Class X intervention: Prove it!

Please read the other two posts
1) Why intervention?
2) First few classes
if you care for continuity.

I'm consolidating the key things that happened over the next two classes.

The next topic within geometry was on properties of chords, tangents with circles. For a transition from triangles to circles and I tried to connect up the two by showing them that the x coordinate of a cow going in a circle at constant speed is a sine wave :). This was way beyond the syllabus and had the teacher browse through the textbook to see where this 'stuff' was. It was a quick transition that touched on trigonometry and gave me a chance to plant a seed that I would water in my electronics class with them as I introduce AC signals...

In my preparation I realized that they were supposed to know some really neat stuff with chords and circles in 9th grade. An important one being given a chord any triangle it makes with any point on the circle (on one side of the chord) has the same angle. I tried proving it myself, I couldn't. I didn't find the 6th grade textbook either and finally went though the proof on khanacademy. It had a proof that the angle (subtended by the arc) with the center is double of the angle at any point. Nice, I didn't remember this.

In class I mentioned the theorem and the kids and went a step further and said that the angle with the center is double of the angle it makes, they said that they didn't know it.  It wasn't enough that they knew now they wanted it proved :). I could not believe these are the same kids who would take a formula without question and were afraid of proofs 5 classes back.

Its a cute proof that involves a specific case of a triangle made with the diameter to make an isosceles triangle and then generalizes it by splitting any angle as a sum (or difference) of two angles involving the diameter.

I proved the diameter and then gave an example of splitting a random angle as a sum of two angles. They were not convinced that this was general enough and came up with a point where the angles could not be added (point A - a limiting case of summation). I gave them the proof by subtracting the angles. The wheels in their heads were really whirling now. Then found a point on the other side of the chord (point B) and I told them that it was indeed not valid on this side o the chord and we would talk about it later.

There are only a few proofs that are going to come in the X class exam. The advantage was that I could still prove everything but didn't have to write it out in full on the board. We looked at the figure and I made my case by walking them through the thought process while marking on the figure. Its a real class and there are kids that zone out.

 The good part is that since I didn't need to write things out it doesn't take time to repeat the proof from where they didn't follow it. There may still be one child with a glazed look and when you ask him/her what he/she missed would come back with 'didn't understand anything'. In such cases you just start at the beginning, there is a circle and a chord. They come back with Dah, yes I know that, and then you build on it. I have done a proof at max 3 times orally before glazed looks are replaced by knowing ones.

I also needed the opposite angles of a cyclic quadrilateral add to 180 which I again mentioned. By now they were trying to get all that they didn't in 9th grade and asked me to prove it. I drew the angles required for proof and told them they could do it themselves.

We found similar triangles in the intersecting chord theorems (it would have been nice if they had asked them to prove it in the book). Here the two triangles APD and BPC are similar and their sides are proportional.

The atmosphere in the class is tremendously different from what we started in the beginning of the sessions. There is a lot of questioning regarding principles. The children are engaged and there is very little idle chatter. When it is there. I pause the class for the children to finish and I have reaffirmed that its not because they are disturbing the class, its because the rest of us don't want them to be left out. The children also listen when another child is talking and don't try to cut them off.

We are done with the 'theory', now to focus on application of the same.

Classroom intervention Grade X: Geometry

The problems:
1) Where do I start?
As I went through what they had for Geometry I realized that although they had done coordinate geometry for a couple of yrs, trigonometry for a couple of yrs, they were now going to encounter the idea that triangles can be similar and their sides would hold ratio between them! I also noted that there was no proof given in the text for the same :). Great, I can start with one of the most powerful concepts in geometry that allows me to link with everything they kida, sorta know of and I don't even need to prove it.

2) The kids may remember the concepts but not the corresponding names e.g. adjacent angles, corresponding angles. They would, of course, remember vertically opposite angles, but who doesn't. This gives rise to the issue that we can't talk the same language.

3) The kids also know me to generally have fun with puzzles and games and I need to have some connection with writing examinations.

