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Everyone saves precious time. ​
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Example 1: 3 + 4

The water starts at 3 degrees. 
4 scoops of warm water are added.  
The temperature increases by 4 degrees. 
The water finishes at 7 degrees. 

 I use this one just as an example of what is happening with the water.  


Example 2: (-6) + 11

The water starts at -6 degrees.  
11 scoops of warm water is added. 
The temperature increases 11 degrees. 
The water finishes at 5 degrees. 

Some students find it easier if they have a thermometer in front of them so they can count up.  (if you don’t have a thermometer you can use a printed version).  


Example 3: 4 + (-6)

The water starts at 4 degrees. 
Six ice cubes are added.  
The temperature drop 6 degrees
The water finishes at -2 degrees.  


Example 4: (-2) + (-4) 

The water starts at -2 degrees.  
Four ice cubes are added. 
The temperature go down 4 degrees
The water finishes at -6 degrees. 

This is by no means scientifically valid but the kids seem to understand the concept of adding negative temperature by adding ice cubes.  

I typically start subtraction by asking them what would happen to the temperature if I have a container of ice water and I start taking out ice cubes.  I have always gotten the answer that the temperature would rise.  So I explain that they are taking away negative temperature.  Here is what it looks like.  


Example 1: 4 - 6 

The water starts at 4 degrees. 
Remove 6 scoops of warm water. 
The temperature goes down 6 degrees. 
The water finishes at -2 degrees.

Some students may ask how you remove scoops of warm water from a container and the answer is you can’t, that we are doing it in theory not practice.


Example 2:  (-3) - 2 

The water starts at -3 degrees.  
Remove 2 scoops of warm water. 
The temperature of the water goes down 2 degrees.  
The water finishes at -5 degrees.  


Example 3: 5 - (-6) 

The water starts at 5 degrees.
Remove 6 six ice cubes. 
The temperature goes up 6 degrees. 
The water finishes at 11 degrees.  


Example 4: (-3) - (-1) 

The container starts at -3.  
Remove 1 ice cube. 
The temperature increase by 1 degree. 
The water finishes at -2 degrees.  



The ice cube method has worked wonders in my classroom with students who are struggling to grasp the concept of adding and subtracting integers.  With the visual aid of a thermometer, printed or physical, there are very few students who were unable to grasp the concepts.  


I have found that after students learn the laws for multiplying and dividing integers.  All to often they want to apply the rules to adding and subtracting integers which by no means works.  After students learn the easy rules for multiplication and division they want to make rules for adding and subtracting and are rarely successful.  I have not yet figured out a way to discourage this confusion.  If you have any ideas, I would love to hear them.   

Each student received a dry erase board and two different coloured dry erase markers.  I would have the students use one of the colours to create their grid that they would use for the game.  I would show the students what their grid should look like on the board and then I would check the students who I knew struggled with creating the grids. 
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If there was a lot of time, I would have them make their grids from -8 to +8 on both the x- and y-axis.  If there was less time, or I wanted them to work through multiple partners I would have the make their grids -4 to 4 on the x- and y-axis.  I would manipulate everything in-between to ensure that the game fit the attention span of the particular class.  

For the students who struggled, I would make sure that they had written the number in each line so that it was easier to follow.  For some students, I would have the grid ready to go for them when they entered the class. 

I would have them trade markers with a  student who had another different colour so their ships were a different colour than their guesses and their parters guesses.  

From their I explained that they could use one colour to mark where they guessed, typically using an X when they hit a ship and an O where they missed and the other colour would be to mark down where their partner had guessed.  

Typically battle ship doesn’t require you to record your partners guesses but it provides a way for me to check and see if anyone is having troubles with their graphing.  It is easy to compare what the two partners have written down and then figure out which one is struggling.  Some of my more tidy students preferred to write down their partner’s guesses as coordinate points on a separate page so their grid didn’t get too messy.  I also had some students who would use two grids, one for their guesses and one for their partners guesses.  

From there they would play until there was a winner and a loser.  I would do a quick check to make sure that the graphing was being done correctly by both parities.  As students finished, I encouraged them to make new partners with another group who had finished and play again.  

I found that the first time through the game with students, they were a little slow and needed a lot of assistance but once they had played once they could do it again with ease.  I would have my grade 9 students play 4 or 5 times a year to make sure that they are mastering their graphing skills.  

After the first run through with the students it was a great activity to leave with a sub because they were relatively self sufficient and just needed to be told what size of grid to use and how many of what size ships.  I personally would use a wet erase marker and give the sub a sample board to show to the kids, ships plotted and all.     

Feel free to send your questions and comments my way!

