help with negative integers

[kristin0713]

My DD is struggling with operations with negative integers.  She's almost 13, using CLE 700s.  She is doing really well with CLE, but this concept is hard for her. Any recommendations?  

[HomeAgain]

What exactly is she struggling with?  The concept of them?  Application in addition/subtraction?  Multiplication/division?

We start out with teaching it as debt.  It's a red number that you owe.  You can take away that debt, which means you have more to work with than you previously thought.  Eventually, though, I show some of the videos from Khan academy about why multiplying and dividing 2 negatives make sense.

[wendyroo]

For negative numbers I always talk piles of dirt and holes.

piles are a positive amount of dirt

holes are a negative amount of dirt

+ means adding something

- means taking something away

 

1 + -1 = 0

1 pile of dirt + 1 hole = the pile fills the hole leaving nothing

 

So, first, an easy one:

-7 + 1 = -6

7 holes + 1 pile = 6 holes

 

Next:

3 + -5 = -2

3 piles + 5 holes = 2 holes

 

Now rephrase that one:

3 - 5 = -2

3 piles - 5 piles = you easily haul away the 3 piles sitting there, and then to "take away" 2 more piles you have to dig them up leaving 2 holes = -2

 

Same problem, different ways of envisioning them.

Then you are ready to move on to subtracting negatives...

 

-6 - -6 = 0

6 holes take away 6 holes

How do we take away those 6 holes?  By filling them in which leaves us with nothing.

 

4 - -6 = 10

4 piles take away 6 holes

How do you take away a hole?  You fill it in.  But how can we take away holes that aren't there?  You haul in a hole's worth of dirt and you put it on the ground.  What does that leave?  A PILE!!

4 piles take away 6 holes leaves you with 10 piles of dirt

 

Wendy

[wendyroo]

It works with multiplication as well.

Just remember when multiplying to always visualize flat ground to start with...no piles, no holes.

 

Multiplying two positives:

3 * 5 = 15

Adding 3 groups of 5 piles each means you have made 15 piles.

 

Multiplying a positive and a negative:

4 * -2 = -8

Adding 4 groups of 2 holes each means you have made 8 holes.

 

Or

-2 * 3 = -6

Taking away 2 groups of 3 piles each...well, since the ground is flat to start with, but you still need to "take away" 6 piles worth of dirt, you will end up with 6 holes.

 

Multiplying two negatives:

-10 * -3 = 30

Taking away 10 groups of 3 holes each...well, since the ground is flat to start with, but you still need to "take away" 30 holes, you will have to haul in 30 unit of dirt and you will find yourself with 30 piles.

 

Wendy

[wendyroo]

5 minutes ago, square_25 said:

That's a cool model! Like debt but more visual. Do you find it works better than just talking about debt? (Where debt can be measured in apples and "things" and not just money.) 

Negative numbers are normally a topic that start getting explored at my house around age 7.  At that point, the kiddos don't really have much personal experience with debt. 

Plus, the money model always starts to get muddled in my head when I think about negative * negative.   It is easy enough to imagine having a debt and then going further into debt (addition: -$3 + -$5 = -$8) or having a debt and then paying some back (subtraction: -$3 - (-$2) = -$1) or having multiple debts of the same size (multiplication: -$3 * 4 = -$12), but how does it work with -$3 * -4...you have -4 debts, each $3?  I have a hard time conceptualizing having a negative number of debts.  Obviously, I can say the right words to make it "work".  "For every negative debt, that means that they owe you the money instead of you owing them.  So if they owe you $3, 4 times over, then they owe you $12, which is positive because it is owed to you instead of you owing it."  But I don't see how I could possibly explain that to small children and have it make intuitive sense. 

So, I stick to piles and holes which we can draw or physically make and move around in the sandbox.  Actually, I'm going through this with my 2nd grader right now, and our favorite manipulative is playdough.  We've been using a mini muffin tin to mold our "piles" of playdough.  Then we cover up any extra "holes" in the muffin tin that we don't need for the problem.  As we work through the problem, the piles truly do slip right into the holes and negate each other, leaving a flat surface.  We can run our hand over a physical manifestation of x + -x = 0.

Wendy

[kristin0713]

2 hours ago, square_25 said:

I'm not familiar with CLE: how do they introduce negative integers? What's the conceptual model? 

CLE teaches the rules.  I believe they did introduce the concept of debt, which I had talked to her about before, but it doesn't make sense to her.  I think they also showed a number line when first introducing negatives.  She will sometimes draw out a number line and count forwards or backwards on the line.  But she's still not really getting it.  It is the one thing she consistently gets wrong in her daily review. 

2 hours ago, wendyroo said:

For negative numbers I always talk piles of dirt and holes.

piles are a positive amount of dirt

holes are a negative amount of dirt

+ means adding something

- means taking something away

 

1 + -1 = 0

1 pile of dirt + 1 hole = the pile fills the hole leaving nothing

 

So, first, an easy one:

-7 + 1 = -6

7 holes + 1 pile = 6 holes

 

Next:

3 + -5 = -2

3 piles + 5 holes = 2 holes

 

Now rephrase that one:

3 - 5 = -2

3 piles - 5 piles = you easily haul away the 3 piles sitting there, and then to "take away" 2 more piles you have to dig them up leaving 2 holes = -2

 

Same problem, different ways of envisioning them.