Class 1
I started with asking kids to make a cheat sheet  (bit paper as the kids call it) for Geometry by putting in all that they know about it. The kids were excited by the prospect though I made it clear I was not encouraging them to take it to examination.

We started with a small list which was enough for our purposes and decided to add more as we went along:

I added one new one of similar triangles, with just one rule (AA) if two angles of two triangles are equal the triangles are similar. In similar triangles there is a constant ratio between any two corresponding sides.

I mentioned to them that the similarity of triangles is the most powerful of the geometrical theorems and now they can rule the world. We started with trying to prove the 'mid-point formula'.

Actually, if you can understand the picture the steps are simple, but I went through many iterations of building up this picture one piece at a time. The biggest bottleneck was that they didn't associate the coordinate in coordinate geometry with a real distance. In a point (x1,y1) they were unclear as to what x1 really was. It had just been a number they needed to substitute in an equation. Well, this was precisely what I wanted to address and I didn't mind going over it again and again. Once that is understood it was actually quite easy for them to take up the new concept of ∆CBD ~ ∆CAE and were even able to supply be the reasons (adjacent angle, right angle). As 

CA/CB=2 (mid point) 

 (x2-x1)/(x2-x)=2

=> x2-x1 = 2*x2 - 2*x 

=> x2-x1-2*x2 = -2*x

=> x2+x1 = 2*x or x = (x1+x2)/2

I proved it for x and I asked them to prove it for y at home. I completed the picture as below.

I asked them to complete a couple of problem sets in the book and using similar triangles prove Thales Theorem as well.

I worked out a couple of questions from two exercises and asked them to do the rest by themselves.

Class 2

I didn't expect everyone to do everything, but I expected someone to do something, and they did.

Four kids (in a class of 16) tried and succeeded in proving the mid point theorem, few others attempted the exercises in the book, as they now looked trivial. Few read the book by themselves for the first time to see what's really there in it :). However, there was still half the class that was waiting for the dust to settle and for me to solve the questions on the board.(ha)

I proved Thales theorem using similar triangles and asked them to add this to their cheat sheet as well.

I asked them to continue to solve the problems in their book. It didn't matter that they had done half the exercise or looked sheepish that they hadn't started. They all had to work and work independently.

Its surprising how noisy the classroom usually is, its not the kind of organized chaos the teachers permit its the kind that comes from boredom of kids who are able to solve faster and waiting time of those who don't follow for the answer to be written on the board.

They could not talk or borrow pencils/pens/erasers/scales or any of the other million things they seem to always need in the classroom. The instruction was, use what you have, do what you can and find out for yourself what you are capable of. It was not easy, they had not done this except in examinations. I needed a more supportive setting than an exam and I gave occasional individual attention, at times needed to remind a couple of kids about what we are doing, but we were able to get some peace and quiet in the class (another first). I went around to see how kids were doing and for once compromised and let kids know when they had got the answers. Almost 4-5 kids other than the ones who had already done some work were very kicked and would say, is that all there is to it.

This class was a very big achievement, kids worked and to their ability succeeded in solving some problems and started believing in their ability to do so. The most questions solved were close to 10, but even the least were 3. Usually 4-5 questions are covered in class when the teacher solves it on the board, not bad.

Class 3

It was much easier this time for the kids to settle and work this time. I addressed a couple of common issues encountered in the previous class. We talked about the common strategy in the questions to have multiple triangles and a common side that was divide in the same ratio as the others. We addressed ideas for cute questions like ratio of sides of a trapezium that needed an additional line to be drawn to see the triangles.

We also compared altitudes of similar triangles and proved that they are the same ratio (of course, using the idea of similar triangles). This brought us to right angled triangles and all the physics they had done of comparison of shadows of objects and their heights. We talked about how these triangles are the same and what would happen if the time or place of one measurement is different from another and what the assumption is to say that the angle of incidence is the same. We concluded that the rays of the sun need to be parallel between the two objects, times and places of the measurement. Our misconceptions are buried deep, so for yucks I drew the sun, as we drew as kids with rays coming out in all directions and pointed out that this in such a picture the rays don't seem parallel and any bright objects gives light in all directions :). It was a fun, after a couple of intense classes this whole exchange seemed just what we needed.

