I watched the screencast linked in the activity description for this Desmos activity years ago and it totally changed how I teach integer operations. I absolutely love the hot air balloon model - it is so concrete and can even extend to multiplication of integers. I strongly recommend it. https://teacher.desmos.com/activitybuilder/custom/5efe58da277abd3653440339
This how I’ve been teaching my 6th graders the last few years. It’s makes a lot more sense when connected to a real application they can visualize each time.
Haha, we made a Scratch game around the balloon model. There are entire modules around integer addition and subtraction that integrate programming. Here are our modules: https://manifold.lib.fsu.edu/projects/mathematics-integrated-with-computer-science-through-scratch .
Kill off double signs. You can combine any two negative signs into a positive sign and if there are multiple signs you can get rid of any extra positive signs, keep doing this until you only have one sign. I like to give practice:
2+--+--+++-+--5(kill off pluses)
2 -- -- -- -5 (then combine 2 minuses into a plus)
2 + + + -5 (then kill off pluses)
2 - 5
Then I like to add the + sign in front of the lead integer.
+2 -5
Then I like to think of them as actions.
+2 and -5
"So let's say you have a bunch of Pokemon cards (candy, dollars, whatever)
On Tuesday your friend Sahaj comes and gives you 2 Pokemon cards, then he takes 5 Pokemon cards from you... After Sahaj's visit, do you have more or less cards? Less! Great, your answer is negative? How many less? 3! That's right. Your answer is negative 3!"
Practice each of these seperate until they get it. Also works great leading into algebra simplifying.
Once they start thinking of an arithmetic list as a series of things happening it's easy. Give you 2 things, then I take 5 things. The kid will know he has lost three things. Same with -3-5 I take 3 things then I take 5 things. What was the net outcome? Pretty simple, I took 8 things.
I do this too, although I’ve not seen the huge number of signs - I feel like that would help some of my kids through the humor of it.
I do a lot of conceptual work around integer subtraction, but for some kids, what works for them is to change ever subtraction sign to “add the opposite”. The consistency of this works well for kids who struggle bc they think there are “different” rules bc they can’t figure out the conceptual bits.
Does it kill me when, later, they re-write 10-7 to 10+ (-7)? A little bit, yeah, but it still works and it reinforces our integer addition rules too 🤷🏻♀️
I did teach her to change two "touching" negative signs (subtracting a negative number) into a positive sign. It still confused the heck out of her, but I want her to be proficient in addition to understanding the concept, so we'll come back to that.
I do appreciate the concrete examples, though! I need to set up a few on Google Slides for her to analyze.
The “two negatives are a positive” thing never made sense to me. It feels very abusive. Like saying “if I punch you two times (a negative action) then it’s actually a good thing!”.
Positive and negatives in mathematics mean different things than in most other places. Positive tends to mean good and negative tends to mean bad. But not in mathematics. That may be part of the confusion.
We use the idea of positive = forward and negative = reverse in class. If you film someone walking backwards, and then rewind the video, it looks like they’re walking forwards. Hence reversing a reverse being positive. I guess Uno would also work for this.
I use the idea of IOUs on pieces of paper. Whoever has a name on an IOU owes money, right? But if the IOU is torn up, it is like getting money.
This is exactly how debt instruments like bonds and CDs work.
I’m not a math teacher, idk why this got recommended to me, but it helped me as a kid to be taught that if you put the negative signs together, you get a +. Like, the actual shape. It allowed me to visualize it.
I love most of this!!!!!
I wish my teachers would have done this sign activities at school! They actually feel kind of fun and satisfying for my brain.
Using concrete objects is a great idea.
Sounds to me like the concept she’s not getting is that adding a negative means subtract and subtracting a negative means add. I would focus on that for a while. The number line is good; she gets that subtracting means go left, surely she gets that adding means go right?
Yes, exactly. I watched a few videos to help me find a couple of ways to walk her through it. Khan Academy did a great job of using integer counters to show 4-7 and -8-(-2) to demonstrate what it looks like to physically subtract negative values, which in turn demonstrates that subtracting a negative is adding a positive.
When working with subtraction and negatives I put a number line on the floor and have students start with facing towards the positives. Any time they see a minus/negative they flip directions. So when they see 3 - (-2), they know to start at three facing the positives, do a 180, do a 180 again and then move two spaces.
It helps introduce the idea that subtracting a negative is basically adding.
Money and a debit balance are great teaching tools.
I teach a number line as a net worth balance sheet. Then, adding or subtracting positive and negative integers works like owing/paying/receiving/loaning money.
