### Example Problems

Below, find some example problems that could serve as jump-off points for discussions and explorations of mathematical concepts. They are split into those for older and younger students.

Each problem is accompanied by a hint -- simply click on the text.

## For older students

### Grid Walking

We have a grid of 3 columns and 4 rows. How many different paths can we draw from the bottom left to the top right (following the grid lines), with only 7 moves?

Hint: How many steps to the right will you have to inevitably take? What if we think of this as purely a problem of deciding the order of the 'up's and 'right's?

### Sum of Consecutive Integers

Suppose you are a farmer, planting one flower the first day, two the second, three the third, and so on. How many flowers in total will you have by the nth day? Can you prove a general formula for the sum 1 + 2 + 3 + ... + (n-1) + n?

Hint: Let's think about rearranging this expression to pair the first and last values in the series, writing it as (1+n) + (2 + (n-1))... and so on. Each pair has the same value -- now there is just one step left.

## For younger students

### Scaling Shapes

If we have a rectangular pool and we decide to make a new one whose side lengths are twice longer, what will happen to the area of the pool? Will it double? Why or why not?

Hint: Let's draw an example, and go from there!

### Venn Diagrams

Let's create our own venn diagram based on characteristics we come up with. How do we define the sets of objects? Are there sets that don't have any objects in common? If we know the number of objects in each set and know the number of total objects, can we figure out how many are in both?

Hint: Let's think about the objects we are overcounting -- adding twice because they are in both categories. What can we do to make sure we just account for them once?

### Fractional Pies

Would you rather have half of five pies, or five halves of pie? Is there a difference? How can we show there is or isn't?

Hint: Let's start by turning everything into halves for easier comparison of the two options.