Level G Lesson 13
Dividing Integers

In Lesson 13 the objective is the students will divide integers. Essential questions that will be guiding us through the lesson will be 1. How do you divide integers that have the same signs? Number 2 how do you divide integers that have different signs? And 3 how does division of integers compare to multiplication of integers?

We will then rewrite our question in our own words. This problem is asking me to find the integer that represents the average temperature loss every hour.

We will now be using our manipulative to show a concrete way to divide integers. There are 2 statements we use when you’re dividing integers. One will be if you divvy up blank into blank equal groups, what will be in each group? And the other way is splitting up blank items into groups of blank, how many groups can you make? Remember that a red chip will be worth a negative 1 and a yellow chip will be worth positive 1. In problem 1 positive 6 is divided by a positive 3. We are going to use the statement if you divvy up. If you divvy up positive 6 into 3 equal groups what will be in each group? So we will begin with our first number a positive 6, so we will place down 6 yellow chips. Now we want to divide it into 3 equal groups. You will have group 1, group 2 and group 3. So we will divide equally into 3 equal groups. Then we ask our question, what will be in each group? So what is in each group? There are 2 yellow chips in each group. So our answer will be a positive 2.

Problem number 2 is a negative 6 divided by a positive 3. We are going to use our statement if we divvy up. If we divvy up negative 6 into 3 equal groups, what will be in each group?

We begin by placing our first number which is negative 6, so we will have 6 red chips laid down, and then we will ask ourselves what will be in each group? So we will need to, we want 3 equal groups. So now we will separate our negative 6 chips into 3 equal groups. Now what will be in each group? There will be 2 red chips in each group. The value of negative 2.

Problem number 3 is negative 6 divided by negative 3. we are going to use our statement of splitting up. We will be splitting up negative 6 items into groups of negative 3. How many groups can you make? Once again we will represent our first value, so we will have 6 red chips and now we are going to need to separate our chips into groups of negative 3 items. We will need groups of negative 3, that will be 1 group of negative 3, that will be another group of negative 3. We need to ask ourselves how man groups can you make? We have 1 group, 2 groups. So there are 2 groups so our answer will be a positive 2.

Problem number 4 we have a positive 6 divided by negative 3. We cannot use either one of our statements because there are no models of a positive divided by a negative. Now we are going to look at each one of the problems we just did with our manipulatives at the concrete level and we are going to move to the pictorial and verbal representation.

Our very first problem that we looked at was a positive 6 divided by a positive 3. The way we write this out in our verbal representation is if you divvy up positive 6 into 3 equal groups what will be in each group? What we started with was we started off with was 6 yellows. And we wanted to know what would be in each group. We had to break it into 3 equal groups . We have 1 group, 2 groups, 3 groups we want to know what will be in each group. We will have 2 yellows in each group which gives us an answer of positive 2.

The second problem we looked at was negative 6 divided by 3. We first want to write our verbal description out, which would be if you divvy up negative 6 into 3 equal groups what will be in each group? So what we need to start with is our first number, so we have our 6 reds and then we need to put that into 3 equal groups and then we need to ask our question what will be in each group? We have 1 group, 2 groups, 3 groups. So what would be in each group? Well there are 2 reds in each group, which will give us the answer of negative 2.

The third problem we looked at was a negative 6 divided by a negative 3. our verbal description of this one was if we are splitting up negative 6 into groups of negative 3 how many groups can you make?

So the first thing we need to do is we need to represent our first number which is negative 6 we will have 6 r’s written down, and now we need to split up negative six into groups of negative 3. There is 1 group of negative 3 and there is another group of negative 3. And our question is how many groups can we make? We have 1 group, 2 groups. So our answer is we have 2 groups which will give us an answer of positive 2.

The last problem that we looked at was a positive 6 divided by a negative 3. There are no models that we can do that represent a positive divided by a negative.

We are now going to fill out your graphic organizer on the bottom of student page 109. So the first one is for the problem above, we have 6 divided by 3. What we got for our answer was a positive 2. The corresponding multiplication problem 6 times 3. We know that we already learned that a positive times a positive was a positive so we have a positive 18. Now we want to ask ourselves 2 questions: How are they similar? If we take a look at both problems the factors are the same, they are both positive. So the signs of the factors are the same. How are they different? Well one is division and one is multiplication.

Now for problem 2 negative 6 divided by positive 3 we had an answer of negative 2. Now multiplication 6 times 3 is 18 and when you multiply a negative times a positive we know we will get a negative 18. So now we want to ask ourselves 2 questions again : How are they similar? And

How are they different? If you take a look at our 2 problems you are going to see the signs and the factors are different one is negative and one is positive . How are they different? Once again one is division and one is multiplication.

On problem 3: You have a negative 6 divided by a negative 3 we had an answer of a positive 2. Now in the problem with the corresponding multiplication negative 6 times a negative 3. 6 times 3 is 18. We know that a negative times a negative is a positive. So now our two questions How are they similar and how are they different? How are they similar? Well the factors are the same sign. How are they different? One is division and one is multiplication. The signs of the factors are the same and one is multiplication and one is division.

