Charged Particle (Chip) Model of Addition and Subtraction of Integers
Exploration # 1: Integer Addition
Use the “Union of Two Disjoint Sets” Model for Addition to solve the following problems:
-5 + 3
Represent –5 with 5 red chips
and 3 with 3 yellow chips.
Take the union of the two disjoint sets.
One red and one yellow = zero, so we have three zeros
and two red chips, giving us an answer of –2.
a) 4 + 3
©Dr Barbara Boschmans/Dr Brian Beaudrie
b) (-3) + (-6)
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c) 3 + (-7)
d) (-8) + 5
Exploration # 2: Integer Subtraction
Use the “Take-Away Set” Model for Subtraction to solve the following problems:
-5 - 3
Represent –5 with 5 red
chips.
We want to take away
three positive chips. We
do not have any positive
chips, so we add in three
zeros. The value of our set
is still –5 (why?).
©Dr Barbara Boschmans/Dr Brian Beaudrie
Now we can take away
three positive chips.
This leaves us with eight
red chips = -8.
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a) 5 – 9
b) (-6) - (-5)
c) 3 - (-4)
d) (-4) - (-8)
©Dr Barbara Boschmans/Dr Brian Beaudrie
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Number Line (Vector) Model of Addition
For this, an introduction is in order: “Class,
meet Mr. Turtle. Mr. Turtle will help us do
number line addition.”
Rules for number line addition:
1) Mr. Turtle always starts at zero, and is always facing the positive (right) direction.
2) If the first addend (number) is positive, Mr. Turtle walks forward. If the first addend is negative, Mr. Turtle
walks backwards.
3) If the second addend is positive, Mr. Turtle walks forward. If the second addend is negative, Mr. Turtle walks
backwards.
Example #1: 2 + 3 = 5
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-2
-1
0
1
2
3
4
5
6
7
8
-2
-1
0
1
2
3
4
5
6
7
8
-2
-1
0
1
2
3
4
5
6
7
8
Example #2: 2 + (-3) = -1
-7
-6
-5
-4
-3
Example #3: -3 + 4 = 1
-7
-6
-5
-4
-3
Example #4: -3 + (-2) = -5
-7
-6
-5
-4
-3
©Dr Barbara Boschmans/Dr Brian Beaudrie
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Number Line (Vector) Addition
Draw out the proper vectors and the Mr. Turtle to show how to compute the following. Circle your answer on
the number line. Write your answer next to the equal sign.
1) 2 + 5 =
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
2) 3 + (-2) =
-7
-6
3) -4 + 6 =
-7
-6
4) -2 + (-3) =
-7
-6
5) -5 + 3 =
-7
-6
6) 4 + (-7) =
-7
-6
-5
©Dr Barbara Boschmans/Dr Brian Beaudrie
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Vector Subtraction on the Number Line
Rules for number line subtraction:
1) Mr. Turtle always starts at zero, and is always facing the positive (right) direction.
2) If the first number is positive, Mr. Turtle walks forward. If the first number is negative, Mr. Turtle walks
backwards.
3) To indicate subtraction (i.e., take away), Mr. Turtle turns around to face the negative (left) direction.
4) If the second number is positive, Mr. Turtle walks forward. If the second number is negative, Mr. Turtle walks
backwards.
Example #1: 2 - 3 = -1
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-2
-1
0
1
2
3
4
5
6
7
8
-2
-1
0
1
2
3
4
5
6
7
8
-2
-1
0
1
2
3
4
5
6
7
8
Example #2: 3 – (-2) = 5
-7
-6
-5
-4
-3
Example #3: -4 –(+3) = -7
-7
-6
-5
-4
-3
Example #4: -2 – (-3) = 1
-7
-6
-5
-4
-3
©Dr Barbara Boschmans/Dr Brian Beaudrie
Page 6 of 8
Number Line (Vector) Subtraction
Draw out the proper vectors and the Mr. Turtle to show how to compute the following. Circle your answer on
the number line. Write your answer next to the equal sign.
1) 5 - 3 =
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
2) 2 – (-2) =
-7
-6
3) -4 – 3 =
-7
-6
4) -2 – (-3) =
-7
-6
5) 3 - 5 =
-7
-6
6) -5 – (-3) =
-7
-6
-5
©Dr Barbara Boschmans/Dr Brian Beaudrie
Page 7 of 8
Example #1: Jenny entered an elevator. She went up three floors, down five floors, and up
six floors. She was then at the top floor of the building. Then she went down five floors, up
two floors, and down eight floors. She was then on the first floor.
a) On what floor did Jenny enter the elevator?
b) What floor is the top floor?
Example #2: Use the vertical line (at right) and mark it in
equal intervals from -175 to +175. Then indicate the
following:
a)
b)
c)
d)
e)
f)
g)
h)
i)
sea level is at zero meters
a boat is on the water (at sea level)
a helicopter is at 125 meters
a sparrow flies at 50 meters
a osprey soars at 160 meters
a school of fish swims at -90 meters
a treasure is at -170 meters
a deep sea diver is at -100 meters
a shark is cruising at -75 meters
Solve the following problems, writing the equation that you
use.
a) How far is it from the osprey to the sparrow?
b) If the sparrow flies 30 meters lower, at what height will it be flying?
c) What is the distance from the helicopter to the boat?
d) How far does the shark have to swim to reach the school of fish?
e) How many meters separate the treasure and the diver?
f) How far is the deep sea diver below the helicopter?
g) How far is it from the shark to the osprey?
©Dr Barbara Boschmans/Dr Brian Beaudrie
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