What common student issues do you find when teaching the addition and subtraction of positive and negative integers?
What student issues and/or misconceptions do you find when teaching the addition and subtraction of positive and negative integers? How do you review this topic to address these issues and misconceptions?
I find that when teaching integer operations, students struggle in large part because of an inundation of information from various teachers and textbooks. The concept of numbers having additive inverses on a side of the number line that for years didn’t exist can be very hard for students to grasp when they first see this concept in late elementary school. Thus, while I find the number line to be helpful in visualizing parts of integer operation problems I think that it can confound too many scholars without other supports. When teaching addition and subtraction of signed numbers I first start by reviewing addition only, and I’m strict about this. In this context I tell students to create a “Minus-Plus Chart,” which is basically a T-chart with one column labeled “negative -- ” and the other “positive +” that is used to organize the numbers in the problem. I have students place each of the terms in the proper column and then ask two basic questions: (1) “Who wins?” and (2) “By how much?”. For instance in the problem 9 + -14 the 9 would naturally go into the positive column and the -14 in the negative. The “negatives” would win, thus establishing the sign of the answer, and the margin would be 5. Thus, the sum is negative 5. This method also works for problems involving a multitude of signed numbers, for questions where students have to add fractions and decimals, and on a more advanced level is vital when combining like algebraic expressions. I also find the linear structure and the decomposition of the problem to be helpful for less-apt learners.
When dealing with subtraction problems I have students reformat each question as an addition problem by having them add the additive inverse of the latter number/term. (If needed, I teach a separate lesson just on additive inverses/opposites.) For example, -8 – 2 would be rewritten as -8 + -2 in this model. When students were faced problems involving subtraction of a negative number, such as 4 – -6 they are reminded to again convert to this to problem that adds the additive inverse of (-6), so the problem becomes 4 + (+6). I also tell students that subtracting a negative number is akin to “cancelling a debt” which is something that always results in a more positive result. Students are then directed to the same exact procedure they used to add integers: create a minus/plus chart, drop in your values, answer two simple questions. After students get an answer using this method, I teach them to use the work they did with these charts to create a number-line diagram if they are asked to. This way, students have a firm starting point and a firm answer/endpoint to the problem to guide their work. Without this I find that students often can’t get a handle on where to begin on the number line, which direction to go, and most importantly—where to end up. Lastly, whenever possible, I have students check their mechanical math using a calculator if one is available. This might sound obvious, but so often I find students who won’t check their arithmetic to ensure they calculated the numbers properly. I always emphasize the difference between the “minus” key and the “negative" key on the calculator to ensure students understand the distinction and when to use which. Teaching operations with signed numbers can be tough, but after eight years in the classroom I think I finally nailed this topic down by using this method.
As a 9th grade Algebra teacher, I find that students still have issues with adding and subtracting integers. Though this is a skill that is taught in elementary school, they still have difficulty remembering the rule. I believe students will remember the rules better if it is taught through song. I teach this skill through song and they remember it better. The song is to the tune of row, row, row your boat . However, I go on youtube and find an instrumental version to a song that is trending with the teenagers to use as background music.
Lyrics to the song-(Author unknown)
Add and Keep
Different sign, Subtract
Take the sign of the higher number
Then it'll be exact
Change the minus to a plus
Change the sign of the next
All you do is Add them up
As if it were a plus
I have found the largest areas of confusion for adding and subtracting integers comes when students want to “add the opposite” (use additive inverse) in order to subtract integers. For example, students see something like –5 – 2 and want to rewrite the expression as –5 + –2. Students will make mistakes like rewriting it as 5 – 2, –5 + 2, or 5 + 2.
Students need to understand what’s going on when they rewrite this expression. The only way for them to identify their own mistake is for them to understnad why they are allowed to use additive inverse. I have used manipulates called “air conditioners and heaters” in the past in order to teach and review this idea. I cut out blocks of blue and red construction paper (“air conditioners” for negative integers and “heaters” for positive). Each “heater” symbolizes 1 unit, each “air conditioner” symbolizes –1 unit. I have found this construct allows students to learn the skill and allows me a conceptual “in” if I want to review it months later.
To start, you can teach adding and subtracting integers by having students model each problem. By having students line up all the air conditioners blocks in a problem under all the heater blocks in a problem, students see that air conditioner and heater cancel each other out. If there are 4 heaters and 4 air conditioners, the temperature is 0. 4 heaters and 5 air conditioners is –1. 6 heaters and 4 air conditioners is 2. In the links below is a picure of 7 heaters and 5 air conditioners (7 + -5).
Once students have mastered addition, you can then talk about how if you want to raise the temperature in the room, by say 4, you have two options: 1.) adding four heaters or 2.) removing four air conditioners. You can have a class discussion where you ask students various ways to increase or decrease the room by various temperatures. Then, show them how to represent this with symbols. A room with a starting temperature of –20 that you want to decrease by 10 could be shown as –20 – 10 or –20 + –10.
Once students have practiced this many times you can formalize their knowledge by teaching them the correct vocabulary (“additive inverse”) and by having them complete the evaluation of subtraction expressions by rewriting them as addition (–20 – 10 = –20 + –10 = –30). A common error is for students to start rewriting addition problems as subtraction (–4 + 7 = –4 – –7). You can redirect them by telling them that while that’s not wrong, it doesn’t make the problem to easier to do.
When reviewing this with students, you can start class with a short do now that makes them recall the simulation. I may have a short class discussion about the rules, where I go through three or four scenarios out loud with them. For example, if –5 – 2, is on the do now ask the class:
“What’s the temperate in the room?” (Correct response: “–5”).
“What am I doing to the room?” ( “removing two heaters”).
“What else could I do that would have the same affect (“adding two air conditioners”).
“How would you write that?” (–5 + –2).
“What is –5 + –2?” (“–7”).
If students are making the mistake of writing –5 – 2 as say, –5 + 2, you can ask them to reflect on their work using air conditioners and heaters. “You were taking away two heaters. What has the same effect as taking away two heaters?” (Response: “Adding two air conditioners”). “Is that what you did?” Ideally, that type of questioning allows students to self-correct their mistakes.
I have using air conditioner and heater blocks lessons posted on the links below.
The most common misconception I have found when teaching addition and subtraction of integers is that students always want to change the operational sign, regardless if it is addition or subtraction. I normally teach addition first, and students will have a good grasp on the rules. We start by using a laminated number line that is taped to their desks, and the students use dry erase markers to manipulate the problems on the number line. After a few lessons, students generally have the rules memorized and use the number line less and less. Once subtraction is introduced, and students begin to “add the opposite,” they then begin trying to use the same rule with addition, and begin changing signs unnecessarily.
Depending on the student, and the depth of their misconception, I try a few different approaches. For many students, I use manipulatives – red and yellow chips representing positive and negative ones. Students that do not respond to the hands on approach often will respond to the “not not” strategy. We talk about “not not” doing something, and they are able to draw a connection with the double negative, which they know from language arts is really a positive. For the lowest of students, I tape out a number line on the floor and have them walk out the problem, and most often walking a vertical (rather than the traditional horizontal) number line has made a lot of difference in their understanding.