We then cover altitudes of right angled triangles that create smaller right angled triangles that (needs rotation carefully) are similar to the original triangle, which further can have altitudes giving similar and smaller right angled triangles. 

This was a fun exercise and we really needed to be sharp about our rotations. We finally figured out the algorithm to get things right. Start with the corner where the angle is unchanged, mention the corner with the right angle next and finally the corner left out.

I found a really cute problem in the book. Try it out:

Find the other diagonal AC given AB=BC, AD=CD and the right angled triangles marked as above.

Classroom intervention in Class X: Why?

At Udavi school I observed math classes for about three weeks. This gave me a good idea of what was missing and which children were having the toughest time. The children in 6 and 7 grade had the most difficult time. They were expected to be comfortable with fractions, decimals, the corresponding arithmetic operations and required to abstract in algebra, etc. They had also started quantifying their learnings in science including knowledge of speed, density, acceleration, light years, etc.

By the time children get to Xth grade abstraction is assumed to such an extent that a teacher would work on coordinate geometry for 1-1/2 hrs without drawing much on the board. When something is drawn is called a rough sketch e.g. to demonstrate mid point of  (x1,y1) and (x2,y2) a straight horizontal line would be drawn with these two points and (x,y) would be the mid point, given by the formula...x = (x1+x2)/2, y = (y1+y2)/2.

I'm not convinced that we can assume that the children have already developed this ability to abstract as well. Here is an example of what I experienced:
The teachers were kind enough to give me 10 -15 mins towards the end of a 1-1/2 hr class to do 'what I wanted'. Generally, I connected concepts they had learnt in different Math classes (e.g. geometry with algebra) and sometimes beyond. 

After a class of coordinate geometry, I had talked about the characteristic of a straight line of holding a ratio i.e. from a point on the line, if you move some distance x away then a point on the line moves a certain distance y away and this ratio holds, so if you move 2x away then point on the line would have moved 2y away. They had been able to co-relate this to what they had learnt from Physics of how the image of an object keeps getting bigger as they moved away from the focal point.

After a little more than a week I enquired about the characteristics of a straight line are and after the customary shortest distance between two points (ahem, not what I'm looking for and that's a line segment), has no beginning and no ending (my Goth! sounds philosophical) one kid says hey it holds a ratio. By now other kids are going, oh yeah that's obvious, is that what you wanted? 

I probed further and let them jog their memory about lenses and their images with distance and one kid even brings up a shadow (whao, more material for later). Then I asked if they knew what this ratio was called. Of course, they came back with, but you haven't told us (and of course, I had, but you can't expect them to remember a name when they just found out that Physics may tally with mathematics in 10 mins!). I let them know that they know this name, now  they give me every name they know associated with geometry till my previous discussion (ray, angle, line-segment, co-linear)...but no slope. 
This was their second full class (1-1/2 hrs each) of learning and solving most the exercises in their text regarding slopes with the formula m = (y2-y1)/(x2-x1). I was talking at the end of the class they had learnt that lines are parallel if m1=m2 and perpendicular if m1*m2=-1.
When I finally told them that this 'ratio' is slope, they were shocked that it was something related to what they were doing in class.

After the observation period I have been working primarily with 6th and 7th graders, but the above instance convinced me to stay involved with the Xth grade math for one slot in the week. I was pleasantly surprised when one of the teachers asked me to take geometry for them. I wondered if it was because I drew so much in algebra classes. He said because the students four it very dry. Whatever the reason, I accepted the offer this became my Math connection with the kids (I also have one slot for electronics)

Public (state board) examinations are conducted for the children of the Xth grade. These examinations are taken seriously and decide if you will be allowed to pursue science, commerce or arts (usually in that order). The questions in Mathematics are picked up as is from the exercises in the book (apparently, verbatim).
Accordingly the present teaching methodology is that the teachers parse the text and peel off any proofs, theory to boil the lesson down to a set of formulae. They sometimes put some context to the formulae (what will be given and what will be asked) and then proceed to solve all the problems in the  exercises. The teacher may even try to give some time for the 'brilliant' students to solve the exercises, but soon the teacher relents and solves the exercise on the board for the benefit of all children. At the time the teacher solves the problem the children are expected to pay attention and not copy (yet). Copying happens soon after, often exceptionally well.

