I’ve never had to be the teacher to introduce my students to negative numbers before; as an 8th grade teacher, I assumed students had at least been exposed to negatives before (and operating with them), even if they weren’t proficient with the skills. Now that I’m going to be teaching 6th grade, I’m going to be (presumably) my students’ first introduction to negative numbers.
I’m trying to figure out how to introduce them. I think I’ve decided on one task that I definitely want to do as a “pre-assessment.” The way that it works is as follows. Give each student a post-it note and have them write their name and their response to an open-ended prompt, like “What do you know about negative numbers? Write down everything you know on this post-it.” This will give me an idea of what they already think they know (if they know anything). Then you ask students to get up and place their post-it note in one of three places: “How well do you think you know it?” “Strangers, say hi in the hallway, besties” – three labeled groups to describe how well they feel they know the topic. I think this will actually give me a really good idea about what my 6th graders are coming in with.
But from there, I’m a bit conflicted about contexts, models, representations, tasks, etc. I’m going to need to teach my students about absolute value, coordinate planes, and all four of the operations in the end (plus presumably, comparing numbers). But I’ve been reading a lot of blogs that are giving me conflicted ideas about what works well for students and what has limitations.
Let’s talk about some of the contexts I’ve heard and how much I like them first:
- Time zones are all plus or minus from GMT-0. We can talk about the idea of people calling people in other time zones and needing to add or subtract time to make the difference. To some degree, this works on a bit of a number line (and we can deal with the ideas of how many hours ahead/behind are different locations relative to other places that are not the zero?). I like this context, and I got it from the Math in Context series.
- Elevator arithmetic from NCTM’s website – https://illuminations.nctm.org/Lesson.aspx?id=4075 I like the context, and I remember using this before. I find the parallels between the floors and the vertical number line to be helpful in seeing the numbers aligned along a y-axis-like line. I don’t think it’s super real-world, because most buildings using basement, sub basement, etc. rather than negative numbers. On the other hand, living in NYC, most of my students will have experience with buildings that are tall and taking elevators.
- Temperature – I like this one because I think it’s pretty realistic (we do actually use negatives to describe temperature), but I think this one may be better for just descriptions, rather than lots of operations with it. I also think because I live in NYC and we use the Fahrenheit system, I’m not sure that my students are super familiar with negative temperatures – we don’t have a lot of very cold days in the winter.
- Altitude above and below sea level – I like this to some degree, because we can talk about submarines (which fascinate me), but it’s not super connected to my students experiences. I do think there’s likely to be a question on the state exam set in this context, so that can be good exposure for them.
- Points in a jeopardy-like game show. I know CMP 3’s accentuate the negative unit starts off by having students analyze some scores from a game, and there are negative scores. I think this one is totally fake-world, because it’s a contrivance that scores go negative like that – on the other hand, I think it’s a perfectly reasonable contrived context.
- Directions on a map – I’m not positive about how to fully visualize this one, but I think I like the idea of vectors corresponding with directions (especially if they correspond with the coordinate plane), but I’m not positive how this makes sense for operations.
- Debt and negative money – I dislike using negatives here because I think it can be confusing about what the negative represents. If I owe money, I have negative dollars, but if I gain money, I have positive money.
Beyond contexts, there are also models and representations that student may use to perform operations on negative numbers:
- Number lines are in my opinion the most important representation, and I like the idea that the symmetry can be emphasized (i.e. if you fold the number line at 0, the 1 and -1 line up, and so do all of the other opposites). I think adding and subtraction on the number line can make sense (though I think subtracting a negative gets tricky to make sense of and there’s an element of needing the signs to indicate direction), but I’m not sure how you could represent multiplying and dividing on a number line…
- I also wonder how the negative and positive chips (or algebra tiles) work out for students. I know a few years ago, I used algebra tiles with my students to deal with “zero pairs” and being able to eliminate the pairs helped my students a lot to see the value. I also like the idea of “adding” zero pairs (to not change a number – i.e. adding 0) to emphasize that you might need to take away something you don’t yet have. But I think there’s also misconceptions to be had from these as well.
- I also think that the coordinate plane provides a rich representation, building on the number line. I wonder if there’s a way to use the two coordinates to help represent multiplication.
- Kate Nowak was drawing some vector type diagrams with arrows (for adding and subtracting with signed numbers) that intrigued me.
I know from having taught 8th grade math that many of my students still struggled with negatives. In fact, two years ago, I had one of my former students doing some volunteer work for me. She was in 10th grade at the time and had been a strong math student even in middle school. She was helping me by making an answer key to a solving equations packet. She was showing all of her work and checking her answers. She came across three or four that didn’t check and she couldn’t figure out her mistake, so she brought them to me. In ALL of the cases, she had made a mistake with her integer operations! FOUR years after first learning about integers, she was still making mistakes. As I recall, most of the mistakes were dividing by negatives and subtracting negatives. I know this is a tough issue for students (and mathematicians in general!), and I think it’s not something I’m going to figure out perfectly the first year around!
Any ideas or thoughts about how to introduce integers in general, absolute value, and operations with integers to my sixth graders this fall?