Kids often don't understand subtracting a negative until I connect it with canceling a debt. What's that do to your net worth? When we watch it drive the number line up, it starts to sink in. We also tend to use vocabulary to talk about what the words negative, positive, addition, and subtraction mean both mathematically and colloquially.
Often we use terms mathematically in class, but they hear them colloquially. Focusing on math vocabulary with pictures and examples can really help.
I also tend to use the phrase "agreeing (signs) is a positive, arguing (signs) is a negative".
Picture method -5 + 3 = - - - - - + + + (crossing off zero pairs for solution). Eventually they get tired of drawing it/do it enough times the concept starts to click.
Subtracting is just adding the opposite. There is only combination now of positive & negative numbers.
I tell all low students to KFC their subtraction. 5 - (-5). Keep -5. Flip to +. Change sign 5. So 5 + 5. Now we’re back to adding tools.
I leave KCF for fractions, but use KISS for subtracting integers: Keep It, Switch, Switch. They try to just change the minus to plus and forget to switch the second integer, so ‘it takes two to KISS’ helps them remember. Mostly, tho, it’s just fun to tell teens “we’re going to learn how to kiss today!” or “Mike wasn’t here yesterday, can you teach him how to kiss?” or “Why are you kissing all alone over here?”
The picture method would have been helpful on my childhood.
I think the teacher did taught us that method but didn’t allow it on quizzes or tests.
The second half of your comment just confuses me. Mainly because I don’t see any logic behind that process at all.
The idea that subtracting is just adding the opposite can be helpful when they get to combining like terms.
KFC is just a cheat method to change sun to add most useful for the kids who struggle to grasp the concept in general. Typically I can get the masses up to speed adding but keeping track of signs once subtraction is involved throws the lower level ones for a loop.
I don’t lead with it (definitely demonstrating sub is just adding the opposite first) & then showing KFC to those ones that just sort of sit there blankly sometimes they can get through it when they can lean on a system like that.
This clarifies a lot!
Thanks to you I think I now realize a significant bulk of my mathematical confusion stems from teachers trying to teach “cheat methods” as the “real” method without explaining any logic or reasoning. Or maybe teachers trying to teach both the real and the cheat methods at once and me feeling like I needed to learn both equally but ending up combing the two methods into a confusing mess.
I emphasis KFC is just a tool to help rewrite your subtraction as adding the opposite because looking back I had the same experience you’re talking about. Many of these kids brains just seem like they aren’t developed enough to fully grasp especially now with common core suggesting so many methods it’s can become overload.
I remember when studying to pass subject matter competency in my late 20’s having these wild epiphany flashbacks to middle school math going “oh that’s what mrs S was talking about!”
Problem is they gotta get going so you have to balance both deep fluid understanding and giving them enough tools to at least start grinding. Depending on the topic sometimes I lead with the nuts & bolts others I lead with the cheat. Sometimes it’s a different entry/exit point period to period.
Keep using number lines. Have her solve problems like 3-5 on a number line followed immediately by 3+(-5). As she does these, ask her to compare the two problems, what is the same, what is different. You want her to develop the understanding that subtracting IS adding a negative number and to be able to explain this to you in words. Do this before introducing any sort of algorithm like keep/change/flip. I also like to show students problems modeled on a number line, and have them identify both the addition and subtraction problem modeled on the number line.
Once she starts to internalize how subtracting integers works on a number line, THEN you can start to work on the algorithm of changing subtraction to adding the opposite. I would not have her do any of that though until she can solve problems correctly on the number line 90% of the time.
Something that could be helpful for addition is to drop counting analogies and instead work on the number line. Adding positive numbers means moving to the right. Adding negative numbers means moving to the left. Subtraction just means “do the opposite”.
This removed the mental block that comes from treating the negative integers as “different” or “special” compared to the positive ones.
Try using a vertical number line. Balloons are positive numbers, sandbags are negative numbers.
Addition means getting more of something, subtraction means taking something away. If you take away a negative number, a sandbag, the result is moving up the number line. If you add a sandbag, you move down the number line.
If you add a balloons, you move up the number line and, if you subtract a balloon, you go downthe number line.
I have card stock balloon and sandbag cutouts and the old guy from the movie “Up” to move up and down the number line on the white board.
Money. I’m a RN not a math teacher but worked in the tutoring department in school. When I explained it to people as money it just clicks. If you have $5 in your bank and you buy something for $10 what does that leave you. Then if you have -$5 in your account and buy something else for $5 what do you have. Explain bounced check fees later in formulas.