When teaching integers, it is important to bring in real world examples as you teach. Numbers are very abstract for some students, but when those numbers have meaning or relevance to their own lives, they become easier to understand. With integers, the most common examples I use that help students visualize what they are doing are money and elevation in regards to sea level. For example, owing one person $4 and another $8, makes sense to a child that they own $12 total. Just the same, they are able to understand if they owe someone $4 and then they are given $12, that they really only have $8. For subtraction, I use diving a lot as an example. If a swimmer dives 8 feet, and then another 9 feet, they are 17 feet below sea level – this type of problem also allows students to see that -8 – 9 is equivalent to -8 + -9.
Some students may think that when working with negative integers they are just subtractions of a sum, for example, the concept of a negative variable cannot be demonstrated by adding 10 negative ice cream cones to 15 positive ice cream cones. It is very important that the student is able to conceptualize this concept before trying to use the numbers in an equation. I would want to share examples that will help them to visualize the process. For example, on a map of the world, I would use simple integers, mark the space above the equator as the positive integers and the numbers marked below the equator as negative integers.
Another example I might use is where the student is riding an elevator up and down the various floors. We would discuss as the elevator rises toward the top floor, the numbers are positive. As they are traveling downward, back towards
the lobby, the numbers are now negative. By using these types of examples when working with positive and negative integers, the students will have a better understanding that negative integers are a symbolic representation of a value in an equation.
One of the biggest issues I have found with teaching students how to add and subtract integers is that students really have no understanding of what addition and subtraction of integers “looks like”, and are forced to remember the rules for each operation. Students should first be taught how to add integers using models (colored chips or a number line). Additive inverses and zero pairs should be explored and discussed so that students understand what happens when a positive integer is added to its opposite. As with any skill, students should be given opportunities for guided practice and independent practice. When teaching students how to subtract integers, they should be allowed to see for themselves that subtraction can be expressed as addition. This can be done using simple examples that the students themselves can verify [e.g. 5 – 3 is the same as 5 + (-3) or 3 – 2 is the same as 3 + (-2)]. Provide students with practice re-writing subtraction as addition before allowing them to move on to actual computation. Since students should have already mastered the skill of adding integers, they should be successful with problems involving subtraction of integers just as long as they re-write the subtraction as addition first. Additionally, it is best to expose students to real-world application problems that involve addition and subtraction of integers. In order for students to master such question items they must be familiar with vocabulary terms synonymous with positive and negative change (i.e. increase/decrease, rise/fall, above sea level/below sea level).
There are numerous misconceptions that can occur when students are adding and subtracting positive and negative integers. Some students might incorrectly use the absolute value and then confuse whether the sum or difference should be negative or positive. For example, 8 + -3 = 5 and 10 + -15 = 5. They may use the sign of the first term as the sign for their answer, which might produce the correct sum or difference at times, therefore reinforcing that they are adding or subtracting integers correctly. I have also witnessed students incorrectly using the number line. Students usually begin with the first addend or minuend in the correct spot, but then “travel” in the wrong direction. Perhaps the most prevalent misconception I have noticed is when students incorrectly apply something they have learned by a procedural memorization strategy. For instance, “two negatives make a positive” or an acronym such as KCC for “Keep, Change, Change” when subtracting integers. Due to our past spiraling curriculum, students had come into my class with these sayings without any real understanding of why these strategies worked, therefore, often incorrectly using them.
In order to address such misconceptions, I would first give students a formative pre-assessment in order to gain access to their prior knowledge and if they are making errors, determining the pattern of their mistakes. I would help students develop a conceptual understanding of addition and subtraction of positive and negative integers by using hands-on manipulatives such as two-colored counters and number lines. I would help students use these models beginning with positive whole numbers and then introducing negative values using a real world context such as money, football or temperature. I would also create learning opportunities in which students would analyze equivalent expressions such as 4 + -7 and 4 – 7 and guide them to make the connection that they are modeled the same way and have the same result. Most importantly, throughout this unit, the emphasis for students is on the rationale behind their answer rather than their pace.
The misconceptions that I see surrounding adding and subtracting positive and negative integers are usually based on students trying to remember and apply rules that they don’t understand. Having them recite that they need to change the operation and the sign of the second number in the subtraction problem does not create understanding of why the rule works so it doesn’t help students apply the rule to similar problems with consistency. Once students reach my classes, they are usually adept at adding and subtracting. Applying that knowledge to positive and negative integers can be difficult if students don’t have an understanding of what those integers mean. When that understanding hasn’t been created, the common misconceptions that I see have been:
- Adding the integers and choosing any sign for the answer;
- If subtracting, performing the subtraction and making the answer negative;
- Or adding the integers and making the answer positive.
To address these misconceptions, I start by reviewing positive and negative integers before we perform any operations. It is important to help students understand what integers actually represent. So, -7 is not just a random number, it actually means something. Using examples they know and see in their daily lives and activities helps them understand what the integers mean. Weather, sports, and financial examples are good starting points that build understanding of integers for students.
Students are familiar with using a number line, so I generally start with a number line to establish a connection with a model they already know. Once they recognize the meaning, it is easier to understand and associate representations of that meaning. Using two-color counters to display that meaning allows students to visualize the concept. Being able to visualize the concept provides a good foundation for performing operations. If students can visualize 9 and -7, and know what the integers represent then we can have discussion about adding and subtracting them. We can use counters or other representations to model the addition and subtraction. I try to help students create an understanding of zero pairs and how they can be used to support the operations.
Generating a solution to an addition or subtraction problem using positive and negative integers becomes easier for my students as their understanding of the concept of integers becomes stronger. Then they can also recognize when an answer doesn’t make sense and take a different approach to solve it. Additionally, I spend time with them developing reasons why the misconceptions they may have will not work. Knowing “why not” also helps solidify why the “why” actually works. Using the visual modeling also helps students understanding the concept of taking something from nothing. Ultimately, I try to strengthen their understanding of the concepts to help them develop the rules for themselves.
Perhaps the most challenging issue that I see arise for students when adding and subtracting positive and negative integers with the relation that occurs when subtracting integers is involved. Many students are able to follow along conceptually when adding integers in same and different signs, however, when the operation of subtraction is involved, students often struggle. Typically, we teach students to solve subtracting integer problems by changing the operation to addition and adding the inverse. This often involves a lot of rule teaching and tricks to remembering how to do this successfully. Inevitably, some students forget the rule, are not sure when to change subtraction to addition, or change only one, or the wrong part of the sentence. I usually combat this challenges using the following methods:
- First, I place significant importance on building up the concept of adding the inverse with students. We explore visual models, work with manipulatives, and have discussion about why adding the inverse is equivalent to the subtraction problem in the first place. I think this piece is important, because I do not want my students just taking my word for it, I want them to know they are equivalent. This helps students to not just think of changing a subtraction problem as a “trick” but they have some understanding of the rationale behind it. (I also demonstrate for students how they can change any division problem into a multiplication problem using the multiplicative inverse).