It is possible to complete this activity in the hallway or out on pavement or concrete.  The supplies vary a little bit for the two variations but are easily accessible.  The most important part of this activity is that the students understand what they are supposed to be practicing which starts with the teachers seamless explanation.  I practiced a lot on my number line at the front of my classroom and showed them on the smart board with a stick man before I ever sent the out to do their trial problems.  I found that with this age group the crazier I could be the closer they paid attention.  So when I was demonstrating the number line dance, it started with the bunny hop I did some crazy leaps and what ever else could make the distance between the numbers.  

Supplies For in the School 

• Masking tape: about 5 m per group 
• dark coloured dry erase markers:1 or 2 per group (this ensures easy clean up, Tip: don’t use red and some greens as for some reason they don’t come off as easy) 
• a question sheet 

Supplies for Outside

• side walk chock: 1 or 2 for each group (any chock will work, in my school we had boxes left over from the ages of the black board and I often used that because it was free) 
• a question sheet 

As you can see there isn’t much required for this activity but it requires a lot of enthusiasm on the part of the teacher.  First I will explain how to do The Number Line Hop for multiplication and then I will give you the instructions that worked for my classes.  The “Hop” itself is very simple but it fosters skills to assist students in understanding multiplication of integers which they struggle so much with.  

We will start with something all your students will know the answer to and progress from there.  

Example 1:  4 x 2

If your students were to see this, they should know to start at 0 on the number line and count up by two, four times which takes them to 8.  

To do the number line hop for this, start by standing on the 0.  When we are multiplying positive numbers, we know that the values get larger.  We face forwards because we have positive four.  

Then we hop forward two units at a time because we have positive two in each of the four groups. 

Resulting in standing on the 8.

 I always begin with simple multiplication because most students know how it is supposed to work if not it is a good review. 

The next example is one that they struggle a little more with, but this gives them a better idea of what is going on.  

Example 2: (-3) x (2)

Again, we start on the zero.  This time we are going to face the negative end of the number line because the first number is a negative.   

This time, we hop forward, because its positive,  2 units three times.

The result is landing on -6.

Logically you can have your student think that if you have negative three groups of 2 they make six.

Or you can look at it the other way, two groups of negative three

Example 3: (2) x (-3)

You begin at zero and face the positive side of the number line to represent the positive of the two.   

Your hop would be three units backwards twice if you conceptualized it that way. 

Resulting in the same negative six as before.

Finally, the hard one to figure out…negative times negative.  

Example 4: (-4) x (-2)

Just as before, we start on zero.  We are going to face the negative end of the number line to represent the negative four.  

Then, we are going to hop backwards, to represent the negative, two units four times.

Resulting on us landing on positive eight.

I intentionally use the same numbers for example 1 and 4 because it helps the students begin to see the patterns when multiplying negative numbers.  I try to have them come up with the pattern them selves of two negatives multiply to make a positive and if the signs are different it will be a negative.  

There is a lot of teacher annotation that needs to happen in order to make this activity meaningful to the students but having them up out of their desks does great things for their memories.

Here are the instructions I sent with the students to work with, after I had given verbal instructions as well as shown them examples both on the number line at the front of my classroom as well as the smart board. 

Instructions: 

1. Lay your tape in a straight line or draw a straight line with your chock. 
2. Find where the centre of your line is approximately and mark this zero.  
3. Use either your foot or your partners foot to measure and label the ticks on the number line (if time constraint is an issue, I have then use a pair of shoes as the measuring tool and one person goes each way)
1. place your heel on zero facing the positive end of the number line 
2. make a mark where your toe is
3. Label the mark one
4. place your heel on one
5. Make a mark where your toe is 
6. Label this mark two
7. continue in the positive direction until you reach … (enough so they can complete the practice problems you have created for them) 
8. Place your heel on zero facing the negative end of the number line 
9. make a mark where you toe is 
10. Label the mark negative one
11. continue in the negative direction until you reach …
4. take turns doing the number line hop to complete your sample problems. 

Tip: If you have students who need a little extra help, it is more efficient if you set up the number lines a head of time for certain groups.

I always follow this activity with practicing on the number line with your fingers as the person hoping the number line.  For student’s who really struggle with the concept, I would often give them a lego man so they could face them the correct direction and hop them along the number line they have in front of them.  (I had a class set of laminated number lines that I could hand out and use with dry erase markers on them to help solve problems) 

I found that they key to multiplying integers was having the students recognize the patters for them selves.  This was a great activity to follow the Number Line Dance for adding and subtracting integers.  As long as it didn’t rain the number lines would stay on the ground they were useable again, as well as the instructions were a lot easier to follow. 

Good luck! Send me a comment if you have any questions.   

