Tuesday, January 10, 2017

Toothpick Task 2017

I had an icy two-hour delay this morning, and when you operate off the a combination of getting to school by 6 every morning and living 40 minutes away from school, I never find out about the delay until after I arrive. I enjoy these mornings because I get to catch up on some work. I had a few students come in 3 hours before school started to work, I submitted a manuscript review for Mathematics Teaching in the Middle School, and worked on questions for an upcoming MTMS Twitter chat (which you should join the 3rd Wednesday of each month at 9ET/8CT with using #MTMSchat).

To the task...

Additional Information 

Q1~How many complete rows of toothpicks can you make with the amount given?
Q2~Will there be any extra? How many?
Q3~How many more would you need to complete the next full row?
Q4~How many toothpicks would the container need to create 20 rows?
Q5~If someone wanted to know how many toothpicks would be in the 75th row and you didn’t want to build that big of a pyramid, what could you do to find that answer fast? How do you know that will work?

The first part of this toothpick task (by @ddmeyer http://bit.ly/2jzv0sZ) begins innocently enough, with students trying to find out how many full rows could be made with the given amount of toothpicks. The interesting part to this is how students choose to solve this first problem. Almost all of the students (I did this with 7th grade, but could work for 5-8 or more). will start by dividing 250 by 3, not fully thinking what the result will mean. Once it is understood what this answer of 83ish means (asking a quick question about what does the 3 mean normally brings light to this problem), students branched off into three different ways to solve: a full blown picture, a chart, or a combination. Within these three options, most students either went with the tooth pick totals, or the total number of triangles. While many of these options will be similar, the different number of ways to solve this problem speaks to the accessibility of this task.

Question 5 is a great question as well because you can get into some generalizing. Some students automatically search for patterns, but some don't think of that as an option. Making it more explicit helps students understand that finding patterns is an option that can always be looked for. Patterns may not always be prevalent, but at least now students know that option exists.

This task will help us move into some tasks with some equation/expression writing built in such as Central Park, In-N-Out Burger, and Detention Buy-Out.

Enjoy the student work below! I especially like the one towards the middle where the students exclusively used triangles instead of individual toothpicks like many of the students used.

Note: As I was typing this, we are now cancelled :-(


Other Task Posts Examples (more available in previous entries)...

Pokemon Go

Man Versus Squirrel

Shrinking Dollar

Friday, December 2, 2016

Sticky Note Task Expanded

With my 7th graders, we recently finished the sticky note task, which is a compilation of tasks by Andrew Stadel and Jon Orr. I have done this in the past, but added the part by Jon this year. Surface area is probably the predominant math concept, but I found question 4 to be particularly interesting. Before we talk about that, here is the task...

Q1: How many sticky notes are needed to cover every exposed face of the cabinet?

Q2: How many pads would you need to have enough sticky notes?

Q3: How many sticky notes fit on the whiteboard?

Q4: If you had 120 of the 1.5 inch by 2 inch sticky notes, what is one size of board that you could cover (dimensions and area)?

Q5: What fraction and percent represents the following categories?
*Pink and blue Combined?
*All colors that are not pink or blue?

For question 4, students had a variety of amounts of sticky notes on the boards...anywhere from 12 notes by 10 notes to more of a ribbon board look of 120 notes by 1 note or 60 notes by 2 notes. The unexpected part for the students came when figuring out how much space the board they created took up. No matter the configuration of the 120 stick notes, the area of the board was always 360 square inches. This makes sense since all of the boards have the same number of sticky notes, and each note has an individual area of 3, but this took some time for everyone to comprehend. A great discussion piece for sure. 

The other thing I think was interesting about this task was if the students chose to find the sticky notes by finding the area of the board/face of the cabinet first, or find the number of sticky notes along the length and the width first. Some student did one method for question 1 and the other for question 3. 

This was a great task that would be appropriate for almost all levels of middle school. I have included some student work below. 






Monday, November 14, 2016

Mathematics Tasks for 2016-2017/The Pokemon Go Task

I've been doing a lousy job of blogging. I've thought to myself many times that I have no idea how I blogged in previous years. Last year at my university job, I always made a point to tell my preservice teachers just how much time and effort it takes to even do an adequate job teaching. I may have even underestimated that point, because clearly I forgot how much time this job takes up.

With that being said, I think it is important to get the tasks that my students have done this year out to other math teachers so they know what is possible. I have had multiple groups of teachers come into my classroom this year, and the comments are always the same...