- Once the conceptual understanding has been laid, I do offer some students some devices to help them remember how to change a subtraction problem to an addition problem. I do this by using the phrase, KEEP-CHANGE-CHANGE, KEEP the first number the same, CHANGE the subtraction to addition, and CHANGE the sign of the second number. This helps students from making the mistake of changing the sign of the first number, and is also a quick phrase they can say to themselves whenever they see a subtraction problem. I usually tell my class that we will be changing subtraction anytime we see it, all year long. It helps take away any confusion for students about knowing when they should use it, the students who get it start to understand when it makes the most since to change a subtraction problem, and the students who need more time to build understanding get it.
- In order to be consistent in helping students to build their understanding, I also teach students how to solve equations using the additive inverse and multiplicative inverse. Using this method, students have to be consistent in changing subtraction problems to addition problems, while also building their understanding of the Inverse Properties of Addition and Multiplication. I present using inverse operations as an additional method they could use, but for students who struggle or need more time to build concepts, using the additive inverse is a nice support for concepts they have already learned.
There are several misconceptions that my 7th graders had when we learned to add and subtract positive and negative integers. One misconception was having a basic example of what a number line is and how numbers are organized on a number line. After a few of days of teaching students how to use number lines to show addition and subtraction, I discovered that students were successful in answering questions when number lines were drawn and given to them. However, if students had to construct their own number line they would get confused on how to organize numbers. For example, when students had to show 5 -7 on a number line, they might construct a number line that starts with 5 on the left side and movement of 7 spaces to the right. This showed me that they did not understand the organization of a number line of numbers decreasing as they move left and numbers increasing as they move right. In addition, it also demonstrated that students were not connecting the concept of subtraction as a movement left on a number line and addition as a movement right on a number line. To address this issue, I had students role play as a number line in the classroom. Each student was given a post it note with a number on it and they had to arrange themselves from least to greatest. The activity led to a discussion about how to organize and create number lines.
The other issue students had with adding and subtracting integers was being able to identify the change in numbers on a graph or number line. For example, students were given a problem where they needed to identify the number of degrees that the temperature changed when it went from -5°F to 35°F. Only about 35% of students were able to correctly answer 40°F when given this problem the first time. The majority of students answered 30°F. When we had a discussion about the problem I discovered that students were struggling to understand that students were getting confused when they had to read a thermometer that had to cross the 0°F marker. To help with this misconception, we practiced problems on thermometers where students had to begin by identifying the temperature change between the negative number and zero. Then they had to identify the temperature change from zero to the positive number. Then add it together. In other words, students need to learn the concept that when finding the change between a negative and positive number, it is best to find the change from the left zero and the change from the right of zero and then combine the two changes.
Probably the greatest misunderstanding my students had when learning how to add and subtract positive and negative numbers was when symbols such as addition, subtraction, positive, and negative signs need to be “changed” to solve the problem. I found students were especially confused about the rule that subtracting a negative is the same as adding a positive. I started by introducing the concept with money problems so that they were familiar, but students would still get confused because of the multiple negative symbols, such as in a problem like: -3 – (-2).
In order to help students understand a problem like -3 – (-2), I introduced two different models. First, we practiced on a number line. I had students begin by modeling -3 – (+2) on a number line and explain their reasoning. The majority of students were able to show that this problem involved moving left from -3 because we were subtracting +2. Then, I gave the students -3 – (-2) and asked them to model it on a number line and write an explanation of how they got the answer. This initially confused many of my students, but about half of students were able to model that they would have to move right of -2. Several groups said that they had to move right because if it was -3 – (+2), they would have to move left along the number line. Other students explained that they had to move right of the number line because the second negative sign told them to switch directions on the number line.
The second way I modeled problems like -3 – (-2) was to use positive and negative chips. The chips were little plastic circles that were yellow on one side for positive and red on the other side for negative. We modeled this by beginning with 3 negative chips. Then we discussed what it meant to subtract -2. Students explained that subtraction meant to take away, so students took away -2 and they were left with 1 negative chip or -1. This model was more successful than the number line model because my students could visually see the addition and subtraction of chips.
The chip model became for difficult for a problem like -3 – (-5) because students begin with -3 chips and they do not have enough negative chips to take away -5. After a brief think, pair, share on what would happen when we added sets of 1 positive and 1 negative chip, most students were able to understand that we were essentially adding zero to our chip model. So on -3 – (-5), students would begin with -3 chips. Then, they would have to add two sets of +1 and -1. Students then had 5 negative chips and 2 positive chips. Finally, they could subtract -5 and be left with an answer of +2. After introducing the chip model, I had students practice problems by using chips and then modeling on a number line. For many of my visual learners, using the chips first, helped them better understand the number line model.
In teaching how to do problems like -3 – (-2), there were also some symbolic notation misunderstandings that would confuse students. For example, we had to conceptually discuss how to tell the difference between the negative symbol and a subtraction sign. The other confusion students had was in a problem like -3 – (-2) that some students thought that the parentheses around (-2) meant multiplication. I addressed both of these issues through a brief role playing activity where students had to play a symbol or number in a problem and then tell the class what they represented. Identifying symbols on an expression has been frequently used as a bell ringer activity throughout the year to give students more practice.
Adding and subtracting integers is one of the hardest concepts for 7th grade students aside from fractions. Most students come to me with a general idea of what a negative number is, but they seem to think negatives have a separate set of “rules” and patterns from positive numbers, when in fact they are a mirror image. If we can use what kids already know about positive numbers to help them learn how to operate with negatives, they will be more likely to retain and apply their knowledge.
There are a few common errors students make when it comes to adding and subtracting integers. The first is when students add integers such as -3 + 4. Students usually never say +7, as they notice the (-), but they will give an answer of -7, as they don’t understand how to deal with opposite signs. They just add the absolute values of each number and slap on a negative sign. Negative 1 is the other wrong answer I see. This is less concerning, however, as a student in this case is just not paying attention to which number had the bigger absolute value, the negative or the positive.
The majority of confusion comes with subtraction. Because of lack of exposure to negatives in elementary school and more importantly the constant message we feed our young kids: “you CAN’T subtract a larger number from a smaller number,” students either reverse problems like 7 – 9 so they get an answer of 2, or they just say “it can’t be done”. This error seems to be fairly easily corrected when students are reminded of the existence of negative numbers, but problems such as – 4 – 3, - 4 - - 3, and – 3 - - 4, cause major problems for students who don’t have a solid mathematical foundation.
Math is all about patterns and making connections. When teaching integers I like to take what students already know about positive numbers and help them appropriately apply this to the negatives. To start off I use a card game I found called Zip, Zilch Zero (http://nctm.org/LessonDetail.aspx?id=L819">http://illuminations.nctm.org/LessonDetail.aspx?id=L819). In this game red cards are negative and black cards are positive and your goal is to lay down combinations of zeros to score. We spend one full period playing (only the addition version) and then I do a follow up mini lesson the next day.
