Adding and Subtracting Integers using colored counters

FCPS Math Team, 2013.

https://youtu.be/hGVm2xs0HEA

This video shows how you can model integer addition and subtraction with counters and unifix cubes on a number line.

All right today we're going to talk about operations with integers and we're going to focus on looking at addition and subtraction by looking at models of ways you can show students how to add and subtract integers. One of the models we're going to look at is using counters and we're going to use the double sided counters. So this problem says 3 plus 5 so we put out 3 yellow counters which are positive and we're adding five yellow counters and then this shows that there is a total of eight. It's really good to start with an example that students understand and can build from their background knowledge with. So if we were to do this on the number line again we would start off with 3 and I'm just using you know flicks unifix blocks . So we have 3 starting at 0 and then we would add a set of 5 more and together students can count on the number line or they can count the individual blocks to see that that's a total of 8 blocks. And then you'd want to start with an example ike that. Another example would then be to add a negative. So 3 plus negative 5. Again you can start with the 3 counters that's those positive ones and then you would add 5 negative counters which are the red side. When you do that you want to talk about how one positive and one negative together make a zero. So we would take out our zero pairs and we'd end up with a total of negative two. You can do that on the number line as well. We'll start off with three unifix blocks. We're going to add five negative blocks to that. So when we add our five negative blocks, we're going to start where the three is and add our five negatives. Negatives go towards the other way, they go towards the left. So we'll have one two three four five negative blocks and that ends us up on the number line at a negative two, which is the same as the counters. This helps connect the way that students have learned to add either through a number line or just adding objects in the past.

This also works for subtraction. I'd want to start with a problem that's very familiar to them again. So we'd have 5, 1 2 3 4 5 counters and it asks us to take away 2. So we're just going to take away 2 and they're left with 3 counters. Do the same with the unifix blocks. We have five blocks and then they can take away two and they're left with three. Again this is to connect them to what they already know then you can move into then you can move into the integers and use a negative 5 minus 2. So we're going to start with negative 5 counters and it asked us to subtract two positives we don't have two positives so we need to add 0 pairs if we add 2 0 pairs that allows us then to take away two positives and we're left with negative 7. What that looks like on the number line with the unifix blocks is we start off with negative five and then we add two, two to make our zero pairs again. We add two positives and two negatives. We're going to take away those two positives we're going to end up with seven negatives. This is just one way to connect to what students already know and help to build the conceptual knowledge around operations with integers.

Ffor more information you can research, you can look at this elementary and middle school mathematics book called Teaching Developmentally by John Vander Waal. The mathematics office has a copy that you could look at.

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Multiplying and Dividing Integers (CRA)

Patton, 2018

This video shows how to model the multiplication and division of integers across a CRA sequence of instruction.

