Stacking Cups Activity (Systems of Equations)

This winter I observed a lesson in an 8th grade math class that used a cup stacking activity to introduce systems of equations. I really like the activity and I thought it provided the students with some rich mathematical exploration. This activity was something that I was very interested in using in my own classroom someday. I made a few changes and decided I would share!

I started planning this lesson with one thought: what do I want my students to be able to do once this activity is done. With that in mind I went to search for the CCSS and Mathematical Practices that aligned with this activity.

Common Core State Standards:
8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to the points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
8. Look for and express regularity in repeated reasoning

I also had some questions that motivated me while working on this activity. Here they are:

• What does understanding look like for this?
• How does this activity help with understanding?
• What will students do?
• How will I check for understanding?
• How will I have students consolidate?

This activity is meant to be an introduction to systems of equations, or an activity to do right after the topic is introduced. So I decided that if a student had understanding of these topics they would be able to write equations that represent real world scenarios, they would be able to graph these equations and find their solution, and they would be able to explain to me what this solution represents. This activity will help with their understanding because it will walk them through writing equations, graphing them, and interpreting their solutions. As far as checking for understanding, I have created a worksheet for the students to do after this activity that I might collect and check. However, this would all depend on my classroom structure at the time. To have my students consolidate I have given room at the end of the activity for a reflection.

Now for the lesson. I have broken it up into four parts: the warm-up, the introduction, the investigation, and the reflection. The warm-up consists of two questions that will brush students up on some skills that will be needed for the investigation. The introduction includes a 101qs video and a group discussion. The investigation is the Cup Stacking Activity. Finally the reflection includes another group discussion and a written reflection.

Warm-Up

For this warm-up I would either put these problems on a projector and have students write their solutions down or else I would pass out this warm-up on a paper. After it seems like most students are done I will pull the class together and we will discuss the warm-up.

Graphing and Writing Equations Warm-Up

Introduction

The purpose of this introduction to to get the students ready to complete the activity. First I would have the students watch Andrew Stadel's 101 questions video called Stacking Cups. The purpose of this is to get the students interested in finding a solution to this activity. 

I would have the students watch this video two times and then I would have a group discussion. I would ask my students what they Notice and what they Wonder about the video. I would record these answers on the board. Then I would have the students watch the video one more time. I would tell them that their task is "How many cups does it take for the stacks to be equal in height?". At this point I would introduce our activity, distribute materials (yardstick, two stacks of different cups, and the activity). 

Before I started I also wanted to anticipate some student strategies. The first thing that I anticipated was that students would assume that the height of two cups is twice the height of only one cup. If I saw any students using this strategy I would remind them to remeasure their stacks of cups and look and see if the height is really doubling. I also anticipated students not knowing where this change in height was coming from, that is why in the first problem of the activity I have the students label the different measurements of the cups. Hopefully by doing this the students can see that the height is changing each time by the height of the lip of the cup. These were some things specifically that I would be looking for throughout the activity. 

Investigation

The students should complete this activity in groups of 2-4 students. I will be walking around the room while the students are working so I can help them if they are struggling. I want a healthy bit of frustration to happen while the students are working. I think that this frustration is where awesome learning happens. 

Cup Stacking Activity 

Reflection

The reflection is the most important part of this lesson. With 15 minutes left in the hour (or once everyone is done with the activity) I would bring the whole class back together and we would discuss the activity. I would reference the Notice and Wonder that we did earlier, and see how many of the "wonders" we answered. I would also share some of the students work and we would share our answers. At this point I would show the video Stacking Cups - Act 3. Finally I would have my students complete the reflection at the bottom of their activity if they haven't done so already. I may or may not collect these depending on how well I think the activity went. 

I also created a follow up worksheet for this activity. This activity uses a lot of the same concepts used in the Cup Stacking Activity. I might have the students do this activity after they have been working on systems for a day or two. 

