The Positives & Negatives of Working with Integers

Hello mathletes,

This week I had the task of presenting an activity dealing with integers and the pleasure to watch the presentations of 2 classmates on the same topic. During my preparation I learned about some common stumbling blocks experienced by students when they are first introduced to integers and also became more aware of integers in real life contexts that hadn't previously occurred to me. We all go through positive and negative experiences every day, and carry positive and negative perspectives about certain things so isn't life just a series of integers?


 Don't Worry Be Happy by Bobby McFerrin...Is this really a song about integers?


Real Life Examples

• Credits and debits to a bank account
• Negative acceleration experienced by a moving object that is slowing down
• Negative time when you count down to the new year or blast off
• Temperatures above and below zero
• Hockey plus/minus ratings
• Golf scores
• Heights above and below sea level
• Buildings with floors above and below ground level 

Big Ideas

Operation vs. Notation
Before being introduced to integers, students will be most familiar with the + and - symbols in an operational sense being used for the operations of addition and subtraction.  Attaching these symbols to the numbers themselves presents them in a notational sense, as a way to describe the number, which students are likely to be unfamiliar with and may cause some confusion, especially when performing the operations of addition and subtraction using negative numbers.

Value and Direction
When it comes to the value of negative and positive numbers, positive numbers are greater than negative numbers. It will be obvious to students that 5 > 3 BUT when those numbers become negative their values change and -3 > -5.  Integers are also associated with a direction on the number line.  Depending on what style of number line is used (vertical or horizontal), getting more negative will result in moving further left or down and getting more positive will result in moving further right or up. Think of the coordinates on a grid.

Opposites
Every positive number has its opposite in the form of a negative (1 and -1 ; 50 and -50, etc.).  Adding  opposites will always equal zero.  This is called the zero property of integers. In addition opposite operations can have the same effect.  I will try to explain this in detail below.

What I really wanted to focus on in my presentation was to try and conceptualize the idea of adding and subtracting negative numbers. While preparing, I read that a common point of confusion for students when adding and subtracting integers is that they are used to numbers getting bigger or greater in value when they add and smaller or lesser in value when they subtract.  For example, adding 5+5 always increases in value to 10 and subtracting 10-5 always decreases in value to 5. This is not the case when adding and subtracting negatives.  Adding -5 + -5 results in -10 which is lesser in value and subtracting (-5) - (-5) results in 0 which is an increase of value.

With this point of confusion in mind, I first found it easier to not even use the words 'plus' and 'take away'. It was easier for me to use the words 'join' and 'remove'. For example, -5 join -5 made it easier for me to understand that the result was a more negative number. For subtraction,  -5 remove -5 helped me understand that I was starting with -5 and by removing it, the resulting number was going to be more positive. This was a minor change but for myself and students like myself, will help them understand what's happening when we add and subtract negative numbers a little more clearly.

This was a big one for me:

• Adding something positive has the same effect as removing something negative, and
• Adding something negative has the same effect as removing something positive

Did I add milk or remove negative milk?
Still tastes good either way.
Shannon, Adam. (2016, October 28). Milk.

The example I used in class was to think of light and dark.  They are opposites the same way that every positive integer has its negative opposite.

Since darkness is just the absence of light, another way to say this is that darkness is 'negative light'. In class we had 6 rows of lights that were on.  I asked Courtney to turn off 2 and we were left with 4.  This can be represented as 6 - 2 = 4.

BUT, if darkness is just the absence of light, couldn't we say that instead of removing 2 rows of light we added 2 rows of negative light? We can represent this as 6 + (-2) = 4.

If we started with 6 rows of darkness aka negative light (-6) and added 2
rows of light (+2), it would look like (-6) + 2 = -4...we've added 2 rows of light and still have 4 rows of darkness aka negative light.

We could also say that we started with 6 rows of darkness aka negative light (-6) and removed 2 rows of negative light (-2) and that would look like (-6) - (-2) = -4.

Stay thirsty, mathletes.

Keep It Simple Students

Hello mathletes,

The biggest thing I took away from last class was that you don't need to complicate things any more than they are already perceived to be with math. In particular, I'm thinking about the discussion we had about using common denominators when dividing fractions and the story we were told about a previous student who ended up in tears because they were, at one point in their own school days, taught that they were wrong for wanting to use common denominators and that the way to divide fractions was to cross multiply and then reduce.  That student later found out (years later) that either way, the answer is still the same and more importantly, they weren't wrong.

KISS: Keep It Simple Students...not these guys
Shannon, Adam. (2011, July 4). Kiss. 

The unfortunate outcome for that particular student is what I'll call a #teacherfail in that it resulted in the teacher closing off a pathway for the student rather than encouraging the exploration of it. One of the things that has been freeing about this class is the discovery of and encouragement to explore  many pathways to arrive at a particular destination. Certain math truths still exist  (2+2 is still 4 as far as I know) but how we arrive at those truths is up to us. If it makes more sense to you to add 1 + 1+ 1 + 1, go nuts.  If you're more comfortable doing 2 x 1 ÷ 2 + 3, have at it.

