I have a lot of discovery activities that I’d like to share with you, and I’ll share them 1-2 at a time. I claim absolutely no ownership of any of these activities as I have seen them around before. I would just like to show you my “twist” to these activities. Here are two of my favorites:

**Howie’s Hypothetical Ice Cream Shop (Permutations and Combinations)**

For the entire period, I pretend that we are in “Howie’s Hypothetical Ice Cream Shop.” I have students come up with 10 ice cream flavors as a class, and I tell them our specials for today: they can choose either a permutation cone or a combination bowl, where they can pick three different flavors each. I first ask if the permutation cone or the combination bowl will have more arrangements and why, then I have them try to find out how many arrangements there are for the cone and the bowl.

Permutation cone is pretty easy to figure out, there are 10 choices for the bottom scoop, 9 choices for the middle, then 8 choices for the top scoop. Showing an example of a tree diagram will show us that we multiply 10x9x8 to get 720. Combination bowls are a little harder, where they will still have 10 choices for one scoop, 9 for the second, and 8 for the third, BUT there’s a slight difference between a permutation cone and a combination bowl: You need to eat the permutation cone in a certain order, top middle, then bottom, but a combination bowl, you can eat in any order you want. If you asked for a combination bowl of strawberry, vanilla, chocolate and your friend orders a bowl of chocolate, vanilla, strawberry, you’re getting the same bowl. I then I have students exhaust the arrangements of chocolate, vanilla, and strawberry, then we see that **every group of 6 turns to 1** when we consider combination bowls. That is why combination will always be smaller than permutation. In this case, it would be 720/6=120.

I have students do this for 4 choices (where for combinations, every group of 24 (4x3x2x1) turns to 1), 5 choices, then 10 choices. Before I discuss the 10 choice option, we will have noticed that our number of arrangements keep increasing for both permutation cones and combination bowls. I ask them, “Do our number of arrangements keep going higher for both the permutation cone and combination bowl?” Sometimes they say yes, sometimes no, but we continue along.

After this, I show them that combinations are found in Pascal’s Triangle as well as ask the students what other patterns they see in Pascal’s Triangle.

**Triangle Inequality (with noodles!):**

Before I go with the manipulatives, I have students draw what these triangles would look like if it had these side lengths: a 1, 1, 2 triangle, a 4, 5, 10 triangle, and a 3, 4, 5 triangle. I purposely do a little quiz with these side lengths to start because I heavily believe that making mistakes will make us learn more in the end. After this, I have students compare their drawings with their group members to see if their triangles look similar. I then hand out about 10 strands of spaghetti to each group (I found bags of spaghetti for 25 cents at FoodMaxx) as well as rulers. I tell them to measure and break these noodles to make a 1, 1, 2 triangle, a 4, 5, 10 triangle, and a 3, 4, 5 triangle. After a couple minutes, they will notice that the first two are impossible and the 3, 4, 5 triangle creates a right triangle (I do this purposely because Pythagorean Theorem is the next lesson). I then have students come up with 5 sets of side lengths that will not create a triangle and 5 sets of side lengths that will create a triangle. I would write a couple of these down and then ask the class what they notice about the sets that do not create a triangle and the sets that create a triangle. Boom. We have just discovered the Triangle Inequality.