Join us as we explore a fun math activity to introduce the factorial function for homeschooling.

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We are getting ready to study some probability and statistics for our seventh grade math, and I thought I’d start out with a review of permutations that we’ve done before. This was a fun little exercise that started with a compound question.

If there are 3 flavor ice creams, and we are going to have triple scoop, what are the different combinations that we can have on our cone? How many if there are only 2 flavors for a double scoop? 4 flavors for a 4-tiered scoop? What’s the formula for this series?

In the video, I go over some of my objectives for this exercise. They really are three fold.

• Further develop mathematical thinking and logic.
• Cultivate the observing for relationships and patterns.
• Introduce the factorial symbol.

In my attempt to cultivate that mathematical thinking and logic, in the beginning, I just set her out on the task without giving any process suggestions or any explanations of any of the math concept. I want to say here that making mistakes is necessary in this process. We want to encourage risk taking and exploration. She has several combinations written down, and then she thinks she has them all, but we notice that two are repeats and one is missing. 

At this point, I say, “Oh, I think we need a system, so we know that we have all the combinations.” She comes up with a system of choosing one flavor for the bottom scoop and doing all combination with that flavor as the bottom scoop. Turns out that we can only make two combination with the bottom scoop. Then we moved to the next flavor as the bottom scoop, and found we could only make two combinations with that flavor as the bottom scoop. The we used the third flavor giving us 2 combinations again. That was all of our flavors giving us a total of 3 x 2, or 6 total combinations.

From this point, we moved down to a combination with 2 flavors for a double scoop of ice cream. I want to point out that this sequence that I’ve chosen is purposeful. I’ve started with three because it’s challenging enough that we need to think logically in order to come up with all our combinations, but not so challenging that we can’t clearly see that system (for instance, if we started out with 4 flavors.) Then we are moving to an easier inquiry. Why? First, we need to fill in the gap so that we can see the pattern of our series. Second, we want to move to something simpler to build confidence before moving to something more challenging.

From here we move to the more challenging combination with 4 flavors on a 4-tiered cone. This also requires a system, but since we have a one in place for the 3-flavor combo, this one comes much easier. We choose one flavor to be the bottom with another flavor as second to bottom and do all combinations in that way. We then leave the first flavor at the bottom and change the second to the bottom flavor and do all combinations. We continue in this way until all the combinations with the bottom flavor are exhausted, and then we switch to another color. 

We are able to get six (3 x 2) combination for each bottom color and we have 4 combinations, so we have 4 x 3 x 2 or 24 total combinations. At this point, we stop to reflect what we have done so far. We are looking for the patterns. This was not obvious at first, so I asked how we might find out how many combinations that we would have with 5 flavors. Here we knew that it as 5 x 24 because there would be all the combinations with the 4 flavors with each of the 5 flavors at the bottom. This helps us to see what is happening. We are multiplying the next term (the position in a mathematical sequence, in this case the number of flavors). 

This exercise could also easily be done with Cuisenaire rods using a different color rod for each flavor. Using these will make the pattern a little easier. In our activity, you can see we changed ice cream flavors for each of our combinations. The pattern is probably easier to see if you just add a new color for the new flavor for each term instead of choosing new flavors as we did. 

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