Follow Up Worksheet 

Variations of the Activity 

Noelani Davis- This is a variation on the cup stacking activity that uses the same 101 questions video. Noelani has a cool idea using Hint Cards with her students too! If her students are struggling she gives them a Hint Card to help them. I also adapted my follow up worksheet from Noelani's. 

Tara Maynard- Tara's class was where I first saw this lesson, her student teacher did this activity with her 8th grade students. Tara's activity has a really cool Part 2 that you should check out!

Dan Meyer- Dan has a variation that has students estimate their teachers height in cups!

Andrew Stadel- Here is the link again for the Stacking Cups video this activity is based on. 

Interview Reflection

This semester I have been observing in an 8th grade math class at Creekside Middle School in Zeeland Michigan. During my time observing I also conducted three student interviews. During these interviews I gave a student a math problem to work on, and my goal was to try and understand their thought process. While the students were working I would ask them questions to try and get them to explain their thinking. The purpose of this post is to reflect on the interview process and talk about what I would do different next time.

Before this assignment I had never done a student interview, so my first goal was to find a couple resources that would help me. I used NRICH to search for challenging problems to give the students I would be interviewing. One of my favorite problems that I found was called What's it Worth. Even though I did not end up using this problem for my interviews it was one that I seriously considered. Then I found a Student Interview article that gave a lot of great guidance on interviews including questions to ask the students, what to look for, and math problems to use.

I ended up giving all of my students the same interview question. Initially I was going to give them a choice of a couple problems but then I realized that I really wanted to be able to compare the thinking of the students I interviewed. I thought the best way to do this was by giving them the same problem. One of the problems that I found in the Student Interview article aligned with what the students had been doing in class. This is the problem that I ended up choosing:

A restaurant has square tables. Four people can sit at 1 table. Six people can sit at 2 tables. Eight people can sit at 3 tables. How many people can sit at 37 tables?

Before I started my interview I wanted to decide what I wanted to look for while the students were working on their problems. I made a list of things I would be happy to see in the students work.

My List of Awesome Mathematical Modeling:

-Table of Values

-Equation

-Picture/Diagram

-Recognizing a Pattern

I am not going to lie, I was very nervous to do these interviews at first. I was worried that I would not know what to ask the students while they were working on the problem. So again I went back to the Student Interview article and I found an AWESOME list of non-leading questions that I could ask during my interviews. 

My List of Awesome Non-Leading Interview Questions:

What do you predict will happen?

Can you solve it in a different way?

How did you figure it out?

Why did you______?

You wrote ______. How did that help you?

I noticed that you stopped what you were doing just now. What were you thinking?

I don't know what you mean by that, will you show me?

Will you draw a picture of that?

You started with _____ and then went to ____. Tell me your thinking.

Can you tell me what ______ means?

Are you right? How do you know?

Is there another way to show me? What is it?

Are there things that you already know that would help you solve this problem?

I was overall very pleased with the way that my interviews went. Two of the three students I interviewed used a table, a picture, and an equation to represent their work. All three of the students identified the pattern (increases by two) in the tables. While only one student had a correct final answer, one other student just made a mental math error that resulted in an incorrect solution. 

The most interesting interview was with my second student. This student did not use tables or an equation to model their thinking. They thought about the problem as adding "sets" of tables. During the interview I was confused why they took this approach, because I knew they had done problems similar to this in class. So then once the interview was almost finished and the student had given me their final solution, I asked them to draw what they thought the tables in this problem looked like. More specifically I asked what they thought "8 people can sit at 3 tables" meant. The student then drew 3 separate tables all with 8 people sitting around the table. Suddenly I realized why I was so confused! I had not stated in the problem that the tables were end to end, so this student thought that 3 tables meant 3 separate tables. If I would have asked this question sooner, I would have explained to the student that I meant for the tables to be end to end. Maybe by doing this the student would have solved the problem using a different method.  Here is the work of the three students that I interviewed.  