In the mini lesson I start with problems like 3 + 7, 4 + 5, and 2 + 9. Students are asked to give an answer and show their solution on a number line, paying specific attention to the direction the problem moves them. I then move to problems like – 3 + -7, - 4 + - 5, and – 2 + -9. I ask students again to show their moves on a number line and think about what direction they are moving. Finally, I end with problems like -7 + 3, 4 + -5, and 2 + -9. We then have a quick class discussion about what kind of problems move you forward and which move you back.
Next I put up a longer expression such as – 9 + 10 + - 8 + -2 + 9 + -7 and ask the students to solve this problem using a method of their choice. Some students use zero pairs, some jump on a number line, and a few will even break apart numbers to create zero pairs; for example – 7 + 10 might become – 7 + 7 + 3. I let the students choose the method they like best, and encourage the number line approach for those who are struggling.
The subtraction problems are a lot more difficult than addition. To introduce this concept I start with another mini lesson. The first string I give the students involves 10 problems like 10 – 4, 5 -3, and 7 – 1. I have them draw these on a number line and think about what direction the problem moves them. I then given them problems like – 10 - - 4, - 5 - -3, and – 7 - - 1. Instead of teaching them the rule that a negative minus a negative is a positive, I play off the idea that if you have 10 negatives and you take away 4 negatives, you should have 6 negatives. The first one or two problems from this string make students think, but after that they jump right in and are fine. Next we move to problems like 4 – 10, 3 – 5, and 1 -7. We talk about what direction these problem move you and that when you run out of positives to subtract you cross over into the negatives. Or that when you are subtracting a positive amount you need to move back that many spaces no matter what side of zero it lands you on. Before I move to the last string that involves problems like – 4 - -10, - 3 - -5, and – 1 - -7, I ask students to predict where you might land if you are subtracting a negative number and you run out of negatives to subtract. It usually takes them a second to respond, but most will answer “in the positives.” The last string of problems I typically show on a number line and if students get stuck I go back to expressions like – 4 - -3, move to – 4 - - 4, and then onto – 4 - -5 until I work my way up to the exact problem given in the string. Once we have explored these problems I ask my students the following questions: What direction does subtracting a positive number move you? What direction does subtracting a negative move you? We then go into the last type of problems that look like – 5 – 9. We talk about how if subtracting a positive moves you backwards, what might the answer to this problem be? Once students have made a prediction I start with the following problem on a number line: 1 – 9, and then move to 0 – 9 and continue backwards till I get to – 5 – 9. We then summarize by stating that it doesn’t matter where you are on a number line, subtracting a positive moves you backward and subtracting a negative moves you forward. I then ask what else moves you forward and backward, looking for the responses of “adding a positive moves you forward and adding a negative moves you backward”.
Once we have these discussions students work on practice problems for the next few days. I encourage them to show their work on number lines or to use + and – signs, where they make or add zero pairs to help them solve. I was also given a great way to tie the forward and backward movement idea to a realistic-like story, without taking away the meaning of the math. It is called “hot and cold cubes.” Essentially you have a glass with a neutral temperature (starts at zero). If you add in a hot cube (positive) your temp/value rises, if you add in a cold cube (negative) your temp/value drops, if you take out a hot cube (positive) your temp/value drops, and if you take out a cold cube (negative) your temp/value rises. We never specifically write the many “rules” of adding and subtracting integers but it fun to see the students figuring them out on their own and talking about them with their peers. I hear them saying things like “…its -8 cause you pretty much subtract the numbers and then the answer is negative cause the negative number was the bigger one,” in response to problems like -12 + 4.
Adding and subtracting integers is always a difficult concept for my seventh grade students. For most of them, this is their first experience working with negative integers. The misconceptions do not stem from their understanding of what a negative integer is but rather, how adding and subtracting with negative integers is different than adding and subtracting with positive integers. For example, many of my students think that 5-3 would produce the same difference as 3-5. So, understanding that the order of the subtraction does make a difference is one misconception. Along with subtraction, the most common misconception I see, is that they do not understand that “taking away” an integer can be the same operation as addition as in [5 - (-3)] = 5 + 3. In their minds, if you take something away, the difference should decrease. They have a difficult time realizing that subtracting a negative integer is actually the same thing as adding a positive integer. To addres these misconceptions, I use a variety of teaching techniques and manipulative materials to enhance my lesson and their learning. I begin by teaching how to add and subtract integers with the use of a number line. Students practice moving left and right according to the operation and integer. I also make my students stand up and physically walk along a number line, this helps those students that learn kinesthetically. I also use red and black number chips. The red chips represent negative integers, and the black chips represent positive integers. By making “zero pairs” with one black chip and one red chip, students gain a deeper understanding of how integers work together. Lastly, I always use “real-life” math examples to help explain the concepts. For example, we talk about their allowance and owing someone money. This strategy usually alleviates the “taking away negative integers” problem. They understand that if you take away debt, you owe less, which is the same thing as earning or “adding” money.
Another misconception that I have with my students is the actual notation of writing positive and negative numbers. When given a number sentence like 5 + 3 - 6 + (-8), my students do not make the connection that "minus 6" would actually be combined with "negative 8." That is, they do not always understand the connection between adding a negative and subtracting a positive. They will ask questions like is that "subtract six" or a "negative six?" When I say both, they are confused. I respond by giving them real life examples like if the temperature drops six degrees, wouldn't that be the same as adding negative six or just subtracting six? Or, I connect it back to our number chips. However, the best way I have found to teach this concept is by simply using the word "combine." We talk about how the word combine means to add or put together. I then have them circle all the positive numbers, or numbers with addition symbols directly in front of them, and put squares around all of the negative numbers or numbers with subtraction symbols directly in front of them. Once they have combined the same sign numbers by addition, the combining of same sign numbers seems much simpler to them. They can then use their strategies learned with the number lines or chips. They have to know what numbers they are working with before they can begin to evaluate. This strategy is also a building block for combining like algebraic terms.
Overall, the best technique I have found that both highlights the misconceptions and corrects the misconceptions is through the use of peer teaching. I have noticed that my class is typically 50/50 in terms of their understanding. I usually have approximately half of my class that is able to grasp the concepts and half of my class that struggles. I teach full inclusion math classes. So, what I do is set up peer teaching round tables. My students that have mastered the concepts will lead a group of 2-3 students through a variety of integer problems. Not only am I reinforcing the "peer teachers" knowledge and understanding, but most of the time, my students that are struggling with the concepts are able to connect with something said or a strategy used by their peers.
Adding and subtracting integers is by far the most difficult concept for middle schoolers to grasp. Every year I have found that my students struggle with making real world connections to this concept. They find it hard to see the usefulness in adding and subtracting intergers.
To address this issue I always start with real life situations like a football game. I ask the students if on the first down their favorite running back runs ten yard, but then the second down he lost 3 yards what is the total amount of yardage for the two downs. Most of my male students (and some of my female students) really seem to understand and can connect with this real life application. I also use a problem with their allowance (money) and a majority of students then can solve some of the problems.
