Generating intuition in algebra (2): story for x+y=10

I split each class that we use computers into a primer and an execution. In the primer I asked them for a story of x+y=10. I got some interesting replies and I wrote them down and we analyzed them together:
1) I have 2 mangoes. My mother gave me 8 mangoes. How many total mangoes do I have?
2) My sister has 5 sweets, I have 5 sweets. Together we have 10 sweets.
3) I have some mangoes. My mother gave me the same mangoes. Total mangoes are 10. How many mangoes did my mother give me?
4) I have some oranges, my sister has 10 more mangoes than me.
5) Total biscuits are 10. The dog ate 6 biscuits and the cat eats something. How many biscuits does the cat eat.
6) Ramu and Ravi collect some sticks. The total number of sticks is 10. Ravi collected 2 more sticks than Ramu.

There was generally a fixation with having one 'answer' which came from the one variable one equation situation they had encountered thus far...

I started listing the specific examples they had given 1) and 2) 
x + y  = 10   
5 + 5 = 10 (possible, but only one of the possibilities)
2 + 8 = 10 (possible, but only one of the possibilities)
this was enough to get them kick started in getting a load of other possibilities
1 + 9 = 10
3 + 7 = 10
4 + 6 = 10
0 +10= 10
and their flips.
I mentioned that there isn't just one answer like they have always been used to there are simply many possibilities. 
I also pointed out that if one value increases then the other decreases as the total is the same. We took an example, if Var received 10 chocolates on Sha's b'day and she shared it with her sister, if she decided to be a generous elder sister and gave them all to her sister, her sister would have 10 and she none, if she kept one her sister would get only 9 and so on. They seemed to think I went to too much effort to state the obvious. 

We then considered each case of the other stories they had come up with and went through what they mean and if they sufficiently cover x+y=10 or not.
3) Is like having twice the number of the mangoes indicating 5+5=10. Doesn't seem to cover everything.
4) You can't add mangoes to oranges unless you start treating them as fruit and forgetting that they are mangoes and oranges. The source of confusion was that x and y needed to be different as mangoes and oranges are! But, if they needed to add up to 10 they needed to be of the same kind. The child came back with, ok, they just need to be different numbers not different fruit. Another child corrected its possible they are different and its possible that they are the same!

By now they figured out that 5) was incomplete as I didn't give the number 6. 
This led us to 6) and that the first part made sense, but where did the second part of the story came from? Arc pointed out that that's how some puzzles were and more condition was needed for an answer. Hmm...I asked them to work with what we have right now and that the reason for the other condition was coming soon.

Back to the possibilities, I drew out the x and y axis and started putting in values for x and y as ordered pairs on the axis starting with (10,0) and (0,10). I put a few more points and then jumped the gun and stated that it turns out that all these points actually lie on just one line. Magic! The kids were impressed and pointed out that it seems believable as every time Var gets a chocolate her sister looses one so the line going down makes sense.

We then handled x-y=10. The kids had of course gotten the hang of it and made the stories. We tried to look at how we could make the stories believable! The best we could come up with were - The fruit vendor delivered some bananas and my sister ate some in the morning and left, now I see that there are 10 bananas left.

We then analyzed that how if the sister took more bananas then there should have been more to start with and how this line (they already guessed that this would be a line) would go up.

Computer assignment, plot 
x+y=10
x+y=20
x+y=30
x-y=10
x-y=20
x-y=30

Generating intuition in algebra (1) : Geogebra : introduction

We had worked on one variable algebra with puzzles and looking at abstracting something real and taking an abstract equation and interpreting it in real terms. I introduced geogebra as part of this effort to get an intuitive feel of what two variable algebraic equations look like.
On the resource front computers were available for all the students of 7th grade a couple of days a week which could be used for some unique learning.

To get comfortable with the tool I worked through practical geometry. I had worked with one grade for 2-1/2 weeks to construct 90, 60, 30 degree angles. After a demo (only geometry) in one class their first assignment was to draw an equilateral triangle for a line segment they drew, measure the sides (to see if they are equal), measure the angles (to see if they are equal) and label the vertices.

I noticed how quickly the children become comfortable with a tool that is marginally intuitive and how differences like lines and line-segments become quite clear as they try out different menu options. The references of constructions become obvious e.g. for a circle what was the center and which point was used as reference for radius. 
The measurements add a self-check so the children notice something is incorrect and try to figure out what could have gone wrong.Perhaps, if I had introduced practical geometry this way I would have saved a lot of paper from having holes and trying to make sure everyone had compass, scale, protractor, etc.

Not everyone was interested in trying out geogebra and in the first class I focused on the early adopters and allowed the rest to work on mathlab (computer games that are based on math concepts) . By the second class everyone was in and some of the students who had been able to complete their assignments well were paired to support the ones who were doing it for the first time.

For those who had completed I asked them to make a square with each side 4 cm using arcs and lines. It was a fun challenge and one child claimed that he did it. Most others were able to get squares, but not of the same length as their original line segment. 

