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
Page 4 of 7
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
Page 5 of 7
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 7
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 7