It is possible to complete this activity in the hallways of the school or out in pavement or concrete courtyard.  The supplies vary a little bit for the two variations but are easily accessible.  The most important part of this activity is that the students understand what they are supposed to be practicing which starts with the teachers seamless explanation.  I gave a lot of examples on my number line at the front of my classroom and on the smart board with a stick man before I ever sent the out to do their trial problems.  I found that with the 10 to 13 age group the crazier I acted the closer they paid attention.  So when I was demonstrating the number line dance, it started with the bunny hop and did some salsa steps and what ever other crazy dance came to my head at that moment when I was moving between the numbers.  

Supplies For in the School 

• Masking tape: about 5 m per group 
• dark coloured dry erase markers:1 or 2 per group (Tip: don’t use red and some greens as for some reason they don’t come off as easy) 
• a question sheet 

Supplies for Outside

• side walk chock: 1 or 2 for each group (any chock will work, in my school we had boxes left over from the ages of the black board and I used that because it was free) 
• a question sheet 

As you can see there isn’t much required for this activity but it requires a lot of enthusiasm on the part of the teacher.  First I will explain how to do The Number Line Dance and then I will give you the instructions that worked for me.  The “Dance” itself is very simple but it fosters skills to assist students in understanding addition and subtraction of integers which they struggle so much with.  

We will start with something all your students should know the answer to and progress from there.  

Example 1:  4 + 2
If your students were to see this, they should know to start at 4 on the number line and count up 2 which takes them to 6.  

So to do the number line dance for this, they start by standing on the 4.  When we are adding and subtracting, we know that we already have the first value that shows up in the expression so that is where we need to start on the number line.  When we are adding things, we know that the sum should be getting larger, hence we face the  positive side of the number line because the operation is addition.  

Resulting in them standing on the 6.  

I always begin with simple addition because students know how it is supposed to work and it assists in the understanding of the number line dance that they are going to be learning.  It is also a question where they can build their confidence to work in this system.  

We will still do another one that they usually find simple, though some still struggle. 

Example 2: (-3) + (-4)
Again, we start on the first number (-3).   The operation is addition, so we are going to face the positive end of the number line because when we add we are getting larger.   

We are adding negative four, so we are going to take 4 steps backwards to represent the negative four.  If students think through this intuitively there is usually a number of different rationalizations they have but I find the most common one is that the number is getting larger by negative four, which has the same result as getting smaller by 4.  

Leaving us standing on negative seven.   

Now lets get into what the students really struggle with…subtracting!  The only real difference with subtraction is that we are getting smaller, so when we position ourselves on the number line we are going to face the smaller end of the number line, or the negative side.  

Example 3: 3 - 5 
We start on the number line at three, but this time we are going to face the negative side of the number line because we are subtracting and as our students should know subtraction means the solution should get smaller.   

We then take 5 steps forwards to represent the positive five that we are getting smaller by. 

Landing at (-2) Which we know to be the answer.    

Finally, we will subtract a negative, the one students find the hardest to understand.  

Example 4: 1 - (-2) 
We begin standing on the number line at one, facing the negative end of the line because we are subtracting.   

Then we take two steps backwards to represent the (-2), which makes sense if you are following the rules of the number line dance however some students are going to need a little more explanation.  I would talk about subtracting negatives as the opposite of subtracting positives.  We know when we subtract a positive we get a smaller number, thus the reverse is true for subtracting negatives.  When we subtract a negative the result should get bigger because we are taking away things that are negative.  

Leaving us standing at 3.  

There is a lot of teacher annotation that needs to happen in order to make this activity meaningful to the students but having them up out of their desks does great things for their memories.  Many students will also need help transitioning from doing the number line dance to being able to add and subtract integers on a number line without walking along it.  

Here are the instructions I sent with the students to work with, after I had given verbal instructions as well as shown them examples both on the number line at the front of my classroom as well as the smart board.    

Instructions: 

1. Lay your tape in a straight line or draw a straight line with your chock. 
2. Find where the centre of your line is approximately and mark this zero.  
3. Use either your foot or your partners foot to measure and label the ticks on the number line (if time constraint is an issue, I have then use a pair of shoes as the measuring tool and one person goes each way)
1. place your heel on zero facing the positive end of the number line 
2. make a mark where your toe is
3. label the mark one
4. place your heel on one
5. make a mark where your toe is 
6. label this mark two
7. continue in the positive direction until you reach … (enough so they can complete the practice problems you have created for them) 
8. place your heel on zero facing the negative end of the number line 
9. make a mark where you toe is 
10. label the mark negative one
11. continue in the negative direction until you reach …
4. take turns doing the number line dance to complete your sample problems. 

Tips:
It is more efficient if you set up the number lines a head of time for certain groups of students.  If you are able to leave the number lines on the ground for a little while, they can be reused for multiplication of integers.  