I learned a lot from conducting these interviews. I now have a couple resources that have great problems that I can use in my classroom someday, and a list of great  questions to ask students while they are working on these problems. The biggest take away I have from this assignment is learning the signs of when a student is struggling and when a student in confident in their work. I noticed that when a student did not feel comfortable with what they were doing they normally slowed down their work, looked to me for validation, or talked quietly to themselves. When one of the students was showing me one of these signs I tried to find a question on my list that could guide the student. Interviews like this could be very valuable to the teacher because by the time the interview is over the teacher will know exactly what their student knows and what they don't. The biggest drawback? Finding time in your plans to do interviews like this. 

Technology and Formative Assessment

Formative Assessment is arguably the most important aspect of successful teaching because it lets us know exactly what our students know and more importantly what they don't. I want to take a look at two online formative assessment tools, thatquiz.org and Socrative.

Both of these programs use quizzes as a way to assess student's understanding. These programs are meant to be used with technology but both also have a print option if a paper copy of a quiz is needed.  There are features of both programs that I like and also some that I find frustrating. 

Socrative:

Is there an app? Yes there is both a student and teacher app. Also quizzes can be accessed online at socrative.com

Do you need an account? Teachers do need an account but students do not. Students will need to enter room number and their name in order to access their quizzes. 

On Socrative you can create quizzes, quick questions, Space Races (a type of group quiz), and exit tickets. Here is what the teachers homepage looks like.

Best Features:

- Can create true/false, multiple choice, and short answer questions

- Can add images to questions

- Can randomize question and answer order

- Can import quizzes from an excel file or if you know the SOC number of an already made quiz you can import a copy of that quiz 

- There are different quiz preferences (Student-Paced with Immediate Feedback, Student-Paced with Student Navigation, and Teacher-Paced) 

- Can get detailed reports of quizzes for feedback 

Some things that I don't care for are that there are not many options are far as math characters (there is a sub/super script button though). I also don't like how there is not a built in gradebook or graphs and shapes cannot be created within the program. 

My favorite part of Socrative are the reports the teacher has access to after the quizzes are completed. Teachers get live results after they send out the quiz, these results give the students' names, question numbers, and whether the question was answered correctly or incorrectly. This is what a teacher would see during the live results. 

Once the quiz is completed teachers have a few opitions of how they want to recieve their reports. 

thatquiz.org

Is there an app? Yes (cost is $3.99) however, you can only use premade quizzes. On the app quizzes can not be created. The online version of the program however allows teachers to create and share their own quizzes. 

Do you need an account? No, but without an account (its free!) you cannot create and browse quizzes you can only use the premade quizzes. 

On thatquiz.org you can create quizzes that are matching, multiple choice, or short answer and multiple choice (called "slides"). Tara Maynard presented about thatquiz.org at this year's Math in Action conference and you can find her presentation here. 

Best Features:

- Manage multiple classes all from the same account. This makes it easier to have quizzes across different subjects for different classes. 

- Browse/Share quizzes (quizzes can be seached by code, subject, or even teacher's name)

- Teachers can set a number questions for each quiz. For example, you can set the quiz up so that it randomly chooses 5 questions out of a possible 20. This allows students to have different quizzes and also students can retake quizzes without getting the some questions. 

- Can create graphs and shapes and upload images directly in the program

- Built in gradebook that allows the teacher to see what their students answered and also how many times they attempted the quiz. 

- Teacher can choose to always save grades or else they can save once a students gets higher than a certain percentage (In Tara's class she only records her students' grades if they are an 80% or above)

Here is what a teacher would see on thatquiz.org while they are editing their quiz.

I think the best part of thatquiz.org is the ability to share and browse quizzes. My biggest compliant about this program is how times consuming it can be to create quizzes. That's why it is so great to be able to share and collobrate with other teachers. You also have the option to not share your quiz as well. 

I have seen both of these programs used for formative assessment in the classroom, and I think both have a lot of benefits. I love the reports that Socrative provides but wish they have a built in gradebook like thatquiz.org. I also wish Socrative made it easier to share quizzes with other teachers. As far as looks, my vote goes to Socrative. 