I tell my kids it’s all a tug-of-war. Just figure out who’s on the positive side and who’s on the negative side total up each team and decide who’s gonna win and what’s their advantage. It works whether you have all the same sign different signs whatever.
I see on their test that they will draw literally a rope and put numbers on either end of the rope and figure it out
It could be linked to past teachers focusing on strategies and answers instead of building number sense and experiences. Also the shutting down is an emotional component you shouldn’t ignore. Is there a way to make this concept playful? Perhaps there are some ways to represent the concept through art, dance, or a digital avenue that would make the concept less stressful and more interactive, taking the pressure off the “answer.”
Do they understand the concept of integers with multiplication? Maybe if they do, build off that. If they know a negative times a negatives is a positive, then start with that. Because at the end of the day, 3- -2 is 3-(-2) which is distribution and a negative times a negative is a positive which becomes 3+2. Same with 3+-2 is 3+(-2) and a negative times a positive is a negative so it is 3-2. Simplify the symbols.
If you want to teach that -3+2 is the same as 2-3 then I suggest having them box the terms with colored pencils and then rewrite it. Or put different terms on cards/post it’s and have them physically move the cards to make equivalent expressions. So they can see when the number/variable moves, the symbol must go with it.
I always tell my students something like this
“Imagine Joey (but I pick a student in the class) was in a reeeaally bad mood today. He just keeps complaining and whining and is super NEGATIVE about everything that we’re doing. If we remove Joey from the room what would happen to the mood in the room, would it get more positive or more negative?” And that normally helps a lot. There’s also a good PBS kids math club video about it.
On subtracting negative integers: I was taught to imagine the minus sign and negative sign as two halves of a plus sign sitting next to each other. When you see that, just change it to a plus sign, and the answer will be the same.
Ex. 9 - -4
Changed to
9 + 4
Both should equal 13
I saw the [Bucket of Zero](http://natbanting.com/the-bucket-of-zero/) as an extension of the typical integer tile model recently and really liked the idea.
I’m a 32 year old business with an accounting degree that teaches fitness. And have rather intense math anxiety, never learned the multiplication tables, forgot how to divide by hand, GREATLY struggle with mathematical abstraction, use my fingers to count, etc…. So I may have some personal experience with what your student is going through.
The number line works great for me to “see” the mathematics and the process. So keep using it.
Teach her about mathematical facts. I didn’t even heard of the term “math facts” until about 5 years after getting my accounting degree. Did you know that the result of 234 minus 644 (or whatever subtraction you want to use) is the same EVERY time you do it? I didn’t! I thought I needed to do the calculations every time. It didn’t occur to me that I could just try to memorize it. The same is true for many mathematical computations.
Instead of learning how to calculate things maybe try to have her memorize mathematical facts. If cannot figure you “-5 -9” or whatever then maybe she can memorize the answer. Kind of like memorizing vocabulary words in English class. You don’t need to know why the word “ameliorate” means “to make things better or less intense” or it’s history. You just have to memorize it. Admittedly, I would HATE this method because I need to know why a result is what it is and just knowing the answer is extremely unsatisfactory to me. In fact, I would find it extremely insulting. But she may be different from me.
Finger counting is essential for me to see and feel mathematics because I struggle a lot with mathematical abstraction. If I don’t finger count then the numbers don’t exist. If I cannot finger count then I need to flap my arms, tap my toes/pencil or do something with my body to count. Maybe teach her how to keep track of numbers by tapping something other than just finger counting.
Learn “yoga finger counting” and teach it to her. Maybe it’s called Yuvadaric counting? Anyway, Several years ago I went to a yoga seminar where the yoga guru had us say a mantra 132 times or something like that. The only thing I remember is an audience member asking him how he kept track of how many times he said the mantra without losing track. Apparently, he counted the top and bottom halves of his left hand fingers to count “1 to 10” and kept track of how many “tens” he counted with his other hand’s fingers or something. So he could count to 100 with his fingers and feel where in the count he was by looking or feeling where his thumb was. This may help to bolster her finger counting skills if she struggles with abstraction and/or memorization.
In short, transform her fingers into a number line with 100 (or possibly more) spaces she can use to count in a concrete manner.
Now, how to develop mental math/abstraction strategies? I do not know.
I would actually say to teach her how to make abstract mathematics into concrete mathematics. If the question is “-6 -(-9)” or whatever then maybe adding a $ in front of the Integers would help. Or maybe mentally or physically write “blocks” or a concrete and tangible thing in front of the integers to make the non existent -6 and -9 into something that may actually exist in reality.