in this video we're going to take a look at multiplying and dividing integers and we're going to do that across a CRA progression. Recall that C stands for Concrete, R stands for Representational, and A stands for Abstract. And we're going to work across that entire progression in this video. So when we think about multiplication and we think about the way students learn to multiply, if they saw three times tw, young kids second graders, third graders, are asked to interpret this as three groups of two. That's the language that teachers use to introduce multiplication “three groups of two.” So if we wanted to model this for integers we would say that this is three groups of two and in that case it's a positive two. So I would do a group of two a group of two and a group of two and that would give me three groups of two or altogether that would give me six. So three groups of two, of positive two, would give me six. We're not going to change that definition just because we're working with integers we're going to stick with it. So if we altered this problem then to be three times negative two, it's still three groups. The only thing that's changed is what we have groups of. So instead of a positive two this time we have groups of negative two. So if I wanted to model that three groups of negative two would look like this. There's one group of negative two a second group of negative two and here are three groups of negative two so three groups of negative two give me a product of negative six okay. So three groups of two or three groups of negative two. Where it gets interesting is when the first number is negative let's say that this was negative 4 times 2. Now we can't do negative 4 groups. What we probably want to do intuitively is utilize the commutative property and say let's think about this as two groups of negative four. But we can't do that yet because that's considered abstract work and students aren't ready for that if they're this is an introductory level to multiplication with integers so again we want to model exactly what's given to us. So how do we handle that negative sign? Well technically what we will do is we will factor off that negative one and we will treat this as negative one times four times two. So now we can still do our four groups of two like we used to and then the way we handle the negative is to do the opposite. So we would interpret negative four times two as the opposite of four groups of two and the way that that would look when we were modeling is we would do four groups of two so that's four groups of positive 2. Which would give me a positive eight, but we want the opposite and so the opposite is going to be a negative eight. Okay so negative four times two we are going to interpret as the opposite of four groups of two and the reason we can do that is because we're factoring off that negative one off of the negative four. If we wanted to look at another example, if we had two negatives that we were multiplying let's say negative three times negative four we would interpret this as the opposite of three groups of negative four. So again what we're thinking about is factoring off that negative one and then multiplying 3 times negative four. So we're gonna handle three groups of negative four and then we'll do the opposite so we'd have 3 groups of negative 4 so that would be a group of negative 4. There's one there's two groups of negative 4 and there is three groups of negative 4 so 3 groups of negative 4 gives me negative 12. But we want the opposite of that which would be a positive 12 and the same is going to hold true. The same language will hold true when we work with the representation. So the way that we would draw this instead of using manipulatives, we would still say the opposite of three groups of negative four but we would draw three groups of negative four which would give us negative twelve. But we want the opposite and what teachers tend to do is circle all that and put a big plus sign over it or I've seen teachers say that this is negative 12 but we want the opposite which is going to make that a positive 12. So let's take a look at a couple more representations for multiplication if we had a problem like 2 times negative 3 we would interpret this as two groups of negative 3. So this would be two groups of negative 3 which would give us negative 6 if we had negative 3 times 1 this would be the opposite of three groups of 1. So we would do three groups of 1 which would give us a positive 3 but we don't want that we want the opposite which would give us a negative 3 and so we can see here how we can generalize rules if I have groups of negatives this would be groups of negatives. My product will be negatives. Here I had a negative times a positive which is the opposite of groups of positives, so the opposite of groups of positives would also be negatives. If we tried another problem where we had two times four this is saying two groups of four so I'm gonna have a group of positive four and another group of positive four and so two groups of positive 4 is going to give me a positive 8. Or another problem like negative 3 times negative 1 would be the opposite of three groups of negative one. So we're gonna do three groups of negative one which would be negative three but we don't want negative three we want the opposite which would be a positive three. Okay so in both of these cases here we have a positive times a positive we would interpret that as groups of positives would give us positives or a negative times a negative would be the opposite of groups of negatives. The opposite of negatives are positives and so we can use these models whether concrete or representation to establish the abstract rules that when we multiply the same sign the product is positive and when we multiply different signs the product is negative division is going to work the same way. I t's just the inverse so if the model says that A times B is equal to C then I interpret A groups of B was going to give me C as the product. Division then we would say if a times B is equal to C I could divide both sides by A and that tells me that B must be equal to C over A or the total group divided into this many groups remember A indicated how many groups you had and B was what was in each group. So if we think about the same meaning for these variables, C is a total number of items and we're going to divide that into a certain number of group. Then what is in each group will be the quotient. So if we have a problem like negative 6 divided by 2 we interpret that as negative 6. We interpret that as negative six and we want to divide that into two groups when we divide that into two groups, what's in each group three negatives or negative three. Same kind of problem except we'll put the negative sign in the denominator. So if we had let's say six divided by negative two remember now the denominator is what tells us our groups how many groups. Well we can't do six things into negative two groups negative two groups doesn't make sense, so officially what we're doing is factoring off that negative again and handling the six divided by two and so we're gonna do the opposite of six and the two groups just like we did with multiplication we're gonna use the language of the opposite. So this would be six, I show six into two groups, there are three in each group, but I don't want that. I want the opposite and so that would be a negative three.So the same language is going to hold for multiplication to help us with division and we can also do that with a representation. So if we had negative 8 divided by negative 2 we would interpret this as negative 1 times negative 8 divided by 2 so it would be the opposite of negative 8 into two groups and again we interpret as the opposite because the denominator is negative. So we would say negative eight one two three four five six seven eight into two groups there are negative four in each group but I want the opposite so that's going to be a positive four. So the opposite of negative eight into two groups gives us four in each group. Negative ten divided by two we interpret as negative ten into two groups, so that's one two three four five six seven eight nine ten. That's ten things and we are going to do two groups and so there are five negatives in each group because the numerator is powered that I'm sorry because the denominator is positive, this is just two groups. Whereas in this problem the denominator was negative, so it was the opposite of two groups and so these rules again. Or the rules can be generalized from these models. So if we wanted to think about a positive divided by a positive okay that would just be a positive into this many groups how many ever that is and that would just give us groups of positives. If we did a negative divided by a negative this would be the opposite of negatives which would give us positives. If we had a positive divided by a negative this would be the opposite of positives in two groups which would give us negatives. And if we had a negative divided by a positive it would just be negatives divided into groups which would give us negatives in each group. So the language for multiplication carries over into the division and can once again be used to establish the rules for why when we're dividing the same signs we get a positive but when we're dividing different signs we get a quotient that is negative.

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