I have also heard somethings about the program Nearpod which allows teachers to be able to create presentations as well as quizzes for formative assessments. I am currently tyring to learn more before I give this program a review. Are there any other formative assessment programs that you use and would reccomend? 

Number Talk

A Brief Background on Number Talks:

Kathy Richardson and Ruth Parker began giving teachers mental math exercises at professional development settings during the 1990s. As these teachers completed more mental math Kathy saw the teachers becoming more proficient. Kathy and Ruth realized that they wanted to develop students' understanding, but they did this best by using concrete models to figure out computation. This idea lead to number talks. 

So...what is a number talk? A number talk is a short daily routine that gives students meaningful practice with computation. Number talks help students develop fluency while solving mental math problems. One great way to introduce a number talk is after a counting circle. In case you haven't heard of a counting circle here are some blogs to check out:

 Sadie Estrella - Counting Circle

 Malke Rosenfeld - First Counting Circle

 John Golden - Sadie's Circles 

I would love to explain in detail what exactly a number talk is but Jo Boaler does a beautiful job (way better than I ever could) in this video, it is worth the watch!

Here are my Top 5 Benefits of Number Talk:

1. Teaches Number Fluency 

This is in my mind the greatest benefit of number talks. The more students are involved in number talks the deeper conceptual understanding they have about numbers. Students will begin to see different fact strategies for addition, subtraction, multiplication, and division the more they practice. They will then use these known facts be able to see things like 6+7=6+6+1 or 6+7=7+7-1. Students show their fluency when they are able to explain why they are doing what they are doing, and also when they know how to solve a problem using different methods. 

2. Number Talks Build Confidence in Students

Once a student has a deeper conceptual understanding of numbers and how they work they will start to solve mental math problems at a faster pace. It might not happen right away, but the more practice a student has the faster they will get during number talks. This gives our students confidence. They can feel that they are becoming more proficient. I know that this confidence will carry over into their mental attitude while solving other math problems. 

3. Students Learn Different Strategies 

Sometimes a student will solve a problem in a way that other students would have never thought of. By sharing these idea in the form of a number talk our students gets to collaborate and may learn a more effective way of solving mental math problems. You may hear students say things like "Oh I never thought of that!" That is one of my favorite sentences! This means that our students are recognizing that other students think differently than them and that they have valuable ideas to share as well. Even the teachers have valuable things to learn from their students. 

4. How to Make a Problem Easier to Solve

Which is the easiest to solve? 

13-7?

12-6?

11-5?

10-4?

Chances are that students would struggle more when presented with 13-7 versus 10-4 or 12-6, but the awesome part is they all equal 6! During number talks students begin to see that they can change the problem they are working on to make it easier on themselves. If a student gets uncomfortable solving a certain problem they can change that problem into something easier using their fact strategies! One student might struggle trying to solve 46-29 mentally, but once they see that 46-29=50-33 they might have an easier time solving this problem mentally. Its important for students to be able to make mental math easier for themselves. 

5. Teaches Students Communication

How do we expect our students to be able to communicate about mathematics if they never practice? At first students will struggle trying to explain their thinking during a number talk but the more practice they get the more articulate they are and the more thorough they are in their explanations. One of the 8 Mathematical Practices of the Common Core is for students to be able to "Construct Viable Arguments and Critique the Reasoning of Others". This means that students need to be able to communicate about math, using mathematical language. Number talks are a great way to get our students on the track of mathematical communication, which leads to deeper conceptual understand and fluency, look at that its like a cycle!

Last semester I wrote an article about math anxiety and counting circles. I talked about how nervous I felt during counting circles and how I basically dreaded doing them in class. I can honestly say now that I enjoy counting circles. I enjoy them because of how much the number talks have helped me. Over the last two semesters I have deeper understanding of numbers and better number fluency. Counting circles no longer make me so anxious because I have different strategies to solve mental math problems. I will 100% do regular number talks in my future classroom because they will do nothing but help my students in the long run and make them better mathematicians. 

