Pascal’s Triangle and Magic Squares

We have been working on patterning in Grade 7 math.  We spent a lot of time looking for patterns in Pascal’s triangle and seeing how the numbers in the triangle work together. I asked my students to each try to find a different pattern in Pascal’s triangle, and they rose to the challenge. They came back to class excited to share what they found, and each student was hoping that no one else had found his/her pattern. At the end of the first day of presentations, most students had claimed a pattern, but there were a few students whose patterns were claimed by others and needed to explore further. The next day I decided to help them out, and gave a short lesson about figurate numbers and asked the students to find tetrahedral and hexagonal numbers in Pascal’s triangle. We then looked into fractals and how the Sierpinski triangle can be created in Pascal’s triangle by blacking out all of the odd numbers. I left them with another challenge – to see what happens when you block out even numbers, and numbers that are multiples of 3 and 4.  I also showed them some of the Pascal patterns discussed in The Number Devil, a book I mentioned in a previous post.

Here are some of the links that I used for this series of lessons:

Pascal’s Triangle and its Patterns

Pascal’s Triangle from Math is Fun

Patterns in Pascal’s Triangle from Cut the Knot

Pascal’s Triangle from Math Forum

Wolfram MathWorld Fractal Page

Wikipedia Fractal Page (Scroll down to see the changing fractal beside the history section.)

Sierpinski’s Triangle from Math Forum

As we were having so much fun with numbers, we went on to look at the Magic Square in Albrecht Durer’s paintings. In his magic square, the sum of all rows and columns is 34. We used the Powerpoint below (source unknown) that was sent to me by a friend. To start, I only showed the first five slides, and then I left it to the students to determine where else they could find the sum of 34 in the square. They made me proud and found all of the sums mentioned in the Powerpoint, as well an additional sum found through a zig zag pattern.

Albrecht Drer’s Magic Square

Hope you have as much fun exploring numbers as we did.
Have a great week.

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