How Learners Can Help Each Other

I was working with a class of students creating a map that featured coordinate grids. One student came in late and I asked him for a suggestion about what I could do so he would feel part of our project. Together we figured out a plan and another student suggested that he could help him get started. Here is what his exit slip said that day:

How Learners Can Help Each Other

By Anthony

When I got in the classroom late, I didn't know if I would be able to finish.  I was a little worried about that.   I figured out that a community of learners can help someone that comes in late, by giving pointers, and not focusing too much on me, so we all were able to complete our projects.    After my friends helped me I felt awesome, because I was able to finish part of what I had to do. If they didn't help me, I probably would have to finish when they were working on something else. I still had some trouble with my project, but not as much as I would have had if they didn't help me. If we didn't have a community of learners, work would be more difficult and less enjoyable.

Math is important, but creating a class that feels like a "community of learners" is the REAL DEAL!

Deborah

Writing About Our Thoughts While Problem Solving in Mathematics

            Problem Solving means engaging in a task for which the solution method is not known in advance. In order to solve the problem, students must draw on their knowledge of the process, and through this process they will often develop new mathematical understandings.

This easy to use graphic organizer was created to assist students in writing down their thoughts as they solve a problem.  The assessment requires students to express not only how a problem is solved, but why each step was taken.  According to the text, “Mathematics Assessment: A Practical Handbook for Grades 6-8”, students “need regular opportunities to write about their mathematical thinking to become better writers and better thinkers. One good writing task is to have students explain strategies and solutions using words, mathematical symbols, graphs, diagrams, or other representations.”

            One of the reasons this particular graphic organizer was developed  was the high percentage of second language learners in our school district. They were so overwhelmed in expressing their ideas and thoughts in English and writing them down in sentence format that they struggled to express their mathematical thoughts. In addition, students participating in our Gifted Program had difficulty expressing why they preformed certain operations or utilized a particular strategy they just "knew it."    

      
           A four quadrant graphic organizer was developed with the work of Dr. George Polya is mind. First, Dr. Polya stated that in order to solve a problem, a student must take time to understand the problem. They must ask themselves: “What is the unknown? What is the data? What is the condition? or Can you restate the problem in your own words?” Therefore the first quadrant begins with understanding the problem. Students usually begin with…First, I know.

            Next, students must understand what they are being asked to find. In order to clearly state their answer with appropriate units, the second quadrant begins…The answer is.  Students will insert a blank where the amount will later be filled in, for example, “The answer is $_________ is the price at store A, and $_________ is the price at Store B.”

            The third quadrant lists what the student does in a step-by-step manner, in order to solve the problem.  Many students find that they like to number the steps numerically. Since Dr. Poly stated that they must begin by devising a plan, teachers in our district teach some of the problem solving strategies described: guess and check, make an organized list, make a picture of diagram, make a table, look for a pattern, use a variable, and use an operation or formula. So the third quadrant usually begins with the student writing…First, I decided to.

            The fourth quadrant lists why I did it that way.  Again students usually want to number the reason why to reflect the step that they took, though it certainly isn’t necessary.  Many of the statements on this side of the organizer seem to start with because, to show, or to find. The “why I did it” is written immediately after they write down the “what I did.”

            An example of two more completed four quadrant problem solving organizer is shown below.  The same problem has been solved using different problem solving strategies. This easy to use graphic organizer is used in Grades 3-8.

Click Here to obtain a copy of the organizer.

Deborah

Also here is the student friendly scoring rubric provided by our state currently to score student's thinking on mathematical problem solving extended responses.

