So I’ve been thinking about how to introduce my 6th graders to negative numbers, and what I’d like to do with them over the course of the first few weeks of school. I’ve basically decided that no one model is sufficient – they all have their limitations, and students are going to need to use negatives in a variety of contexts. Certain things will be more complex than other skills – for example, subtraction of negatives is harder than adding negatives, and explaining why the product or quotient of two negative numbers has challenged many.
I’m also thinking about how to teach for conceptual understanding and how to avoid tricks and shortcuts. One of the things that I find so difficult to wrap my mind around is teaching, even on the MTBoS, who devote tons of instructional time to teaching songs and other mnemonics to remember procedures and algorithms! I want things to make sense! (And I will admit, the operations with integers are not easy to wrap our minds around – especially when you consider how long it took humans to accept their existence!).
Anyway, here’s the overview of how I think I’m going to launch negatives with 6th graders:
- Start with a situation sort according to some categories (to be defined by the students): http://math4teaching.com/2010/04/27/a-problem-solving-approach-for-teaching-positive-and-negative-numbers/ I like the idea that the students can debate and discuss the categories here too, which can help reinforce the norms of how we do math class in my room.
- Then I think we start sorting numerical expressions (with some that are positive and some that are negative): http://math4teaching.com/2011/03/15/introducing-negative-numbers/
- After having done some number talks with dot patterns during the first week of school, I hope to begin arithmetic with subtraction, and I know in the Building Numeracy book, there are some good strings around negatives within subtraction (basics like 8 – 5 vs 5 – 8). I also think using the number strings to highlight the differences between thinking about subtraction as “take away” vs “distance” can support the idea of subtracting negatives later on, so I want to be sure to include those types of number strings early in the school year (those probably come first, though I thought of them second). I read a powerful article about how mathematicians (in the history of developing the concept of a negative number) were ready to accept a negative as a solution (the answer) long before they were ready to accept a negative number within the question. I think this will be a good technique to build in.
- I can’t decide whether those first two activities should be one class period or two, nor can I decide how long they’ll actually take to do in a group. I’ll need to sit down and plan the lesson in more detail at some point. But based on how long it takes us, I want to do an exit ticket early on, where I ask students “What do you know about negative numbers?” give them a few minutes to write, and then rank their understanding in one of three groups: “Strangers, Say Hi in the Hallway, or Besties” – this will give me some sense of what they already learned in their elementary schools and how proficient they feel about it.
- The next thing, I think might be to use the launch video from CMP 3’s DASH website for Comparing Bits and Pieces Investigation 3.1, and ask students what they think goes to the left of zero on the number line. The investigation can be tweaked to emphasize the symmetric structure of the integers, the positives and negatives (how zero is neither, but instead the base), and the idea of opposites being the same distance, but opposite direction. This also can include the idea of absolute value and the idea of distance.
- Somewhere in here is where I’d like to brainstorm situations with students where negative numbers could make sense. Sometimes, I think negatives make sense as representing change, even when they don’t make sense as having negative of something. For example, you can’t have a negative population for a city, but you could have “negative population growth” – however, I think this is probably more complex than we need to discuss at this stage. It seems like the ideas of temperature, elevation, credit/debit, electric charge are specifically mentioned in the standards, and I know both my school’s unit test and old state exam problems use the context of jeopardy-style games with negative points. I think there’s some work to be done here with translating between the idea of the situation and the numerical way to represent it.
- I think in this section is also probably where we’ll do some work with comparisons, especially with regards to numbers having bigger absolute values vs. being bigger numbers (i.e. farther from zero). I’d like to discuss the number line’s structure (bigger numbers go further to the right and smaller numbers go further to the left) along with ideas about placement of opposites (i.e. equal distance from zero but in opposite directions). I’m going to need to create activities for these or find some good ones. I also think developing some kind of foldable numberline for the students to use and touch and feel is important, as it will help to emphasize the symmetry around 0, and the pairing of the opposites.
- I like the idea of using time zones next as a context (even though I don’t see it used often) to begin to think about shifting and adding/subtracting. My cousins really do live in Israel, and I have to figure out how many hours apart we are. In EST, we’re at UTC-5, whereas in Israel, they’re at UTC+2, so they’re a total of 7 hours ahead of me, but I’m 7 hours behind them. I have some problems on this context from the Math in Context series from Encyclopedia Britannica.
- We’ll need to write some inequalities using negative and positive numbers as boundaries. I don’t have any good ideas for this section of the unit yet.
- My school does all of the operations in 6th grade (for better or worse). We’ll probably move from the basics of representation to discussion of some models and the beginning of operations. I think addition is probably the simplest to understand, so I’ll probably begin there. I want to make connections with models like the number line, algebra tiles/integer chips, and story problems. I’ve got some resources for this from CMP3, but I’m not convinced that is sufficient. I’ve also got the Engage NY resources from 6th & 7th grade.
- I’ve got some links to various websites within the MTBoS, but there’s so much to review, I don’t know that I could decide among it all “in time.” This does a pretty good summary of some major ones: https://alg1blog.wordpress.com/2012/07/29/some-approaches-to-negative-numbers/
- I think I’ll probably do addition, subtraction, then both, and then multiplication and then division, and I’ll probably chunk the different operations, taking up several day on each. I know I want to use a combination of story-telling, sense-making, model-use and representations, and then we’ll also use patterns to convince ourselves/deduce what should be the product or sum, etc. I want to work on predicting the SIGNs of the answers first, before we delve too deeply into the nitty gritty or coming up with a procedure. I have to think more and evaluate the resources more before I make a conclusion about this phase of the unit.
- Once we’ve done some integer operations, we’ll develop an understanding of the four quadrant coordinate plane. I know from teaching 8th grade that it’s not intuitive that after 0 we always have a symmetric 1 or -1, so I’ll want to be sure we emphasize that. I’m also not positive how to motivate the NEED for four quadrants yet! I’ll have to think about that too. I HATE HATE HATE the overemphasis on drawing pictures from plotting points because when they get to 8th grade, and we just wanted to know whether they could follow the coordinates, I had students who wrote “Where’s the picture?” when it was just literally random dots. Plus, we’ll do plenty of that for the unit project.
- The students will also do some work with reflecting points over the axes and finding distances on the coordinate plane when it’s across quadrants so we can connect the idea of subtraction to distance between two points again!
I know how long I’ve spent thinking about this first unit, and how lucky I’ve been that I have all of this summer time to study up on the topic, find best practices for teaching it, and ensure that I know the skills embedded within the unit itself. I’m nervous because this is only the first unit, and I haven’t done this level of investigation on any of the other units yet! And once the school year starts, I know I won’t have as much time to do all of that! 😐 At least I’ll have more regular contact with other teachers who’re teaching the same/similar content, and I’ll be able to pick their minds or use their resources.
With visions of negatives dancing in my head, I’m off to bed!