Module 2 Lesson 5: Understanding Subtraction of Integers and Other Rational Numbers

Problem Set

1.       On a number line, find the difference of each number and 4?  Complete the table to support your answers.  The first example is provided.

Number	Subtraction Number Sentence	Addition Number Sentence	Distance (units)
10	10 - 4 = 6	10 + (-4) = 6	6
2	4 - 2 = 2	4 +  (-2) = 2	2
-4	4 - (-4) = 8	4 + 4 = 8	8
-6	4 - (-6) = 10	4 + 6 = 10	10
1	4 - 1 = 3	4 + (-1) = 3	3

2.       You and your partner were playing the Integer Game in class.  Here are the cards in both hands.

         Your Hand:     -8,     6,      1,      -2        Your Partner’s Hand:     9,     -5,     2,     -7

a.         Find the value of each hand.  Who would win based on the current scores? (The score closest to 0 wins.)

     Your Hand: (-8) + 6 + 1 + -2            (Hint: Add two integers together at a time.)

                              (-2) + 1 + (-2)         (I added (-8) + 6 to get -2.)
                             (-1) + (-2)         (I added (-2) + 1 to get -1.)
                                    -3              (I added (-1) + (-2) to get -3.)

Your Partner's Hand: 9 + (-5) + 2 + (-7) 

                                                  4 + 2 + (-7)              (I added 9 + (-5) to get 4.)
                                                 6 + (-7)              (I added 4 + 2 to get 6.)
                                                     -1                  (I added 6 + (-7) to get -1.)

My partner would win because the value of my partner's hand is -1 and the value of my hand is -3. My partner would win because -1 is closer to 0 than -3 is. 

b.         Find the value of each hand if you discarded the -2 and selected a 5, and your partner discarded the -5 and selected a 5.  Show your work to support your answer.

 Your New Hand: (-8) + 6 + 1 + 5
                                 (-2) + 1 + 5             (I added (-8) + 6 to get -2.)
                                       (-1) + 5             (I added (-2) + 1 to get -1.)
                                              4                (I added (-1) + 5 to get 4.)


Your Partner's New Hand: 9 + 5 + 2 + (-7)       

                                      14 + 2 + (-7)            (I added 9 + 5 to get 14.)
                                     16 + (-7)            (I added 14 + 2 to get 16.)
                                           9                  (I added 16 + (-7) to get 9.)

c.         Use your score values from part (b) to determine who would win the game now.

I would win the game now because the value of my hand is 4 and the value of my partner's hand is 9. I would win because 4 is closer to 0 than 9 is.

  
3.       Solve the following problems.

      a. (-2) + 16
         14


When two integers are added together with different signs, subtract the two integers. So we subtract 16-2. The number with the bigger absolute value tells us the sign. Since 16 has the bigger absolute value and it is positive, then our answer is positive 14. 

b. (-2) - (-16)
    (-2) + 16 
       14


Subtracting is the same thing as adding the opposite of a quantity. Subtracting a negative integer is the same thing as adding a positive integer. So we change the problem to (-2) + 16, which is the same as the problem in part a. 

     c. 18 – 26  
18 + (-26)
   -8 

Subtracting a positive integer is the same thing as adding a negative integer. So we change the problem to 18 + (-26). Since 18 and -26 have different signs, we subtract 26 - 18 and we get 8. Since the absolute value of -26 is bigger, our answer must be -8.

     d. -14 – 23 
(-14) + (-23)
      -37 

Subtracting a positive integer is the same thing as adding a negative integer so we change the problem to (-14) + (-23). Since (-14) and (-23) have the same sign, we add 14 + 23 to get 37. Since both numbers are negative, our answer must be -37.

     e. 30 – (-45)
  30 + 45
     75 

Subtracting a negative integer is the same thing as adding a positive integer so we change the problem to 30 + 45, which equals 75. 


4. Explain what is meant by the following an illustrate with an example:
"For any real numbers, p and q, p - q = p + (-q)."

Subtracting a positive integer is the same as adding a negative integer.
For example, 10 - 4 = 6 and 10 + (-4) = 6.


5. Choose an integer between -1 and -5 on the number line and label it point P. Locate and label the following points on the number line. Show your work.

a. Point A: P - 5
Example: I choose -4.
P - 5
-4 - 5                 
-4 + (-5)             
-9                      

I plugged in -4 for P. I changed -4 - 5 to -4 + (-5) since subtracting is the same as adding a negative. Since -4 and -5 are both negative, I add 4 + 5 to get 9. Since -4 and -5 are both negative, my answer is -9.


b. Point B: (P - 4) + 4
Example: I choose -4.
(P - 4) + 4
(-4 - 4) + 4
[-4 + (-4)]+4
(-8) + 4
-4

I plugged in -4 for P. I changed-4 - 4 to -4 + (-4) since subtracting is the same thing as adding a negative. Since -4 and -4 are both negative, I add 4 + 4 to get 8. Since -4 and -4 are both negative, then I get -8. I now have -8 + 4. Since -8 and 4 have different signs, I subtract 8 - 4 to get 4. Since -8 has the bigger absolute value, then my answer is -4.


c. Point C: -P - (-7)
Example: I choose - 4.
-P - (-7)
-(-4) - (-7)
4 - (-7)
4 + 7
11

I plugged in -4 for P. Since -P means the opposite of P, I have to take the opposite of -4 to get 4. Since subtracting a negative is the same as adding a positive, 4 - (-7) is the same as 4 + 7 = 11.


6. CHALLENGE PROBLEM: Write two equivalent expressions that represent the situation. What is the difference of their elevations?
"An airplane flies at an altitude of 26,000 feet. A submarine dives to depth of 700 feet below sea level."

26,000 - (-700) is equivalent to 26,000 + 700.