                                               

Score Level	Mathematical Knowledge (Do you know it?)	Strategic Knowledge (How do you plan?)	Explanation (Can you explain it?)
4	I get the right answer. I label my answer correctly. I use the right math words. (For example, I know when to add or multiple.) I work it with no mistakes.	I find all the important parts of the problem, and I know how they go together. I show a good plan about how I got my answer. I show all of the steps I use to solve the problems.	I write what  I did and why I did it. If I use a drawing, I can explain all of it in writing.
3	I do the problem, but I make small mistakes.	I find most of the important parts of the problem. I show most of the steps I use to solve the problem.	I write mostly about what I did. I write a little about why I did it. If I use a drawing, I can explain most of it in writing.
2	I understand a little, but I make a lot of big mistakes. I only give part of the answer.	I find some of the important parts of the problem. I show some of the steps I use to solve the problem.	I write some about what I did or why I did it but not both. If I use a drawing, I can explain some of it in writing.
1	I try to do the problem, but I don’t understand it.	I find almost no important parts of the problem.	I write or draw something that doesn’t go with my answer. I write an answer that is not clear.
0	I don’t try to answer the problem.	I don’t show any steps.	I don’t explain anything in writing.

Student Desktopper for Order of Operations

This handout provides 4 desktoppers that you can run off on a heavier paper and then tape at the top of a student's desk to serve as a scaffold when learning the order of operation or while working on their math assignments.



Deborah



8 x 8 = 64

In a recent post about my math fact strategies flip book I had written down our strategy on how to remember that 8 X 8 = 64. I've had several emails that asked me to explain that more.

When you multiply the TWO 8's together, count backwards by 2. 8,6,4 and there is the pattern that will help you remember the fact! You might think... well that's simple...it is, and that's what makes it so simple to remember the fact.

Are there any other ideas in the math fact flip book that you would like me to explain?

Do you have a pattern that helped you remember a multiplication fact that I didn't mention in my flip book?

Deborah

What Common Referents Should Students Understand in Order to See the Relationships between Fractions, Decimals, and Percents?

Since I've been creating all of these fraction, decimal, and percent games recently, I began to think about this question:

What basic fractional measures should all students be able to picture in their minds so that they can compare different fractions, decimals, and percents?

Well, I like to think about a measuring cup set to help me answer that question.

If a student can compare and visualize: 1 whole, 3/4,  2/3, ½, 1/3, and ¼ , then they can use this information to compare other quantities.  For example, if I was trying to make sense out of 5/8, I would think:

•4/8 is the same as 1/2

•5/8 is a little more than 4/8 or 1/2

•So 5/8 is a bit more than 1/2 or 50% or .5

•I bet it is about 60% or .6 something (actual = .625 or 62%)

Now... compare 5/8 and .42.  Could you do it?

Deborah

P.S. This is the perfect reason why you should have your children cooking and measuring with you in the kitchen... building number sense!

Probability of Spinning Blue

In this activity the students created their own spinners that were created to produce certain results. For example, the first spinner must be designed so that the probablity of landing on blue is CERTAIN.


 After creating the spinner, the student then conducts an experiment and spins the spinner 20 times and records the results. Next, the students looks over their data and records the fraction ?/20 and percent that expresses how many times the spinner actually landed on blue.


To create a spinner, a small paperclip is placed on the brad in the center of the circle.

Lastly, a summary statement is written by the student. The starter prompts are : "I noticed ... " and "This made me think..."


Here are the two pages that you will use to create the final product as shown above on a piece of
 12 x 18 piece of construction paper.

I like this activity because it places the responsibility for creating the appropriate spinner on the student, and is an authentic way to collect data about your own creation. Click Here to obtain your copy of the documents above.

Deborah

Math Fact Flip Book

When working with students who have difficulty just "Memorizing the Multiplication Facts", I use this Math Strategy Flipbook to help see patterns in the multiplication tables.  Each page represents a lesson or mini-lesson with a group of students. These lessons look for patterns or strategies to help them remember their facts.  Take a look at a finished product.

The completed flipbook

The END.  We kept track of the facts are we learned them FLUENTLY!



0's and 1's. Used base unit blocks to help understand the concept.
 They formed the rule... I did not just tell it to them.

We noticed a lot of patterns with the 9's



This flipbook is then kept as a resource by the student that created it. 
Why take the time to create the flipbook?

It is a scaffold for students to use UNTIL they no longer need it.

