What I've Learned About Teaching... Negative/Directed Numbers with Counters

Two sided counters have massively impacted how I teach negative numbers. Below are suggestions of what I found beneficial for students and reflections on what to improve on in the future. I hope it may be of help to anyone wanting to try this approach to negative numbers. Before I dive in, I must thank Bernie Westacott and Craig Barton whose discussion on negatives I watched avidly in lockdown and ultimately inspired me to be brave and try this.

1. Allow time for students to look at counters and work out the number they represent.

1. Starting with a negative counter (red) representing negative one and a positive counter (yellow) representing positive one respectively, ensuring students know which side is which.
   
   

      positive  or    +1

   
      negative    or   -1

   
   

   Then have small numbers of positives, small numbers of negatives. Then have examples of some of each. This is great on mini whiteboards as a show me the value of this diagram, draw a diagram that represents -2 etc.

2. Develop the concept of "zero pairs" and use the term often

At this point you need to introduce the concept of zero pairs - students will likely be familiar with it but naming it and being clear with what they are helps adding and subtracting later. If you have the same number of negatives and positives you have a total of zero. The negatives negate the positives - you can discuss the word negate and how it relates to negatives in this context. Show some diagrams and ask are they equal to zero?

Why is it easier to spot with some than with others (when they line up)? Might it be helpful to arrange our counters in a certain way? What do I need to change to make the diagram equal to zero?

Make sure at this stage you show how a number can be represented in numerous ways depending on the number of zero pairs (e.g. start with 2 then add zero pairs, is it still 2?). Make sure they understand you can add as many zero pairs as they like to a number and the value doesn't change.

Again this is another important concept particularly when subtracting, so worth spending time with doing some questions with the counters and mini whiteboard work with diagrams. When students are fluent in evaluating counters from looking, confident with zero pairs and happy moving and arranging the physical counters and drawing diagrams of the counters they are ready for the next step!

3. Focus just on ADDING

Adding is when you start with a line of counters and then add more counters, collecting them together and evaluating the result. It is the easier concept for students to work with. Start with positives add positives using the counters to get used to the process.
e.g. 3 + 2

There are five positives altogether so (+3) + (+2) = (+5)  (read as positive 3 add positive 2 equals positive 5).

As students are confident doing this WITHOUT the counters, it is good to get them modelling with what they are confident with before moving on to the concepts they tend to find more difficult. Do a couple and explain to the students it is an important part of the process even though you know that they know the answer.

Next introduce negatives add negatives.
e.g.(-3) + (-4)

There are seven negatives altogether so (-3) + (-4) = (-7)  (read as negative 3 add negative 4 equals negative 7).

Which follows the same kind of logic as above with all positives. Practice a couple of these. Students will start to notice positives add positives mean you have more positives so you have a greater positive pile and that negatives add negatives mean you have more negatives so you have a greater negative pile. Avoid you making these generalisations and try not to encourage them yet, we want them using counters until they are ready to move on from them, so don't rush the counters/diagrams stage.

Then look at a mixture of adding negatives and positives in various forms. Make sure they see that the sum can either have a positive or negative answer.

e.g. 3 + (-2)

There are two zero pairs, and one extra positive counter, so (+3) + (-2) = (+1)   (read as positive 3 add negative 2 equals positive 1)

At this point, instead of continuing in the line, if you are adding a different type of counter (negative/positive) then start a new line underneath. You can ask the students why we might do it like that instead (so they can line up zero pairs and see the answer easily). With counters you can actively take a positive and negative and say they are a zero pair and physically remove them out of the way. But when drawing them, you don't have that luxury - you could cross them out instead like this:

It depends which you/your team prefer.

You also need some examples where the answer is negative. You may need more examples when you are adding two different kinds of counters.

e.g. (-4) + 2

There are two zero pairs, two extra negative counters, so (-4) + (+2) = (-2)   (read as negative 4 add positive 2 equals negative 2)

At this stage avoid generalising – students will want to find a quick and easy way to do it or find a phrase like “a negative and a negative make a negative” which can lead to misconceptions. Allowing students to perform the calculations in a structured way with the concrete manipulatives and/or diagrams will build a concrete understanding. With time and enough practice they will build these connections in a way that means they do not need the counters and can perform bigger calculations more fluently, but do not rush to remove the counters!