Then you are ready to move on to subtracting negatives...

 

-6 - -6 = 0

6 holes take away 6 holes

How do we take away those 6 holes?  By filling them in which leaves us with nothing.

 

4 - -6 = 10

4 piles take away 6 holes

How do you take away a hole?  You fill it in.  But how can we take away holes that aren't there?  You haul in a hole's worth of dirt and you put it on the ground.  What does that leave?  A PILE!!

4 piles take away 6 holes leaves you with 10 piles of dirt

 

Wendy

 

The holes and piles imagery is amazing!!  I think that might work for her!

[kristin0713]

14 hours ago, Plum Crazy said:

Key to Algebra Book 1 Integers uses a football field and teams gaining and losing yards as their example. They always turn a subtraction problem into adding the opposite. -5 - 4 = -5 + -4 and -5 - -4 = -5 + +4

 

CLE does say to add the opposite when it's subtraction.  That has sort of worked for her,  but she's still not getting them right consistently.  I have avoided the Key To series because we hated Key to Fractions.  But if the holes and piles doesn't do the trick, I'll look into that.  

Thanks so much for all the replies!  This has been very helpful to me!

[Lori D.]

Wendyroo -- Fantastic visual of holes and piles (and the muffin tin for physically understanding) -- thanks for sharing!

re: where negative integers get used in real life
Love Plum Crazy's football example of yards gained/lost! I've also seen real-life examples talk about temperature -- above (positive) or below (negative) 0˚, and about measurement of geographical altitude -- above (positive) or below (negative) sea level.

[Sarah0000]

I introduced the concept first with temperature. Then DS did Math Mammoth Integers which is only a couple dollars. IIRC, MM shows the concept with blue and red colored circles that can cancel each other out and with a number line. 

[Violet Crown]

Never found a a better conceptual method than the number line. Seconding the endorsement of Key To Algebra.

[wendyroo]

2 minutes ago, square_25 said:

So the idea with the number line is that addition is just following one integer, then the other integer, away from 0? Like 3+ (-4) is you go 3 to the right (since that's how you get to 3 from 0) and then 4 to the left (since that's how you get to -4 from 0)? And taking away is doing it backwards? 

I'm interested how well this clicks conceptually for kids, on average. I know when I've taught vectors (which is the same thing but in two dimensions), the kids didn't find this particularly intuitive. Not so much with addition but with subtraction because it became a two-step thing and they had trouble reasoning about it, somehow. Not sure why: I'm very visual and it's not hard for me. 

I have had success with a number line, but only by actually marking one on the floor and having the child physically stand and move on it.

I mark the numbers (normally from -12 to +12), and I also prominently mark the "positive direction" and the "negative direction" on the ends of the line.

The important thing to realize is that you have to consider two factors: whether you should face in a positive or negative direction, and if you should move forward or backward.

So, 2 + 3 = start at 2, the plus tells you to face in the positive direction, the +3 tells you to move three spaces forward.

2 + -3 = start at 2, the plus tells you to face in the positive direction, the -3 tells you to move three spaces backward (negative steps).

2 - (-3) = start at 2, the minus tells you to face in the negative direction (you are "taking away" or reducing), the -3 tells you to move three spaces backward (negative steps).

-2 - 3 = start at -2, the minus tells you to face in the negative direction (you are "taking away" or reducing), the +3 tells you to move three spaces forward

2 * 3 = start at zero, the +2 tells you to face in the positive direction and do whatever you are going to do twice, the +3 tells you each move should be forward three spaces

-2 * 3 = start at zero, the -2 tells you to face in the negative direction and do whatever you are going to do twice, the +3 tells you each move should be forward three spaces

2 * -3 = start at zero, the +2 tells you to face in the positive direction and do whatever you are going to do twice, the -3 tells you each move should be backward three spaces

-2 * -3 = start at zero, the -2 tells you to face in the negative direction and do whatever you are going to do twice, the -3 tells you each move should be backward three spaces

I find this method is fine for allowing children to accurately calculate with negative numbers, but it still feels a bit inexplicable.  When calculating -2 * -3, why does one minus tell you the direction to face and the other the direction to move.  I can obviously demonstrate that it doesn't matter which is which, and that -3 * -2 leads to the same answers, but intrinsically why is that the correct way to handle the problem?

I do normally set up a number line like this a few times for each child.  I do it after we have worked with holes and piles extensively and they are very comfortable with negatives.  I want them to be exposed to many different visuals and models to expand and challenge their understanding.  But instead of thinking of it as a teaching tool, I actually see it as an "exploratory manipulative."  I set up the number line and challenge the child to use it to model more and more difficult problems.

Wendy

[Lori D.]

re: using a number line

I may be mis-remembering, but I thought that the concept was all based around zero as the center point, and you move RIGHT or LEFT (rather than forward/backward), in relationship to zero. So adding means moving in the direction of the RIGHT-facing arrow of the number line, and subtracting means moving in the direction of the LEFT-facing number line...