1. I wish my students communicated like that...
2. I wish my students showed understanding like that...
3. I wish my students stayed on task like that...

I am very proud to hear these comments, and I always pass the kind words along to my students. One of the comments I don't hear are that my students are geniuses or rocket scientists. I am glad this is recognized by teachers as well, because teachers think that you can only do these tasks with the highest level of student academically (I've heard that comment many times at different presentations I have done). This certainly isn't true, because frankly a lot of my "lower level" students figure these tasks out faster than my traditional "successful" students. My students struggle in general with these tasks, but given time and the allowance of communication, everyone figures them out.

This process of getting students to communicate, show justification, and stay focused doesn't happen the first time you do one of these tasks. It's a process...sometimes even a slog. I think that problems happen when a teacher gives up on these tasks after the first try. You must keep plugging away. The benefits will far outweigh any struggles you have. All of these positives that teachers pick out carry over into everything else you do. These tasks aren't novelties, but an expectation of how everything should go in class. Even on days we don't do these tasks, my students display these desirable traits (for the most part...they are middle school students :-) ).

I think from here on out, I will post one task every so often, talk very briefly about it, and post some student work. Feel free to use these tasks, ask questions about these tasks, give suggestions to make these tasks better, etc. I will make sure to link where each task was adapted from so you can go right to the source if you wish.

I did the Pokemon Go task with my 8th grade students. I adapted this from Dan Meyer. I polled my students before I did this task, and only a small percentage had ever played Pokemon Go, so I was initially worried that some students wouldn't understand what I was going for. I ended up having very few question about the mechanics of the game, so don't worry about that. One idea that many student brought up after the fact was that we didn't calculate for the middle of the fan. I did not take this into account when I went through my task either, so I will have to account for it next time. As sad as the last two question are, they are both true depictions of my weekends these days. I have included a few pieces of student work along with my notes that I took the first time I prepared for this task. Hopefully I will have time to include these tasks a little more frequently, but here are links to both of my task pages for the year (updated as we do more tasks) in case you are curious what else we have done.

7th Tasks

8th Tasks

Task Nine ~ Pokemon Go (Dan Meyer)

Info Two ~  100000 cm = 1 km

Info Three ~ 1 mile = 1.61 km

Info Four ~ 1 mile = 5280 feet

Info Five ~ Each of the fan’s blades is 45 cm long

Info Six ~ The fan spins at a rate of 10 rotations for every 8 seconds

Info Seven Target Map

Q1 ~ How long would it take to hatch the 5 km egg using the ceiling fan?

Q2 ~ My daughter was sleeping in the car while my wife went into Target. If I stop the car, my daughter will wake up, so I kept driving around the Target building until my wife was finished. If I kept at an average speed of around 6 miles per hour, how many laps did I make in the time she was in Target if she was in the store for 20 minutes?

Q3 ~ I had my Pokemon Go app on as I was driving around Target (I wasn’t playing...it was just on). How many 2km eggs (I started with 4) did I hatch in the 20 minutes if all of my 2km eggs started at 0km?

Wednesday, June 29, 2016

What to Expect From the Blog and Me in 2016-2017

It is still June, but I am very excited to get the school year started. I had a great year at UNI as the middle level coordinator. UNI is a first class institution with great faculty, staff, and most importantly, wonderful students. I felt like I was able to make a difference in my classes and with the middle level program, but I missed working with middle school students, so I was lucky enough to be able to get my job back.

I am busy revamping my classroom website, locating and testing new tasks, and getting prepared for the grind that is the school year. The great thing about summer as a teacher is I can still work 3 or 4 hours a day (compared to the 12 or 13 during the school year) and still have time to spend doing summer type activities.

If you are new to my blog, I normally post about mathematical tasks I have done in the classroom. I post student work, and let you know what went well and what could be improved so if you try these tasks, you will know what to expect.

Here are a few of my previous posts to give you an idea...

In-N-Out Burger Task

Man Versus Squirrel

Doritos Roulette

I look forward to doing these tasks again, along with the many tasks on my spreadsheet I have tested...

Mathematical Task Spreadsheet

I found some new tasks as well that I know my students will enjoy...

Vroom Vroom

The Nardo Ring

Gas Station Ripoff

Marble Slide

The nice things about these tasks is that many come with base questions, but you can expand your questioning to include what students are wondering about the situation. By the way, a big thanks to the #MTBoS and other math professionals on Twitter for consistently pumping out these tasks for my students to enjoy.