Think carefully about the types of question you are asking when practicing too - help guide the students to identify important relationships, such as (-3) + 4 will be the same as 4 + (-3) and ask students why (the diagrams are essentially the same but the other way around).

4. Think of subtraction as removing

Students find subtracting a bit trickier. Start what they should be comfortable with with bigger positives take away smaller positives.

e.g. 4 - 2   which is (+4) - (+2) and you might want to phrase it that way. You might also want to phrase it as "positive 4 REMOVE positive 2"

There are two positives, so (+4) - (+2) = (+2)   (read as positive 4 subtract/remove positive 2 equals positive 2)

Start with one line of counters, remove the ones you are subtracting from the line, and see what is left. With physical counters you can literally remove them, but with the diagrams it is worth crossing them out (as above). Again reinforce we know they can do that calculation without the counters but it is important for the next stages to practice a couple of these.

Then look at more negative numbers subtract less negative numbers

e.g. (-5) - (-2)  which you could rephrase as "negative 5 REMOVE negative 2"

There are three negatives, so (-5) - (-2) = (-3)   (read as negative 5 subtract negative 2 equals negative 3)

The most difficult subtractions to do are when you do not have the counters needed to remove them.

e.g. 2 - (-3)   or (+2) - (-3)  which you could phrase as "positive 2 remove negative 3"

There are five positives, so (+2) - (-3) = (+5)   (read as positive 2 subtract negative 3 equals positive 5)

Highlight the fact that the middle diagram is still worth positive 2, that we can add as many zero pairs as we like and the value doesn't change. To enable you to have the negatives there to remove, you need to use zero pairs. You will likely need quite a few examples of these for students to get their heads around.

e.g. (-3) - 2  or (-3) - (+2)  which you could phrase as "negative 3 remove positive 2"

There are five negatives, so (-3) - (+2) = (-5)   (read as negative 3 subtract positive 2 equals negative 5)

Give loads of opportunity to practice just subtraction, again ensuring you select questions you know they need to see, for example (-3) - 3 is NOT zero.

6.  Practice a mixture of adding and subtracting

Develop some fluency by practicing a mixture of questions, using the counters physically, drawing the counters and when students are ready you can remove the scaffold for them. You can also relate questions to number lines and the direction in which you are moving along the number line. To lead them into generalising when they are ready, questions such as these brilliant ones from Chris McCrane: Alternative representation of Integers – Starting Points Maths 

Top Tips:

·        A visualiser will be your best friend! I love writing the question on the desk with a whiteboard pen and then modelling with the counters. Students can build the question with you and if you can, get students to sit under the visualiser to show how they would do the calculations.

• Don't be afraid of using them with older students - it is a bit more difficult but actually I've done with Y9 and Y10 and found it to have great impact.

·        Use lots of mini whiteboards to check for understanding! Get them drawing the diagrams on their boards, pick out some to discuss. The next step after the counters is diagrams so they need to develop that skill too (they can always draw a diagram in an exam, but they can't use counters!).

·        Don’t rush – it is worth investing time in this early on so you can reap the benefits for years to come. How many Y11/sixth formers still make silly errors due to negatives? Worth the time investment early on.

·        If possible, use this approach as a whole department so there is consensus and students have the consistency of using the same method.

·        Avoid ‘simplifying’ with phrases like ‘adding a negative is the same as subtracting’ but instead encourage students to either do the calculation with counters or diagrams instead. They will then build their own connections based on understanding.

·        Play with the counters yourself first, completing questions and getting really used to them.

• The counters are a scaffold, so they do need removing eventually (imagine doing (-345) + (-45) with counters) but they can always go back to a diagram as a bit of a nudge by doing something smaller like (-4) + (-3) what happens? How can we apply that to (-345) + (-45)?