.

18) What is the Greatest Common Factor of 24 and 18? ____6_______

    Make a factor tree for both:
      24 : 2*2*2*3
      18:  2*3*3
   Look at what is in common: 2*3 = 6

19) What is the Least Common Multiple of 9 and 15? ___45________

     Make a factor tree for both:
       9 :  3 * 3
     15 :  3 * 5
 
    Multiply all the factors together except one of the 3's which are in common.
     So  3 * 3 * 5 = 45.

     Another way to do it is to look at the multiples of 9 (18, 27, 36, 45) and 15 (30, 45) and stop when you find
   the smallest one that is in common.

20) A dance choreographer requests enough dancers so that they can be evenly divided into groups of 4, 6, or 9 with none left over. What is the fewest number of dancers that must be hired?    ____36____

   This is exactly the same as asking "what is the LCM of 4, 6, 9?"
   Consider the multiples:
      4 (8, 12, 16, 20, 24, 28, 32, 36)
     6 (12, 18, 24, 30,  36)
     9  (18,  27,  36)