Thank you 101qs.com

This past week was a humbling reminder that I still have a long way to go before I'm satisfied with my ability to be an effective teacher.

Nonetheless, I am feeling more comfortable with my ability to implement problem solving tasks with my students. I am very grateful for the wonderful community of resources that exists through the twitters and all of the blogs, and one in particular that I am excited to have started using is http://www.101qs.com/. 

I had started my seventh graders working on representing percentage using equivalent expressions, when I came across a picture that led to a really interesting question about deciding between two coupons. The picture led to students noticing and wondering, which led to them thinking.

Which coupons is better, 30% off the entire purchase or 40% off one regular item?

Their first inclination was that, "it depends." Now came for the fun part, because they realized that they would be figuring out on what it would depend. What prices would cause you to choose one coupon over another? Does it matter how many items are in your cart?

They were split up into groups of three, and most every group started working through different purchase situations. Most began to find that the 30% off coupon consistently won out, and this led to a prompt for them to find a situation when the 30% off coupon would not be the better deal.

As groups worked through this portion, I could see the wheels really start turning. They were noticing different patterns, they were being strategic in the prices that they assigned to the one item on which the 40% off coupon would be used, they were thinking about the problem in creative and insightful ways.

When a student first gave me his answer, I didn't quite know how to respond. I expected that the groups would find an answer, but I anticipated solutions to start popping up later based on the progress they were making at the time. For several of the groups to start noticing that the one item had to be more than 75% of the total purchase in order for the 40% off coupon to be a better deal, really impressed me.

I was excited, and I was very proud to see them start with no more than a picture and some questions only to end with a powerful solution.

The best part of the solution was the extension afterwards. Having the conclusion that the one item had to be worth more than three fourths of the total in order for the 40% off coupon to be worth using, led to students observing that 30/40 reduces to three fourths. Their problem now became one of verifying whether that result would continue for other combinations of discounts, or if it was merely a coincidence.

All in all the problem was rich with great thinking and exploration, and so far it served as one of my best class sessions this year.

I am still struggling to present the problems more consistently and effectively, but I won't be able to improve if I don't try new things. Here is to my own exploration and discovery.  

Making "Real World" Problems Feel More Real

Third week of school, and I am feeling more and more satisfied with the amount of problems on which students have been working. Starting with my Geometry classes working on the great transversal tape and sticky activity that I found on the wonderful problem based learning curriculum maps (that I adore exploring). The activity didn't go quite as planned because I wasn't able to make copies of the handouts, but the students worked well none the less.

Those Geometry classes also had fun tackling some logic puzzles as we started talking about forming proofs and deductive reasoning. I was very thankful that my wife and I were able to visit her grandparents last weekend, and while there her grandmother gave me a book titled "The Brain Game." The book has 27 intelligence tests "that will reveal your unique abilities" like an IQ test. 

When I first saw it, I instantly thought of the possible problems and tasks I could pull from for the students, and thinking about the proof lessons coming up, I was look at the logic tests for some gems. One of the puzzles involved blue wombats, and it was a nice starter before we started proving angle relationships.

Moving to the Algebra 1 classes, I tried my hand at implementing a 3 Act task today with a Pepsi Points task  that I also found using those wonderful curriculum maps. I am looking forward to talking with the kids on Monday about the court case involving the person who tried to redeem pepsi points for a jet, and today they were really into estimating how much things cost and starting to figure out how many points it would take to get the jet. This was also a fun task because I got to talk about my experience with my dad riding in his van and stopping on the side of the rode to pick up pepsi bottles for the caps, so that he could redeem the points for stuff.

The fun did not end with Algebra, thankfully, as my seventh graders got to work on a problem about designing a model of the solar system, a problem about McDonalds, and a problem about the Curiosity Rover. They were working with exponent properties and scientific notation, and problems about space and the McDonald's serving 1% of the world's population seemed to fit. I loved showing them pictures of some really cool planets and stars, and they were really interested in how the Curiosity Rover landed. Infecting kids with the wonders of space explorations is my favorite past time. The fun again came in when I was able to relate the problems to experiences from my life. They seemed to enjoy some funny stories from my childhood when my mom worked as a manager at McDonald's, and I would play around the store in the freezer, in the drive through (scrounging for dropped change for a small fry), and wearing the headsets listening to the drive through orders. One particular group of girls found the idea of a car coming through the drive through while I was looking for change quite funny.

I started thinking after having the chance to share the stories about myself to my students that being able to connect these problems to my life really seemed to help make the work we were doing more meaningful. The students weren't just working through problems from some foreign book, or even good problems that aren't related to anything. Not only were students given an opportunity to work on interesting and challenging problems, they were able to see these problems as something more than just a math problem. Through telling my little stories, I felt that I was able to make these "real world problems" more real.