I will also be involved in a few professional development and speaking opportunities where I will be discussing 1:1 Mathematics, and Task-Based Mathematics.

Finally, I am in year two of being on NCTM's MTMS Editorial Panel. This is such a great opportunity for me to get involved with middle level math on a national level and meet other math professionals I wouldn't normally get the chance to meet.

I look forward to blogging this year and hopefully helping you feel good about implementing tasks into your classrooms! Have a great summer!

Dr. Clayton M. Edwards
Grundy Center Middle School
Middle School Mathematics Instructor

National Council of Teachers of Mathematics Editorial Panel
Former Middle Level Coordinator (UNI)
Ed. D. Curriculum and Instruction (UNI)
MA Middle Level Mathematics (UNI) 

Yager Exemplary Teaching Award 2014
Awarded Outstanding Doctoral Dissertation at the University of Northern Iowa 2014

Monday, January 4, 2016

Cube Net Task and the Excitement of Substitute Teaching

I miss working with middle school students on an everyday basis. There, I said it. Don't get me wrong, I enjoy my new job as a middle level coordinator in higher education. I get to work closely with middle level preservice teachers and make an impact on their future teaching and the successes of all the students they will ever have in a classroom. I had a wonderful group of students for my first semester who I thought were genuinely curious to learn about what really goes on in a middle level classroom and how to best serve middle level students.

One of the preservice teachers asked me point-blank "if you liked teaching middle school students so much, why did you leave?" That was a great question, and of course the answer was to have a more widespread impact on middle level education and NOT because I was trying to get away from middle school students. I took that question as making sure I had a genuine connection to middle school students, which I appreciated. Just like middle school students, college students will be brutally honest...and that is why we get along well.

I still have a strong connection to my previous school district, and I am very grateful for that. I enjoyed working with the students and staff. I go back frequently for many reasons, with the most important being access to the classroom...making sure I don't lose touch with the reality that is middle school. I feel like this gives me a lot credibility with my preservice teachers, and I enjoy it as well. I also went back for a few open gyms over break...I am only 30 something, but I felt like I was near the end after that experience. I didn't know my feet could fit so many blisters.

Recently, I had a unique opportunity to actually be a substitute teacher in my old classroom. I didn't have class scheduled in the morning when I was needed to sub, so I jumped on the opportunity. The day I was there was the start of a new trimester, so the current teacher left it up to me what to plan for the day. As you can tell from my previous blog posts, I am into robust problem-solving tasks, so I thought I would go that route.

From my past experiences, my sixth grade students always seemed to lack in justification, collaboration, and problem-solving skills (not so much in math content knowledge). I don't know why that was (maybe the middle school transition or using a more traditional curriculum?), but when I gave them one of these tasks, you would have thought I gave them something in an unfamiliar language.  For example, when I first get my 6th graders, I would ask them to justify an answer, and many would write, "I used a calculator" or "mental math." They would give up at the drop of a hat when something was difficult (which it always was). It generally took a month or so for most of the 6th graders to be comfortable with what justification and perseverance, but when they did, we were able to reach some truly amazing mathematical heights.

I decided to select a task dealing with nets and if they could be folded into cubes. The closest 6th grade standard to this would be 6.G.A.4, but that is probably a stretch. You would only focus on the solving problems with nets and ignoring the rest. I wasn't too worried about the standard anyway, because I was more concerned with the mathematical understanding and collaboration. This task is very accessible to many different grade levels and that helps to focus on the process standards without worrying about the content so much

Here are the directions for the task (note: this task comes from @NCTMillum):


Before we started, I put up a simplified version of my classroom expectations. My expectations are centered on students improving and not worry about their current standing compared to others...just getting better all the time. This is a strange concept for most students who I haven't had. I can keep making mistakes and correcting until I understand? You won't leave me behind? I still get full credit if I end up showing understanding even if it isn't on the first try? I had no idea another person with this philosophy existed until I ran into Rick Wormeli at AMLE.

They are always louder than a whisper, but if you are going for a 2 on a 1-10 scale with 10 being the loudest they could be, and they want a 10, I'll happily take a 4 so everyone can have a meaningful discussion and hear themselves think.

I generally let students work with others they were comfortable talking with so there aren't a lot of  conversational barriers. I printed off two copies of this sheet off of the @NCTMillum website so students had plenty of room to write, draw, and explain. 

