Select and solve a good problem
A good math problem can be approached in a variety of ways. They also lead to new problems.

Of course, a good math problem doesn't do these things on its own. It's your thinking that does it!

Keep an open mind. Use your imagination. Don't be afraid to make mistakes. Have fun!

Here are a few examples that you might try:

PROBLEM 1. How many blocks to create the first 100 stages?

Imagine a pattern like the one below.
The above pattern has 5 stages. How many blocks would you need to create the first 100 stages?
Method 1
You might start by counting the blocks.
You might notice that the first stage has 1 block, the second 3 blocks, the third 5 blocks, the fourth 7 blocks, and the fifth 9 blocks.
Aha! They are all odd numbers!
If we add the first 5 odd numbers we get 25.
How many blocks would we need for the first 10 stages?
We might notice the following pattern and get an answer of 100.

So for the first 100 stages we need to find the sum of the first 100 odd numbers.
Can you see how you might solve this problem using the method in the diagram above?
Method 2
If you play with the first 5 stages you might notice that they fit together to form a square.

How does this help you find the sum of the first 5 odd numbers? the first 10? the first 100? the first 1000?
Extension
What if we took 1 block away from each stage?

Now we have the even numbers!!
How do we find the sum of the first 100 even numbers? Does the following pattern help?

PROBLEM 2. The answer is 10. What was the question?

Here's one possible way to think about this problem:
Let's assume that the question involved adding two numbers.
We can form a number sentence like this: __ + __ = 10
If we look at the possible solutions in an organized list, we might get:
0 + 10 = 10
1 + 9 = 10
2 + 8 = 10
3 + 7 = 10
4 + 6 = 10
5 + 5 = 10
6 + 4 = 10
7 + 3 = 10
8 + 2 = 10
9 + 1 = 10
10 + 0 = 10
We could make ordered pairs out of the pairs of numbers that give us 10:
(1,10), (1,9), (2,8), (3,7), 4,6), (5,5), (6,4), (7,3), (8,2), (9,1), (10,0)
If we plot these ordered pairs on a grid, we get a neat pattern: all the points line up!!
This might lead you to wonder about the following:
What if we used negative numbers as well?
What if we used decimals?
What if the answer was 6 or 12? How would the graph change?
Can we make up a number sentence whose graph sloped in a different direction? Or curve?
Perform your Problem
Here are some elements of good mathematical performances:

They connect mathematical ideas.
The first problem connected physical patterns to odd and even numbers, and to different ways of finding their sums.
The second problem connected the sum of numbers to number sentences, to ordered pairs, and to patterns in graphs.
They offer some surprise.
In the first problem, it was surprising that the stages of the pattern formed a square, which helps us solve the problem.
In the second problem, it was surprising that the points lined up!
They express feelings and emotions.
Mathematics, like all human activities and experiences, involves both thinking and feeling.
See for example the videos excerpts on the right. How might you improve on these videos, to make the mathematics, the surprises, and the feelings come to life?
Perhaps you could create a dialogue between two or three people.
Perhaps you could write a poem about this problem and perform it.
Perhaps you might create a drawing or painting to accompany your performance.
What else?
Share your Problem
Share a performance of your problem in our Math Performance Festival.