<---- -3 ---- -2 ---- -1---- 0 ---- +1 ---- +2 ---- +3 ---->

      adding --->
<--- subtracting    

[wendyroo]

20 minutes ago, Lori D. said:

re: using a number line

I may be mis-remembering, but I thought that the concept was all based around zero as the center point, and you move RIGHT or LEFT (rather than forward/backward), in relationship to zero. So adding means moving in the direction of the RIGHT-facing arrow of the number line, and subtracting means moving in the direction of the LEFT-facing number line...

<---- -3 ---- -2 ---- -1---- 0 ---- +1 ---- +2 ---- +3 ---->

      adding --->
<--- subtracting    

Sort of, but adding obviously doesn't always mean moving in the direction of the right facing arrow.  (-5) + (-3) is an addition, but moves more and more negative.  And subtraction doesn't always mean moving in the direction of the left facing arrow.  6 - (-4) is a subtraction, but it clearly moves "right" on the number line.

"Plus means right" and "minus means left" are fine if you are only dealing with positive numbers, but it confuses the issue when negatives come into play.  Adding and subtracting fundamentally tell you which way to "face", and then the sign of the second addend tells you not if you should move right or left, but if you should move forward or backward.

Wendy

[Sarah0000]

I just asked my 7yo how he does it and he says he imagines a number line and if the operation is subtraction he imagines a minus sign over it and if it's addition he imagines a plus sign so he'll remember which direction to move. 

[Sarah0000]

14 minutes ago, square_25 said:

 

Does he do negative numbers and just flip sign, then? 

No I don't think so. I gave him a few problems like -9-2 and -4+6. I asked him to describe what he's doing when he evaluates an expression like that and the above is what he told me. It sounds like he's thinking of the signs as directions. 

ETA: He's also done work with coordinate graphing and translation so thinking of operations in movement is not new to him.

Edited February 18, 2019 by Sarah0000

[Violet Crown]

3 hours ago, square_25 said:

 

So the idea with the number line is that addition is just following one integer, then the other integer, away from 0? Like 3+ (-4) is you go 3 to the right (since that's how you get to 3 from 0) and then 4 to the left (since that's how you get to -4 from 0)? And taking away is doing it backwards? 

I'm interested how well this clicks conceptually for kids, on average. I know when I've taught vectors (which is the same thing but in two dimensions), the kids didn't find this particularly intuitive. Not so much with addition but with subtraction because it became a two-step thing and they had trouble reasoning about it, somehow. Not sure why: I'm very visual and it's not hard for me. 

Well, we did this.
For 3 + 4, start on the 3, then hop four towards the "plus end."
For 3 + (-4) start on the 3, then hop four towards the "minus end."
For 3 - 4, start on the 3, then hop four away from the plus end (4 is a plus number).
For 3 - (-4), we started on the 3, then hop four away from the minus end.
So "plussing" was going toward; "minusing" was going away from. They had no trouble with the idea of "go away from the minus end of the line." Also we didn't use "left" and "right" because they were little and that was one more thing to keep track of unreliably; we just labelled the line.

We also avoided the terms "positive" and "negative," "add" and "subtract." Everything was "plus" and "minus." (Part of dh's Grand Theory of Mathematics Teaching (TM); don't blame me.) Everything seemed to make sense to them, so I suppose it worked. YMMV naturally.

ETA: We used Miquon, which is heavy on the number line stuff. I especially liked their pages where the number lines are at wonky angles on the page, to get away from the "left" and "right" business. We didn't much use the rods however.

Edited February 18, 2019 by Violet Crown

[Sarah0000]

1 hour ago, square_25 said:

 

Those both involve adding and subtracting a positive number, though. You'd need to do something like 4 - (-6) to see what he does there. 

You're right. I asked him and after thinking a moment he came up with ten. I asked how he got that and he said taking away from a negative is going the opposite direction.

[Creekside5]

Hands on algebra teaches this  really well in the second and third booklet. It’s also kinesthetic using pawns.  I appreciate how it’s concrete, pictorial then abstract. 

[lmrich]

Use the number line but turn it to be a vertical line - so helpful to go up and down.

[kristin0713]

Old thread!  She's in Algebra 1 now and she gets it 😊 

[Not_a_Number]

45 minutes ago, kristin0713 said:

Old thread!  She's in Algebra 1 now and she gets it 😊 

And we’ve been talking negatives in another thread!

[carrierocha]

I know the OP is solved, but in case someone comes across this in the archive...

We use a number line with right and left movement from 0 as addition or subtraction.

But, we use a car and the car faces right or left depending on whether it is addition or subtraction.

Then, the car itself goes forward or backward depending on whether the number is positive or negative.

2 + (-5) = 

the car faces to the right since it is an addition problem

The car starts on +2 (facing to the right as I said), then goes in reverse 5 steps because (-5) means (reverse 5)

This was *super helpful* for my kids. I let them make their own number line and pick a car, yes even in late elementary they loved that, then they could use it until they had internalized the concept.