Keep in mind that the order of the nets shifts around every time you start the online portion of the activity over, but it is easy to find which net the students are talking about. We did not use their 1:1 laptops for this activity even though the activity was online. I like to save that for the discussion at the end. There are 11 nets that make cubes and 13 nets where something is incorrect (not enough faces, too many tops, etc). I did not tell them how many did or did not

This activity really takes 60+ minutes to do it right. I was there on an early out day with 40 minute classes, and by the time we did a few introductions and went over expectations/task objectives, we had about 30 minutes of student work time/discussion. I had the students put the sheets in a safe place in case I got an opportunity to come back and finish the task (I know what you are thinking...fat chance a 6th grader could hold onto a piece of paper for 2 months...it could happen :-)
Right away, I could tell that some of the students were not strong with spatial thinking. I try not to butt in too much, but listen to the ongoing conversations. One of the students who was confident in spatial thinking was trying to help a student by explaining that it made thinking about the cube easier if you picked out one of the faces to be the bottom before starting, and go from there. I thought this was sound advice, so I let this student explain to the class what she was thinking. I think this idea helped jump start a few individuals. 

I enjoy using tasks from @jstevens009 and his site, www.wouldyourathermath.com, and I saw a lot of that argumentative spirit come out in these students when I am not sure they had ever thought about math this way before. Two boys were paired up, and they were not getting very far in making a decision on this net...


This seems pretty cut and dry to me, but from a 6th graders perspective who has been used to going with a gut feeling, I could see why he thought it was a cube. The conversation between these two boys was basically a yes it was, no it wasn't back and forth, which was going no where. I noticed this as I was walking around and stopped to talk with them. I reiterated the need for justifications, and told them to separate for a few minutes, come up with their best arguments, and come back together to make a decision. I gave no indication of who was correct and who was not. When they got back together, they had actually both came up with the same reasoning for it not being a cube, meaning that after taking the time to think and justify, one of the boys changed positions. Both boys actually pictured the top/middle square as being the bottom and stated that there could never be a back side and the top would overlap. This is a great example of the power of individual think time before talking with a group. It arms everyone with ideas instead of just accepting what someone else has or denying someone's suggestions just because a gut feeling told you so.

I had been in this classroom before helping out, and like any 6th grade classroom, there are some student who just need to be doing something. Listening to someone talk at them for 40 minutes doesn't work for them. I noticed those students really shining. They were free to talk and discuss, and they were getting to create the knowledge. These types of tasks are perfect for middle school students and their unique needs.

I wanted to keep letting them work, but we were running low on time. I put the computer version of this task on the screen and called on random students to select one of the nets and tell me yes it makes a cube, or no it does not and why. The students had plenty of time to work through justification with each other, so I did not feel bad at all calling on different individuals. They were prepared because I gave them plenty of time. Some of the answers were simplistic but to the point. "This cannot be a cube because it only has 5 sides (note: we did have a conversation about what a face is to clarify)." Some of the students thought it would be best to show a model of what they were doing, so one student actually grabbed some textbooks and started folding them up like each book was a face. After the student explained, I click on yes or know and the animation folded up. In one instance, a student was incorrect, but we had a discussion on what their misconception was based off of the animation. This is another value of the improvement mindset. It's more of an environment, really. Sometimes you feel bad if someone has a faulty explanation and you have to correct the student, but when the goal is improvement, students don't see it as being so bad. It gives them a chance to learn and it gives other students a chance to further explain misconceptions. I only knew these students for a short time, and to them, this concept of improvement was an epiphany.

Overall, this is a very accessible task to start addressing the concept of justification with your students. This would be one of the first tasks I would do to make that point. If your students are not used to justifying, you will get weak justifications at first, but you have to understand that your students probably don't know any better. It is your job to get them to that point. Trust me, it will get better. Here is a collection of student work from that day. Hopefully you will notice how quickly we were able to go from "this is a cube because it folds into a cube," to some of the justifications listed on the student work.

(Notes: not all of the student work is correct...but I hope to have more time in the future to discuss, and some of the incorrect work was discussed during our discussion portion of class. Some of the justifications have letters like B to represent bottom or back and so forth...you have to look closely. There was a wide variance in how far the student got, which was alright. I would allow for as much time as needed normally).

Dr. Clayton M. Edwards
Assistant Professor of Curriculum and Instruction 
Middle Level Program Coordinator
University of Northern Iowa
Curriculum Vita
Mathematical Tasks Spreadsheet
1:1 Mathematical Philosophy