Prime square minus one

I had a few interesting classes with 6th and 7th graders around the idea that a prime number (beyond 3) square minus one is divisible by 24.

The biggest difficulty in approaching this topic is that many children get confused between double and squares. I indicated the difference using an area of a square of a side of a certain length (avoiding 2 so as not to add to the confusion) vs a rectangle of the same length and breath fixed as 2. This, however, was not enough for all children and I introduced a short cut to get squares of 2 digit numbers. Even though this was a diversion, it reiterated the squares of small numbers and helped them see squares of large numbers not just as random numbers.

The method is
1) To write the two digit number say 24 in two columns. Now you have single digit numbers in each column. 2) Take the units place which is 4 and get 4x4=16 and write it as a two digit number in the right side of the column.
3) Then you take the tens place 2 and square it to get 2x2 = 4 and write it on the other side of the column.
4) Then you take both the digits 2 and 4 multiply it to get 2x4 =8 and then double it to get 16 and write it in the bottom with the units digit in the first column.
5) Add the two rows you created in steps 2) though 4) as a regular addition.

2	4
4	16
1	6
5	76

One trick in the method is when the units place is 0,1,2,3 when the squares are also digit numbers. Here, you need to continue writing them as a two digit numbers - 0x0=00, 1x1=01 2x2=04 and 3^2 = 3x3 = 09. For example 63^2. 2) 3x3=09, 3)6x6=36, 4)6x3=18 double 36.

6	3
36	09
3	6
39	69

In the first class I tried this with the children had asked why this was. Being a rookie teacher I thought they meant they wanted to know why you write single digit squares as two digits and tried to explain that the '2' is 20 and its square is 400. Luckly, I also listen to the kids and realized that they only meant that I should repeat this step and give examples for them to master it. (Though it gave me an idea to build on the real why later).

The teacher had displayed prime number all around the class and we picked up prime numbers and squared them (subtracted one) and checked if this was divisible by 24. This is, of course, easier said than it was done. I tried to explain that the factors of 24 were 8 and 3 and we need to do a divisibility test of these two numbers and all these ideas fell flat on their but. Finally, we just wrote the 24 tables and did long division. The issue the children had with the divisibility test is that it doesn't tell you exactly by how much 24 divides the square and till they have this number the division is not real!

This itself was fun and helped as a way of learning squares and gave a practice of long division. What was more fun was to answer two questions:

1) Why does the squaring method work

2) Why is a prime number square minus one (p^2-1) divisible by 24.

I posed the two questions to the children and asked them which one they really wanted to know. If you are an engineer and not a teacher you would be surprised to know that the children only had interest in 1) and 2) was well, a nice side dish.

I realized that I think in algebra and it was a challenge to think of solutions that were visual and didn't require algebra. I was able to address 1) in this way, but that's for another blog.

Solving puzzles: Quiet time

I have been putting up mathematical puzzles at Udavi for about 3 months in the area for sections 7th to 10th grade. I have been getting responses mainly from a few kids in the 10th grade. The other kids have not been as involved in solving puzzles. I noticed that in the younger grades (6th) the teacher does put up puzzles occasionally and thought it would be interesting to pursue the solving puzzles there.

I created a couple of puzzles that were visual (match stick puzzles) and put them up in the 6th grade. Again I found that 2-3 children were keen on solving it, but the rest were not able to. We then had a class to solve the puzzles.

I had been working with these kids for a couple of months now and had noticed that the classroom gets pretty noisy as the kids who 'got it' were too eager to sprout out the answer and the kids who had 'not yet got it' were all to comfortable not having to think. I had been working with the kids to do individual work in the notebooks without having to raise their hands or talk allowing me and the other teacher to go around and look at their work. The classroom had been marginally quieter at times. Of course there is an equal mix of group activities (games) or discussion time (sometimes moderated) and hearing about each others work.

In this class, we took a step further. There were enough puzzles for someone to continue solving if they finished one and they were asked to work on this and try to create a quiet space to allow them to think. The teacher and I went around to see how the kids were doing. Occasionally, we encouraged the class that they could do it and just needed to relax and find their quiet space.

That class was magical, the children were able to find their space and over 80% of them were able to solve the puzzles on their own. The other 20% needed individual time from us to ask them questions that led them to think of the answer and elliminate the impossible that they were stuck with.

Unfortunately, with everything magical, it has been difficult to replicate. But, there are now more children that attempt puzzles and also ask for it when I am in class.

We have now been slowly moving from visual puzzles to ones with numbers and in a more recent class we all created puzzles of our own.

An exam that didn't end

One of the attitudes that some of the children have is 'finished', not an idea that something was completed, but just that something was done and effort can cease. This is particularly an issue with math when sums are finished and often nothing is learnt in the process. At the end of the first term exams are conducted at Isai Ambalam School. I decided to challenge this attitude...

A week before the exam I announced to the children that it would be a 'cheat sheet' examination i.e. they can bring one sheet of small/large notebook paper with whatever they needed help with to remember to the examination. To me this was groundwork for them to assimilate the information they had (which most of them didn't do).

I also created a question paper that favored word problems vs 'sums'. I created a paper for 120 marks to and in addition, I created bonus questions that would build on the understanding of a question. The children didn't attempt the bonus questions, but it gave me an opportunity to talk about it later and point to where our knowledge will grow in time.

I let them see the paper for a short time on a couple of days before the examination and I went over any questions with the English a day before the examination (as comprehension of English text is also a challenge).

The examination itself lasted for the standard 2-1/2 hrs, but many children were unsatisfied with this morning session, so they got another 1-1/2 hrs in the afternoon class as well. While the kids were happy they got extra time, I was smiling inside as this was the first time they did math for 4 hrs in a day and that too after they 'finished' with the examination! By the end of it though a child had reached the limit of his mental stamina and said he didn't need extra time even if it was available.

Nope, we were not done. Self-evaluation! I asked the children to grade their work and write down 'Yes', 'No' or 'Maybe' indicating they had confidence they had done it right, wrong or didn't know on their question papers. They would get an additional 2 marks for every correct assessment of their attempts with 'Yes' and 'No' and loose 2 marks for every incorrect assessment. They could play it safe with 'Maybe' and neither loose or gain marks.
This exercise was really good to understand attitude of children to their work. It also brought up some interesting introspection from the children regarding how they perceive their own work.

The examinations were week long and some of the days they would study for the next exam. But the next day I solved all the questions on the board and they did a self evaluation.

For the week long holiday after the exams their only homework was to understand and be in a position to answer the question papers (which they didn't do).

The context of word problems help revisit these questions even in the classes we do now when we revisit decimals or need to clarify a topic by making a story out of it.

Here is some of the feedback from the kids:
"I have never been able to solve word problems and this gave me the confidence that I can approach them."
"I guessed that I did things wrong when I got them right and right when I got them wrong. I don't trust myself enough."

Of course, two kids didn't do well and their performance was stark both in ability and in self assessment. One of them left her morning work of 2-1/2 hrs and restarted the paper in the 1-1/2 hrs I gave in the afternoon. Both did what they knew repeatedly without attempting something new. But, the fake confidence she displayed in class was broken and she has been more awake and engaged in class since...