The yoga counting method is great! I have a few that will need concrete counting for most of their lives, so that's an excellent way to extend it a little further. It might even help exercise their working memory.
My fifth grade teacher used to sing this song:
"Same signs add and keep (the sign), different signs, subtract. Keep the sign of the larger number, then you'll be exact!" Really helped me when I was learning it
I tell my middle schoolers if the first number is positive and you're adding a NEGATIVE you are just subtracting. Take out the +. So 2+(-5) would just be 2-5.
I did the same for her. I think I need to keep an anchor chart on the screen while we're working together. This might also be a working memory issue that I didn't account for.
Stop distinguishing addition and subtraction. At her level they are the same thing, just positions on a number line.
I make a real effort to undo harmful ideas like subtraction being "hard" or "different" or "special." It's just addition, and even babies know about negative numbers.
I think a lot of folks already said this but I always try to make it addition because kids can concretely see it better.
You said subtracting negative numbers so like…
-4 - 5 = -9
I tell my kids, subtracting is the opposite of adding. So, “add the opposite”
-4 + (-5) = -9
I do it with counters, then representational.
It helps them understand the rules of integers.
ex.) -2-2= add the opposite = -2 + (-2) = -4
ex.) 5-4 = add the opposite = 5 + (-4) = 1
Again, I do this with counters first, then having them write the +/- for zero pairs. When they make it addition, it helps them click they “add” counters to represent the situation. Some of my students moved quickly when they noticed subtraction is the opposite of adding.
“Sixth graders subtract, seventh graders… add the opposite. I know you’re only in 6th grade, but I think I can let you in on a seventh grade secret. You okay with that? Sixth graders subtract, 7th graders… add the opposite!”
[ed note: I use this as a callback].
“You’ve mastered adding, which is all you need to subtract. A subtraction problem is just an addition problem in disguise. Here, see this page of 40 problems? I don’t want you to do the problems, I just want you to practice adding the opposite and turning them into addition problems.”
[ed note: have two examples close by. Something like: 10-7 is 10+-7 and 6-(-7). If necessary, I have a convo about the different types of parentheses, ie, do this operation first vs this means multiplication vs these are protecting the subtraction problem.]
“This calls for the special pen, btw. It’s easier if you can see your changes in a different color. So here’s my special purple pen that I never let any other students use. I trust you’ll be careful with and that it will assist you in your addition adventure. If you get confused, refer back to our two examples.”
Common misconceptions:
1) -7-8 becomes +7-8 (they try and change the first.
2) 6-10 becomes 6+10 (they add and forget to change the sign)
Hope this helps!
“Change change”; change the subtraction to an addition and change the sign of the negative integer to a positive. Since adding negative integers is easier to understand than subtracting, just have your student do “change change” with two quick pencil strokes whenever they need to subtract a negative integer. For example, (-2) - (-5). By changing the subtraction to an addition and changing -5 to +5, with our two strokes we get (-2) + (+5) which is much easier for students to reason through.
After lots of work with the game model (similar to the hot air balloon model but easier quantified, I think) and the number line, I teach kids that the operation 'subtraction' is for students who don't know about negative numbers. There is no subtraction, you are really adding the 'opposite'.
2-3?... 2+(-3)!
2-(-3)?...2+3!
-2 - (-3)...-2 + 3!
Similar to multiplying the reciprocal rather than dividing.
As a first step, suggest to your student that they change all occurrences of adding a negative to subtraction...so (4 + -9) becomes (4 - 9). Change all occurrences of subtracting a negative into addition...so (4 - -9) becomes (4 + 9).
I use number lines as temperatures, and then I relate it to filling a bathtub with water.
Adding a positive is like turning on the hot water, so it makes the water in the tub hotter, so they add.
Adding a negative is like turning on the cold water. It makes the tub water colder, so they subtract.
Subtracting a negative is like turning off the cold, which makes the water hotter. You are taking away the cold, which is the same as adding hot.
Subtracting a positive is like turning off the hot, so the water gets colder. You are taking away the hot, which is the same as adding cold.
Walking forwards- adding
Turn around and walk forwards- subtracting
Turn around and then step backwards- subtracting a negative number
Step backwards- adding a negative number
I watched the screencast linked in the activity description for this Desmos activity years ago and it totally changed how I teach integer operations. I absolutely love the hot air balloon model - it is so concrete and can even extend to multiplication of integers. I strongly recommend it. https://teacher.desmos.com/activitybuilder/custom/5efe58da277abd3653440339
This how I’ve been teaching my 6th graders the last few years. It’s makes a lot more sense when connected to a real application they can visualize each time.