           Math truth: 2+2=4:

Keeping it simple doesn't have to mean finding an answer in the least number of or most basic steps. It can also mean doing what you're most comfortable with or what makes the most sense to you. If you're okay with flipping numbers and multiplying as the main operation to find the answer to a division question, power to you. I think we and our future students will gravitate to what is simplest for them to do in order to answer a math question or problem, but it's up to us as teachers to give them the opportunity to figure out what that is for them by providing them with multiple pathways while encouraging and supporting their exploration of them. 

If old school math used 'drill and kill' as a teaching method, consider referring to the new way of math as 'seek and destroy' - seek out negative math preconceptions/experiences and destroy them by encouraging the discovery and exploration of multiple pathways, and teaching math in a way that is doable, sensible, useful. In other words, keep it simple students...and it's okay to make it fun too.

                Seek and Destroy...negative math preconceptions and experiences

Make Math Great Again

Hello mathletes,

One thing we are regularly reminded that educators need to do is work on changing attitudes towards math, and including chocolate is a good place to start.  Our most recent class focused on fractions and I became considerably more engaged when the Hershey's bars were placed on our tables.  The Hershey's book and chocolate bar activity would be a great introduction to fractions for students since it was engaging and memorable. I thought the coloured plates were another great idea from one of our presenters for including manipulatives. 

1 half + 1 half = chocolatey goodness 

Fraction Plates

Another thing we touched on in class was that it is important to teach kids about the big ideas around a topic in addition to just the topic itself.  I personally think this makes sense because, for me,  knowing about what I'm doing makes it more relevant to me and therefore more interested in doing it.  Imagine if you were given a shovel and just told to just dig  without being told anything about the hole you're digging or why you're digging it.  If you were told that the hole you're digging is for the purpose of laying the foundation of a house and that your hole is going to be the start of what will become the basement of the house, you'd understand the purpose of why you're digging and likely be more willing to keep going.  Knowing more about something in general makes it more relevant to me and automatically I become more interested in it.  If i can relate to that thing in some way, my interest increases.  We've talked about it before but I feel like this is so important to do with math, especially because it isn't something that comes easily to everyone.  For those who it doesn't come easy to and have a hard time with it, helping them to understand not just how to add or subtract fractions but also things about fractions - what they are, what they're not, how they are relevant to the students and how they can be applicable in the real world - are ways to engage students more than they would be engaged learning math the traditional way where we all sat in our seats and learned 'the rules' and formulas of math.

A graphic organizer for fractions

One of the ways to teach kids about a topic is the graphic organizer.  In the middle is the topic, fractions in this case.  At the top left is where a usable definition can be determined.  At the top right, characteristics of fractions are identified.  The bottom left is for examples so students can see what a fraction looks like and the different types.  The bottom right is for non-examples.  I think knowing what something is not is just as important as knowing what it is.  This graphic organizer is a good jumping off point for students to become familiar with the math topic they will be learning.

These strategies are a departure from the traditional methods of learning math and I think would be effective to make math more fun, engaging, doable and sensible to students. They help communicate what's important about fractions in different ways and allow students to experience the same principles from different vantage points in an attempt to develop for them a more rounded and full understanding of the topic at hand. 

Good start, math, at trying to become great again.

I Don't Always Do Math but When I Do, I Use Alternative Algorithms

Hello mathletes, 

Last class was a pretty solid lesson on alternative algorithms for addition, subtraction, multiplication & division. Just to quickly recap, there was skip counting to add or subtract, partial addition/subtraction, compensation addition/subtraction, constant sum addition and constant difference subtraction.  I have no idea how I went through my entire years of schooling without being taught these methods. All I could think during class was "why am I learning this for the first time?".  

My favourite concept was the idea of 'friendly' numbers.  Numbers have not been friendly to me in the past so my first thought when I heard this term was "whatchu talkin' bout, Willis?".  After being enlightened, at one point in class we were told to multiply 28 x 25 mentally.  At first, I looked at that and had a brief panic attack.  Once composed, I first turned 25 into the friendly number of 100 by multiplying it by 4.  From there I knew that 28 x 100 was 2800.  After that, I just needed to divide by 4 since I multiplied by 4 earlier in the question and bam! I arrived at the correct answer of 700.  Mind...blown. 

A live action shot of skip counting to add

What we are being encouraged to do, and I completely agree with it, is to encourage the use of alternative algorithms.  They can be thought of as tools in a toolbox.  Some equations lend themselves to being solved more easily with certain algorithms and it's up to us to recognize which tool fits the situation.  For me, trying to do 28 x 25 was most easily done by finding the friendly number and going from there.  In class, each person who spoke up used a different algorithm to arrive at the answer.  So, you could say...'different strokes' for different folks. 

This point leads into something that we also talked about this week in class and which I think is important: respect for diversity of learning.  Not every student will be able to comprehend the world of math with the same clarity.  For some it will be crystal clear & for others it will be clear as mud. Encouraging the use of alternative algorithms gives students the choice to use the method(s) that make the most sense to them.  Teaching them principles like these will do more for them than trying to drill into their heads "the correct formula".  It also inspires a creativity of thought that math isn't especially known for.  It can be a great confidence booster for them once they learn to use these strategies effectively and the confidence that comes from being able to succeed in math is will be what keeps them from running away from it.   

Stay hungry, mathletes...hungry for numbers.