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
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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 a 6 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
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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
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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 equals a 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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