Transcript Negative numbers 1

Why Bother. I have a
Calculator!
 You might press a button wrong. It is good to have an
idea of what your answer should be.
 These visualizations and reasoning can be used to
estimate, not just find exact numbers. This can help
you see if your exact answer from a calculator makes
sense with the problem.
 On some tests a teacher may not allow a calculator.
In the red or in the black?
 One very common way to make sense of negative
numbers is to think about money.
 When I see negative numbers I sometimes think about
the problem as if the numbers were dollar amounts
 Positive numbers are what you have
 Negative numbers are what you owe
Debt
 If I have a negative amount of money, that means I
owe money. That is I am in debt.
 Net worth – how much money I have once I pay all my
debts, and I am payed all the money other people owe
me.
 I think of negative numbers as “net worth problems”
Examples (1)
 Positive plus positive: 4+ 5 = 9
 My mini word problem: I have 4 dollars in my wallet
and my pays me 5 dollars for gas money.
 Now I have 9 dollars in my wallet
Examples (2)
 Positive minus positive: 4- 5 = -1
 My mini word problem: I have 4 dollars. I owe my
friend 5 dollars, what is my net worth?
 I have 4, but I owe more than that, so I know my net
worth is going to be negative (eep!). But how negative?
I subtract 4, I have zero left, but still need to subtract
one more. So I have -1 dollars.
Examples (3)
 negative plus positive: (-4)+ 5 = 1
 My mini word problem: I owe my friend 4 dollars. But
this same friend pays me 5 dollars for gas money.
 What is my net worth now
 Well I had a debt. After he gives me $4 I have no more
debt, but he give me $1 more than that. So I have
positive 1 dollars.
Examples (4)
 positive plus a negative: 4+ (-5) = -1
 I think of this as “adding to my debt” I owe 5 more
dollars now. In this case $4 was my net worth. Then I
borrowed $5 dollars which puts me in debt.
 So my new net worth is $-1
Examples (5)
 negative plus a negative: (-4)+ (-5) = -1
 I think of this as “adding to my debt” I owe 5 more
dollars now. In this case $4 was my net worth. Then I
borrowed $5 dollars which puts me in debt.
 So my new net worth is $-1
Examples (6)
 negative minus a positive: (-4)- 5 = -1
 My net worth is $-4. that means I owe 4 dollars. Then
I buy lunch on my credit card. So I take away 5 more
dollars from my net worth. Now I owe more.
 So my new net worth is $-9
Examples (7)
 positive minus a negative: 4- (-5) = 9
 I think of this as “taking away my debt” I’m getting
rid of 5 dollars of debt I had. In this case $4 was my
net worth. Then my friend payed my $5 dollar debt.
Since I don’t have to pay it now my net worth goes up
and I have more money
 So my new net worth is $9
Examples (8)
 This works with negative minus a negative: (-4)- (-5) =
1
 I’m still “taking away my debt”, but my original net
worth is negative! In this case $-4 is my net worth.
Then my friend payed my $5 dollar debt. Since I don’t
have to pay it now my net worth goes up and I have
more money
 So my new net worth is $1
Number Line
-5 -4 -3 -2 -1
0
1 2 3 4 5
•Lets start with positive numbers so we get the
idea.
• There is an extra step than what you might be
used to. But I’m hoping that step will help make
sense of the negative numbers
•Lets try it!.
Number Line
-5 -4 -3 -2 -1
0
1 2 3 4 5
• Addition 3+2= 5
•We start at 3( )
•We look at where 2 is , (
). It is to the right of 0
•We add that shape starting at the 3 (
). So we
move to the right of 3.
•We “land” on 5
Number Line
-5 -4 -3 -2 -1
0
1 2 3 4 5
• adding a negative to a positive 3+(-2)= 1
•We start at 3( )
•We look at where -2 is , (
). This time it is to the
left of 0
•We add that shape starting at the 3 (
). That
means we move to the left of the 3
•We “land” on 1
Number Line
-5 -4 -3 -2 -1
0
1 2 3 4 5
• adding a positive to a negative (- 3)+2=( -1)
•We start at -3( )
•We look at where 2 is , (
). This time it is to the
right of 0
•We add that shape starting at the 3 (
). That
means we move to the right of the 3
•We “land” on -1
Number Line
-5 -4 -3 -2 -1
0
1 2 3 4 5
• Adding a negative to a negative (-3)+(-2)= -5
•We start at -3( )
•We look at where -2 is , (
). It is to the left 0
•We add that shape starting at the -3 (
). So we
move to the left of -3.
•We “land” on -5.
Number Line
_
+
-5 -4 -3 -2 -1
0
1 2 3 4 5
•Addition summary
When we add a positive we start at the first number,
then move right (the positive direction)
When we add a negative we start at the
first number, then move left (the negative direction)
Number Line
-5 -4 -3 -2 -1
• Subtraction 4-1= 3
0
1 2 3 4 5
•We start at 4( )
•We look at where 1 is, (
). It is to the right of 0.
•To take away, we go the other direction from 4 (
).
So in this case we move to the left.
•We “land” on 3 (this makes sense, we took away
something from 4 so it should be smaller)
Number Line
-5 -4 -3 -2 -1
1 2 3 4 5
0
• Now it gets tricky.
•subtracting a negative from a positive 4-(-1)= 5
•We start at 4( )
•We look at where -1 is, (
). It is to the left of 0.
•To take away, we go the other direction from 4 (
).
So in this case we move to the right.
•We “land” on 5
Number Line
-5 -4 -3 -2 -1
1 2 3 4 5
0
• Still Tricky.
•subtracting a negative from a negative (-4)-(-1)= -3
•We start at -4( )
•We look at where -1 is, (
). It is to the left of 0.
•To take away, we go the other direction from -4 (
).
So in this case we move to the right.
•We “land” on -3
Number Line
_
+
-5 -4 -3 -2 -1
0
1 2 3 4 5
•Subtraction summary
When we subtract a positive we start at the first
number, then move left (the negative direction)
When we subtract a negative we start at the
first number, then move right (the positive direction)
Number Line
_
+
-5 -4 -3 -2 -1
0
1 2 3 4 5
•In General:
 We can start with our arrows pointing right. When
we have a negative sign or a subtraction sign (both
written as “-”), we need to flip our arrows.
When we subtract a negative, we flip the arrow twice
(once for each minus sign) and end up facing left again.
My general strategy
 I combine these methods
 I normally think about debt to figure out if my net
worth is increasing or decreasing overall.
 I then use this with the number line. I know if I need
to go positive or negative, then visualize (or even draw)
number line to get the exact answer.
 For larger numbers I might jump by 10s or 100s on the
number line until I’m closer to the answer so it doesn’t
take as long.