Subtracting Integers

Lesson 3

In lesson 3 our warm up is on adding integers. We want to review this concept because adding integers is very important to subtract integers. We are going to begin lesson 3 with our SOLVE problem. The high temperature in Hopedale, Ohio was -2 F. The low temperature was -7 F. What was the difference between the high and low temperature for the day? The first step in S study the problem, is to underline the questions, our question is what was the difference between the high and low temperatures for the day?

The second step in S is to answer the questions what is the problem asking me to find? This problem is asking me to find the difference between the high and low temperatures.

In problem 1 we are going to model the problem 5 minus 2. Subtraction is different than addition, in subtraction we model the first number and we take away the second number. We will represent our 5 with five yellows. Our problem is 5 minus 2 which mean we want to take away a 2 or take away a positive 2, because we started with 5 yellows it is possible for us to take away 2 yellows., and this leaves us with a difference of positive 3. To represent this problem pictorial, we will represent our five with 5 yellows and we will take away the two yellows by crossing them out. This leaves us with a final answer of three yellows or positive three.

In problem 2 negative 5 minus negative 2, we are going to subtract a negative 2. We will begin this problem by representing the negative 5 with 5 red integer chips. To subtract negative 2 or take away negative 2, it is possible because we have 2 reds that we can take away. So we take them away and that leaves us with an answer 3 reds or negative 3. To represent this problem pictorially, we will begin by representing our negative 5 with 5 R’s and we want to take away two reds and we will do this by crossing out 2 R’s, which leaves us with a final answer of 3 R’s or negative 3.

To model problem 5 we have 8 minus negative 3, we will begin by representing our 8 with 8 yellow chips. We want to take away negative three, this is impossible right now because we do not have any negatives to take away, however we can create this possibility by using our zero pairs, our value right now is 8, if we add one zero pair ,one yellow and one red, our value is still 8, because there is 8 here plus zero here, so our value stays at 8. But now we have the possibility of taking away one red. We don’t have enough reds to take away yet so we going to add another zero pair , is it possible to take away three reds not yet, so we add another zero pair and now we have created the possibility to take away negative three, we take away the three reds and our answer is all yellow. 1,2,3,4,5,6,7,8,9,10,11 because their yellow our answer is positive 11. To represent this pictorially we will begin , by representing our positive 8, once again we try to take away negative three, it is impossible to take away negative three, so must add a zero pair we still must take away negative 3, we have not changed our value we still have a value of 8, but now have created the possibility of taking away our negative three, and it leaves us with a final answer of 11 Y’s or positive 11.

To model problem 6, we have negative 8 minus positive 3. We will first represent the negative 8 with 8 red integer chips. We want to take away a positive 3 once again we do not have any positive to take away in order to be able to take away positives, we must create the possibility using zero pairs. If we add one zero pair we are still maintaining our value of negative 8 creating the possibility to take away one positive, if we add a second zero pair are still , but we can now take away two positive, and when we add our third zero pair we still maintaining a value of negative 8, but we have created the possibility to take away three. Our final answer is in all one color and it is 8 plus 3 which is 11, and because they are red our answer is negative 11. To represent this problem pictorially we will first represent our negative 8 with R’s, from here we will represent the 3 zero pairs that we must have in order to take away that positive 3, one yellow one red we still cannot take away positive 3, now we can take away positive 2 and when we add one more zero pair, one yellow one red we can take away positive three, which leaves us with 1,2,3,4,5,6,7,8,9,10,11 reds which is negative 11.

To the problem 4 minus one we will begin by representing our 4 with 4 yellow integer chips. The problem tell us we want to take away 1 we are able to take away one positive, so our answer is positive three. To represent this pictorially we will represent our 4 with 4 Y’s and we will take away 1 by crossing one of the Y’s leaving us with 3 Y’s or positive 3.

