Wednesday, November 30, 2016

Do Two Minuses Make a Plus? (1)

I've been into a few Grade 7, 8 and 9 classes over the past few weeks and one of the areas of focus has been integers. It's no big secret that integers can be problematic for many students (and, I would add, adults too). Chatting with some students, I heard things like "Two minuses make a plus": is this really true? I saw students use this rule to then say that ¯3+¯3=6. My concern is that students have developed misconceptions because they have skipped straight to the abstract part of the concrete-diagrammatic-abstract part of that continuum. So here are some activities I used to help them through the concrete and diagrammatic phases to allow them to make sense of the abstract:

1) My first goal is to get students to be able to add integers. Like many teachers, I use integer tiles to represent integers concretely:

I tell students that this is the most important concept: if the red tile represents negative 1 and the yellow represents positive 1, then together they make the eponymous 'zero pair'. Anytime they see a zero pair, it represents a value of 0 (even though you see two tiles).
I then ask students to use the tiles to represent quantities such as ¯3 and 5:

Then I ask them to use these two sets to work out ¯3+5. What is useful here is by the actual act of addition (i.e. joining the two groups) we get something like this:
I then give other examples for the students to work:
One of the things I'm aiming to do here is to get students to visualise the solution. For example, in the fourth question above, students (who were using tiles for the first three questions) were able to picture (mentally or with a bar model) 19 yellow tiles pairing with 19 red tiles (giving 19 zero pairs) leaving 4 red tiles, or negative 4. 
After enough of these examples, I ask students to complete this statement:
Adding a negative is the same as ...
Before too long, there is consensus that it is the same as subtracting a positive. This is much better than saying 'a minus and a plus makes a minus'.

2) Once we have established that adding a negative is the same as subtracting a positive, I then like to show how we can use a number line to show this:

Some students really like this way, others less so but I let them choose what method works best for them.

3) Subtracting integers can also be modelled with integer tiles but I've always liked to introduce it in this way. First, I call up three students and give each of them four cards. Two of them have cards numbered 1 to 4, whilst the third has cards numbered ¯4 to ¯1. I then tell a really, really corny joke such as "What do you call a fly with no wings? A walk!" and ask the students to give me a mark for my joke. For example, I might get this:

I feign disappointment at the student who has given me a negative score and the class has a good laugh about this. We then get my total, in this case 4+2+¯3 gives 3. At this point, I make a strong protest against the student who has given me a negative score to such an extent that I demand that this score is removed and I 'escort' them to a corner of the room. Again, laughter all round. I then go back to the remaining students:
I then ask the students what my 'new' score is and they tell me 6. I then say, "We started with 3 and ended with 6. But what happened in between?" The response is usually "You took away the negative 3." I then write down what they have said:
I repeat this for a couple more bad jokes then ask them to look at the results and ask them to complete this sentence:
Subtracting a negative is the same as...
Again, before too long, the consensus is that subtracting a negative is the same as adding a positive. This is mind-blowing for some students and I tell them that I can understand why they might find this counter-intuitive but this is because they have always believed that subtraction makes smaller. However, the activities have shown that subtracting a negative will make a number larger: like it or not, we cannot argue with this beautiful mathematical logic!
It is possible to model this with integer tiles:
Notice that we cannot 'take away' negative 3 because we do not have three red tiles in our model. We can overcome this though, by adding enough zero pairs:
Now we can take away the negative 3:
Some students like this approach, others less so. Either way it reinforces the big idea that subtracting a negative is the same as adding a positive. This is so much better than saying 'two minuses make a plus'. Once again, I follow this up by getting students to try some questions that will wean them off the concrete and getting them to think diagrammatically and abstractly.

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