We're going to hold a three-hour math workshop for some middle school students. It'll about the Pascal's triangle.
Well, we can ask the students to find patterns in the triangle, or try to prove some of its easiest properties. But on the one hand they'll soon get to properties which are not easy enough to be left to them to be proved, and on the other hand we want the workshop to be fun and exciting.
So we need
- Some activities (building something, a game) related to the triangle,
- An interesting short movie about the triangle,
- Some tasks about the triangles to be given to the students to do , on their own.
I'd be grateful if you could share your ideas with me, and help me with finding what is needed (namely, the items above).
P.S. These are some of interesting properties of the triangle, which are not likely to be easy to be proved by the students on their own:
- The rows are symmetric.
- Each element is the sum of the two elements above it.
- Elements of each row sum up to a power of 2.
- Elements in a row related to a prime number are all divisible by it - except the two 1's.
A property that combines two interesting kinds of number:
$$\newcommand\br{\color{brown}}\begin{array}{cc} &&1\\ &\swarrow&\br{1}&1\\ 1&\swarrow&1&\br{2}&1\\ \br{1}&\swarrow&\br{1}&3&\br{3}&1\\ 2&\swarrow&1&\br{4}&6&\br{4}&1\\ \br{3}&\swarrow&\br{1}&5&\br{10}&10&\br{5}&1\\ 5&\swarrow&1&\br{6}&15&\br{20}&15&\br{6}&1\\ \br{8}&\swarrow&\br{1}&7&\br{21}&35&\br{35}&21&\br{7}&1\\ 13&\swarrow\\ \br{21} \end{array}$$
The diagonal sums are the consecutive Fibonacci numbers. (The fact isn’t even all that hard to explain informally: the construction rule for Pascal’s triangle naturally produces the Fibonacci recurrence.)
There are the hockey stick (or Christmas stocking) patterns:
$$\begin{array}{cc} \begin{array}{cc} x\\x\\x\\\vdots\\x\\x\\&\searrow\\ &&\text{sum} \end{array}&\quad&\text{and}&\quad&\begin{array}{cc} x\\&x\\&&x\\&&&\ddots\\&&&&x\\&&&&&x\\&&&&&\downarrow\\&&&&&\text{sum} \end{array} \end{array}$$
Reducing Pascal’s triangle mod $2$ produces the Sierpiński triangle, which has all sorts of patterns:
$$\newcommand\0{\color{magenta}0}\begin{array}{r|cc} \underline{0}&\underline{1}\\ \underline{1}&\underline{1}&\underline{1}\\ 2&1&\0&1\\ \underline{3}&\underline{1}&\underline{1}&\underline{1}&\underline{1}\\ 4&1&\0&\0&\0&1\\ 5&1&1&\0&\0&1&1\\ 6&1&\0&1&\0&1&\0&1\\ \underline{7}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}\\ 8&1&\0&\0&\0&\0&\0&\0&\0&1\\ 9&1&1&\0&\0&\0&\0&\0&\0&1&1\\ 10&1&\0&1&\0&\0&\0&\0&\0&1&\0&1\\ 11&1&1&1&1&\0&\0&\0&\0&1&1&1&1\\ 12&1&\0&\0&\0&1&\0&\0&\0&1&\0&\0&\0&1\\ 13&1&1&\0&\0&1&1&\0&\0&1&1&\0&\0&1&1\\ 14&1&\0&1&\0&1&\0&1&\0&1&\0&1&\0&1&\0&1\\ \underline{15}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}&\underline{1}\\ \end{array}$$