Mem Fox starts kiddos thinking about numbers with her well known and loved book, "Let's Count Goats". Kids love to participate in telling you why something is going to happen the way it does. You'd think that number sense would be well developed by the time my 8th graders get to me, but there's still more to do.
Phase One: Equivalent ways to express answer
This is a pretty simple idea but it makes a big difference. When we are working with writing equations, for example, I will always play the straight man. So if someone says they have 1.5x, I will say, "Oh, I don't think that's right because I have 3/2x." And then wait to see what happens...
It used to be that students would dutifully mark their 1.5x answer wrong!
Now they're smart enough to stop and consider what I've said....at least a little bit. Once they caught onto that I then had to intermittantly make some conversions that weren't equivalent. After almost 4 months of this kind of embedded practice, students are getting pretty strong in their number sense and I can hardly even stump them....and now they love to ask the question, "Mrs. Ratzel I don't have 5.2 for that answer, I have 5 and 4/10. " And they wait to see what I say. Sometimes we have pretty good conversations about simplifying answers and what makes the most sense in that particular situation. But the real deal is that I have them sucked into mathematical conversation and dare I say, mathematical thinking.
Phase Two---Extending this idea to distributive and additive identity property
Once I saw this was effective, I started looking for other places to embed number sense practice...mostly with the distributive property. I picked it because it is the property where we are the weakest. I know you can already predict where I'm going to go with this.
Right now we're just starting functions....and we're using images of function machines that say things like "multiply by 5" and "then add 2". Simple enough to write the rule 5x + 2. Then we move onto function machines like "add 2" and then "multiple the result by 5". So students immediately write (x+2)*5. So we work a bit on rewriting it as 5(x+2)....and then I start to throw in my distributive property jabber. "Oh, I got 5x+10 not 5(x+2)".
Sadly they dutifully marked their 5(x+2) wrong. But this time, it only took me doing that one time and they were wise to be on the lookout. And we've had loads of opportunities to practice the distributive property. Won't it be fun when we are going to be able to go the other way?
The other variation of this is writing answers. Since we're sort of in the second year of their writing equations experience, they are very apt to write things like y=4x+0 when the y-intercept is 0. It's taken lots of experience where I tell them that I had the answer as y=4x. I really waited to start doing this until I was sure they had the idea that +0 was really there, but just didn't need to be written because of the additive identity property.
Maybe not the most thrilling way to work on number sense, but very effective
OK...so this isn't the most thrilling idea or biggest ah-ha moment any one has ever shared with you. But the real deal for me is that it has been incredibly effective...much more than I think I'd ever get with loads of practice problems that weren't attached to the other work we were doing. So....if haven't already caught onto this idea, I'd really recommend give it a try.