To model the problem negative 7 minus negative 5 we will first represent our negative 7 with 7 red integer chips. The problem tell us that we want to take away negative five, five red chips that we can take away 1,2,3,4,5 and when we take them away we are left with negative 2 our answer is negative 2. To represent this problem pictorially we will begin by representing our negative 7 with 7 R’s, we want to take away negative 5 so we will cross out 5 R’s, our final answer is negative 2 because we are left 2 R’s

To model the problem negative 3 minus 4, we will first represent our negative 3 with three reds, the problem tell us we want to take away positive 4 because we start with negative 3, it is impossible to take away negative 4, so we must create the possible by using zero pairs. One yellow one red is a zero pair, our value stays negative three but now it is possible for us to take away, one yellow. One yellow, one red, we can take away two yellows, one yellow one red, we are still maintaining a value of negative 3 but by adding zero pairs we’ve created the possibility to take away a positive 4,.. 1,2,3,4, when we take them away we are left with our answer in all one color, and our answer is seven reds or negative 7. To model this problem pictorially we will begin by representing our negative 3 with 3 reds because we want to take away, a positive 4 we must add zero pairs to create the possibility of taking away four yellows and when we take away our positive four we are left with 7 R’s or negative 7.

To represent the problem 6 minus negative 2 we will first represent our 6 with 6 yellows , the problem tell us we want to take away negative 2 but we do not have any negative that we can take away, so must create this possibility by adding zero pairs. When I add one zero pair my value of positive six does not change, it creates the possibility for me to take away one negative, when I add a second zero pair, my value of 6 is still 6 but we have the have the possibility to take away negative 2. When we take away negative 2 we are left with 8 yellows our answer is positive 8. To represent this problem pictorially we will first represent our six with 6 yellows we want to take away negative two but we do not have negatives to take away, so we will add zero pairs to create the possibility of taking away negative 2, our value is still six, but now we can take away negative 2, our final answer will be 8 Y’s or positive 8.

Once we have developed all of our rules for subtractions we are going back to our graphic organizer. Our rules for subtraction,subtraction is the same as adding the opposite. We make two changes to the problem, change subtraction sign to addition sign, and we change the sign of second number. After these two steps are completed we follow the rules for addition

Now that we know the rules for subtraction we are going to go back and complete the SOLVE problem we began with at the beginning of the lesson. We had already underlined our question, which was , what is the difference in the high and low temperature for the day, we had also found what the problem was asking us to find, which is the difference between the high and low temperatures.

In O we organize the facts, we are now going to identify the facts in the problem. The high temperature in Hopedale, Ohio was -2 F, one fact . The low temperature was -7 F, another fact. We must decide if these two facts are necessary or unnecessary. The first fact the high temperature in Hopedale, Ohio was -2 F is necessary. We list all necessary facts. The second fact the low temperature was –7 degree F, also needs to be listed.

Now we are going to line up our plan. We are finding the difference between the high and low temperature for the day so we are going to use subtraction. Our operation was subtractions, we are going to write in words what your plan of action will be. We’re going take away the low temperature from the high temperature. Change subtraction problem to an addition problem, then follow the rules for addition.

In V verify our plan with actions, we will first estimate, we know our temperature has gone from-2 to -7 we are going to estimate that the difference between the two temperature is about 5. Now we are going to carry out our plan, our plan was to take away the low temperature from the high temperature. We are going to change the subtraction problem to an addition problem and follow the rules for addition. We are going to change our subtraction to addition and we are going to add the opposite. Now our problem is negative 2 plus 7. When we have two integers with different signs we subtract and take the sign of the larger group. 7 minus 2 is 5, and 7 is the larger group so it will be a positive 5.

In E we are going to examine our results, we will first answer the question does your answer make sense, if you go back to what your original question was we were looking for the difference between the high and low temperature. So we ended up with an answer positive 5 F. so our answer does make sense. Is your answer reasonable? We look at our estimate, which was about 5, so yes our answer is reasonable. Is our answer accurate, for accuracy you will have your students rework the problem or use a calculator to check their answer. The difference between the high and low temperature was 5, we will write our answer as a complete sentence.

To close the lesson we will review the essential questions. How do you subtract integers using manipulative, represent the first value with a red or yellow chip and take away the second value. What do you do if you can not take away the second number from the first number? You create the possibility by using zero pairs.