Family-style lesson in discovering patterns with Pascal’s Triangle.

Skills required: 2-digit addition or calculator.

Pascal’s triangle has fascinated mathematicians for centuries. Western civilizations give credit to Blaise Pascal, who discovered it in the 17th Century, but it was independently discovered by Chinese and Indian mathematicians centuries before that.

## Create Pascal’s Triangle

As a family, lets each create our own Pascal’s triangle and see if we can discover some of its patterns. Teachers/parents, jump in and make one with your students too.

**PRO TIP**: Be really tidy as you draw your triangle and leave lots of space at first. Neatness will help with finding patterns later. The space will help as numbers get larger.

To begin, take out a sheet of lined paper. Label the top “Pascal’s Triangle.” This can be a loose sheet of paper or a page in the student’s science journal.

Next, draw a series of 1’s diagonally down to the right and left like this.

Starting on the third row, add the two 1’s and write the sum on the line below in the space between the two numbers.

Continue adding two numbers on a row, and and writing the sum between them on the line below.

This animation shows how to add the rows.

By Hersfold on the English Wikipedia – Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=3902538

Ask students “How long could we keep on doing this?” (Forever)

For younger children, stop after about 8 rows. For older children, have a race to see who can keep adding rows to the triangle and find the first number with a 1,000’s place value. It’s okay for kids use a calculator for the larger numbers. This is not an exercise in adding skills. It’s an exercise in pattern-finding.

## Discovering Patterns

Once the triangles are complete, ask students to look for patterns and share what they noticed. A “pattern” is any trend or cool thing they see. I like to use John Muir Laws’ questions here. Ask children, “What do you notice?” “What are you wondering about?” “What does this remind you of?” (I notice, I wonder, This reminds me).

Here are some ways we could look for patterns.

- Look at the diagonal rows.
- Add up the rows.
- Look for symmetry.
- Color the sections of odd numbers red and the even numbers yellow. (Hint: The sections make triangle shapes.)

When students share a pattern with me, I like to write it or draw it on my own triangle. When I write their discovery, it helps kids feel like their pattern is important and worth writing down.

## Another Cool Pattern

Something I learned yesterday while studying about Pascal’s Triangle is the hockey stick pattern. Starting at any 1, draw a line down one of the diagonals. When you are ready (at any point), veer off toward the center in the opposite diagonal. The shape you made looks like a hockey stick. The number you veered to will be the sum of all the other numbers.

In this example, we drew a box around 1, 4, 10, 20, and 35. When you add these up, they make 70. Pretty cool, right? Try this on other diagonals. Use a calculator to see if it is true.

## Conclusion

Mathematicians spend a lot of time studying Pascal’s Triangle and searching for elegant patterns. This type creative thinking and noticing patterns is what mathematicians do all day.

## Learn More

To learn more about Pascal’s Triangle, check out these videos. Some of the more advanced math in these videos will be over their heads, but the big picture idea of the depth of Pascal’s triangle is most important. Kids may see patterns in a video that they didn’t notice yet.