2. Operations (N)
  3. Lesson 1: Adding Integers

Answers and Solutions


    1. \(8+5=13\)
      The first arrow starts at 0 and extends 8 units to the right, the second arrow then starts at 8 and extends 5 units the right and ends at 13.
    2. \((-3)+7=4\)
      The first arrow starts at 0 and extends 3 units to the left, the second arrow then starts at -3 and extends 7 units to the right and stops at 4.
    3. \(2+(-8)=(-6)\)
      The first arrow starts at 0 and extends 2 units to the right, the second arrow then starts at 2 and extends 8 units the left and ends at -6.
    4. \((-4)+(-6)=(-10)\)
      The first arrow starts at 0 and extends 4 units to the left, the second arrow then starts at -4 and extends 6 units the left and ends -10.
    5. \(4+(-4)=0\)
      The first arrow starts at 0 and extends 4 units to the right, the second arrow then starts at 4 and extends 4 units the left and ends at 0.
    1. Up one floor is to add \(1\).  Down one floor is to add \((-1)\). In summary, The floors are 7, 6, 5, 4, 3, 2, G, P1, P2, and P3.\[(-4)+3+5+(-6)+(-3) = (-5)\] which means that the elevator is \(5\) floors below its starting position, or on P3.
      • \((-4)\) tells us the elevator moved down \(4\) floors. If it's starting at the \(3\)rd floor it is now on P2.
      • \(3\) tells us the elevator moved up \(3\) floors. If it's now on P2, then it will move to \(2\).
      • \(5\) tells us the elevator moved up \(5\) floors. If it's now on \(2\), then it will move to \(7\).
      • \((-6)\) tells us the elevator moved down \(6\) floors. If it's now on \(7\) then it will move to G.
      • \((-3)\) tells us the elevator moved down \(3\) floors. If it's now on G then it will move to P3.
    2. Up one floor is to add \(1\).  Down one floor is to add \((-1)\). In summary, the addition statement representing this movement is:\[3 + 3 + (-6) + 2 + (-3) = (-1)\]  The answer \((-1)\) means that the elevator is one floor below its starting position.
      • Travelling to floor \(4\) from its current position, G, means the elevator has travelled up \(3\) floors.
      • Travelling to floor \(7\) from floor \(3\) means the elevator has travelled up another \(3\) floors.
      • Travelling to floor G from floor \(7\) means the elevator has travelled down \(6\) floors.
      • Travelling to floor \(3\) from floor G means the elevator has travelled up \(2\) floors.
      • Travelling to floor P1 from floor \(3\) means the elevator has travelled down \(3\) floors.
    3. Answers may vary.  One possible scenario is:
      If the elevator starts on the ground floor, G, it could move to the \(7\)th floor, travelling \(6\) floors.
      From there, the elevator could move to P3, travelling \(9\) floors.
      In total, the elevator has travelled \(15\) floors.
  3. The two integers are opposites.  For example, \((-4)\) and \(4\) are opposite integers and \((-4)+4=0\).
    1. Starting from the right and moving left, we have that the number being displayed is the sum of these values.\[(-1)+2+0+(-8)+16=9\]
      • The first bead represents \((-1)\) because the column has a value of \(1\) and the bead is below the line, meaning we make that value negative.
      • The second bead represents \(2\) because the column has a value of \(2\) and the bead is above the line, meaning we make that value positive.
      • The third bead represents \(0\) because the bead is on the line.
      • The fourth bead represents \((-8)\) because the column has a value of \(8\) and the bead is below the line, meaning we make that value negative.
      • The fifth bead represents \(16\) because the column has a value of \(16\) and the bead is above the line, meaning we make that value positive.
    2. Yes there is more than one way to represent \((-11)\) on the abacus.  Two possible examples are given.
      From right to left, the beads in the 2nd and 3rd column are above 0, at zero in the 4th column, and below 0 in the 1st and 5th columns.      From right to left, the beads in the 3rd and 5th columns are at zero and the beads in the 1st, 2nd, and 4th columns are below 0.
    1. The next seven numbers in the list are \(\ldots,8,~ 13,~ 21,~ 34,~ 55,~ 89,~ 144, \ldots\)
    2. The third negative Fibonacci number is \((-2)\) since \((-1)+(-1)=(-2)\).
    3. \((-1), ~(-1),~ (-2),~ (-3),~ (-5),~ (-8),~ (-13),~ (-21),~ (-34),~ (-55), \ldots\)
    4. The numbers in the list of negative Fibonacci numbers are the opposite of their counterparts in the usual list of Fibonacci numbers.
  6. Sign of Integer 1 Sign of Integer 2 Sign of Integer Sum
    positive positive positive
    positive negative either
    negative positive either
    negative negative negative

    The sum of a positive integer and a negative integer is positive when the first integer is larger than the opposite of the second integer.  For example, \(10 + (-2)\) is positive because \(10\) is larger than the opposite of \((-2)\) which is \(2\).

    The sum of a positive integer and a negative integer is negative when the first integer is smaller than the opposite of the second integer.  For example, \(3 + (-7)\) is negative because \(3\) is less than the opposite of \((-7)\) which is \(7\).

    The sum of a positive integer and a negative integer is \(0\) when the first integer is the opposite of the second integer.  For example, \(4+(-4)=0\).

    Since addition is commutative, the sum of a negative integer and a positive integer follows the same rules.
    1. They both count the number of games in inventory and on order. Since they are counts, they should be represented using integers.
    2. The merchant has unsold games on hand.
    3.   Day 1 Day 2 Day 3 Day 4 Day 5
      Starting Inventory 15 12 10 6 1
      Total Daily Orders 3 2 4 5 2
      Updated Inventory 12 10 6 1 -1
    4. The number of orders exceeds the number of games.
  8. The frog's landing spot on the number line can be represented using the list: \[100, ~(-100), ~99,~ (-99), ~98, ~(-98), \ldots\]
    1. To find the frog after \(10\) moves, we must extend the list to include the first \(10\) numbers.\[100,~ (-100), ~99, ~(-99), ~98, ~(-98), ~97, ~(-97), ~96, ~(-96) \ldots\]  Therefore, the frog is at \((-96)\) on the number line after \(10\) moves.
    2. Instead of extending the list above to include \(51\) numbers, consider the list\[100, ~99,~ 98,~ 97, \ldots\] which represents the odd numbered movements (i.e., movement 1, movement 3, etc.) of the frog.  The \(51^{st}\) jump of the frog would be the \(26^{th}\) number in this list, which is \(75\).
    3. Consider the list from part b.  \[100,~ 99, ~98,~ 97, \ldots\] The number \(50\) would be the \(51^{st}\) number in this list.  However, remember that this list only represents the odd numbered movements, so if we were to add the even movements back in (which would be every other number), then this number would become the \(101^{st}\) number.  Therefore, the frog lands on the number \(50\) after \(101\) movements.
    4. The frog lands on \((-50)\) immediately after he lands on \(50\).  As a result, the frog would land on \((-50)\) on the \(102^{nd}\) movement.