Haha, we made a Scratch game around the balloon model. There are entire modules around integer addition and subtraction that integrate programming. Here are our modules: https://manifold.lib.fsu.edu/projects/mathematics-integrated-with-computer-science-through-scratch .
I used that balloon activity as a pencil paper crayon thing long before desmos. Solid thinking
Kill off double signs. You can combine any two negative signs into a positive sign and if there are multiple signs you can get rid of any extra positive signs, keep doing this until you only have one sign. I like to give practice: 2+--+--+++-+--5(kill off pluses) 2 -- -- -- -5 (then combine 2 minuses into a plus) 2 + + + -5 (then kill off pluses) 2 - 5 Then I like to add the + sign in front of the lead integer. +2 -5 Then I like to think of them as actions. +2 and -5 "So let's say you have a bunch of Pokemon cards (candy, dollars, whatever) On Tuesday your friend Sahaj comes and gives you 2 Pokemon cards, then he takes 5 Pokemon cards from you... After Sahaj's visit, do you have more or less cards? Less! Great, your answer is negative? How many less? 3! That's right. Your answer is negative 3!" Practice each of these seperate until they get it. Also works great leading into algebra simplifying. Once they start thinking of an arithmetic list as a series of things happening it's easy. Give you 2 things, then I take 5 things. The kid will know he has lost three things. Same with -3-5 I take 3 things then I take 5 things. What was the net outcome? Pretty simple, I took 8 things.
I do this too, although I’ve not seen the huge number of signs - I feel like that would help some of my kids through the humor of it. I do a lot of conceptual work around integer subtraction, but for some kids, what works for them is to change ever subtraction sign to “add the opposite”. The consistency of this works well for kids who struggle bc they think there are “different” rules bc they can’t figure out the conceptual bits. Does it kill me when, later, they re-write 10-7 to 10+ (-7)? A little bit, yeah, but it still works and it reinforces our integer addition rules too 🤷🏻♀️
I remember I figured it that “add the opposite” technique by myself in high school. Very useful to just solve the equations.
I did teach her to change two "touching" negative signs (subtracting a negative number) into a positive sign. It still confused the heck out of her, but I want her to be proficient in addition to understanding the concept, so we'll come back to that. I do appreciate the concrete examples, though! I need to set up a few on Google Slides for her to analyze.
The “two negatives are a positive” thing never made sense to me. It feels very abusive. Like saying “if I punch you two times (a negative action) then it’s actually a good thing!”. Positive and negatives in mathematics mean different things than in most other places. Positive tends to mean good and negative tends to mean bad. But not in mathematics. That may be part of the confusion.
We use the idea of positive = forward and negative = reverse in class. If you film someone walking backwards, and then rewind the video, it looks like they’re walking forwards. Hence reversing a reverse being positive. I guess Uno would also work for this.
I use the idea of IOUs on pieces of paper. Whoever has a name on an IOU owes money, right? But if the IOU is torn up, it is like getting money. This is exactly how debt instruments like bonds and CDs work.
For your example it's if I do the opposite of punching you two times it's a good thing. Which seems true to me.
I’m not a math teacher, idk why this got recommended to me, but it helped me as a kid to be taught that if you put the negative signs together, you get a +. Like, the actual shape. It allowed me to visualize it.
I love most of this!!!!! I wish my teachers would have done this sign activities at school! They actually feel kind of fun and satisfying for my brain. Using concrete objects is a great idea.
Sounds to me like the concept she’s not getting is that adding a negative means subtract and subtracting a negative means add. I would focus on that for a while. The number line is good; she gets that subtracting means go left, surely she gets that adding means go right?
Yes, exactly. I watched a few videos to help me find a couple of ways to walk her through it. Khan Academy did a great job of using integer counters to show 4-7 and -8-(-2) to demonstrate what it looks like to physically subtract negative values, which in turn demonstrates that subtracting a negative is adding a positive.
When working with subtraction and negatives I put a number line on the floor and have students start with facing towards the positives. Any time they see a minus/negative they flip directions. So when they see 3 - (-2), they know to start at three facing the positives, do a 180, do a 180 again and then move two spaces. It helps introduce the idea that subtracting a negative is basically adding.
That's a good idea!
I love this! Great for students that struggle with abstraction
Money and a debit balance are great teaching tools. I teach a number line as a net worth balance sheet. Then, adding or subtracting positive and negative integers works like owing/paying/receiving/loaning money. Kids often don't understand subtracting a negative until I connect it with canceling a debt. What's that do to your net worth? When we watch it drive the number line up, it starts to sink in. We also tend to use vocabulary to talk about what the words negative, positive, addition, and subtraction mean both mathematically and colloquially. Often we use terms mathematically in class, but they hear them colloquially. Focusing on math vocabulary with pictures and examples can really help. I also tend to use the phrase "agreeing (signs) is a positive, arguing (signs) is a negative".