I always follow this activity with practicing on the number line with your fingers as the person dancing the number line.  For student’s who really struggle with the concept, I would often give them a lego man so they could face them the correct direction and walk them along the number line they have in front of them.  (I had a number of laminated number lines that I could hand out to the kids at any time so that they could use then without destroying them. They could also use dry erase markers on them to help solve problems) 

The number line dance allowed me to get my students out of their desks and when weather permitted, out of the school doing math which in itself was a novelty.  They had fun and there was a lot of great learning that went on.  I would see them a year or two later pulling out their number lines to add and subtract integers.  Good luck and feel free to send me any questions.  

The Simple Things - Here are a few of the little things that I did in my classroom that ensured students had the tools they needed at the same time as having to take some responsibility for themselves.  

1. White boards and Dry Erase markers: I had a class set of small dry erase boards with a grid on one side and the other side was blank.  The students love the non-perminance of the whiteboard and would complete more work on the whiteboards than they would on paper.

           TIP: The markers last longer if you keep them tip down in a container.  
           MONEY SAVER: I had each student bring in a pack of 5 thin dry erase markers at the beginning of the   
           year.  I took them as stock for a working class set and left the rest in my desk drawer to replace the 
           ones that died.  

• Acrylic-board as white board: I hung sheets of acrylic board on the walls in my classroom, it was like having extra whiteboard space.  When my students were working on long problems in groups, I would have them standing at the whiteboard and acrylic-board working through the problem.  They liked to work standing up, in fact some of them would choose to do their practice problems on the acrylic-boards, and it ensured that the whole group could see what was going on in the problem solving process.  If you don't have enough wall space for the acrylic boards, it is possible to get the board cut into desk size pieces and use it in a similar manner.  

           TIP: once a week I would scrub the acrylic-boards with a magic eraser to remove the left over colour.

• Extra writing utensils: I kept pencils and pens in a cup on my desk and I made students sign them out.  It didn’t work perfectly but they usually were signed back in at the end of class epically if I reminded them.  To keep my cup stocked, I made a deal with the cleaners and they would pick up any pens and pencils they thought would be useful in the hallways for me.  I made them a batch of cookies at the end of the year to say thanks!
• Calculators for loan: I always had a extra few calculators in my classroom that I would put on loan, in exchange for something I knew the students wouldn’t go far without.  Car keys and cell phones were always good collateral.  

Things to Laminate - Laminating makes your work last longer and things look more valuable and it makes them a great surface for dry erase markers.   

• Number lines: l always kept a class set of laminated number lines in a cupboard.  The students knew where they were and were allowed to use them when ever they chose.  Students who were struggling with adding, subtracting or multiplying integers would often use the number lines and a dry erase marker to aid them.  Students also found them very useful when they were trying to order numbers.  
• Formula sheets: For each level I taught I copied a class set of formula sheets one colour of paper then laminated them. Each class had a different colour for their formula sheet.  I allowed the students to write on the formula sheets with dry erase markers during exams and class as long as they erased it before handing it back.  Writing on the formula sheets was a great memory aid for many students when it came to the uses of their formulas.    
•  Nets of 3D Shapes: I copy all the different 3D shapes used in my classes, laminate and folded them for students who were learning or working on surface area.  The students would fold the shape up, write the dimensions on the shape, (again using dry erase markers) and then unfold it to find the area of each of the 2D shapes.  This worked really well for any students who were struggling with surface area of 3D shapes.   

Transparencies - Transparencies are a great way to make sets of some of the tools we use so regularly in math.   

• Photocopy Protractors: At the beginning of the year I would photocopy about 40 protractors on tho transparencies.  If you look in black line masters books there is usually a  page of protractors that are meant to be photocopied.  Copying your protractors ensures that they are all the same because sometimes the ones that students buy at the stores are not all the same and don’t all work the same way.  

           TIP: the protractors need to be cut out if they are going to be used to draw angles.  

• Photocopy rulers: I kept a class set of photocopied (onto transparencies) rulers.  This ensured when I was teaching my class to measure using a ruler that they all had the same measurement and the same measures on their ruler.  Any ruler is photocopiable, however depending on your photocopier there will be better and worse rulers to copy. 

           TIP: Photocopied rulers are only good for measurement, nor drawing.  

• Transparencies for transformations: I cut transparencies sheets into quarters and handed them out along with dry erase markers when we were working on transformations.  Students lavished in the ability to see the original and the transformation at the same time.  This was especially good for rotations when some students really struggle.   

          TIP: make sure the students erase when they are done because the markers will stick permanently if 
          it sits on the transparency too long.  

Some of the tips and tools I have shared are time consuming to start but once they are completed they made my classroom run much more efficiently.  If you have any ideas that you use in your classroom leave them in the comments section. I know I am always looking for new ideas to improve efficiency.