Apples, Apples, Everywhere!

Exploration #1: The Leveling Model. Joan, Jane, John, Jen, Jack, and Jill went up the hill to fetch a pail of apples. Joan found 9 apples, Jane picked 5 apples, John got 3 apples, Jen fetched 4 apples, Jack found 7 apples, and Jill got 8 apples. The six friends decide to share the apples equally among them. Use stacks of unifix cubes to model the apples that each friend picked (use a different color of cubes for each friend, if possible). Then use the cubes to distribute the apples among the friends. (That is, find the arithmetic mean number of apples for the friends.) How many cubes (apples) are in each stack initially? How many cubes are in each stack after distributing apples? What is the arithmetic mean number of apples?

To begin I had 6 different stacks with a different number of cubes in each stack. I then moved cubes around until each stack had the same number of cubes, 6. 

Exploration #2: Balance Model. Joan, Jane, John, Jen, Jack, and Jill went up the hill to fetch a pail of apples. Joan found 9 apples, Jane picked 5 apples, John got 3 apples, Jen fetched 4 apples, Jack found 7 apples, and Jill got 8 apples. Use a balance (Links to an external site.) to determine the arithmetic mean number of apples for the six friends. On one side of the balance write an addition equation to represent the number of apples. On the other side of the balance write a multiplication equation where one of your other factors represents the number of people. You need to determine what the other factor is that will make the balance level. What missing factor proved to be the one that balanced the original side? Why did the missing factor prove to be the balance point? What is the arithmetic mean number of apples?

To begin the task, I placed 9+5+3+4+7+8 on one side of the balance. 

I then needed to find out the multiplication equation that would equal 36. I knew that one factor would be 6 because there were six people in the problem. Knowing my facts, I knew that 6 X 6 is 36 and the resulted in a balance. The missing factor is the balance point because it is the same as finding the mean using 36 divided by 6. It is like finding the mean using the inverse operation. 

Exploration #3: Collect a number of pencils from students. The aim is to use eight pencils, but collect more so that from this number, eight pencils of different lengths can be used. Lay the pencils end to end (this works really well in the grooved tray of some marker boards or chalkboards. With the pencils laid end-to-end, measure the total length of the pencils with adding machine tape or similar strip of paper. Now fold the paper in half; fold this fold length a second time, and then this length a third time. Unfold the paper to observe eight sections of equal length, each the arithmetic mean length of the pencils. What mathematical process summarizes the laying of the pencils end to end? What mathematical process summarizes the folding of the adding machine tape? Why does the length of any of the eight pieces of paper represent the arithmetic mean of the length of the pencils?

I collected pencils of 8 different lengths, measured in inches: 3, 4, 5, 6, 7, 8, 9, 10. When I laid all the pencils out end to end they measured a total of 52 inches. When I measured one section of the receipt tape, it measured to be 6.5 inches. When you fold the paper into 8 section, that represents the 8 different pencils. They are all folded the same size, which represents dividing them all equally. The mean is the average, or the amount of 1 section. 

Exploration #4: Joan, Jane, John, Jen, Jack, and Jill went up the hill to fetch a pail of apples. Joan found 9 apples, Jane picked 5 apples, John got 3 apples, Jen fetched 4 apples, Jack found 7 apples, and Jill got 8 apples. When they got back down the hill, they wanted to share the apples equally. So they dumped their apples in a bushel basket and proceeded to distribute them fairly. Model this situation with unifix cubes and use them to find the arithmetic mean. What mathematical process summarizes the dumping of apples into the bushel basket? What mathematical process summarizes the distribution of the apples? Why does the number of apples each friend eventually gets represent the arithmetic mean of the apples?

Initial apples 

Apples all in one bucket

Apples when sorted out of the bucket

Putting all the apples into the bucket is like finding the sum of the apples, adding all the different peoples apples together to find out the total. When dividing by 6, or the number of people, that is the same as dividing the apples among the people. If they knew there were 36 apples and 6 people they would automatically know they each received 6 apples. This can also be represented by giving each person 1 apple until all the apples are gone. The number of apples each person gets the mean of the apples because it's the average. On average, each person picked 6 apples. Not everyone actually picked six apples but on average. 

5 for Friday

For the first time in a long time, I'm linking up for Five for Friday! 