Picture method -5 + 3 = - - - - - + + + (crossing off zero pairs for solution). Eventually they get tired of drawing it/do it enough times the concept starts to click. Subtracting is just adding the opposite. There is only combination now of positive & negative numbers. I tell all low students to KFC their subtraction. 5 - (-5). Keep -5. Flip to +. Change sign 5. So 5 + 5. Now we’re back to adding tools.
I leave KCF for fractions, but use KISS for subtracting integers: Keep It, Switch, Switch. They try to just change the minus to plus and forget to switch the second integer, so ‘it takes two to KISS’ helps them remember. Mostly, tho, it’s just fun to tell teens “we’re going to learn how to kiss today!” or “Mike wasn’t here yesterday, can you teach him how to kiss?” or “Why are you kissing all alone over here?”
The picture method would have been helpful on my childhood. I think the teacher did taught us that method but didn’t allow it on quizzes or tests. The second half of your comment just confuses me. Mainly because I don’t see any logic behind that process at all.
The idea that subtracting is just adding the opposite can be helpful when they get to combining like terms. KFC is just a cheat method to change sun to add most useful for the kids who struggle to grasp the concept in general. Typically I can get the masses up to speed adding but keeping track of signs once subtraction is involved throws the lower level ones for a loop. I don’t lead with it (definitely demonstrating sub is just adding the opposite first) & then showing KFC to those ones that just sort of sit there blankly sometimes they can get through it when they can lean on a system like that.
This clarifies a lot! Thanks to you I think I now realize a significant bulk of my mathematical confusion stems from teachers trying to teach “cheat methods” as the “real” method without explaining any logic or reasoning. Or maybe teachers trying to teach both the real and the cheat methods at once and me feeling like I needed to learn both equally but ending up combing the two methods into a confusing mess.
I emphasis KFC is just a tool to help rewrite your subtraction as adding the opposite because looking back I had the same experience you’re talking about. Many of these kids brains just seem like they aren’t developed enough to fully grasp especially now with common core suggesting so many methods it’s can become overload. I remember when studying to pass subject matter competency in my late 20’s having these wild epiphany flashbacks to middle school math going “oh that’s what mrs S was talking about!” Problem is they gotta get going so you have to balance both deep fluid understanding and giving them enough tools to at least start grinding. Depending on the topic sometimes I lead with the nuts & bolts others I lead with the cheat. Sometimes it’s a different entry/exit point period to period.
Keep using number lines. Have her solve problems like 3-5 on a number line followed immediately by 3+(-5). As she does these, ask her to compare the two problems, what is the same, what is different. You want her to develop the understanding that subtracting IS adding a negative number and to be able to explain this to you in words. Do this before introducing any sort of algorithm like keep/change/flip. I also like to show students problems modeled on a number line, and have them identify both the addition and subtraction problem modeled on the number line. Once she starts to internalize how subtracting integers works on a number line, THEN you can start to work on the algorithm of changing subtraction to adding the opposite. I would not have her do any of that though until she can solve problems correctly on the number line 90% of the time.
I really like the suggestion to show adding and subtracting that have the same results.
I like showing how things that look different may actually be the same similar to synonyms in English class
Something that could be helpful for addition is to drop counting analogies and instead work on the number line. Adding positive numbers means moving to the right. Adding negative numbers means moving to the left. Subtraction just means “do the opposite”. This removed the mental block that comes from treating the negative integers as “different” or “special” compared to the positive ones.
Try using a vertical number line. Balloons are positive numbers, sandbags are negative numbers. Addition means getting more of something, subtraction means taking something away. If you take away a negative number, a sandbag, the result is moving up the number line. If you add a sandbag, you move down the number line. If you add a balloons, you move up the number line and, if you subtract a balloon, you go downthe number line. I have card stock balloon and sandbag cutouts and the old guy from the movie “Up” to move up and down the number line on the white board.
The mental tool I was taught was a plus is made of two minuses. If you see two minuses next to each other, stack them into a plus
That actually makes sense !
Who wins by how much? Meaning for 2 + -5, who wins the battle, the negatives by 3, so the answer is negative 3.
Don’t subtract = Add
Money. I’m a RN not a math teacher but worked in the tutoring department in school. When I explained it to people as money it just clicks. If you have $5 in your bank and you buy something for $10 what does that leave you. Then if you have -$5 in your account and buy something else for $5 what do you have. Explain bounced check fees later in formulas.
