First Day of School Handshake!

On the first day of school my two goals are to make every child feel welcome and excited about learning math in my classroom. In the book The First Days of School: How to be an Effective Teacher, Dr. Wong recommends standing at the door and shaking each child's hand as they enter the classroom. So I stand outside Room 108, look each student in the eye and give them a firm handshake and a big smile as they come in and join the class.

And then once the students are seated, I want to generate some enthusiasm for mathematical problem solving by giving my students an opportunity to do some interesting math on the very first day of school by posing an intriguing and mathematically rich problem for the students to consider and solve. The handshake problem is a fun way to combine these two first day of school goals--it goes nicely with the idea of meeting and greeting new people, it is pretty easy to set up and explain the problem, it is accessible for most students, there are multiple ways to solve this problem and it contains some intriguing and elegant mathematical patterns. 

After I shake every student's hand at the door, I ask the students to get up and shake the hand of every other person in the room. I play James Taylor's version of Getting Know You while students walk around the room shaking each others' hands. Thank You For Being a Friend would also be good background music to play. Then I ask "I wonder how many handshakes it took for every person in this room to shake hands with each person in the room."

Here are some ideas I would like to try this year to improve my presentation of the problem and to get students more engaged in the problem by asking them what they notice and wonder about this activity. I will have a slide with a photo of each child on the screen and I will get the discussion started by asking several students if they shook hands with all the students in the room. I could also ask a child if they shook hands with another particular child and I will start to draw lines on the slide connecting any two students that acknowledge that they did shake hands with each other. This would be a great time to discuss the question about how we count handshakes:   if two people shake hands does this count as one handshake or two handshakes? After I have drawn several lines connecting various students in the classroom, I will ask some general questions to get the students thinking about this problem:  What are you wondering about? What mathematical questions could we ask about this activity? What is the problem? What do you find interesting about this problem? What do you notice about this problem?, etc. Hopefully through this discussion, the students will come up with the mathematical question posed by the classic handshake problem:  "if you have a room full of people and everyone shakes hands, how many total handshake are there?" and suggest some possible ways to think about and solve this problem. Then I will give the students some time to discuss this problem in small groups. Other possibilities for class discussion, either before or after the students work on the problem in small groups, could include showing a slide of all the students, but this time the photos would be shown in a circle instead of in rows. In addition, I could show a slide with one student and ask how many handshakes?, two students:  how many handshakes?, three students: how many handshakes?, etc. and then discuss possible patterns that emerge.

I  plan to use this problem for the first Problem of the Week (POTW) for the 2015-2016 school year:) And I am hoping that it will get the class off to a good start in terms of working on challenging math problems in small groups, discussing and sharing our mathematical thinking with others and finding multiple ways to solve problems.

Activities for Exploring Number Sets

1) Who Has a Number Bingo Board:

Who Has a Number Bingo Board and Vocabulary Review (below) available here.

2) Students work in groups to review and learn vocabulary associated with number sets. (I plan to do the vocabulary review first to help student activate prior knowledge before playing the Bingo game above.)

3) Number Sets Search found on End of Year Resources post from Resourceaholic which came from here.

4) Number Experts: a Bitesize Gem from Resourceaholic.com: Assign a number to each student. They become an expert in that number, creating a display and giving a presentation about its interesting properties.

5) Find the Factors: I think these puzzles are brilliant and plan to use these during the first week of school to help my students practice their multiplication facts.

6) Stand Up Sit Down (Quick Physical Games for the Math Classroom)

• Stand when the number is prime; sit if it is composite
• Stand when the number is even; sit if it is odd
• Stand when the number is a multiple of 3; sit if it is not a multiple of 3
• Stand when the number is a factor of 24; sit if it is not a factor of 24
• Stand when the number is a perfect square; sit  if it is not a perfect square

Try to "trick" the students by standing up or sitting down when they should be doing the opposite.

6) I Have Who Has Games

Deposit and Withdrawal Cards

When we study negative numbers, we have the opportunity to learn about real world applications of this mathematical concept. In general, sixth grade students are unfamiliar with banks and how bank accounts work. The Bank Account Game is a fun way for students to explore these ideas and familiarize themselves with the vocabulary used in basic personal finance. Before we play the game in class, I ask the students to make two deposit cards and two withdrawal cards that we can use for the game. Making the cards themselves is a valuable learning experience and makes the game more fun for the students to play.

Here are some examples of student made cards:

Open Questions and Parallel Tasks for Sixth Grade Math

This book is a wonderful resource for good math questions. It contains questions:

• that are perplexing so that students are engaged and want to use mathematics to explore the topic
• that have multiple entry points for students of varying abilities
• that are rich enough so that the exploration can drive the entire lesson
• that allow students to solve a problem in many different ways and then explain and prove their results.

As I read this book it really made me think about what makes a good math question and how I could adapt and improve anchor questions that I have used in order to make them more open ended and accessible to all students. I decided to go through the book and pull out all the questions that I could use in my sixth grade math classroom and organize them by the topics that we cover, in the order that we cover them, giving me ready access to some model questions that I can use or adapt throughout the school year. This was a very valuable process for me because by reviewing all the questions, especially those aligned with the sixth grade common core standards, I was able to more clearly understand what makes a good question and how to create these type of questions for other topics we cover.

