negative numbers

Why Bother. I have a Why Bother. I have a Calculator!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 In the red or in the black?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

DebtDebt

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)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


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.


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.


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


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.



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.



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


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 LineNumber Line

01 2 43-5 5-4 -3 -2 -1

•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!.


01 2 43-5 5-4 -3 -2 -1

• 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


01 2 43-5 5-4 -3 -2 -1

• 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


01 2 43-5 5-4 -3 -2 -1

• 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


01 2 43-5 5-4 -3 -2 -1

• 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.


01 2 43-5 5-4 -3 -2 -1

•Addition summaryWhen 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)

+_


01 2 43-5 5-4 -3 -2 -1

• Subtraction 4-1= 3•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)


0 1 2 43-5 5-4 -3 -2 -1

• 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


0 1 2 43-5 5-4 -3 -2 -1

• 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


01 2 43-5 5-4 -3 -2 -1

•Subtraction summaryWhen 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)

+_


01 2 43-5 5-4 -3 -2 -1

•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 strategyMy 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.

negative numbers

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