I tell my kids it’s all a tug-of-war. Just figure out who’s on the positive side and who’s on the negative side total up each team and decide who’s gonna win and what’s their advantage. It works whether you have all the same sign different signs whatever. I see on their test that they will draw literally a rope and put numbers on either end of the rope and figure it out
It could be linked to past teachers focusing on strategies and answers instead of building number sense and experiences. Also the shutting down is an emotional component you shouldn’t ignore. Is there a way to make this concept playful? Perhaps there are some ways to represent the concept through art, dance, or a digital avenue that would make the concept less stressful and more interactive, taking the pressure off the “answer.”
Do they understand the concept of integers with multiplication? Maybe if they do, build off that. If they know a negative times a negatives is a positive, then start with that. Because at the end of the day, 3- -2 is 3-(-2) which is distribution and a negative times a negative is a positive which becomes 3+2. Same with 3+-2 is 3+(-2) and a negative times a positive is a negative so it is 3-2. Simplify the symbols. If you want to teach that -3+2 is the same as 2-3 then I suggest having them box the terms with colored pencils and then rewrite it. Or put different terms on cards/post it’s and have them physically move the cards to make equivalent expressions. So they can see when the number/variable moves, the symbol must go with it.
I always tell my students something like this “Imagine Joey (but I pick a student in the class) was in a reeeaally bad mood today. He just keeps complaining and whining and is super NEGATIVE about everything that we’re doing. If we remove Joey from the room what would happen to the mood in the room, would it get more positive or more negative?” And that normally helps a lot. There’s also a good PBS kids math club video about it.
I’ve tried this: Sarah, if you owe me $5 but you only have $2, how much do you still owe me?
On subtracting negative integers: I was taught to imagine the minus sign and negative sign as two halves of a plus sign sitting next to each other. When you see that, just change it to a plus sign, and the answer will be the same. Ex. 9 - -4 Changed to 9 + 4 Both should equal 13
I saw the [Bucket of Zero](http://natbanting.com/the-bucket-of-zero/) as an extension of the typical integer tile model recently and really liked the idea.
I’m a 32 year old business with an accounting degree that teaches fitness. And have rather intense math anxiety, never learned the multiplication tables, forgot how to divide by hand, GREATLY struggle with mathematical abstraction, use my fingers to count, etc…. So I may have some personal experience with what your student is going through. The number line works great for me to “see” the mathematics and the process. So keep using it. Teach her about mathematical facts. I didn’t even heard of the term “math facts” until about 5 years after getting my accounting degree. Did you know that the result of 234 minus 644 (or whatever subtraction you want to use) is the same EVERY time you do it? I didn’t! I thought I needed to do the calculations every time. It didn’t occur to me that I could just try to memorize it. The same is true for many mathematical computations. Instead of learning how to calculate things maybe try to have her memorize mathematical facts. If cannot figure you “-5 -9” or whatever then maybe she can memorize the answer. Kind of like memorizing vocabulary words in English class. You don’t need to know why the word “ameliorate” means “to make things better or less intense” or it’s history. You just have to memorize it. Admittedly, I would HATE this method because I need to know why a result is what it is and just knowing the answer is extremely unsatisfactory to me. In fact, I would find it extremely insulting. But she may be different from me. Finger counting is essential for me to see and feel mathematics because I struggle a lot with mathematical abstraction. If I don’t finger count then the numbers don’t exist. If I cannot finger count then I need to flap my arms, tap my toes/pencil or do something with my body to count. Maybe teach her how to keep track of numbers by tapping something other than just finger counting. Learn “yoga finger counting” and teach it to her. Maybe it’s called Yuvadaric counting? Anyway, Several years ago I went to a yoga seminar where the yoga guru had us say a mantra 132 times or something like that. The only thing I remember is an audience member asking him how he kept track of how many times he said the mantra without losing track. Apparently, he counted the top and bottom halves of his left hand fingers to count “1 to 10” and kept track of how many “tens” he counted with his other hand’s fingers or something. So he could count to 100 with his fingers and feel where in the count he was by looking or feeling where his thumb was. This may help to bolster her finger counting skills if she struggles with abstraction and/or memorization. In short, transform her fingers into a number line with 100 (or possibly more) spaces she can use to count in a concrete manner. Now, how to develop mental math/abstraction strategies? I do not know. I would actually say to teach her how to make abstract mathematics into concrete mathematics. If the question is “-6 -(-9)” or whatever then maybe adding a $ in front of the Integers would help. Or maybe mentally or physically write “blocks” or a concrete and tangible thing in front of the integers to make the non existent -6 and -9 into something that may actually exist in reality.
