Transcript L1 Adding and Subtracting Rational Numbers

Adding and Subtracting
Rational Numbers
Whole Numbers Positive numbers that represent a "whole."
ex. "0, 1, 2, 3, 4, ...
Zero is a special case.
Integers Positive and negative numbers that represent a "whole."
ex. "... -4, -3, -2, -1, 0, 1, 2, 3, 4, ..."
So all Whole Numbers are included in the
Integers group.
Are ALL numbers included in the integers group?
Rational Numbers Any positive or negative whole numbers that can be written
as a fraction.
ex. "-1" "0.5"
"-0.5"
Whole Numbers
Integers
Rational Numbers
Opposite Numbers Numbers that are the same distance from 0 on a number
line, but have different signs.
*Opposite numbers always add up to 0.
*Opposite numbers have the same Absolute Value.
Find the Opposite
0
1)
10 =
0
2)
-5 =
0
3)
0.3 =
Absolute Value The distance from 0 that a number is on a
number line. (In either direction)
ex.
5
0
-5
5
0
5
Review:
What are Whole Numbers?
What are Integers?
What are Rational Numbers?
*Give an example of each.
What are opposites?
*Give an example.
What is absolute value?
Homework: Wksht 4-1 Rational Numbers, Opposites and
Absolute Value # 1-10
Adding Positive and Negative Numbers
using a Number Line
* Always start at 0.
* Negatives move LEFT. Positives move RIGHT.
-3 + (-7) =
?
-3
0
* Always start at 0.
* Negatives move LEFT. Positives move RIGHT.
5 + (-3) =
0
* Always start at 0.
* Negatives move LEFT. Positives move RIGHT.
4 + (-8) =
0
* Always start at 0.
* Negatives move LEFT. Positives move RIGHT.
-3 + 9 =
0
More?
My Friends
Neil
Pete
Anytime I hang out with Neil, I have a real
negative time. He really brings me down.
My mood = -1
The only thing worse than hanging out with Neil, is
hanging out with TWO Neils!
My mood = ?
Pete, however, is pretty cool. He always knows how to
cheer me up.
What does Pete do to my
mood?
My mood = ?
How do you think I feel when my mom makes me
play with Pete AND Neil at the same time?
So obviously, Pete and Neil are OPPOSITES.
They cancel out.
When they are balanced, they
make my mood 0. Meh.
1 + (-1) = 0
But when they are unbalanced,
my mood can change.
1 + (-3) =
BrainPop
Adding/Subtracting Integers
5 + (-6) =
-3 + 4 =
5 + (-2) =
-4 + (-4) + 5 =
Additive Inverses Two opposites added together to make 0 are called
Additive Inverses
ex.
=0
-10 + 10 = 0
4 + (-4)
Review:
Number Lines:
Positives go?
Negatives go?
-10 + (-3) =
Using Pete and Neil.
-15 + 10 =
5 + (-3) =
150 + (-15) =
HW: Wksht 4-2 Adding Integers #1-9
Extra Review
Subtracting Integers
If I am hanging out with 6 Petes, and 5 Neils show up, what
will happen to my mood?
Write this number sentence:
What do we see?
If I am hanging out with 6 Petes, and 5 Petes go home, what
will happen to my mood?
Write this number sentence:
Adding negatives is the same as
subtracting positives. They both
affect our answer the same.
The Chef’s Hot and Cold Cubes
(Interactive Mathematics Program)
In a far-off place, there was once a team of amazing chefs
who cooked up the most marvelous food ever imagined.
They prepared their meals over a huge cauldron, and their
work was very delicate and complex. During the cooking process, they
frequently had to change the temperature of the cauldron in order to
bring out the flavors and cook the food to perfection.
They adjusted the temperature of the cooking either by
adding special hot cubes or cold cubes to the cauldron or by removing
some of the hot or cold cubes that were already in the cauldron.
The cold cubes were similar to ice cubes except that they
didn’t melt, and the hot cubes were similar to charcoal
briquettes, except they didn’t lose their heat.
+
If the number of cold cubes in the cauldron was the same
as the number of hot cubes, the temperature of the
cauldron was 0 degrees on their temperature scale.
=
Additive Inverses
+
=
0
0
For each hot cube that was put in the cauldron, the temperature went up one
degree; for each hot cube removed, the temperature went down one degree. Similarly, each
cold cube put in lowered the temperature one degree and each cold cube removed raised it
one degree.
The chefs used positive and negative numbers to keep track of their
changes they were making to the temperature.
=
=
Think about how the hot and cold cubes affect
the temperature.
Which cubes should represent positive numbers
and which cubes should represent negative
numbers?
For example, suppose 4 hot cubes and 10 cold cubes were dumped into the cauldron. Then
the temperature would be lowered by 6 degrees altogether, since 4 of the 10 cold cubes
would balance out the 4 hot cubes, leaving 6 cold cubes to lower the temperature 6
degrees. They would write
4 + (-10) = -6
to represent these actions and their overall result.
Adding Zero Pairs
Similarly, if they added 3 hot cubes and then removed 2
cold cubes, the combined result would be to raise the
temperature 5 degrees. In that case, they would write
3 - (-2) = 5
How can we take away 2
cold cubes?
And if they wrote -5 - 6 = -11, it would mean that first
5 cold cubes were added and then 6 hot cubes were
removed, and that the combined result was to lower
the temperature 11 degrees.
Show the expression using
Zero Pairs.
Sometimes the chefs wanted to change the temperature of the
soup without adding any cubes in. To do this they simply used
their large ladle and scooped out hot or cold cubes. If they
wanted to warm the soup up, they scooped out a ___________
cube. If they wanted to cool the soup down they scooped out a
_____________ cube.
Suppose one of the chefs accidentally tossed in 5 cold cubes and then removed
2 hot cubes to adjust the temperature. How would these actions have changed
the temperature of the soup?
Write the number sentence and solve.
-5 - 2 = -7
Cooking with Integers
In pairs, use your manipulatives to model each problem. Then, write
and solve a number sentence for each. Finally, write what the "Overall
Temperature Change" would be for each problem.
When finished, come up with a rule to describe how to subtract
integers. We will discuss possible answers as a class.
Subtracting Integers Rule:
Keep Change Change or simply turn the two -'s into a +
Give Example
Review:
Adding Negatives
is the same as
Subtracting Positives.
3 + (-5) =
3-5 =
ALSO
Adding Positives
is the same as
Subtracting Negatives.
3+5=
3 - (-5) =
HW:
Subtracting Integers Worksheet #1-20
Easy Rules:
Addition:
If the signs are the same: Add/The sign stays.
If the signs are different: Subtract/ Use highest
Absolute Value to find sign.
Subtraction:
Double Negatives: Change to positive.
Regular: Add the opposite of 2nd number.
Adding and Subtracting Rational Numbers
6-8=
__
11
3 - (-4) =
-0.5 + 2 =
22
1
__
2
+2=
__
5
+ ( - 22 ) =
1.2 + (-0.5) =
1
__
5
6
- (
__
2
3
+
__
3
4
)
1
__
3
8
- (
__
1
5
+
__
4
10
)
End Zone
A football team gains 4 yards on each of 3 consecutive plays. Find the total
change in yards from where the team started.
10
20
30
40
50
End Zone
A football team gains 12 yards on their first play. Loses 2 yards on their second play.
And loses 15 yards on their third play. Find the total change in yards from where the
team started.
10
20
30
40
50
End Zone
A football team loses 7 yards on each of 3 consecutive plays. Find the total
change in yards from where the team started.
10
20
30
40
50
Review:
HW:
Comparing, Adding, and Subtracting Integers
worksheet