To Flip or Not to Flip

Flipped classrooms are a hot topic in education today. Many teachers believe that they can do more meaningful mathematics when they use technology to support their students' learning. The idea of a flipped classroom is that the students will learn the content, that normally would be taught with a lecture style, outside of the classroom with the use of technology. This then allows the teacher to be able to use class time doing activites that apply the students' knowlege. Normally students will learn the material by using a combination of videos and coinciding worksheets. These worksheets are normally checked and counted as a "homework" grade. Other teachers may use quizzes at the beginning of class to keep their students accountable for watching the videos the night before.

One of the struggles of teaching is that not all students learn at the same speed or in the same way. While some students may succeed in a traditonal classroom setting, other students may fall behind. The flipped classroom model allows students to rewatch the videos multiple times if they are not grasping a concept on the first try. Also by watching the videos before the next class students can come to class with questions, some teachers even put a spot on their guided worksheets for such questions. 

Having students learn the material independently also allows teachers to give students more individual support. Since class time in a flipped classroom is used to do activites, the teacher is then able to walk around the room and get immediate feedback about where their students are as far as understanding the content. Therefore, if a particular student is struggling then their teacher can spend one-on-one time addressing their questions.

In the Februrary 2014 issue of Mathematics Teacher there was an article about flipped classrooms. Two teachers, Amanda Moore a 7th grade math teacher from Battle Creek, MI and Matthew Gillett an 8th grade math teacher from Armada, MI, observed the effects of flipping their classrooms. Both teachers saw homework (watching instructional videos and completing guided notes) completion rate rise, with one class even increasing by 19%. After doing a unit of flipped instruction both teachers agreeded that by flipping their classrooms student engagement increased in class and they also said they had more time to do meaningful mathematics. The homework completion rate indicated that more students were doing mathematical work outside of school, and students were also engaging in more problem solving in class that previously was provided through homework. 

The last two semesters I have volunteered in Tara Maynard's 8th grade math classes in Zeeland Public Schools. Mrs. Maynard uses a flipped instruction model in her classroom, so I have been able to see the benefits of flipping first hand. Mrs. Maynard uses videos that she makes (with her own voice using an app called ShowMe) for her students' instruction. The students in Mrs. Maynard's class can access these videos without the use of internet at home through their iBooks app on their iPads (Zeeland Public Schools are 1:1). Mrs. Maynard also provides the students with a note packet to complete while they watch the videos. 

Here is what Mrs. Maynard says are the benefits of flipped classrooms:

- She gets to have conversations with everyone of her students everyday

-Truly know what the students understand and what they haven't yet grasped

-There is more time to do explorations, activites, demonstrations, labs, and deeper questions

-Students are able to move at their own pace and learn with their own style

-Supports anywhere anytime learning

-Parents can't give the excuse of "I can't help"

As with everything in life there are also some negatives to flipped classrooms as well. Here are some of the things Mrs. Maynard says she struggles with:

-Time (creating videos, creating activities etc.)

-Still giving paper and pencil assessments

-If its just a video and a worksheet it is not flipped instruction

-Videos can be done wrong (bad lecture, too long, boring, etc.)

-Students still come to class unprepared and waste time

Here is a video of a typical day in Mrs. Maynards flipped classroom.

Mrs. Maynard also asked her students what they thought of learning in a flipped classroom, here are their thoughts.

Almost all of the teachers who have flipped classrooms can agree that students are able to engage more in the mathematics they are learning. While flipped classrooms might have some downfalls I believe the positives outweigh the negatives. I would love to some day use a flipped model for my own classroom. I think that flipped instruction can provide so many learning opportunities for students and isn't that why we are teaching to begin with? 

Fears of a Future Teacher

Well I am going to be honest, I have had a very hard time writing this blog post. Each idea I came up with just didn't feel right, I would start writing and then lose interest in what I was writing about. I think my writing block came from some stress I have been experiencing lately. This last Thursday I turned in my application for the College of Education. All of a sudden everything felt so real to me. I realized soon I will be responsible for teaching students. This terrifies me. I worry whether I will be prepared for the challanges that are ahead of me. Then I realized I wasn't alone.

