## Trick with 27

I finished watching another fantastic video from Numberphile. The title is perfect “Beautiful Card Trick”. After you watch it I think you will agree.

The video has some wonderful math and I cannot wait to try out different versions of the trick. Here are some examples that come to my mind.

- If you have a class of 27 students or more, then you could have each student put their name on a card, making a deck of 27 names. I think this could be a good first day activity.
- Do one of those activities where you use four fours to make the numbers 1-27. Put each one a card, making a deck of 27. Then do the trick.
- Have students type in “random city” into WolframAlpha. A lot of math could be done with each city. Then collect 27 cities and do the trick.
- This would be a great introduction into bases. The trick could then be followed with base four and other bases.
- Have students create a function and list out 27 inputs-outputs in coordinate form. Do the trick with those 27 coordinate points.

Overall, there are this card trick is simple and it can easily be modified, probably into at least 27 different ways. Hopefully, you find at least one way to share the trick.

Bonus video! This video goes beyond only 27 cards. You will see that (the number of cards) = (number of piles) ^ (number of times shuffled).

## What is tangent?

Trig functions have a lot of acronyms. Is this good or bad, well I don’t know. What I do know is this: tangent is much more than just sine over cosine or opposite over adjacent. However, the text books I teach from usually focus on that. So, I thought I would make an investigation that allows students to see that tangent can also be seen as length of the tangent line to the unit circle at Wikipedia gives some good pictures. Click here to see the article.

I also like in the article when it writes about “Slope Definitions”.

“Tangent combines the rise and run” meaning that Tangent takes the angle of the line segment and tells its slope; or alternatively, tells the vertical rise when the line segment’s horizontal run is 1.

So, next time you talk about the trig functions be sure to expand on the idea of tangent. Here is a Desmos graph I made that you can use. Click on the image to see the graph.

## Angle Side Side

Recently, I was at the 2012 AMATYC conference and heard many great ideas for my classroom. One idea was this,

Don’t use Law of Sines for Angle Side Side, just use Law of Cosines and then apply the quadratic formula.

What a revelation that was! My students never seem to get the ambiguous case. Maybe it is the way I try to explain it. Overall, it seems like a bunch of hoops to them. But by using the quadratic formula students refresh their memory of the quadratic formula and also apply some basic trig. So, let’s look at an example.

Let the angle be the adjacent side be 31 and the opposite side be 12. Going straight to the Law of Cosines we get the following:

now set the equation equal to zero and we get

now find the values for the quad formula

. Since the discriminant is greater than zero there will be two solutions or in other words, two lengths for the third missing side. Once you have the two sides, then use the Law of Sines to find the angles. Voila, you are done!

I have created a Desmos graph that can be used to further model the idea of Angle Side Side. (See below) In the graph, you can move the opposite side, which is the ‘c slider”, to try and connect with the missing side. Through careful observation you will notice that the value of the ‘c slider’ that makes a triangle actually is the +/- discriminant/(2a) part of the quadratic formula! Having students discover this correlates to the idea how the quad formula is the vertex +/- distance to x-intercepts. In this case, the quadratic formula is the distance to the height +/- distance to opposite side. So when the discriminant is imaginary this means there is not a distance to the opposite side. If the discriminant is zero, then the opposite side is actually the height, altitude, of the triangle. Finally, when the discriminant is greater than zero there are two distances to the opposite. I think this is a great bridge back to x-intercepts and reinforces the importance of the quadratic formula.

Check out the desmos graph by clicking on the below picture. It is the Angle Side Side tutorial I mentioned above. Let me know what you think. Will you still use the Law of Sines?

## Dartboard and Equations

While reading Dan Pearcy’s blog entry about Rich Questions I could not agree more about his statement

I love it when people share rich questions. Generally because I find that I can’t keep track of all the resources which come my way but I do seem to remember great questions.

For me, piling up resources is so easy to do, while actually implementing them is the difficult part. So, I am trying a new strategy next semester. Instead of having the “oh this would be a cool problem and oh this would be a cool problem too, oh-oh-M-G and I have to do this”, which ends up in only doing a few problems, I will focus on one problem and see how much I can get out it. Can a whole chapter be focused around one problem? I don’t know but if I find one, it will be placed in the “Rich Question” Category.

After seeing Dan’s post about linear equations and shapes being shaded, I wondered if the two could be combined. Here is my idea. First, I would have students look at this graph made in Desmos and ask the basic question, “What part is shaded in each square?”

(Click on the above picture to see graph at desmos.com.) After we agree that 50% is shaded in each region, I would ask the main question of the project.

If each square is an individual dartboard, then which square is the easiest to hit a shaded region? Which square is the hardest to hit a shaded region?

Hopefully, this would hook their attention to the project. Next, each student would create four different dartboards. One dartboard they think is too easy, one dartboard that is too hard, and one dartboard that is just right (fair). The fourth would be one that is a cheat – that is it looks like 50% shaded but actually it is less. Once each student completes their four graphs they can go around assessing each other graphs with HARD, EASY, FAIR, and CHEAT. Side note: The student that gets aways the least amount of shaded region should earn extra points. This would give more of an incentive to create unique ways of shading 50% of a square. Finally, the general case be examined. How should one shade 50% of a square so that it is the easiest/hardest dartboard?

Where are the linear equations? The students would create the dartboards in Desmos by inputing linear equations. For example would graph the linear equation x+9 with a domain restriction -9 to -1. I think this could provide some good opportunity to solve equations, create equations, and practice domain restrictions. Of course the shading part, which uses a dampening function, would have to be explained. But once that is shown, that would create a bridge of transformations, a concept that seems only start with quadratics and not linear. Also, students might find a reason to know how to find the distance between two points.

Overall, this is still really fresh in my head. There seems to be lots potential I have heard that less is more but with so much great stuff on the internet, focusing on less is tough. But, I think finding a Rich Question can help me do more with less. If you land a bullseye with a Rich Question then let me know. Thanks!