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.
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.
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

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^21) 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.
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