My middle-schoolers are coming along nicely in their conceptual understanding. I would like to share one project that my 7th-graders are working on currently. It's an old ratios assignment from when I taught middle-schoolers way back when, but still one that gets kids all riled up and excited, apparently.
First, I want to share a neat little trick with teaching ratios. My first and second years of teaching, I had a lot of trouble getting kids to "see" how to convert part-to-part ratios (such as girls:boys = 3:2) to part-to-whole ratios (such as girls:total = 3:5). I used all these hands-on manipulatives and guided exercises, but to no avail! My third year, I finally figured it out. All this time I had been trying to explain ratios, when instead I should have been letting kids observe ratios, because ratios are an intuitive concept!
So, nowadays I teach ratios problems using tables of values. The tables always start with zeros, because I want to continuously reinforce the connection between direct variations and ratios problems.
I've noticed that this way, no matter if the kids are starting with a part-to-part or part-to-whole ratio as given in the problem, they can always manage to find the other missing parts. And also, they can see how every part (or column) scales equally, thereby leading naturally to the idea of a scale factor. When the problem then asks, "How many boys would there be, if there are 966 girls?" they can figure out easily which column 966 goes into, and then extend (fairly easily, with a bit of practice) the idea of scale factors to find the other column values. --And this is with my Grade 7s, who had never before seen any ratios problems! In the past, when I used the same method with my 8th-graders, I didn't have to teach them anything at all; they could just observe the pattern and figure out the scale factor shortcut when they felt tired of extending the table tediously.
Anyway, following some practice of the table method, I finally introduced cross multiplication. The kids were mad at me; they thought that setting up and solving proportions was way more difficult than making the tables! I made them practice both methods so that in the future, should a teacher require cross multiplication and/or the problem involves decimals, it'd still be in their arsenal. But, I am happy that their conceptual understanding is strong enough to want to replace the cross multiplication.
In any case, back to the ratios project. The kids are given the specs of an original shape and partial specs of a new shape. They need to use their understanding of part-to-part and part-to-whole ratios in order to find the scale factor and to scale all sides appropriately. In the end, they measure and cut out both versions (initial and final), and in groups of 3 write out an explanation of the entire process -- which everyone has to agree upon -- and create an explanatory poster.
I like this project, because it allows me to differentiate easily. Some groups have scale factors that are whole numbers, and others have scale factors that are unit fractions (ie. 1/n where n is an integer), while others have scale factors that are non-unit fractions. Since they're not told what the scale factors are, finding it can be a bit tricky when it is a non-unit fraction.
Anyway, so far, it's going swimmingly! The kids have almost all finished measuring and cutting out their two shapes and are working on the final written explanations. If that sounds like fun to you, here are the project prompts I used this year (excluding the instructions for the posters) and the warmup exercise I used to introduce the idea of geometric scaling.
For Grade 8, we are wrapping up two application assignments of the functions we have been learning. I did Dan's cup-stacking activity with them and they enjoyed it immensely -- 3 or so groups came within 1 cup of the final result! I also did a traditional profit and revenue assignment with them. It was the first time I felt like we were getting into complicated application, and although it surely challenged them conceptually, I could see that some of them, at least, enjoyed the discussions about why it makes sense that profit and revenue are both shaped like a parabola.
Today, I taught them to graph linear functions. But I did so by tying it firmly into the meaning of slope and y-intercept inside a word problem, so that every time they look at an equation like y=3x/5 + 2 they will think to themselves that 2 is the starting value (such as # vacation days at the beginning of the year) and 3/5 is a rate -- 5 describing "how often" it happens and 3 describing "what actually happens." (For example, every 5 months you gain 3 vacation days.) It was the first time I did graphing by connecting it to my Mad Libbs worksheet on interpreting linear functions, and it worked really beautifully!!! Not a single kid was lost in the graphing of equations, when they were just thinking about the meanings of the rate and the starting value and putting that into graphical form. Kids were even figuring out for themselves the idea of rise and run in a fractional slope, even though they had never learned this before!!
:) Yay to experimenting. Small changes, big impact!
Addendum 11/16/11: Today I gave my kids a bunch of graphs for them to write equations for. Using just their understanding of the meanings of parameters inside the slope-intercept equation as related to word problems, they were able to easily look at graphs and to write those equations without messing up! This has been the smoothest teaching of writing equations EVER.