I feel like every night this week I have had something to do. From graduate school to conference night I am EXHAUSTED!! I am definitely using tonight to sit back and relax.

This week I received a new guided reading/small group table and I couldn't have been more excited! All of last year and up until this week, my guided reading table has been a semi circle with no cut out in the middle. This made it impossible for me to reach all my kids as they were reading! Not to mention the table was huge and took up a TON of room!  Now that I have my new table, I have a lot of extra space and I am so excited about rearranging!! 

Sunday is my grandma's birthday and I am making a strawberry cake for the very first time! Advise from you long time bakers ?!?!

With conferences being on Thursday night I did a lot of progress monitoring this week. I am so thrilled to say about 95% of my students grew in all the assessments I gave them! Some of them have not grown a full reading level yet, but have mastered all their digraphs and most of the s, l, and r blends we have been working on !! Hooray ! 

With conferences and the fall festival these past two weeks, my school had a door decorating contest. Many teachers did a Halloween theme, but my teammate and I joined up together for a Fall scene! Lucky for us we have 2 amazing teacher assistants that worked extremely hard! We WON !! 

Under the pumpkins our students sequenced a story! We adapted the cute activity from A Cupcake for the Teacher !! 

Little Late November Currently

This sinus infection, in combination with all the work I have going on, has me a little behind on the November Currently. Late is better than never though ... right? 

Listening: Nothing really, but Big Bang Theory is on the TV as I'm working away

Loving: it's almost time for Thanksgiving and Christmas! I love love love the Holiday season! Spending time with family, having a nice break from school, and having a pretty good excuse to make crafty things in my classroom!!

Thinking: I'm still getting over this sinus infection and I'm exhausted! If I didn't have report cards, and graduate school work to do, I'd go to bed right this minute. 

Wanting: Other than wanting to go to bed, I'd like the time to curl up with a good book and not think about a single other thing! 

Needing: Finishing up report cards and grad school assignments, both of which are due tomorrow :(

Yummy: Corn Puddin' -- more specifically my grandmas! She passed away last September and no one can make it like her! Aww I miss her so much :( 

Proficiency and Fluency Part 1

Task: Addition & Subtraction Game 

1. You are designing a game for your first grade students to play to give them some practice on: CCSS.Math.Content.1.OA.6 - Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.
2. In order to design this game you want students to generate 2 numbers - either by pulling number cards or rolling dice

Addition/Subtraction War: 

      In this game, you and your partner will get a deck of cards. You are to divide the cards between the two of you. Each of you should have the same number of cards. To play the game, each person will turn over two cards. You will then add your two cards together to get a sum, or subtract your two cards to get a difference. The partner with the highest sum, or lowest difference, gets to take all four cards. Next, you will repeat. Turn over two cards add/subtract your cards and the person with the largest sum or smallest difference will take all the cards. Continue doing this until one person has all the cards. Divide up the cards and play again. 

Part 2:
Design a game that would develop fluency for multiplication or division. If you modify one that you have seen somewhere that is fine. Just let us know where it came from.

This same time of game can be used for multiplication. Instead of adding or subtracting the cards, students can multiply the cards and the student with the greatest product would take all the cards. 

To differentiate for students of all levels, a multi sided dice could be used. Students who are working on basic facts can use regular 6 sided dice, while students who are ready can use 9 sided dice. 

Skeleton Tower

1. Note the figure below. It is made of cubes that form stair steps. The center of the tower is 6 cubes high. On all four sides of the top tower are stair steps that extend from the center.
2. How many cubes are needed to build a tower like this, but that is 3 cubes high? -- 3 + 8 + 4 = 15
3. How many cubes are needed to build a tower like this, but that is 4 cubes high? -- 4 + 12 + 8 + 4 = 28
4. How many cubes are needed to build a tower like this, but that is 5 cubes high? - 5 + 16 + 12 + 8 + 4 = 45
5. How would you calculate the number of cubes needed for a tower N cubes high?
6. How do the answers in parts 1-5 help you?

