Holiday Fractal Design Project

"A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales.They are created by repeating a simple process over and over in an ongoing feedback loop." This definition of fractals comes from the  Fractal Foundation.

This is one of my favorite math activities to do before the holiday break. I like this simple project because it gives students a hands on experience with self-similar pattern making and also after completing only 2 recursions, the students already get a very strong sense of how dramatically the scale has changed.

This project is open-ended enough to allow for colors and designs from many different cultures and the students enjoy coming up with their own designs. And I love the calm mindful work of children coloring their repeating patterns and experiencing mathematics.

Here are the steps for the project with illustrations below:

1) Cut out a 27x27 grid from graph paper and draw lines making 3 rows and 3 columns so that each square is a 9x9 grid
2) Draw a simple, colorful design for the four corner squares and the middle square.
3) With the four remaining 9x9 squares, draw lines making 3 rows and columns so that each square is a 3x3 grid.
4) Shrink the five original designs by 1/9 and draw them in the five 3x3 squares that correspond to their original locations in the 27x27 grid.
5) In the four remaining 3x3 squares, draw lines making 3 rows and columns where each square is a 1x1 grid.
6) Shrink the five original designs by 1/9 again and draw them in the five 1x1 squares that correspond to their locations in the original 27x27 grid (and also in the 9x9 grids).

Below are more designs that were completed by students.

Anchor Task Graphic Organizer

Recently I made a poster of this quote from George Polya and displayed it in the front of my classroom. To go along with this, I created the graphic organizer below and used it to introduce anchor tasks for a new topic. The students wrote the problem in the middle and then tried to figure out four ways to represent and solve the problem. This is new for my students and so I have been giving them prompts for each of the rectangles like: draw a model, write a word problem, explain in words, computation, etc.

Anchor Task Graphic Organizer (Full Page)
Anchor Task Graphic Organizer (Half Page)
Anchor Task Graphic Organizer (in Google Slides)

Shown below are examples of student work on three anchor tasks:

1) Dividing Fractions (2 divided by 1/4)

I love the exploration that this student did to figure out that as the divisor decreases, the quotient increases.

This student was exploring many ways to do the computation.

This student added some ideas to their graphic organizer from another student in the group and gave them credit. I also like how this student wrote out their thinking about how the quotient changes with different divisors.

This student shows a strong understanding of the meaning of reciprocal by explaining that since 4 1/4s go into 1, then you simply multiply by the reciprocal to find out how many 1/4s go into 2.

This student tried so many ways to understand fraction division (models, decimals, percents, etc). What I was most impressed by was the reflection that occurred throughout the class period during small group and whole group discussion and is shown clearly in the work. Student wrote "changed my thinking after discussion" and then indicated on the graphic organizer where he needed to make revisions.

2) Absolute Value (Distance from zero--Find the total distance to touch the rock)

3) Multiplying Decimals (4 x 0.3 and 0.3 x 0.5)

Teaching Effective Math Lessons

Teaching Effective Math Lessons

by Dr. Yeap Ban Har

I was very fortunate to have the opportunity to attend a district math training with Dr. Yeap Ban Har. Here is a summary of some of the things that I learned:

1) Anchor Tasks are used at the beginning of the lesson to engage students in the new learning for the day. An anchor task should be an interesting and rich problem that allows students to explore the mathematical concepts and ideas that are the most integral part of the lesson. This part of the class period should last about 15 minutes which gives students the processing time needed to allow for new learning. 

2) Dr. Yeap encourages his students to find as many ways as possible to solve math problems. He tells his students that it is a clever day when they are able to find more than two ways to solve a problem. He rewards his students when they have 20 or more clever days in one month.

3) Anchor Task Example:  Instead of how can we solve x + 6 = 9, Dr. Yeap said:

• Can you guess what number I am thinking of?
• If I increase that number by six then the result will be nine.
• Can we figure out this number?
• How can we show it mathematically?
• Equation
• Guess and Check (substitution)
• Balance Scale
• Mental Math
• Number Line

4) Students are encouraged to write in their journals throughout the lesson to work out their mathematical thinking and at the end of the anchor task, students are asked to: 

• Explain in their own words the problem they were solving
• Show one method to solve it and
• Give the answer to the problem.

5) Dr. Yeap reads his students' journals frequently. He has a system where he is able to read all of his students' math journals once a week. He has two stamps for checking and commenting in the journals:

• I like this journal entry because it is creative
• I like this journal entry because it shows initiative.

6) He used a very large pencil pointer. He uses this in his classes to constantly remind the students to WRITE in their journals!

7) Questions Dr. Yeap asks throughout the lesson:

• Can you imagine that? (He uses this question frequently in his teaching which encourages the students to see the problem in their minds.)
• Does anyone want to challenge this?
• Do you think this method works all of the time?
• Do you accept that?
• Is my diagram reasonable?
• How is it the same? How is it different?
• Do you notice anything?

8) Theories of Piaget and Vygotsky were mentioned frequently:

9) Differentiation: When students finish their work early he has three possible assignments for them:

• Write a story for the equation or problem that was solved
• Write a note in their journal explaining how to solve the problem that they worked on class to a student that was absent. He will then copy the note and give it to the absent student upon their return to class.
• Can you come up with a new method, an original method, one that no one on earth has ever used before.? If so, you can name it after yourself!

10) Homework:  Pick five problems that you don't think that you can solve (from the exercises page at the end of the section) and then do those five for homework. If they are all too easy, write a paragraph about why they are all too easy and then write three equations that are difficult for you and solve them.

Math App Reviews

This year I have some bulletin board space available for students to post reviews of math apps and website that they have tried. I created the following template for them to write and post their reviews in the classroom.

There are so many apps and website for students to practice procedural mathematics. I am hoping that this opportunity will encourage students to try new apps and websites and share the ones they like with their classmates.

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.