Trying to Improve this Question

This summer I have been working with a few other teachers to help shape the math curriculum for our county. Part of that process has involved compiling tasks for each unit that are meant to serve as representations of what students should be able to do within that unit of content.

I like that a lot of these tasks have a focus on student discovery, and all in all most of the tasks that we have put together have the potential to establish good learning opportunities. One, however, stood out as a candidate for fixing. 

In the spirit of Dan Meyer's Makeover Monday posts, I set out to improve the task below.  

The main reason this problem stuck out as a candidate for fixing is it makes poor use of the cabin idea. My first inkling was that this could do much better in engaging the connection between triangles and roofs. From that motivation, my goal became getting at the same congruence content with something that got away from the step by step set up. 

Below is what I came up with. I did some searching about roof structure and trusses, and I found a visual that made use of triangles complete with all of the terminology.

I'm happy with the facts about common trusses that the students will use to support their proof, but I'm struggling with the task portion. I think that it is a good problem that would require students to make a number of connections as they built their proof, and I like that it's to the point and the pictures can speak for themselves. My trouble comes in establishing a driving motive for why anyone would care that those two triangles are congruent.

The fact that the triangles are congruent does appear to be obvious, and the main challenge is in structuring the reasoning for why that is true. But why does it matter that they are congruent? Why would anyone care that they are congruent? And if no one cares, then how could this problem be saved?

As a bonus, I did find an interesting resource related to the pitch of a roof and how to calculate the angle measures for the truss cuts, but that didn't relate to congruence. It relates more to slope, and so I will save it for then.

In the meantime, what would you do to improve this task? How would you have improved the original problem?

Exploring Quento

I found this game a few months ago, and since playing it, I have been thinking of a lot of questions that I think would be fun for students to explore. 

If you're unfamiliar, Quento is a math game that tests the user's ability to do basic arithmetic. As you can see from the image, there are varying levels of difficulty, and the objective of the game is to combine the numbers in such a way so as to get the desired number. The trick is that the user can only use the specified number of numbers to get the result.

I have had quite a bit of fun playing this game, and I was glad that I was able to show my students the game on the last day of this past school year. While playing it as much as I have, I have been thinking of some fun questions about the math around the game. 

Aside from the obvious arithmetic component, Quento has some interesting design questions too. 

Question 1:

How many rounds could the game have? 

I like this counting problem because it requires the distinction between having numbers repeating and not. 

Question 2: 

How long will it take before I reach the final round? 

Having played through over 300 rounds, I wonder if I'll ever reach the end of the game. 

Question 3:
How many positive (and negative solutions) are there for each section of a round?

The fun part of this question is finding multiple ways to get 5 (for example) using only two or three numbers. Also, while negative numbers can be used to get a result, there have not been any negative solutions in the entire game. What if there were? (I guess that's question 3b).

Question  4:
By how much does the difficulty increase as more numbers need to be used to get a solution?

This is interesting because, while Quento does not have a timer, difficult could be measured in how long it takes to reach a solution. It could also be measured in the number of possible arrangements that arrive at the solution, which is connected to question 3.

Question 5: 
Could there be a level that requires using six numbers?

Quento has a "Hard" setting that I have not yet tried, but it only goes up to requiring four and five numbers. Would a round of six fall into the "Easy" or "Hard" category? (5b)

What do you think? 
I encourage you to try out the game, and ask your own questions. My dad and I are having a competition to see who can get further in the game, and it is a nice game for keeping your arithmetic skills sharp. I also encourage you to show your students the game, and ask them what questions they have about it. It comes on many platforms for free, so let me know what you come up with.

Ordering Pizza

Once again, a problem idea comes from West Virginia. While I was waiting in line at a general store, I noticed the store's menu for pizza. Unfortunately, I forgot to snap a picture, but I did write the prices down.

Pepperoni and Cheese is $10.99

Each additional topping is $1.59

Toppings
Sausage, Onions, Green Peppers, Ham, Mushrooms, Bacon, Banana Peppers, Olives, Sardines.

A pizza with everything is ???

The problem is straightforward for a student. Multiply 1.59 by nine, and add that to 10.99. However, the resulting price is not what the store advertised. Once students found the obvious answer, I plan on showing them that the store is selling an everything pizza for $17.99.

Now the question is, why would they sell an everything pizza for $17.99, when it would make sense to sell it for $25.30?

Why they would charge $1.59 for each topping when their everything price indicates that they can afford to only charge about $0.78 cents for each topping?

I am glad that I found this problem because I'm not sure of the answer, and it would be interesting to see what students came up with with regards to the possible reasons for the stores pricing. I would be interested to find out what they would charge for an everything pizza, and it would be interesting to see how they support their decisions.

This would also be extended to other businesses and how they go about deciding on their prices, and I could see a connection to systems of equations and tall about weighing cost versus revenue.

What do you think about this problem? Have you seen stores price items in perplexing ways?

Train Speed

While working next to the train tracks in West Virginia, I was lucky enough to see three trains go by. It's possible that it was the same train going by three times.

At any rate, I was wondering, how fast was the train moving, so I took a video as it went down the tracks. 

That would be my Act 1, and from here the students would go about figuring out how fast the train was moving.

I could see this problem being used in a unit about rates and unit co versions. I think it could seve as a nice quick activity to get students thinking and working. 

There are a lot of angles to solving this problem. They could find the carts per second and convert to miles per hour, or they could see how long the driveway is and find out how long it takes a cart to travel that distance.

I might try to give them the original video file and let them manipulate to find the time from there.  

What would you do with this video, and what else have you done that's similar?