I struggled to find the equation for "n" layers. Using the work to find the the other answers did not help me much. I tried to see a pattern and found that you are adding four to each layer starting from the top but was not able to come up with an equation that worked to show that for "n" levels. It has been quite a while since algebra class! I pulled from way back and used all the strategies I could remember, but still couldn't make an equation happen :( 

Curriculum Evaulation

My county does not have a set math textbook or resource that we all use. In fact, in the four years that I have been teaching, I have never seen a math textbook or teachers guide. For one of my masters classes I was asked to evaluate a textbook that I had access to. Since I do not have a textbook that I use daily, I chose to closely examine the "EngageNY" resource that my teammate and I use for our math block. 

This resource does a great job of linking all the lessons together. Students are not jumping around from one concept to another, but flowing smoothly. However, the lessons do not provide much for differentiated instruction across various development levels. Students are all doing the same tasks as a whole group. (This is one reason I chose to teach my whole group lessons in my small groups, I can now provide the extra scaffolding or advanced questioning that my students need.) Students do have an opportunity to practice independently and then talk with the class about how they solved their problems or the strategies they used. This part does lend itself to great discussions. 

This curriculum follows a very modern style of teaching. Students are using number bonds and relating facts, they are encouraged to draw pictures or use manipulatives. The lessons do not follow the same format in which I was taught. Students are encouraged to think for themselves and help the teacher move the lesson. This resource does not have the teacher up talking for 30 straight minutes, but instead has the students talking with a partner and answering questions to help themselves gain knowledge. 

My ideal math curriculum would give students a variety of examples, as as experiences. Students should have the opportunity to chose the method they would like to solve a problem instead of having to do it a certain way. The ideal curriculum would introduce multiple strategies, but not force a student into using one particular strategy. Just like in reading we do not always make predictions, or chunk words. We know many strategies to help us read but we must decide which one works best in a given situation. The same applies for math, students need all the tools, but must decide which one is the most useful for a given problem. The ideal curriculum would also have students up moving around. Students would not simply be sitting on the carpet or at their seats working problem after problem. They need to be working with partners or group members on a common task. This could be real world application problem or even a simple word problem. Having students working together allows for discussion and brain storming. Another aspect of an ideal curriculum would be differentiation, suggestions for tasks for students who are struggling, as well as for students who are above level. This does not mean that the teacher must do those tasks, but examples to help the teacher design his/her own task to meet the needs of all students. The final thing I would put into the ideal curriculum is writing after each unit or every few lessons. There is a big push for writing across curriculum's, not only in reading. Students being able to put their thoughts and ideas about math down in writing is just as important as them being able to express their ideas in any other curriculum. This type of curriculum is most suitable for mathematics teaching and learning because it can scaffold student learning while also encouraging students to use critical thinking skills. Teamwork also allows students to develop those communication skills that are necessary outside of the classroom. 

Tuesday Ramblings

I have just finished reviewing my plans and materials for my observation tomorrow. In my county, teachers in years 1-4 have 3 observations a year, one announced and two that are a surprise. I'm feeling pretty good this year and told my principal she could come whenever she wanted... not knowing she would choose two days later. Although I was able to pick the date, she decided she wanted to see a whole group math lesson and at least one guided reading lesson. Moving down from third grade, I feel pretty great about my whole group lessons.. but some days I struggle with guided reading and what these little babies need. 

I've spent most of the night cutting and preparing centers from Cara Carroll and I can't wait to see how engaged my students are. 

Another thing I have been concentrating on a lot lately is my whole group math time. My team decided to use the EngageNY math lessons for our whole group math time (our county doesn't have a set math book). This is going quite well but I am having to eliminate bits and pieces due to time constraints. I have strongly considered only doing a 5-7 minute whole group math talk and then breaking into small groups to teach the meat of the lesson.

 Does anyone else do this?!? 

I feel like I can really see which of my students are getting it and which are still struggling! The other great thing about math groups is my assistant also pulls groups during this time. He always sees the one group that I do not see, plus two others. I want to think that my students are benefiting from this smaller instruction, but I don't want them to be shafted from their whole group time either. 

Incorporating math talks has been a rocky start, but my students are picking up on it quickly. For those of you who have seen or who use the EngageNY program, I'm using problems similar to those in the "application" section as our number talk. These are usually a review from previous lessons and allow my students to think deeper about the problem. 

What do you think? Suggestions about whole group/small group? 

Anybody else use EngageNY?? Thoughts? Advice?