I find that using a number line and counters do work initially because of its concrete nature, but students don't always readily remember to use a number line or counters to solve these problems.
Lastly, when my students can see the connection between their own life and using integers I give them these rules:
If signs are the same add and take the sign of the larger number
if signs are different, subtract and take the sign of the larger number
This has helped students who really enjoy a short cut.
One major misconception that students have is that when you add the answer will always be bigger and when you subtract your answer is always going to be smaller so the bigger number always goes on top. This is only true if you’re working with whole numbers. I understand why this may be said in younger grades because they’re only working within the whole number system but this isn’t a true statement when you get into different number systems such as integers. I know I might perpetuate misconceptions myself when I say you can’t take a square root of a negative number. BUT this is only true in the real number system. When I say something like this I will try to add you can’t take the square root of a negative number in the real number system.
I’m sure our students feel like the rules change all the time in math and I can understand why students become frustrated, disengaged and disenfranchised. But the rules change because of the different numbers systems. One way we can first address the misconceptions is by addressing these rule changes. Perhaps we can compare the progression of baseball from T-ball to the major league to computation in the different number systems. Just like rules change in baseball the older you get so do the rules of computation.
One method I have used in the past when teaching addition and subtraction of integers is using the imagery of light and dark. I have a lab sheet that shows big and little squares being added to and subtracted from each other. Each box represents either light or dark. For instance if small square and big square is being added to each other this could represent a lot of dark is added to a little light . Then we discuss what the result would be. In this case it would be less dark. As we go through this lab students use color pencils to fill in the squares with appropriate colored pencils. We go through all the possible situations of different sizes of light and dark that could be added to each other and record the result on the lab sheet. Then the connection is made that light represents positive numbers and dark represents negative numbers. Similarly, we do this for subtraction as well. If a lot of light is subtracted from a little light the net result is a little dark. We match up addition and subtraction problems that yielded the same results from our light/dark lab. From here the relationship between integer subtraction and addition can be made.
This lab gives an example without numbers of how you can add and end up with a smaller result and subtract and end up with a bigger result because of opposites . The rules change with integers because integers are opposites.
One of the biggest misconceptions that I found with operations with integers is the idea that there is a set rule for adding and subtracting positive and negative numbers similar to the rules of multiplying and dividing integers. Students will often add the two absolute values and then give a sign to the answer based on whether there are two integers with the same sign or one positive and one negative integer. I have addressed this with my students in a couple of ways.
First, students need a strong understanding of the number line and the significance of opposites. As students look at combining opposites they then look at strings of problems related to those opposites and how the changes in the two addends affect the sum. For example: -3 + 3= 0, -4 + 3= -1, -3 + 4 = 1, -5 + 3= -2, etc. Helping students identify patterns though the work of strings is a powerful way to help them internalize the concept of adding and subtracting integers.
Second, we discuss the power of being able to change any subtraction problem to an addition problem. It is often easier for kids (and humans in general) to think about combining quantities rather than look for the difference between the two, so students explore how changing problems to addition can be useful. This is also a place where the discussion of the commutative property can come in to play. While we’re changing problems to be what we want (addition) would it be helpful to switch the order of the addends?
There are many misconceptions when it comes to dealing with positive and negative numbers. Firstly, I think at the lower school level, students are not as exposed to the possibility of subtracting a bigger number from a smaller number. Although we are trying to cater to teaching students on how to understand the concept of subtraction and calculate "5 minus 3", I feel that we are sometimes guilty of dismissing students when they ask, however accidentally, how to calculate "3 minus 5" and are told, "no, it's 5 minus 3. 3 minus 5 is impossible."
Usually, when my students get to me, they’ve seen addition and subtraction on a number line. I do review that and talk about real-world situations such as temperature, yards lost or gained, and debits and credits. I usually teach addition and subtraction of integers with two-color counters. It’s important that students can move from the concrete to the abstract so I use two-color counters to help them develop the rules themselves. Students often have trouble understanding that adding two negatives produces a negative. I find that addition goes pretty smoothly, subtraction less so, but once they learn multiplication and division of integers, they start to confuse things. They get it in their heads that a negative and a negative makes a positive because of multiplication and division, so they start to lose their understanding. This is why I use two-color counters.
We model the problems, and talk about the concept of zero pairs. We talk about how if you give something and then take it back, the person is left with nothing. This helps drive home the concept of the zero pair. Then we use two color counters to model problems, look for patterns, and develop rules. At this stage I give a variety of problems, and we discuss the relationships and patterns and then rewrite like problems in the same group to draw conclusions. When we’re ready for subtraction, I have a series of exploration problems to help students discover the rules. I draw pictures of sample problems and make groups of the same type of problem. For example, I will give them a set of problems all involving the subtraction of a larger number from a smaller number. I make groups of each type of problem, and have the students use counters (I’ve also used “eggs” which are ovals cut in half with one being positive and one being negative- when you put them together, they look like a zero). Once the students have solved groups of like problems, we look for patterns.
After discussion of the patterns, we develop rules for subtracting integers. Then, I give them equivalent expressions such as 6-9 and 6+(-9) which they can solve with the counters or by applying the rules we’ve developed. We check them with a calculator before moving on to a discussion of what they’ve noticed. From there, I ask them questions about the equivalent expressions to help them develop the concept that subtraction is the same thing as adding the opposite. I find that special attention needs to be paid to the concept of subtracting a negative.
Once students understand that subtraction is the same thing as adding the opposite, they seem to have no trouble rewriting problems involving subtracting a positive number, but they struggle with subtracting a negative number because it is difficult to grasp that this becomes addition of a positive, especially if you’re subtracting from a positive. I do two things to help students continue to understand this concept. I refer back to money examples because students seem to be able to relate well to money, but I also go back to drawings. We’ve usually moved away from the counters, but for students who are still struggling, I encourage them to use plus and minus signs to help them recreate the concrete color counters. I also teach the strategy of solving a simpler problem if they forget how it works. If they need to add or subtract integers that are too large to represent with counters or pictures, I tell them to use two smaller numbers to solve the same type of problem so they can remind themselves of the rules.
I like to help students understand that signs are attached to the number so I don’t require that students always change subtraction to adding the opposite. I encourage them to use a strategy that matches their cognitive level. The reason I don’t force changing subtraction to adding the opposite is that when they start learning algebra and concepts like combining like terms, chances are good that they won’t write the changes. They need the conceptual understanding that they are the same, but they don’t need to make the written changes. When we talk about using the commutative property to simplify expressions, I like to help them separate the terms using the addition and subtraction signs and rearrange the terms with their signs. They need that conceptual understanding, but I work with the students individually or in small groups to determine what they’re ready for. Students with a deeper understanding don’t need to be forced to rewrite subtraction as addition of the opposite if they understand the concept. For students who are not ready for that, I encourage them to change subtraction to the equivalent addition expression. When we need refreshers, I go back to a combination of real-world and concrete representations.
Integer operations are always a sticky subject for middle schoolers. Personally, I find the number line to be the best model to represent this concept. It is exciting to experience students bewilderment when they are first exposed to the fact that the number line extends, forever, in many different directions. For the addition and subtraction of integers, we will stick to the vertical and horizontal number lines.
