Transcript Subtracting Integers

ADDITION AND
SUBTRACTION of
INTEGERS
The University of Texas at Dallas
There are three ways to think of
adding and subtracting with integers:
• Use Algebra Tiles
• Use a Number Line
• Use Mathematical Rules
• We will go over all three methods. You
may use any method – all will give the
correct answer.
Algebra Tiles
ADDING INTEGERS
• We can model integer addition with tiles.
• Represent -2 with the fewest number of
tiles
• Represent +5 with the fewest number of
tiles.
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ADDING INTEGERS
• What number is represented by combining
the 2 groups of tiles?
+3
• Write the number sentence that is
illustrated.
-2 + +5 = +3
The University of Texas at Dallas
ADDING INTEGERS
• Use your red and yellow tiles to find each sum.
-2 + -3 = ?
ANSWER
+
-2 + -3 = -5
= -5
The University of Texas at Dallas
ADDING INTEGERS
-6
+
+2
=?
+
= -4
-6
-3
+
+
-3
+ +4 = +1
ANSWER
+4
+ +2 = -4
=?
=
ANSWER
+1
The University of Texas at Dallas
INTRODUCTION TO INTEGERS
• The diagrams below show 2 ways to
represent -3.
-3
-3
• Represent -3 in 2 more ways.
NOTE TO TUTOR
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SUBTRACTING INTEGERS
• We often think of subtraction as a “take away” operation.
• Which diagram could be used to compute
+3 - +5 = ?
The University of Texas at Dallas
SUBTRACTING INTEGERS
• Use your red and yellow tiles to model each
subtraction problem.
• -2 - -4 = ?
ANSWER
The University of Texas at Dallas
SUBTRACTING INTEGERS
Now you can
take away 4 red
tiles.
2 yellow tiles are
left, so the
answer is…
-2
-
-4
=
+2
The University of Texas at Dallas
SUBTRACTING INTEGERS
• Work this problem.
• +3 - -5 = ?
ANSWER
The University of Texas at Dallas
SUBTRACTING INTEGERS
• Add enough red and yellow
pairs so you can take away 5
red tiles.
• Take away 5 red tiles, you
have 8 yellow tiles left.
+3
- -5 = + 8
The University of Texas at Dallas
SUBTRACTING INTEGERS
• Work this problem.
• -3 - +2 = ?
ANSWER
The University of Texas at Dallas
SUBTRACTING INTEGERS
•Add two pairs of red and
yellow tiles so you can take
away 2 yellow tiles.
• Take away 2 yellow tiles,
you have 5 red tiles left.
-3
- + 2 = -5
The University of Texas at Dallas
Number Line
Addition and Subtraction of Integers
In this tutorial, we will learn how to add and subtract
signed numbers with the help of a toy car.
The line the car sitting on is called the number line,
where the positive numbers are on the right and the
negative numbers are on the left.
Addition and Subtraction of Integers
The car will stick to the following rules:
1. It always starts at 0 (its home) facing right.
2. If it sees a positive number, it moves forward.
3. If it sees a negative number, it backs up.
4. If it sees an addition sign, it continues to read the next
number.
5. If it sees a subtraction sign, it turns around and then
continues to read the next number.
Click to see the first example.
Example 1: 2 + 4
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right
(click to see animation)
Example 1: 2 + 4
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right
Example 1: 2 + 4
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right
Example 1: 2 + 4
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
Example 1: 2 + 4
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Next the car will move 4 more
units forward because it sees the
number 4.
(click to see next animation.)
Example 1: 2 + 4
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Next the car will move 4 more
units forward because it sees the
number 4.
(click to see next animation.)
Example 1: 2 + 4
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Next the car will move 4 more
units forward because it sees the
number 4.
Since the car now stops at 6, the
answer to 2 + 4 is 6.
(click to see the next example)
Example 2: (-2) + 5
1. Our car starts from 0 facing right.
2. It then backs up 2 units (to the
left) because it sees the - sign.
(Click to see animation)
Example 2: (-2) + 5
1. Our car starts from 0 facing right.
2. It then backs up 2 units (to the
left) because it sees the - sign.
Example 2: (-2) + 5
1. Our car starts from 0 facing right.
2. It then backs up 2 units (to the
left) because it sees the - sign.
Example 2: (-2) + 5
1. Our car starts from 0 facing right.
2. It then backs up 2 units (to the
left) because it sees the - sign.
3. Next it will move forward by 5
units.
(click to see animation)
Example 2: (-2) + 5
1. Our car starts from 0 facing right.
2. It then backs up 2 units (to the
left) because it sees the - sign.
3. Next it will move forward by 5
units.
Example 2: (-2) + 5
1. Our car starts from 0 facing right.
2. It then backs up 2 units (to the
left) because it sees the - sign.
3. Next it will move forward by 5
units.
Example 2: (-2) + 5
1. Our car starts from 0 facing right.
2. It then backs up 2 units (to the
left) because it sees the - sign.
3. Next it will move forward by 5
units.
Now it stops at +3, therefore the
answer to (-2) + 5 is 3.
Please go to the next tutorial for subtractions.
Subtraction
1. There is a big difference between
addition and subtraction.
2. In addition, our car is always
facing right, because that is the
positive direction,
3. but in subtraction, the car has to
turn around (180 deg) first.
Click when you are ready.
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
(Click to start animation)
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
3. Next it will turn around because it
sees the subtraction symbol – .
Click to see animation.
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
3. Next it will turn around because it
sees the subtraction symbol – .
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
3. Next it will turn around because it
sees the subtraction symbol – .
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
3. Next it will turn around because it
sees the subtraction symbol – .
4. Finally it will move forward (to
the left) by 3 units.
click to see animation.