I, along with over 11 million other people, read the the blog Humans of New York. (In case you haven't read Brandon's blog I would suggest you take a look https://www.facebook.com/humansofnewyork) The other day I read a post that stopped me in my tracks. Over the past week Brandon has been focusing on a school called Mott Hall Bridges Academy. He has talked to the principle Ms. Lopez, students in the school, and also the Mott Hall Bridges teachers. Below is the post about a converstaion a teacher, Ms. Powell, had with Brandon that really resionated with me. 

All of a sudden I realized that Ms. Powell had summed up all of my fears about teaching. Recently I have heard all about the gaps in the education system, how students can be multiple grade levels behind by the time they reach your classroom. My biggest fear is that I will not be able to help my students be successful. How can I teach my students to factor polynomials if they have limited multiplication skills? As a teacher I know I will always have more material to cover than I have time to cover it in. These students who come to me grade levels behind need so desperatly to be successful, but where will I find the time to fill in the gaps? I am sure this is one of the biggest stuggles amoung teachers, and it cannot be fixed overnight.

I think the work of a prepared teacher starts long before they ever step foot in a classroom. My work as a teacher begins now. I am learning the skills and practices of successful teachers, but in my opinion the most important tool I can have as a teacher is collaboration. There are thousands of other teachers in this country who have the same fears I do, and they face them on a daily basis. They are my most valuable resource. Its important that I learn to ask them questions and learn from them. 

We teachers are a team, that needs to work together. We all have the same goal, we want our students to be successful. At times my fears about becoming a teacher can be overwhelming, and I think that I am not cut out for this. Then I realize all of all of teachers I look up to, that inspire me, have all had the same fears. They have had sleepless nights worrying about how they are going to fit in all of their material, and they have experienced the joy of watching a student finally understand something they were stuggling with. I am terrified to become a teacher, but I know as long as I have thousands of other teachers ready to help me in my journey, I will be alright. 

'Twas the Night Before Exams

‘Twas the night before Exams and all through my room,

I sat quietly awaiting my ultimate doom.

The pencils were placed in the backpack with care,

In hope that at the paper I just wouldn't stare.

The students were nervous all stiff in their chairs

While visions of failure hung in the air.

And the teachers with their Scantrons and I with my note cards,

Prepared for a day that was sure to be hard.

When books began to fall creating a clatter,

My teacher looked up asking what was the matter?

Frustration had built inside of my body,

I responded to her trying not to be snobby.

Then what with my wandering eyes should I see?

My clock blinking bright, twelve thirty-three.

A full night’s rest, surely not ahead,

I crawl to the couch, resting my head.

Hours passed by filled with cartoons and news,

But this was not curing my exam day blues.

Go math! Go english! Now physics! Spanish two!

Everything chemistry and anatomy too!

All Gen Ed classes, no matter how small,

Go away! Go away! Go away all!

All folders filled with useless review

All protractors, pens, and everything due.

As I began to drift off, finally calming down,

Thinking of the future made me frown.

I prayed I would not sleep through my alarm,

Surely a snow day would cause no harm.

My eyes-how dull! My hair balled in a mess!

As morning drew near I tried hard not to obsess.

I tumbled out of my cave, my stomach filled with fear.

I am not ready for this oh my, oh dear.

And grabbing my backpack with nerves in my belly,

I embarked on my journey, my legs just like jelly.

All of this panic filling my head,

Soon developing a new feeling of dread.

Last minute cramming would do nothing to alter,

Should have studied but knew I would falter.

With a switch of the light and a quick little sigh,

I trudged to school, kissing my good grades good-bye.

The teachers passed out the intimidating tests,

All I could think is, I should have gotten some rest.

Then I yelled to my peers, with a loud hearty cry,

Happy exams to all, kiss your GPA good-bye!