Before adding and subtracting integers, students must first become comfortable with moving along the number line. Students must practice starting with one integer, and moving a designated number of spaces to end at a second integer. From here, students can practice the concept that you increase and decrease value as you move along the number line. Often, students find that the vertical number line is an, for a lack of a better word, easier way to conceptualize the increase and decrease of value.
From here, students can practice finding the distance between numbers. Students can count the units between two integers, to discover that there is a distance (difference) between them (this practice can also prove opposite integers and absolute value). A favorite of my students is to use temperature contexts. Students can find the “difference” (distance between integers) between two given temperatures.
When explicitly teaching the addition and subtraction of integers on the number line, one must pay attention to the language used. Students need a context to visualize the movement and change happening amongst the integers.
For addition, “adding more negative” can be communicated as adding something “bad” or “negative” to a situation. If you are adding negative to a situation, then everything must become “more negative”. This directs students to move left/down toward the negative end of the number line. For example, 2 + -3, the student must start at 2, and add 3 negative units to the left/down.
For subtraction, “taking away positivity” could be communicated in the same way as adding a negative. You are making a situation more negative or sad. Therefore you need to more left/down on the number line. “Taking away negativity” could be communicated as “taking away sadness or something bad”. This must mean that you are getting more positive and you must move right/up on the number line toward the positive direction.
When being mindful of you language, and using a familiar model, students build confidence to operate with integers. Eventually, rules can be pulled from these situations, and students can build more efficient methods to add and subtract integers.
One misconception students have when learning how to add or subtract integers is that they many times fail to comprehend that subtraction is defined in terms of addition . In other words a – b = a + (-b). They don’t understand that subtracting 3, for example, is equivalent to adding -3. I find that when students “bottle up” rules for adding and subtracting sign numbers, they tend to confuse operations like 2 – 5 because they see the 3 as being positive and separate from the minus sign, so the think, positive 2 and positive 3….add and keep the sign….and they write – 5.
They also confuse operations like –12 – 4, for example. Here they say the answer is –8. Again, they follow a memorized “rule”, they see the 4 as being positive, they subtract and keep the negative sign (from the 12) has, so they answer –8.
Yet another mistake made frequently is when given -5 + 7. Here they say the answer is -12. They confuse the multiplication rule (if signs are different, product is negative), they add 5 and 7……so they write -12.
In order to resolve this issue I first make sure that students understand a basic property, the property of opposites, a + -a = 0.
I make certain all students comprehend this using numbers. I then show with the overhead and manipulatives that an open circle or ring represents a negative, and that a full circle represents a positive. So this means that an open circle (-1) cancels out a full circle (+1) and vice versa. Two open circles cancel out two full circles and so forth.
I then give simple exercises like -3 + 5. I ask them to place the corresponding circles at their desks and they usually quickly see what the answer is when they start canceling. I ask them to represent the circles result with a value, in this case +2.
Then I show them on the overhead. I do this with different operations till they get good at it. I also indicate that 3 – 5 can be re-written as 3 + (-5), which means 3 full circles with 5 open circles, which leaves 2 open circles, or -2, after canceling.
● ● ● each = a positive
○ ○ ○ ○ ○ each = a negative After canceling, answer is – 2
I also indicate that 5 – (-2) means that we are canceling two open circles by replacing them for two full circles. This leaves us with 7 circles, or positive 7.
The next step obviously is to see if they understand what to do with larger numbers like -50 + 58. By this time they’ve figured that 50 open circles will cancel with 50 of the 58 full circles, leaving 8 full circles.
Throughout the years, I’ve tried many things, the number line, thermometers, profit and dept, and so far this has been most effective for me.
What I've found is that students rely too hevily on their previous knowledge of how addition and subtraction "works". A great example is that when students are subtracting, they often want to put the bigger number first, and subtract the smaller number from it. As we know, this practice can lead to mistakes if it is not corrected.
Because of their reliance on previous knowledge, often times the students answer the problems too quickly, before they have thought it through. I like to provide my students with multiple contexts through which to conceptualize the addition or subtraction problems. When the students take their time and use the context, they more often than not arrive at the correct solution. However, when assessment time rolls around, they "fly through" and get questions wrong because they don't both to think of the problem in context.
I've found that the best way to combat all of this is to require students to WRITE a description of how they are solving the problem, before they provide the answer. This forces them to slow down and use the techniques that I have provided them with, and it leads to much better results.
My students are typically high achievers, so the mistakes they make in combining integers sometimes surprise me. Of course they can add positive numbers fluently, but occasionally I have a student who, given the problem 529 -357, for example, will give the answer 232, not understanding the need for regrouping to subtract the tens, or considering the absolute rather than the numeric values when subtracting the digits in the tens place. To address the misconception of this student, I use base ten blocks. When a student sees in the model that there are not enough tens to remove five, they understand the need to borrow (or “make change for a hundred”) in order to subtract the required number of tens. Even so, one student I had still insisted that the base ten blocks didn’t prove him wrong. He seemed to understand better, however, when I told him that if he subtracted the tens digit in the minuend from the tens digit in the subtrahend, he created a hole the size of 3 tens that had to be filled from the hundreds column, and that after that hole was filled, there were still 7 tens left over from the hundred he borrowed. Those tens remained in the tens place of the answer. Happily, he stopped making that kind of arithmetic error once he understood.
The other common errors my students make in combining integers all involve negative numbers. Most students come to understand that adding two negative numbers gives you a larger negative value, but some have trouble with that. A few forget how to choose the sign of the final answer (or even whether to add or subtract) when combining integers with different signs. The most common mistake by far is misunderstanding that subtracting a negative is the same as adding the same positive numeric value. I have used the number line, in the past, to address this problem, but most of my students didn't find it helpful. Over the past few years, I have found that two-colored counters are helpful with all of these errors, mainly because they so handily demonstrate and allow students to apply the idea of the zero pair. Using the yellow side to represent +1 and the red side to represent -1, the students and I can model the addition and subtraction of negative integers. When adding integers of the same sign, the students can use counters to quickly see the sum. If they are adding integers with different signs, they only need to place the correct number of counters to represent their positive and negative addends, then remove the zero pairs until there are none left. What remains of the counters will give the answer to the question, and with enough practice will help students develop conceptual understanding. Using counters is especially effective in showing how (if not why) subtracting a negative is the same as adding the positive; given an expression, I add enough zero pairs (a red-facing counter and a yellow-facing counter) until I am able to subtract the negative number I need. What remains on the desk afterward is proof that subtracting a negative results in an increased overall value of the expression.