Now for problem 4 we have a6 divided by a negative 3. We didn’t have a model to show for that., but on 6 times a negative 3, we know that 6 times 3 is 18 and we know that when we have a positive times a negative which will give us a negative answer. How are our problems similar? Well a positive divided by a negative and a positive times a negative. So the signs of the factors are different. How are they different? Well one is division and one is multiplication.

Now we want to ask ourselves; what conclusion can you make about the problems above? Well if we take a look at what we had, we had the signs are the same and one is multiplication and one is division, so we had both positive answers. When the signs of the factors are different our answers were negative. The signs of the factors are the same we had positive. So what conclusion can you make? You can make that the same rules apply for division as they did for multiplication.

Now we are going to look at our chart, our graphic organizer for two problems. We are going to look at the verbal, pictorial and we are going to write out our answer for each one. Problem 1 we have a negative 4 divided by a negative 2. When you have a negative divided by a negative we started with our splitting up. So our verbal will be splitting up negative 4 items into groups of negative two and we ask ourselves how many groups can you make? We will start off with our first value which is negative 4, so we will have 4 Red. Now we are going to put them into groups of negative two. Here is one group and here is 2 groups and we ask ourselves how many groups can you make? So we have 1 group, 2 groups. So our answers we have 2 groups which would give us an answer of positive 2.

Our second problem: is 12 divided by 3. You have a positive divided by a positive we are going to be using our divvy up method. So our verbal will be If you divvy up positive 12 into 3 equal groups what will be in each group? So we will represent our first number, so we will have 12 yellows. Now we want to put them into 3 equal groups. So we have 12 of them so we will have 3 equal groups and what will be in each group? We will have 4 yellows in each group which will give us a value of positive 4.

Now we are going to go back and look at our graphic organizer from lesson 10. Rules for operations with integers. We are going to fill out the ones for division. So the first question is two integers with the same signs, what do you do?

We will divide and our answer will be positive.

Next one if 2 integers with different signs what will you do? You will divide and your answer will be negative.

You are now going to take out your foldable that you have created. We are now going to be filling out the division section. On the left side you will be writing examples. On the right side you will be writing out how you divide with integers.

Now we are going to finish our solved problem from the beginning of the lesson. We have already asked our problem. We underlined our question. And we have already answered the question what is the problem asking me to find, which was the integer that represents the average temperature loss every hour.
Now we are going to organize our facts. So we are first going to identify our facts. So the temperature in the northern part of Wyoming is very cold in the winter, that is a fact. So we will strike there. One night the temperature had a decrease of 30 degrees in 5 hours that is also another fact. Now remember you can also do a fact after a decrease of 30 degrees in 5 hours it is your preference whether you strike both places or in one. Next one if the temperature fell at steady rate throughout the 5 hour period, that is also another fact. Now we are going to eliminate our unnecessary facts. So the temperature in the northern part of Wyoming is very cold in the winter that is not necessary. So we will cross out that fact. One night the temperature had a decrease of 30 degrees that is a necessary fact. So we will write that down. Decrease of 30 degrees which is negative 30. In 5 hours which is also a necessary fact. So if the temperature fell at a steady rate throughout the 5 hour period we had that down also as our fact. The next thing we are going to do is line up our plan. So the first thing we need to do is choose an operation or operations. Well what we need to do is look at our facts, decrease of 30 degrees over 5 hours. We want to know what integer represents average temperature loss every hour. So we will need division.
The next step of L is to write in words what your plan of action would be. So we will need to divide the decrease in temperature by the number of hours.
The next step of our solve processes is our V step. We first need to give an estimate of our answer. We know we had a decrease of 30 degrees over 5 hours. So we know that it’s going to have to be less than 10 degrees. Now we are going to carry out our plan. So if we are need to divide the decrease in temperature by the number of hours. We have negative 30 divided by 5 which equalsa negative 6. Remember that a negative divided by a positive is a negative.

Now we are onto the E step of our solved paradigm. Does your answer make sense? We want to compare your answer to the question. Our question was what integer could you use to represent the average loss every hour. Yes because we are looking for the number of degrees of temperature loss. The second question is is your answer reasonable? We want to compare your answer to the estimate. Our estimate was less than 10 degrees. So yes it is close to our estimate of negative 10 because we came up with an answer of negative 6. Is your answer accurate? Check your our work. Negative 30 divided by 5 is negative 6. IS your answer accurate? So our we are going to check our work. 30 divided by 5 is 6 and a negative divided by a positive does give me a negative answer, so our answer is accurate. The last thing we need to do is write our answer in a complete sentence. So the integer that represents the average temperature loss every hour is negative 6 degrees.

Now we are going to go back and answer the essential questions from the beginning of the lesson. The first one is How do you divide integers that have the same sign? We are going to divide and our answer will be positive. The next one is how do you divide integers that have different signs.? We will divide and our answer will be negative.

The last one is how does division of integers compare to multiplication of integers? The same rules apply.

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