I went through the geometry text and realized that it had 'theory' and proofs. The teachers are a little squeamish about proofs, the children freeze on the same. Each theorem has corollaries, converse theorems and builds on what is done before. I was going to enjoy myself and the children are going to be blank, unless,...

Math/Science with 7th graders (2): Pendulums

I decided to introduce pendulums this week and see what Math and Science it throws up. I found a small section on pendulums as part of the chapter on measurement of time (p.g.184-186 7th Std) and decided to pursue it. Any measurements or activity take a lot of time and I decided to let it take its time, find the doubts about different aspects and see if we can talk about them some more.

On the first day we spent some time figuring out how to go about our experiments. We collected stones, found some thread and tied it to a meter long scale that would rest across two tables. It gave us around 70 cm from the floor and that determined the limit of our experiments. We started our experiments with a 60 cm long thread.

I clarified the goal of the measurement i.e. a period which is the time the bob takes to go from one end to the other and get back. I asked them to make guesses in the experiments to exercise their intuition and explain why they picked up these numbers to explore their assumptions. Its interesting that everyone assumes that the period of a pendulum will be one second irrespective of the length...

Kids	Small Amplitude	Large Amplitude	Large Weight
Arc	2	2	1.3
Sha	2	2	1.2
Var	1	1.9	1
Sub	1	2.5	1
Pri	2	2.5	1.2
Actual	1.6	1.62	1.56

(**All measurements and guesses in seconds)

The experiments we were able to complete on the first day were the impact on the amplitude on the period and the weight of the bob on the period. They had anticipated that the larger the amplitude the longer the time period, since longer the bob needs to travel. They had also anticipated that a heavier bob will have a shorter period as it has more 'strength'.

We did the measurements using my digital watch. Apparently, it can count to a resolution of 1/1000 s, though I doubt if our triggering accuracy was any better than 1/10 s. I made the first couple of measurements, but following that most of the measurements were made by the kids themselves. To squeeze in division of decimals I suggested that we could 5 periods, the logic being that the amplitude of the oscillations may change significantly beyond that. The children throughly enjoyed the discussions, guessing and even the decimal calculations that came as a result of their experiments.

Its a effort to tie thread around a stone and it sometimes breaks or loosens up. The kids really winded up around a big stone for quite some time till it became our staple for the next day measurement as well.

The second day we reiterated what we learnt namely, bob weight and amplitude does not matter. Now we went after the length to see what we get. Again we went for guesses.

Name	50 cm	40 cm	31 cm	11 cm
Arc	1.6	1.3	1.25	0.5
Sha	1.7	1.2	1.2	0.4
Var	1.6	1.4	1.25	0.52
Shu	1.3	1.3	1.25	0.5
Pri	1.4	1.3	1.25	0.6
Sanjeev	1.45	1.35	1.18	0.7
Actual	1.4885	1.297	1.22	0.839

As we had concluded that the amplitude does not have much of an effect (not true unless it is 'small') we decided to count 10 periods instead of 5 to make calculations easier. We were further able to refresh our decimals by talking about how close the guesses were to the Actual and who was closer. I also participated in the guessing and didn't cheat or use calculations, just gut the way the kids were doing. It was fun to play on a level playing field.

This class went faster and I split a graph sheet to see if we could plot it and see that it did not lie on a straight line. Also I wanted to re-emphasize the decimal numbers on the graph sheet with different scales.