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
3. Next it will turn around because it
sees the subtraction symbol – .
4. Finally it will move forward (to
the left) by 3 units.
Example 3: 5 – 3
1. Our car still starts at 0 facing
right.
2. It then moves forward 5 units.
3. Next it will turn around because it
sees the subtraction symbol – .
4. Finally it will move forward (to
the left) by 3 units.
Since the car stops at 2, the
answer to 5 – 3 must be 2.
Click to see the next example.
Example 4: (-4) – 2
1. Our car still starts at 0 facing
right.
2. It then backs up 4 units because it
sees the negative symbol - in front
of 4.
(Click to see animation.)
Example 4: (-4) – 2
1. Our car still starts at 0 facing
right.
2. It then backs up 4 units because it
sees the negative symbol - in front
of 4.
Example 4: (-4) – 2
1. Our car still starts at 0 facing
right.
2. It then backs up 4 units because it
sees the negative symbol - in front
of 4.
Example 4: (-4) – 2
1. Our car still starts at 0 facing
right.
2. It then backs up 4 units because it
sees the negative symbol - in front
of 4.
Example 4: (-4) – 2
1. Our car still starts at 0 facing
right.
2. It then backs up 4 units because it
sees the negative symbol - in front
of 4.
3. Now it has turn around because
of the subtraction symbol –.
(click to go on)
Example 4: (-4) – 2
1. Our car still starts at 0 facing
right.
2. It then backs up 4 units because it
sees the negative symbol - in front
of 4.
3. Now it has turn around because
of the subtraction symbol –.
Example 4: (-4) – 2
1. Our car still starts at 0 facing
right.
2. It then backs up 4 units because it
sees the negative symbol - in front
of 4.
3. Now it has turn around because
of the subtraction symbol –.
Example 4: (-4) – 2
4. Finally it moves forward by 2
units.
(click to go on)
Example 4: (-4) – 2
4. Finally it moves forward by 2
units.
Example 4: (-4) – 2
4. Finally it moves forward by 2
units.
Example 4: (-4) – 2
4. Finally it moves forward by 2
units.
Example 4: (-4) – 2
4. Finally it moves forward by 2
units.
Example 4: (-4) – 2
4. Finally it moves forward by 2
units.
The car now stops at -6, therefore
the answer to (-4) – 2 is -6.
Click to see the next example.
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
(Click to see animation)
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Now the car has to turn around
because it sees the subtraction
sign.
(Click to see the next animation)
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Now the car has to turn around
because it sees the subtraction
sign.
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Now the car has to turn around
because it sees the subtraction
sign.
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Now the car has to turn around
because it sees the subtraction
sign.
4. It then backs up 3 steps because it
sees the negative sign.
(click to see animation)
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Now the car has to turn around
because it sees the subtraction
sign.
4. It then backs up 3 steps because it
sees the negative sign.
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Now the car has to turn around
because it sees the subtraction
sign.
4. It then backs up 3 steps because it
sees the negative sign.
Example 5: 2 – (-3)
1. Our car starts from 0 facing right.
2. It then moves 2 units to the right.
3. Now the car has to turn around
because it sees the subtraction
sign.
4. It then backs up 3 steps because it
sees the negative sign.
Since the car stops at 5, the
answer to 2 – (-3) is 5.
Click to see the next example.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning
because it sees a - sign in front of the 3.
(click to see animation)
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning
because it sees a - sign in front of the 3.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning
because it sees a - sign in front of the 3.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning
because it sees a - sign in front of the 3.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning
because it sees a - sign in front of the 3.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning because it sees a sign in front of the 3.
3. Next it has to turn around because it sees the – sign.
(click to see animation)
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning because it
sees a - sign in front of the 3.
3. Next it has to turn around because it sees the
– sign.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning because it
sees a - sign in front of the 3.
3. Next it has to turn around because it sees the –
sign.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning because it sees a sign in front of the 3.
3. Next it has to turn around because it sees the – sign.
4. Finally it has to back up 5 units because the next number
is -5.
(click to see animation)
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning because it
sees a - sign in front of the 3.
3. Next it has to turn around because it sees the –
sign.
4. Finally it has to back up 5 units because the next
number is -5.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning because it
sees a - sign in front of the 3.
3. Next it has to turn around because it sees the –
sign.
4. Finally it has to back up 5 units because the next
number is -5.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning
because it sees a - sign in front of the 3.
3. Next it has to turn around because it sees
the – sign.
4. Finally it has to back up 5 units because
the next number is -5.
Example 6: (-3) – (-5)
1. Our car starts from 0 facing right.
2. It backs up 3 units in the beginning
because it sees a - sign in front of the 3.
3. Next it has to turn around because it sees
the – sign.
4. Finally it has to back up 5 units because
the next number is -5.
The answer is therefore +2.
Mathematical Rules
Adding Integers
• Two positive numbers: Add the numbers,
sign stays positive
• Two negative numbers: Add the absolute
value of the numbers, sign stays negative
• A positive and a negative number:
Subtract the absolute values of the
numbers – sign is the same as the number
with the largest absolute value
Examples
• 3+4=7
– Add the numbers, sign stays positive
• -3 + -4 = -7
– Add the numbers, sign stays negative
• -3 + 4 = 1
– Subtract 4-3 to get 1, sign is positive, since |4|>|-3|
• 3 + -4 = -1
– Subtract 4-3 to get 1, sign is negative, since |-4|>|3|
Subtracting Mathematically
• Change the minus to a plus
• Change the sign of the number that follows
• Follow the rules for addition
- (+5)
4 +
(-5)
- (-6)
(+6)
-3 +