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ADDING AND SUBTRACTING POSITIVE AND NEGATIVE NUMBERS
Even though I have taught adding and subtracting positive and negative numbers using black and red manipulative pieces for several years, I have previously exhibited how to use the pieces when representing a specific math problem. I was uncomfortable with allowing students to struggle with solving adding an subtracting positive and negative problems without prior demonstration for a couple of reasons. One reason is that students refuse to think through a problem if it requires more than a couple of minutes to solve. They don't possess the determination or the desire to devote the time or the effort required to solve the math problem. It is easier for them to quit. Another reason for my uneasiness is the dreaded factor of time. There are many concepts that are essential to teach my math students in order to prepare them for the C.S.A.P. Urging them to think through problem-solving situations consumes a great deal of class time. None the less, I decided to spend the mandatory time to teach this concept in accordance with discovery learning.
I initially arranged my pre-algebra class with three students per group. Each group consisted of an above average learner, average, and a little below average learner. I wanted to try and balance the groups so there was a least one person in each group that could lend assistance to the others during confusing moments in the lesson. The introduction of adding and subtracting positive and negative numbers began with asking students to share their previous knowledge on the subject. Students relayed that positive numbers were on the right side of the number line, and the negative numbers were on the left. I asked them if they could relate to any situations or tasks that required the knowledge of positive and negative integers. Answers provided were checking accounts, thermostats, and reporting weather temperatures.
My lesson continued with the definition of the mathematical term 'integer'. This term was added to our growing word web which covers the partial west wall in our room. We also discussed the attributes of a number line including graphing the number, coordinate of the point, and the definitions of the aforementioned. These terms were also added to the math word web. I explained to my students that we were going to add positive and negative integers using black and red squares and also show the movement of the sums on the number line as we solved various mathematical problems. I divulged that the black side of the two-colored tiles represented a positive integer, and the red side was a negative integer. The use of black to represent a profit in businesses and red to indicate that the company was losing money was also shared.
The number 5 was then written on the overhead, and I asked my students to show me a positive five with their pieces. I repeated the same procedure for a (-5). The students responded by displaying first the black five and then the red five. We once again discussed the term 'integer' and its definition as a positive number and its opposite. Students represented several examples of a positive integer and then its opposite. We expanded our definition to include that integers were also call 'signed numbers'. They likewise graphed a few of the previous examples on the number line.
After everyone was comfortable with displaying integers and their opposites, I asked students to use the manipulative pieces to perform some simple sums. The students exhibited sums of 5+3, 6+10, (-5) + (-3), and (-3) + (-4). These simple sums were also added via the use of a number line. The movement of adding a positive to a positive and a negative to a negative was discussed. The number line in addition was turned to a vertical position and was related to an airplane or helicopter moving skyward. Zero was ground level. The movement of negative sums was correlated to a submarine, and zero was the ocean's surface. I did this for a couple of reasons. One being that students comprehend the movement of aircraft and submarines. The second reason is I was hoping to plant this seed for a subsequent lesson involving writing situations by the students incorporating the addition and subtraction of positive and negative numbers. Students were comfortable with this level of knowledge due to their previous experience with the number line.
Observations concerning adding a positive integer to a positive integer and adding a negative integer to a negative integer and the resulting sums resulted in a short class discussion. Students also drew the black/positive and red/negative manipulative squares representing these sums using graph paper and red and black colored pencils. These simple sums were therefore represented numerically, with a number line, and pictorially on graph paper.
My class was then asked to ponder and present what they thought zero would look like. Faces fell throughout the classroom and several shouts of "What?" were vocalized. I again repeated my request. Students were truly perplexed as they knew what zero was but had no clue how to show it using the manipulatives. I had several students make a circle with the pieces, and many students verbalized that a positive one and a negative one would make zero. However, even with the last statement voiced by some students, construction a zero took almost ten minutes. As I moved to each of my groups of students, I reiterated several times to listen to their own definition of zero. Finally one by one several students started using the manipulatives in correlation with their knowledge of zero. To the delight of many students, they realized that one black square and one red square signified zero.