Plot from measurements

Ideal Plot from T=2*pi*sqrt(l/g)

I plot the measurements we got and also the plot with the ideal equation. I noticed that with the ideal curves the error for the measurement with the smaller length was quite a bit. I realized that our stone was not only heavy but large and we needed to take the CG of the stone into account. I wasn't sure how I was going to convey it to my students as this fell in an area where my language skills may prove insufficient.

I started thinking about alternatives to the large stone and thought of where I could get hold of metal bolts. I went to the local two wheeler repair shop and I poked around in his waste/spare and found a goldmine of washers, bolts, etc.

I also wondered how to convey the non-linear nature of the curve to the kids...

I wasn't sure if pendulums were very useful anymore and decided to look around. Turns out the pendulums with temperature compensation were used as a standard of time till not so long ago and they did keep time for humanity for a good part of three centuries. I found a bunch of videos on youtube both on things that pendulums are useful for e.g. ballistic pendulum and things that I thought would be of interest to the kids and what they would analyze.

The first video was a series of pendulums of various lengths that created very interesting patterns. The second was a video of a coupled pendulum that explains the idea of resonance. A cute trick with the idea of pendulum that takes intuition a few seconds to catch up. I found a home made Barton's pendulum that explained how the coupled pendulums work. I took a snippet from an MIT class where the teacher sits on his 15 kg pendulum to show that weight indeed doesn't matter for the period of the pendulum. I also had one of Walter talking about potential, kinetic energy conversion and the experiment where he puts his face on the line to make his point. The newton's cradle also used the pendulum in a different way than we had seen in the class. I also took a video of a science lab with the experiment on the ballistic pendulum and a couple of people who were trying to make physical swings work as coupled pendulums! I also refreshed my idea of the Foucault's pendulum, but decided only to mention it rather than show it.

On the third day we went through all the videos and tried out the trick with the pendulum to start with. The only video that was a flop was the science experiment where the person was describing how to use the ballistic pendulum (science teachers can make a gun being shot boring)!

We talked about what in these we understood immediately, this was the patterns created by the series of pendulums of various lengths. The children knew that the lengths would determine the period and that things would not be in sync, they were now not looking at science, but at the art and beauty created in the exercise. They want to build one of this.

Armed with the new found bolts, washers we ventured on trying the 'cute trick' even as we had just finished watching the video. We spent some time building the coupled pendulum's that had really caught their attention.

We started having a customary circle time to talk about what we had learnt so far at the beginning of a class and in the fourth day we started talking about why the pendulum swings at all, what are the forces on it and what is the speed of the pendulum at different places of the period. To help them remember how it takes time to speed up I had the most energetic kid ran back and forth in the room (I told him it only counts if he doesn't bump into anything to stop). He said that he was fastest when he was in the middle of the room, everyone agreed to his assessment.

We also talked about how the period varies with the length. They all remembered that it decreases with the length. I asked them how much the period of a pendulum of 60 cm was, they remembered 1.6s. I asked them how much the period of a 15 cm pendulum was they said 1/4th. Of course I do not let go of any opportunity in which they need to do any calculations with fractions or decimals and boy this had both! They came back with 0.4 s. I asked them how confident they were and then asked them to try to remember what happened when we tried to measure the 11 cm pendulum and compare it to 0.4 s. They were now convinced that they had no clue and wanted to do the experiment. Yes, it turned out well with the period only going to 0.8 s. I told them that they now knew enough Math to understand how this works. 

We had already gone over powers of a number like 5^4 and I talked about the two special ones squares being area and cubes being volume i.e. if the side of a square was 3 then its area was 3^2=9. I then talked about the complementary question (like subtraction for addition and division for multiplication as I explained) if you had the area of a square what would be one side of it. This helped me introduce a square root. I asked them to now try to figure out by how much the period will change if the period is related to the square root of the length. Yes, it took them a little time, but they arrived at 1/2 and then correlated it to what they had observed.

I shared that we had done something special this week, they agreed (perhaps, they didn't have much choice :)) and that we should showcase the week by building something for the teachers for the meeting the next day.