If students still don’t understand the math after use of counters, I use an analogy to the classroom. We create two imaginary students (Wonderful William and Wacky Waldo). William is always positive; he works hard and keeps everyone around him on task, so when William is in class our productivity level (overall value) increases. Waldo is negative; always disruptive, he talks nonstop and keeps everyone around him from progressing, so his presence in class makes the productivity level go down. William and Waldo are equally strong, so when both are in class, the productivity level is unchanged. However, if Waldo is absent, his negativity is removed (subtracting a negative) and productivity increased again. This parallel helped several students to understand combining integers better. Of course, we use other visuals to help the students, such as counting on a number line according to the sign. We also use a poster of magnetic balloons with sandbags attached, the students come to know that positives (blasts of helium) cause the balloon to rise and additional sandbags cause it to sink. They also understand that dropping a sandbag will also cause the balloon to rise (removing a negative value increases positivity). Students that make more progress conceptually from the allegory are invited to write a brief paragraph about other "negatives" they can remove from their lives to make them more positive. I also review the concept of positive and negative numbers by watching an excerpt from the movie Stand and Deliver that shows Jaime Escalante's brief comparison of positive numbers to the sand you dig up while playing with sand at the beach, and the negative numbers beign the hole you create while digging. More than anything, however, the most beneficial and irreplaceable strategy for combining integers, in my experience, has always been practice.
One of the major topics that students have trouble with is adding and subtracting integers. Some issues or misconceptions I have found when teaching addition and subtraction of positive and negative integers are: 1) students are confused on how to use the number line to find the answer; 2) students have trouble understanding a positive and negative cancel one another; 3) students are confused if the minus sign represents the subtraction operation or a negative sign; and 4)students have trouble applying the integer rules for adding and subtracting integers – for example, they have trouble on when to add or subtract the absolute values of the integers.
To help overcome these misconceptions, when teaching adding and subtracting integers, I would start off using an interactive number line and demonstrate how to move on a number line to represent a problem. Then I have students come to the board to model how to solve other problems using the interactive number line. At the same time, I would also show the students how to draw visual pictures of positive and negatives to represent the same problems. This way, students can see they get the same answer, both from the number line and from their visual picture. These visual pictures, would also help clarify why a positive and negative cancel one another. To help clarify if a minus sign is the subtraction operation versus a negative sign, I would tell my students to read the problem from left to right and keep in mind they always need an operation between numbers, so if it doesn’t make sense to say 6 negative 3, then it must be the operation 6 minus 3.
To ensure a better understanding of how to add and subtract integers, I would use manipulatives such as a red/yellow counter chips to demonstrate the meaning of positives and negatives and how to use the chips to model examples. I would incorporate games into my lesson, in which students would gain a better understanding of the integer rules for addition and subtraction, and at the same time have fun playing the games.
While many students have trouble with adding and subtracting integers, I have found that by incorporating different teaching styles or methods, they have gained a better understanding of how to apply the rules for these operations. Although the manipulatives helped them initially grasp the topic, over time students were able to lessen their dependence on such methods, and apply the rules for adding and subtracting integers.
Students are introduced to integers prior to seventh grade and often overgeneralize one set of integer rules to all operations. When beginning this unit, I begin by trying to assess the student’s current misconceptions. Most commonly, I have observed students using the multiplication and division integer rules for all integer operations. They believe that any operation involving two negatives will always yield a positive solution or any operation involving a positive and a negative will always yield negative solution. In order for students to begin correcting these misconceptions, they first need to realize there is a misconception.
Students begin by responding with “always, sometimes, or never” to the following four statements:
1) Two positives equal a negative
2) Two negatives equal a positive
3) A positive and a negative equal a negative
4) Two positives equal a positive
Next, I provide several contextual situations which indirectly addresses the most common misconceptions. For example, the temperature outside was -2 degrees F at 8:00 a.m. If the temperature dropped a total of 3 degrees over the next 5 hours, what was the temperature at 1:00 p.m.? A submarine was 200 feet below sea level and ascended 45 feet. What is the submarine’s new elevation?
I then have students come to the board and work out the two problems. Class discourse is used to ensure all students agree on and understand the correct responses to each question. After this, we look back at the original four conjectures. Again we engage in class discourse to ensure everyone agrees and understands why each statement is always, sometimes, or never. Counterexamples must be provided by students when sometimes or never are chosen. During this activity, students will bring up the multiplication and division rules. Since the goal is to address the misconceptions they have, which often include the rules for these operations, this is acceptable. This activity is used as an introduction to integer operations.
In following lessons, when addition and subtraction of integers is the focus, I begin by making sure students understand “zero pairs” and “combining terms.” Note, I am not differentiating between adding and subtracting integers, but instead putting the focus on “combining terms.” Again, I begin this with a contextual example: A Hydrogen atom has one proton(+) and one electron(-), represent it’s charge with an integer. During this phase, we will use counters as a manipulative to help establish a visual as we move from concrete to abstract.
The next issue I put out there for students is 2 symbols beside each other. Students often stop in their tracks when dealing with simple integer problems that have 2 symbols beside each other (i.e. 3 – (-2) or 3 + -2). For this type of problem, I instruct students to simplify the symbols into one symbol. We connect this to double negatives in English class and thinking of the negative sign as “the opposite of.” By combining the symbols, students feel less overwhelmed and this allows them to look at simple integer problems through the lenses of combing terms. Students can begin to think of 3 + -2 as 3 and -2. They can visualize 3 positives and 2 negatives with the 2 positives and 2 negatives creating zero pairs and the 1 positive remaining. This type of reasoning makes sense to students because they can visualize it.
The Essential Understanding for adding and subtracting integers are as follows:
1) the symbol directly in front of a term is part of the term ( ex. 3 – 2 can be interpreted as 3 and -2)
2) when combining terms, an equal number of positives and negatives create zero pairs
3) when two symbols are beside each other, simplify to one symbol (once students move into more complex problems where the distributive property is involved, they must know to distribute before simplifying to one symbol)
Since operations with integers in one of the fluency expectations for 7th grade within the Model Content Frameworks for Common Core, I also feel this is one area where substantial drill and practice is appropriate. I meet the expectation for fluency by including integer operations as one of the components on our daily opener. The questions vary on a daily basis. In the beginning of the year, the questions are simple computation, but as the year progresses, integer operations with the context of an equation or expression are included. This spiraling review enables students multiple entry points for the concept and multiple opportunities for mastery and fluency.
In collaborating with middle school and high school math teachers across my school's network, it is clear that comfort and fluency with integer operations is severely lacking. I believe this is due to a few factors. First and foremost, when teaching integer operations, students need to make meaning of a variety of scenarios involving positive and negative values rather than memorizing rules about the outcome of operations with signed numbers. Since the concept of a value that is less than zero is almost as vague as the concept of values between zero and one (fractions and decimals), students need to be provided with a variety of tools to fully grasp what happens when scenarios involve negatives.
Over the course of several years of teaching middle school math, I have observed a number of misconceptions about combining integers. Here are a few of the most common areas of confusion:
When introducing integers, students occasionally fail to recognize that negative numbers are a part of the set of integers and/or students believe that fractions and decimals are integers. Students may believe that 3 and (+3) are different numbers and that -3 and (-3) are different numbers because of the parentheses.