As the TIMSS video study found, Japanese teachers use a structured problem solving format for most of their lessons where the teacher poses a complex thought-provoking problem, students struggle with the problem, various students present ideas or solutions to the class and then the teacher leads a class discussion using the various solution methods provided by the students and summarizing the class' conclusions (from The Teaching Gap by James Hiebert and James Stigler and CT Regional School District #10 Math Program FAQ Page). This lesson format has proven to be an extremely effective method of teaching mathematics and a model I have tried to use in my classroom. But finding rich problems that create the set up for a student centered lesson for each of the topics covered in our curriculum can be a challenge. Japanese educators are part of a teaching community that uses "lesson study" as a way for teachers to share and refine the problems and questions they pose at the beginning of a lesson that create the conditions in the classroom where students are engaged in deep mathematical thinking. This book is one resource that teachers in the US can use to find good thought provoking questions.

Here are a few of my favorite open questions and parallel tasks from the book:

1) Percent:  72 is _____% of ______

2) Percent:  Choose something you have been wanting to buy that costs more than $50. Imagine you have $30 saved. What discount does the store need to offer before you can afford it?

3) Algebraic Expressions: An expression involving the variable k has the value 10 when k = 4. What could the expression be?

4) Algebraic Expressions: Order these values from least to greatest. Will your order be the same no matter what the value of n is? Explain.

• Choice 1:   n/2     3n     n^2     3n + 1     10 - n
• Choice 2:   4n     3n     10n      3n + 1     5n + 2     -n

5) Geometry: You start with a parallelogram. You increase its height by the same amount as you decrease its base length. How does the area change?

Math T-Shirt Friday

On Friday, teachers at our school can "dress down" and wear jeans. I have used this opportunity to also wear Math T-Shirts.  Over the last several years I have been collecting T-shirts that have a math theme and my goal is to have 40+ T-shirts so that I can wear a different one every week! Some of my favorite shirts came from Woot.com and Savers (thrift shop) and CafePress.

        

The Bank Account Game

When I first started teaching high school math, a colleague made a simple card game that the students LOVED to play. It was kind of like the Game of Life without the board and the little cars with removable blue and pink pegs.

Each student had a blank check register and they would keep track of deposits and withdrawals to their checking account by recording these amounts in the deposit or withdrawal column of the check register based on the cards that they drew and then add or subtract to find the current balance in their account. In this game, each player had a job that earned them an annual salary that was paid out to all the players every few rotations. Hence the students had practice keeping an accurate checking account balance but never had to worry about bouncing a check or having a negative balance. The student who had the highest balance in their account at the end of the game was the winner.

Throughout the school year, I found myself going to her regularly to borrow her card game for a few days and always wanted to make my own set of cards, but never found the time.

Five years later, when I started teaching sixth grade, I realized that this game structure would be a great way for my students to practice working with negative numbers. I still didn't have the time to make a set of card myself, so I got my students involved by asking them to make two deposit cards and two withdrawal cards. I told the students to only use increments of $10 with a maximum of $50 for each deposit or withdrawal. These friendly numbers allowed the students to play the game using mostly mental math by referencing a number line drawn in increments of 10. They loved playing the game with cards they had created themselves and now I have an abundance of student made cards--enough for several groups to be playing at the same time.  Here are some examples that I showed them.

And find some examples of student made deposit and withdrawal cards here.

The goal of the game was not to practice integer computation following a set of memorized rules, but instead for them to get a feeling for moving along the number line especially when the balance changes from positive to negative or vice versa.

I encouraged the students to draw number lines to work with during the game. Next year I will provide large number lines in sleeve protectors that they can draw on with dry erase markers as they calculate the balance in their account.

Here is the link to the Bank Account Game Template.

Number Line Gallery Walk

My students did poorly last year on number line questions because I did not emphasize the precision and attention to detail necessary to accurately depict a set of numbers on a number line. I was determined this year to find a way to teach this topic more effectively.

This year I started out by showing the students examples of number lines:

After studying these examples, we discussed what elements these numbers lines had and from this discussion we created a number line checklist that students would use to make sure that their numbers lines contained all the elements needed for accurate numbers lines.

Here are some examples of student created number lines. There are some small errors in these number lines, which I did not correct as I circulated during class. During the Number Line Gallery Walk, I wanted students to have the chance to go through the number line check list for each number line to find any errors.

1) Draw a horizontal number line to represent the set of numbers:  Even numbers between 30 and 45.

2) Draw a vertical number line to represent the set of numbers:  Whole numbers greater than 20 but less than 27.

3) Draw a vertical number line to represent the set of numbers: Decimals between 9.4 to 10.5 with an interval of 0.1 between each pair of decimal.

4) Draw a horizontal number line to represent the set of numbers: Mixed number from 6 to 7, with an interval of 1/5 between each pair of mixed numbers.

                    

5) Draw a vertical number line to represent the set of numbers: Mixed numbers between 5 and 7, with an interval of 1/3 between each pair of mixed numbers.

After the students completed the number lines, groups displayed their number lines around the classroom and then using their number line check list did a gallery walk and evaluated each number line in small groups. There was great discussion in the groups about whether or not a number line met all the criteria for an accurate number line. 

I prepared a PowerPoint presentation for the next day by taking photographs of the number lines so they could be displayed in front for all to see and this was tremendously helpful to facilitate discussion. First, I asked the group who made the number line to decide whether their number line was accurate, and then I opened it up for class discussion.

The construction of number lines on large graph paper, small group work, the gallery walk including peer evaluation and error analysis and finally a class discussion with number lines projected for all to see, helped turn a somewhat dry and tedious topic into a very productive and engaging lesson for my students.

These see through colored dots that I found at Staples were great because it allowed students to add dots to their number lines and still see the tick marks below. The difference between tick marks and dots, has been difficult for some of my students and I think the stickers dots helped them make the distinction between tick marks and dots and why all numbers on the number line need tick marks and only some numbers get dots.