The fifth day was rained out and only three kids came. One of who had been sick for most of the week and had only come on the 4th day. There were very few kids at school and taking classes was difficult so an independence day movie was played for the older kids. The kids got out with less than an hour left before the teachers meeting. We found a stand that was used as a screen to do puppet shows and I carried it with the students to the staff room. Then we sat down and planned about what we could do in the time we had. They decided to do both the coupled pendulum and the Barton's pendulum to explain that resonance is important for the coupled pendulum. I pretty much left them to themselves and once they were done they also gave a demo of what they had made that I recorded these videos for the teachers.

Their effort helped have a good discussion at the meeting among the teachers... Anyway, I had a fun week, it was challenging to let go of what they should learn in a certain time, but in the end they were able something about a physical phenomenon, build stuff that made them question how things work, see math as a way of estimating, guessing and analyzing physical phenomenon, work with decimals and put the numbers and decimals on a graph and think about fractions visually.

We also played a pizza party fraction game on the fourth day, but that's for a different blog post. For now I leave you with a short video of a kid trying to demo the Barton's Pendulum experiment.

Math for 7th Graders (1)

I decided to take up Math education at Isai Ambalam for the 7th Grade (5 and 1 in 6th Grade). I got myself a block of 1-1/2 hr in the afternoon to give enough time to actually do something.

I started with trying to connect Math to real life. The question was what in real life!

While helping teachers with learning gaps they had they brought up that they had trouble connecting volume in Litres (L) to measurements e.g. how much water in L (or a fraction of it) can you put in a rectangular box of 

5 cm x 5 cm x 10 cm. 

The only gap for the teachers was that 1 cubic cm (cc) = 1 ml, but it got me thinking about what the children think of volume, estimates of volume and measurements.

In my first class, we started with very simple question - how much water does the school consume in a day.

The first thing that came to the children's mind was of course drinking water. One child said that he drank 3 glasses of water in a day. There was some debate on how much water was there in one glass of water. One said 250 ml another 500 ml. One kid went off to get a glass and another pulled out the measuring jars that the school has. Soon we found out that the glass he was referring to was only 200 ml. Then I initiated a discussion as to whether everyone only drink 2 glasses of water in a day. This started a short survey with the people around and they found different children and adults were drinking different amounts of water going up to 6 glasses in a day.

I wondered if it was a good place to bring up averages, but my limitation in Tamil came forward and I could not explain this. I went on to getting an estimate of the water of the whole school with around 100 kids (actually 109+teachers, but ballpark) if they just drank 2 glasses of water. With some difficulty, prodding the conversion from ml to L finally happened and we got to 40 L. I asked them to picture 40 L of water and if this really was the only water used in the school.

One kid bashfully mentioned that the bathrooms use some water. I asked how much water is there in the buckets used. I will skip the guesses and the measurement process, but we finally arrived at 8 L of water in one small bucket. Well, and how many buckets of water do you use? The kids had enough of this step by step calculation of water and started asking for short cuts. I asked them to ask around. They asked a few and finally figured that the coordinator will know and asked her, she estimated 1500-2000 L in a day. I think it left the children more confused than when they started, but given the popular concept among children of 'finished' they said they were done.

I told them that they need to figure out if and how this number makes sense. They came back with a gem the large tank on the school premises from where the water goes to everyone. Perfect, my opening to Math! We were going to measure some dimensions of the tank and find its capacity.

I brought my measuring tape and we tried using it a few times to make sure that the boys going up the tank knew what they were doing.

Measuring the tank that is placed 4 m high turned out to be much more of a challenge than I had estimated and involved some acrobatics that made me a little uncomfortable, but the children seemed very comfortable and confident with what they were doing. We got done with the outer perimeter of 6 m, hieght of 170 cm and an approximate diameter (the top is not flat) of 190 cm. Can you figure out the volume of our tank? :). 

I realized that children love activity, but not to process what was learnt from the activity. 
The next day was a shocker, some kids (there are only 5 in the class) had not bothered to write what they found and when they did they still wrote the the glass could hold 200 L of water and the tank perimeter was 6 cm...Needless to say my idea of giving them a sense of what they are doing was looking like an uphill task. 