After that difficult first step of representing zero, students were requested to show me the expression of 5 + 0. To my surprise, this took several minutes. I sincerely expected them to make this connection quickly; however, this was not the case. Inwardly, I was questioning the wisdom of continuing with the discovery learning as at that moment I came to the horrific realization that what I had planned as maybe one or two lessons would take the remainder of the week to accomplish. I had to repeat several times to my groups of students to once again place out a positive five and to reflect on what they had just seconds before concluded was the visualization of zero. The funny part came when I asked for the sum of five and zero. Some students stated "Six". Questioning looks surfaced and insults of "Yeah, right!" invaded peoples thoughts as students tried to determine which side to join in the ever looming horror of joining the wrong side. It took students a lengthy amount of time to adjust their knowledge of the numerical representation of 5 + 0 with the visual manipulative display of 5 + 0. Many students whined, "I don't get this!" However, we continued with different visual sums of a positive integer plus zero until their frustration level decreased a little. To my utter amazement, these few mathematical steps devoured my class time for the day and the lesson was forced to resume the subsequent day.
The following day students rejoined their groups and we reviewed the previous day's information concerning the sum of positive integers, negative integers, as well as the visual display of the sum of a positive integer and zero. On the overhead I again wrote five and encouraged my class to display that integer. When each group had completed this task, I asked them to add a negative three and explain to me how they derived the sum. As I rotated through the groups, the majority of students placed out 5 + (-3) = 2 with their squares exactly as the equation was written. Some had even gone to great lengths to make little equal signs out of paper to place between the (-3) and the 2. When I explained that they had given me the equation and not how to use the pieces to derive the sum of 2, the statements of "I don't get this!", "This is impossible!", and "This is stupid!" once more reverberated off the walls. I continued attending to each group and encouraging them to think about what we had previously learned. I tried to carefully phrase my thoughts in the following manner. "Think about what we have discussed today. Ponder on what you know and use the pieces to show me how to obtain the sum of a positive 2." I was determined not to remind them of the visual representation of zero for fear of robbing them of the "ah-ha" moment.
This aforementioned scene continued for almost twenty minutes and the frustration levels were soaring. I was even beginning to question the sanity of this mathematical endeavor! I flatly stated that this crabby, Scot-Irish teacher refused to show them the solution. Students slowly calmed down realizing that I was not going to spoon-feed them, and they would have to continue to think of methods to solve the problem. A couple of students shouted, "Mizz, Come here quick! I think I've got it!" These students proceeded to show me the zeros I was praying they'd notice and beamed as they revealed their trophy sum of two.
I joyfully exclaimed, "I've a couple of groups that have discovered the answer!" Upon learning that others had found the treasured secret, my quitters had a renewed hope. Eventually, students caught on to constructing the zero pairs. However, the discovery learning was not accomplished by all students, and there were beggars and thieves of the treasured secret.
We continued using the manipulatives for several problems while also discussing the movement on the number line in terms of aircraft and submarines. The diagrams on graph paper took on a new look as I required students to circle the zero pairs with green colored pencil and draw an arrow away from the collection of pieces. Observations were requested from the groups concerning the mathematical process that was taking place. I was hoping they would connect the green circle and arrow drawn away from the collection of pieces as an indication of the subtraction process. Once again, this observation took more time than predicted. Apparently, my class was totally relying on their previous knowledge and use of the number line to derive the sums. Several simple sums of adding a positive to a negative integer were practiced before students came to the conclusion that they were actually subtracting.
Questions were further directed to my class as to why some sums were negative and some were positive. This observation by students was answered in a moderate amount of time as they visually relied more on the black and red pieces instead of solely on the number line. Answers stated were based on the manupulatives as some students looking at their pieces replied, "Because there are more black than red pieces or there are more red than black pieces." At this junction, I felt like they had overcome an obstacle because they were now utilizing some thinking skills plus the manipulatives to solve the posed problems.
As class resumed the subsequent day, my hopes were dashed as my class took a couple of steps in reverse in terms of using previously acquired knowledge and their black and red squares. The lesson resumed with the task of subtracting positive and negative integers. We first reviewed the 'take-away' model. Students placed six black squares in front of them and were requested to 'take-away' from that collection four black/positive pieces. They were a little hesitant, but quickly understood after I stated for them to physically take away the four black pieces. We continued with a few basic subtraction problems using positive integers so I could return them to their comfort zone.