Without fully understanding that a negative value is less than a positive value, there will likely be confusion when adding and subtracting integers. This is especially the case when student rely on estimation prior to solving to determine the reasonableness of their solution. Students tend to generalize the idea that a “bigger” number is always greater when thinking about negative numbers, instead of thinking that numbers are smaller, the further to the left they are on the number line, for instance, –15 < 13 vs. 13 > 15.
When combining terms in an expression students fail to circle the addition or subtraction sign in front of the number, because they believe that these only act as operation signs when in reality, they signal whether the number that follows is a positive value or a negative value.
When teaching students to "add" a negative integer to a term such as 3 + (-8), students may indiscriminately add whenever they see a plus sign because they fail to understand that a negative sign means to do the opposite.
When students learn that subtracting a negative value becomes additon, I have seen students believe that a double negative is just a bigger subtraction sign and fail to recognize that the double negative tells you to do the opposite twice, which is why a double negative turns into a positive. It is important to make the language of "negative" is the same as "the opposite" present within integer discussions.
Lastly, I have seen students undervalue the importance of signs. Every term or value has a sign! Students may solve -8 + 3 and get 5, leaving out the negative sign because they might think it is unimportant. Students sometimes fail to understand that 5 and -5 are completely different numbers and it isn’t just a matter of “forgetting the negative sign.” Over time this also becomes an issue of carelessness.
In order to solidify the concepts of integers and their opposites, and particularly adding and subtracting integers, I truly believe that students must exclusively use number lines, manipulatives, models and diagrams until they are able to explain orally and in writing the WHY behind the results of combining integers. Investigations to prove that properties of real numbers (identity, zero, inverse, commutative) hold true for all rational numbers are also helpful when proving integer addition and subtraction outcomes. Thorough explanations must be used with consistency and regularity in whole class discussions, think-pair-shares, summarizers and in writing. Lastly, including integer computation as well as adding and subtracting integers in the form of a scenario is a topic that is spiralled in Do Nows and on homework assignments 4-6 times per week in an effort to build comfort, fluency and lead my students toward success when faced with negative decimals and fractions as well - not because of luck, but because they deeply understand that they have been working with positive (addition) and negative (subtraction) terms for as long as they can remember!
When first working with integers, I find that students have a difficult time determining whether the sign of a sum or difference will be positive or negative. This is especially problematic if they are taught a procedural method early on. I have seen students have no problem initially understanding that the sum of two negative integers is negative, but then after learning the rule that the product of two negative integers is positive, they confuse this with sums. They will insist that the sum of two integers is also positive. To mitigate the negative impact of these misconceptions, my approach is to avoid teaching a procedure for adding and subtracting integers; I only teach the students how to use models. It is my intention to give each student enough time to develop their own method, even if their fellow classmates have already found a procedure that works.
When first introducing the subject, I will spend four days on these topics. One day for integer sums on a number line and another day for integer sums using counters. The idea of the zero pair is crucial for working with counters. I then spend a day on subtracting integers on the number line and another day on subtracting integers using counters. During these lessons, I ask questions about the absolute values of the addends, the minuend, and the subtrahend. I ask them whether they think a sum or difference will be positive or negative and why. I want students to find a rule that works for them. It is pretty amazing to see students discover how to subtract differences like -5 - (-2) without using the additive inverse! I do however develop problems and questions so that students can see the equivalence of a difference and a sum using the additive inverse property.
Once students have a grasp of sums and differences I can then teach products and quotients by first talking about a product as repeated addition. Students will see that 2 * (-3) equals -3 + -3 or -6. We can then use the commutative property of multiplication to know that -3 * 2 also equals -6. Once these concepts have been established, I can bring in multiplication and division fact families to discover the signs of quotients. All of these concepts are rooted in a firm understanding of sums and differences.
After teaching an integer unit, students will start to see integer problems show up on a daily review. At my school, we have developed a daily review component of class that takes the place of a "do now." Students work on up to six review problems of different types. Here are a few topics that we are currently reviewing: 1) products of fractions in word problems; 2) dividing two decimal values; 3) integer sums and differences; 4) solving 2-step equations & inequalities with fractional coefficients; 5) evaluating algebraic expressions; 6) placing rational numbers on a number line. I assess these weekly to see if the students need more practice on a problem type. Generally, I wait until well over 80% of the students have mastered a problem type before removing it from the daily review. That way I am left with a much smaller set of students with which to intervene. This gives students who are still developing an understanding of integer operations more time to develop solutions. I only discuss the models to help struggling students. As the numbers become "larger,” I encourage students to use more abstract models. I teach them to represent large numbers in multiples of +5, +10, +20, +50, +100, and so on as needed (zero pairs can be made from these too) so they can solve problems like -158 + 78 without having to draw 158 negative counters and 78 positive counters. I also show students how to use the number line to help solve problems without having to draw each consecutive integer on the number line. Either way, I am constantly asking them to explain to me whether a sum will be greater than or less than the first addend, or if a difference will be greater than or less than the minuend. Once students have firmly constructed their own methods for working with integer sums and differences they will be able to reference the standard procedures - yet, they may never need to.
Heather, Gary, Katie, and Tara all hit great points with their reflections. Gary is right on target when he explains that using real life situations diminishes many misconceptions. Some misconceptions that I have seen with integers are, students thinking integers are only negative numbers. Students must realize integers are negative and positive including zero. I use a number line a lot to get the students to realize the value of integers first. I say the farther left from zero the less the number is in value. This helps with comparing integers. Some other misconceptions I have found that students have with negative numbers is that negative numbers, not speaking in terms of positive and negative integers, but negative numbers, do not just include integers less than zero. Students need to know that proper fractions and mixed numbers can be negative as well. Students aslo need to see rational numbers in many different ways for example, -10/5 is aslo -2.
When we are doing adding and subtracting of integers we completely throw out all rules! This is crucial. I created a cool strategy for subtraction of integers, but we work on addition of integers by simply using money in any situation. Negative integers represents money that I owe, always. Positive integers represents money that I have, always. No matter the equation your students will get an "a ha" moment doing it this way. -5 + 3 ( If I have three dollars and I owe 5 dollars, after I give what I have, what is the end result? I still owe 2 dollars. (-2). 5 + (-3) If I have 5 dollars and owe 3 dollars, after I pay what I owe, what is the end result? I have 2 dollars left. My students truly grasp addition of integers with this strategy. We also use the number line and zero pairs, however this is most effective. With subtraction of integers we use the K.F.C method. K.F.C is a well known chicken fast food place here in Ohio so that is the hook that keeps them reminded. K.F.C stands for Keep it, Flip it, Change it. You will keep the first number the same, flip the subtraction to addition, and change the second number to it's opposite. Once you have converted it to an addition problem you will use the money strategy. Students even write the K.F.C over each part of the equation to remind them what to do. This is highly effective. As far as multiplication and division we K.I.S!! Keep it Simple!! Students are told like signs equal positive answers, and unlike signs equal negative answers. The most common mistake made is when they are multiplying or dividing two negative integers and their product or quotient will be negative. I reinforce like and unlike. Accurate feedback on all student work is essential.
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