When I tried to analyze the data with them I realized that the students were yet to encounter area and perimeter of a circle. I attempted to save the class by giving them the 'formula', but I realized that had difficulty understanding the formula and applying it. I mean, of course I arm twisted pi and made it '3'  and got them to find the diameter, but it seemed like getting an 'answer' rather than learning anything from it and I dropped it...

The next day we took measurements of various circular objects perimeter (with a rope) and their diameter for half a class and tried to analyze the data. We found measurement inaccuracies and once I color coded the xls they decided that they were going to remeasure the outliers like 2.2 and 5.5. We fixed the table below. Can you guess which measurements were 2.3 and 5.33 (Hint, they had a scale that was 60 cm long)?

S.No	Things	Perimeter (P) cm	Diameter(D) cm	Ratio (P/D)	Area
1	Shifu Tire	61	19.6	3.112244898	301.84
2	Iron Rod	15	4.6	3.260869565	16.62571429
3	Tree Trunk 2	114.5	39	2.935897436	1195.071429
4	Bangle	19	5.9	3.220338983	27.35071429
5	Van Tire	232.5	74	3.141891892	4302.571429
6	Tree Trunk 1	25.5	8	3.1875	50.28571429
7	Pillar	49	15	3.266666667	176.7857143
8	Bike Light	50.7	15.4	3.292207792	186.34
9	Ring	6	2	3	3.142857143
10	Tank	6	1.9	3.157894737	2.836428571
11	Handle	8	2.5	3.2	4.910714286

When measuring the van tire they had missed out one set of 60 cm giving a low ratio and when measuring the handle they had erred with the decimal and written 2.5 cm as 1.5 cm.

At this point I would like to introduce (anyone who is not familiar with) Marvin (the depressd robot from the Hickhiker's Guide to the Galaxy) to the following:

Marvin: I think you ought to know I'm feeling very depressed.

Trillian: Well, we have something that may take your mind off it.

Marvin: [depressed] It won't work, I have an exceptionally large mind.

...

Zaphod: Its all part of life you know...

Marvin: [even more depressed] Life? Don't talk to me about life!

...

Marvin: The first ten million years were the worst. And the second ten million: they were the worst, too. The third ten million I didn't enjoy at all. After that, I went into a bit of a decline.

From then on, as Marvin would say, things went a bit on the decline. I tried to get the kids to analyze the data themselves and found that they could not do any calculations with fractions (22/7), decimals (3.14) and needed to approximate everything to an integer division. I gave up, got depressed, hit rock bottom, shook it off and decided to fix the fractions, decimals...

I told then that in order to do some experimental math they need to be comfortable with fractions and decimals and we will cover it well and quickly. In 3 days most of the kids really picked up on fractions, they could do calculations, but they still didn't have a 'feel for it'. I thought I could fix this in the following week. I had asked them to make their own problems, try the exercises in the book, but come Monday morning they came like a clean slate. They could not even follow what they had done on their own the previous week! 

The next 5 days they worked harder than they have worked ever before. We recovered (pun intended) fractions and calculations. I figured the issue each kid was having, but made them work in the class and at home. We plunged into mixed fractions, decimals, threw in the number line and negative numbers and then placing positive and negative fractions on the number line. I even threw in powers of 10 as it was the next topic in the textbook. I gave them textbook problems to do and a special problem set over the weekend.

But, as the weekend has come it has given some time to contemplate and look through what is to be covered by the sciences and how it can be related to Math.  I also need to reconsider what I am becoming :), a classical classroom teacher trying to cover the textbook! The kids have actually become much better, they can recognize a fraction and at least think for a second if it is proper (less than 1) or improper. Most of them are able to place them on the number line and tell a story about it...but then let's see Monday morning.

I will start working with a pendulum and see what we can learn from it about Math and the world around us (and maybe sometime when I am brave enough we can get back to the water audit of the school).