Pupils were then directed to place before them a positive six and 'take-away' a negative/red four. Again confusion filled their eyes and comments such as, "What?!" sailed through the air. Thoughts of, "Oh good grief, here we go again!" flooded my mind. I passed from group to group and questioned students how they could 'take-away' a negative/red four. They immediately added the negative four. I further questioned them if they had changed the 'net value' when they added the red four. Students, to my surprise, realized from the pieces that they had changed the value of the collection but were still confused. They were then asked to consider a method of altering the collection of pieces without changing the 'net value'. I believed they would quickly recall the previous day's learning concerning zero, but disappointingly they didn't. It took groups another fifteen minutes or so to remember that adding zero pairs didn't change the 'net value'. Students questioned me, "Do we do that zero thing again?" I replied that if they added zero pairs to their collection of positive/black six, would it change the value? They stated, "No." A few students began adding zeros, but weren't certain of how many to add. "How many zeros can we add?", they questioned.
I replied, "How many zeros can you add without changing the value?"
To this question they responded, "As many as we want." They were again directed to reflect on what they were trying to accomplish. After verbalizing that they were trying to 'take-away' a negative four, I noticed a little light beginning to dawn at the recesses of their minds.
Finally, one student shouted, "Mizz, Come here!" I moved to this group and the student showed me the initial black six and the addition of four zero pairs. He then proceeded to physically take away the negative four. I told him that was correct and asked for the difference. He immediately looked from the pieces to me and exclaimed, "That can't be right!" My beggars and thieves who were watching this scene tried the same procedure and all agreed that the resulting difference didn't make any sense. We did several more problems, and I encouraged them to discuss within their groups what was taking place when they added the zero pairs and then subtracted the negative collection. This continued for the remainder of class and students quizzed me, "Is this that 'leave, change, change,' thing?" The period concluded with them being incapable of explaining to me what was taking place mathematically and answering their own questions in regards to their 'leave, change, change' rule.
A quick review of this week's concept continued before I posed another problem of 10 - (-2). This time I had them recall their own rule of 'leave, change, change' and discuss in their groups what they thought the correlation between the mathematical problem and their rule may be. My intention was to direct and encourage their thinking between the visual model and their previously learned rule as frustrations remained high as a result of the previous day's failure. I traveled between groups quizzing students concerning their observations and the possibility of linking them to their rule of 'leave, change, change'. I found it necessary to review the concept of the opposite of a negative two before one of the students stated aloud that after taking away the negative two, they were adding the opposite of the red two which was a black/positive two. The remainder of the class still didn't comprehend and as an entire class, a lengthy discussion of adding the opposite collection when subtracting integers took place accompanied by several more math problems. It wasn't until after conquering this mental mountain that the same math problems were solved using the number line, and illustrations were drawn using graph paper and red and black colored pencils. I once again had students circle the red/negative pieces in green and draw an arrow directing the movement away from the collection.
The subsequent two days were used to practice similar problems such as 10 - (-4) and -14 - (-4). At this point in time the majority of students understood the mathematical concepts of adding zeros, making zero pairs, the result would be based on if the negative or positive collection had more pieces, and the concept of adding the opposite collection of negative or positive pieces when adding or subtracting negative and positive integers.
My assessment of students' learning was based on a couple of key performances. Students' verbal explanations in conjuntion with utilizing the manipulatives during this acitiviy, I have found to possess merit. As I query them concerning their explanations, this exchange requires students to express their thoughts using explicit details and incorporate correct mathematical terminology. This is a vital skill that is required in written form for the state exams. I also require students to draw illustrations of the various adding and subtracting negative and positive problems in addition to the numerical representation on subsequent quizzes and exams. This provides further substantiation of students' comprehension of these concepts and dispels their reliance on memorized mathematical rules.
In reflection, I thought the lesson went fairly well with the exception of implementation time. I succeeded my personal goal of answering students' questions with a question of my own in hopes as not to provide answers, but keep the level of instruction in the discovery mode. I would also change my implementation of the lesson to cover a week instead of only one or two days because discovery learning requires copious amounts of time in comparison to modeling concepts simultaneously incorporating the use of manipulatives. I believe that I would also save solving adding and subtracting negative and positive integers using the number line for another lesson. Its inclusion may have been too much information all at the same time.
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