Newton's Cradle: momentum

We built a Newton's Cradle at Isai Ambalam. In case, the name doesn't ring a bell you can look at the video below:

We were re-connecting with the pendulum experiments for periodicity and resonance (Barton's pendulum) as a reminder for our work ongoing work on sound. The 7th graders explained the experiments they did before to the 5th graders with videos. Along with showing the videos of the experiments they did they also wanted to show some videos we had seen of stuff they didn't get around doing. One of them that had really caught their attention was the Newton's Cradle.

I have been promoting the kids to be completely comfortable with multiplication and its two associated division stories and told them that a lot of things in science follow this. The cradle served as an introduction to momentum (mass*speed) to continue on our journey (speed, density).

One of the things that limits what you can do with the cradle is that if you try too many tricks the balls no longer stay in the same plane and then the tricks do not work well. We wondered what we could do, possibly use thin enamel wire and solder it to itself to keep the balls in place, etc.

We followed this up with other momentum experiments - heavy ball rolling into light ball and vice versa, what happens to a tennis ball kept on a football when they both land on the floor and wrapped it up with a video on momentum online of stuff we can't actually do in the classroom.

Shifu presses the pause button

Shifu decided that our life was too hectic and even Ani was getting back from her Somatics course she fell quite ill.

The extent of her illness was initially unclear as we called the local vet to come and take a look at her. He felt that there was an infection and gave her an antibiotic shot. She showed no progress, wasn't eating. By the third day she was completely listless and Ani and I went to the vet clinic in Pondy to have her treated. The first day they were closed but made an exception and gave her 100 ml of drip.
The next day they diagnosed her as having a version of tick fever. On the second and third day to the clinic her poop started to show blood. 
Unfortunately, one assumption (hemoglobin low) and one inaccurate lab test (platelets were 1/20 and it showed a normal number) led them to the conclusion that her kidneys were chronically failing and the infection was a secondary issue. It had already been 6 days since she had been sick and I called Shifu's vet in Bangalore. 
Having known Shifu, the doctor felt that she she was young (5 yrs) and generally healthy and a kidney condition is likely to be temporary in response to an illness and she needed continued support.

This started Shifu and my adventure to Bangalore. At least that's what I called it for Arham as I occasionally took pictures of what was going on with us... partly, for him to to have some idea no matter what happened. We reached Bangalore on 21st night.

On 22nd morning was a long drip session and I just sat and pet shifu for 4-1/2 hrs as she lay getting drips. They also took blood for various tests including a new kit that detects if its one of the big four:
Escherichia
Anaplasma
Heart Worm
Borrelia/Lime disease (not sure which of these two)

That evening from the blood tests the doctors felt that the kidneys were under stress primarily due to the illness and should recover in time, but they remained tentative given the discrepancies in the report. They felt that we would be in the clear only when we did a test a couple of days later to see the numbers improving or at least consistent with either report.

They also gave some medicated dog food (k/d diet) that does not require much digestion to see if Shifu can eat on her own. Shifu had not been eating for a while and the drips + visitors + smell of the dog food caught her fancy. She was able to eat what was prescribed (and wanted more)

 

The vets were very happy with her progress with eating and on 23 reduced the drips to a single slow one with her medicines and asked us to slowly increase food intake. They were also feeling a little more confident that the kidneys had not failed and started the antibiotic treatment (imizol).

In the evening Ajay and Neha visited us. Neha overcame her fear of dogs and sat next to Shifu. Shifu dosed off and we also saw her first dream since she had been sick and wondered what  she was chasing in her dream.

Shifu also felt comfortable enough to venture into mom's house (abet briefly) and checked out the bed.

By 24th Shifu was looking better and initially refused to sit down for her single drip but eventually let it go. The doctors also took a blood sample to check the consistency of the reports and to see if there was any improvement in Shifu.

 

Today on 25th the reports confirmed that her hemoglobin in holding and the total cell count has gone up consistent with recovery. The platelets continue to be low that the doctors feel will improve as the month long treatment for tick fever continues.

We still need to go to see the vet for a day or two, but hope its primarily down to medication at home from that point and head back home to Auroville by the weekend.

Thank you all for your emails/messages about Shifu, please consider this post as an update :). Special thanks to Arham, for who, the pics on this blog were taken.

I'm exhausted, but getting better.

Cartesian divers (3): Making it our own

As part of the science exhibition the kids were using Cartesian divers and I felt that we were missing the part of making one ourselves. That morning on my walk with Shifu (I found a straw on the road), a shampoo bottle cap from the trash and borrowed modeling clay from Arham and set out to make some Cartesian divers.

Looking at these divers the kids felt that they could really make it their own and returned my (purchased and self made divers) and committed to make one themselves for their presentation.

In the straw model I put a hole on one of the ends (I remember seeing something like this on instructables). In both cases you add enough clay to just make the object float.

 

 

An interesting conversation started in the class regarding the relative speeds of the different divers.

At my other school the kids made a bunch of divers with pen caps and by the end of the class, there was a litter of caps floating (and sinking) in the bottle we had.

When I got home I thought I would show Arham what Appa made. He found it on his own and set things right, calmly removing the clay that had accidentally made it there and put back in its proper place, his clay box :).