Really Fun Moments from the First Week of School

The first week of school has been a blast. This is my first year at a middle school, and I am really enjoying these seventh and eighth graders. They are an impressive bunch, and I am very excited for what is sure to be a great year.

Starting off the year, I got the idea from a friend of mine to present a really nice counting problem. We started looking at a soda can pyramid picture that I found. The bottom row of the pyramid has 11 cans with each successive row having one less can until the single can at the top.

The goal was to count the number of cans, and after students went through and added them all together, we tried to come up with a better way of counting that would be more efficient. A few students picked up on the trick commonly credited to Gauss, which was fun because I was able to show them a nice numberphile video about the counting problem after they worked it out.

The fun began when they were faced with a pyramid that has 50 cans at the bottom. They thought about it independently for a couple of minutes, then they were split into six groups to work out the problem on big white boards. Below are some of the great ideas they came up with and presented to the class.


This idea was about turning the pyramid into a square and subtracting off the area. The coolest part about this idea was that the group accounted for the single can at the top of the pyramid.


This idea was about adding up groups of tens, nines, eights, and so on. The group noticed that each successive group of numbers was reduced by five, which was very cool, and they were able to use that to quickly get to multiplying their sums. 


This idea was very impressive to me. This group was trying to find a formula to use to find the number of cans. They went through a few different ideas, tried them out to see if they worked, and they finally landed on this beauty for any pyramid with base B. 

The other three groups came up with similar solutions, and this served as a great problem for the start of the year. They loved working in groups, they really enjoyed using the white boards, and I enjoyed being able to go to each group and help them make sure that each member of the group was on board and asking good questions. 

I look forward to posting more of these fun activities throughout the year. I am very excited about the start of the new year.

Count Down to the First Day of School: 2 days and 9 hours

Students arrive this Wednesday, and I am starting to get more and more excited  about starting a new year with a new set of students. Tomorrow will serve as my day to finish getting my classroom ready, and Tuesday will be the day I get to meet students and parents for the first time. The summer has flown by, and I find myself reflecting on all that I have learned from last year and from this summer.

My goals for this year include making better use of technology to help reduce the time it takes for me to give feedback to the students, being effective in challenging students to figure things out with tough questions while providing just enough supports to keep them from feeling discouraged, and I want to execute effective problem based learning this year.

Some challenges I'm anticipating include the space limitations in my classroom, the time it will take me to plan and set up the technology, and my ability to manage my classes when they work with each other.

Another goal is to journal at least every week on my progress, so we will see how that goes.

Wish me luck, and tell me what your goals are?

Bottle Holder Problem Idea

Last weekend I had the privilege of having dinner with my grandparents. While I was there, my grandma showed me a gift she received on the form of this wine bottle holder. Upon seeing it, I instantly thought of a number of questions. 

 

1) What is the angle of the base of the holder? 

2) At what angle is the bottle in relation to the holder? 

3) Are those angles the same? 

4) Is there a way to prove the angle congruence?

5) If the bottle angle is equal to the angle of the base is that a requirement for the holder to work?

6) At what level of liquid will the bottle fail to stay up? 

7) Would it be possible to dispense liquid from the bottle while it is in the holder? If so, for how long? 

These are all questions that my family and I brought up during dinner, and I think it would be interesting for students to think about these questions and formulate a way to answer them. 

What questions would you ask, and do you know the answers to any of the ones above? 

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? 

Giving this blogging stuff a shot

I'm about to enter my second year of teaching, and recently I have found an immense amount of fellow math teachers who have a plethora of great stuff online. As my goal is to make what I teach more relevant and thoughtful, I have spent the past month consuming a lot of great ideas from all over the math blogging and twitter community. I enjoy thinking about interesting new ways to create thought provoking learning experiences, and I'd like to start contributing some of my thoughts and ideas. 

This year I'll be teaching Geometry, Algebra, and 7th grade math at a middle school close to home, and I'm excited to use some engaging  projects and problems to get my future students thinking and and working with math in a way that will stick with them. 

The planning starts now, as I try to answer a ton of questions ranging from the logistics of setting up the problems, how am I goi g to asses student work in a meaningful way, and how am I going to make sure that the projects they work on uncover the content that they need to know. 

I welcome all of the feedback that you have to offer, and I look forward to building my toolbox as my career moves on. 

Octagon Table

While visiting family in West Virginia, I came across this table at their river lot. I'm wondering the angle at which the wood was cut. 

I can see this as being a good problem for getting into polygons and how they're built from triangles, and I like the link to carpentry. 

I can see having the students build their own table top with a varying number of sides, and figuring out what angle to cut the wood cross sections.

I'm thinking that this would fall under common core standard G-MG-A.1 or G-SRT-B.5. What do you think, and what would do with this problem?