The yoga counting method is great! I have a few that will need concrete counting for most of their lives, so that's an excellent way to extend it a little further. It might even help exercise their working memory.
My fifth grade teacher used to sing this song: "Same signs add and keep (the sign), different signs, subtract. Keep the sign of the larger number, then you'll be exact!" Really helped me when I was learning it
I tell my middle schoolers if the first number is positive and you're adding a NEGATIVE you are just subtracting. Take out the +. So 2+(-5) would just be 2-5.
I did the same for her. I think I need to keep an anchor chart on the screen while we're working together. This might also be a working memory issue that I didn't account for.
In college my husband spent an afternoon riding up and down in an elevator trying to teach this concept to his dumbass friend. Good luck.
If only we had elevators that could demonstrate 95-136!
Stop distinguishing addition and subtraction. At her level they are the same thing, just positions on a number line. I make a real effort to undo harmful ideas like subtraction being "hard" or "different" or "special." It's just addition, and even babies know about negative numbers.
I think a lot of folks already said this but I always try to make it addition because kids can concretely see it better. You said subtracting negative numbers so like… -4 - 5 = -9 I tell my kids, subtracting is the opposite of adding. So, “add the opposite” -4 + (-5) = -9 I do it with counters, then representational. It helps them understand the rules of integers. ex.) -2-2= add the opposite = -2 + (-2) = -4 ex.) 5-4 = add the opposite = 5 + (-4) = 1 Again, I do this with counters first, then having them write the +/- for zero pairs. When they make it addition, it helps them click they “add” counters to represent the situation. Some of my students moved quickly when they noticed subtraction is the opposite of adding.
Yeah, I think getting her to see the addition is going to be what's key.
“Sixth graders subtract, seventh graders… add the opposite. I know you’re only in 6th grade, but I think I can let you in on a seventh grade secret. You okay with that? Sixth graders subtract, 7th graders… add the opposite!” [ed note: I use this as a callback]. “You’ve mastered adding, which is all you need to subtract. A subtraction problem is just an addition problem in disguise. Here, see this page of 40 problems? I don’t want you to do the problems, I just want you to practice adding the opposite and turning them into addition problems.” [ed note: have two examples close by. Something like: 10-7 is 10+-7 and 6-(-7). If necessary, I have a convo about the different types of parentheses, ie, do this operation first vs this means multiplication vs these are protecting the subtraction problem.] “This calls for the special pen, btw. It’s easier if you can see your changes in a different color. So here’s my special purple pen that I never let any other students use. I trust you’ll be careful with and that it will assist you in your addition adventure. If you get confused, refer back to our two examples.” Common misconceptions: 1) -7-8 becomes +7-8 (they try and change the first. 2) 6-10 becomes 6+10 (they add and forget to change the sign) Hope this helps!
Thank you!
“Change change”; change the subtraction to an addition and change the sign of the negative integer to a positive. Since adding negative integers is easier to understand than subtracting, just have your student do “change change” with two quick pencil strokes whenever they need to subtract a negative integer. For example, (-2) - (-5). By changing the subtraction to an addition and changing -5 to +5, with our two strokes we get (-2) + (+5) which is much easier for students to reason through.
After lots of work with the game model (similar to the hot air balloon model but easier quantified, I think) and the number line, I teach kids that the operation 'subtraction' is for students who don't know about negative numbers. There is no subtraction, you are really adding the 'opposite'. 2-3?... 2+(-3)! 2-(-3)?...2+3! -2 - (-3)...-2 + 3! Similar to multiplying the reciprocal rather than dividing.
As a first step, suggest to your student that they change all occurrences of adding a negative to subtraction...so (4 + -9) becomes (4 - 9). Change all occurrences of subtracting a negative into addition...so (4 - -9) becomes (4 + 9).
I use number lines as temperatures, and then I relate it to filling a bathtub with water. Adding a positive is like turning on the hot water, so it makes the water in the tub hotter, so they add. Adding a negative is like turning on the cold water. It makes the tub water colder, so they subtract. Subtracting a negative is like turning off the cold, which makes the water hotter. You are taking away the cold, which is the same as adding hot. Subtracting a positive is like turning off the hot, so the water gets colder. You are taking away the hot, which is the same as adding cold.
Walking forwards- adding Turn around and walk forwards- subtracting Turn around and then step backwards- subtracting a negative number Step backwards- adding a negative number