Deborah

Readers Theatre Script: The First Annual Planet Awards

I adapted a storybook from NASA's "Storybook for You" series, by turning it into a reader theatre script as shown below.  I imagined students from the 4, 5, or 6th Grade levels  presenting it to a lower grade as a special treat.  In the script all the planets are winners, but you could have the audience vote for their favorite planet, and graph the results on a  colorful premade graph template, that the presenters would provide, to make a connection between math and science.

Here is the beginning of the script:

The First Annual

Planet Awards!

Readers Theatre Script

 Host: Welcome! Today we will see which planets are the best in the solar system. We have eight contestants—eight planets who will compete for awards. Each planet will tell us about itself and why it is unique among the planets. When all the planets have spoken, the judges will decide which planets will receive a Best Planet Award. Let’s meet our first contestant.

ANNOUNCER 1:  Let’s give a big welcome to Mercury.

 HOST: Tell us about yourself, Mercury.

MERCURY: OK. I am the closest planet to the Sun. I am the second smallest planet in the solar system. I rotate so slowly that my day is 58-1/2 Earth days long! I am rocky, nearly airless, and covered with craters.

  HOST: Do you have any moons?

 MERCURY: Nope, no moons.

 HOST: What makes you special?

  MERCURY: I am the fastest of all the planets when it comes to orbiting the Sun. It takes me only 88 Earth days, which is the length of my year. Also, I have the most extreme temperatures of any planet. I can be as cold as –300° Fahrenheit at night and as hot as 870° Fahrenheit during the day!

 HOST: Whew! Thank you, Mercury. Now, on to our next contestant.

This Reader's Theater script can be obtained by CLICKING HERE. This lesson is from a Multi-grade Planet unit for Grades 3/4 and 4/5.

Deborah

Getting To Know You...Fraction, Decimal, and Percent Game

This is the 3rd Game using the Everyday Math Fraction/Decimal/Percent Card Deck. This game would be a great introduction to the study of the number sense idea that all three values represent the same amount.  As you can see on the illustration shown on the game board, all three values of 3/4, .75 and 75% have 75 out of the 100 squares colored in to represent the three amounts.

Why take the class time to play this game?  Many students have difficulty making the connection that the values expressed all represent the same value. By using the same 10 x 10 grid to express all 3 values, that helps the students see the connection too.
  Click here to get a copy of the gameboard.

Please know that you can also create your own fraction/decimal/percent cards in place of those created and sold by Everyday Math.  I'm just highlighting the Everyday Math product because many teachers already have them, but don't utilize them fully in their classrooms.  I found a great internet site that allows you to create your own cards and will discuss it in a future post.

Deborah

Fraction/Decimal/Percent Card Game: Value Up Fractions

Here is the second game using the Everyday Math Fraction/Decimal/Percent Card Deck from the series of games in which I am creating the accompaning gameboards, recording sheets or game mats to use with the card deck. It's called "Value Up Fractions." You only use the fraction cards from the deck. On the game board is the I CAN statement: "I AM learning to compare fractions with different numerators and different denominators."

I like this game because you have the option of drawing 2 new cards to replace up to two cards already on the game board to try to fix the order of the cards from the least value to the greatest value.  That's where the STRATEGY and THINKING comes into play.

Click on the here to go to my Google Doc account to download.

Deborah

"Fraction/Decimal/Percent Card Deck Games from Everyday Math

The Everyday Math  "Fraction/Decimal/Percent Card Deck"  contains cards showing 18 different fractions, decimals, and percents. The fraction, decimals, and percents are equivalent to each other. Even though Everyday Math created an activity booklet to go along with the cards, I find that many teachers are not aware of the suggested activities because they are not in the Teacher Resource Guide manual, or don't play them because there are no formal game boards to accompany the suggestions.

So I am planning to host a workshop in my district to play 10 New Games that use the Fraction/Decimal/Percent Card Deck  in January 2012.  I've been busy creating game boards or playing mats for the games, and I thought I would share some of them with you in a series of  posts.

The first game is called "Bar Graph Addition" and the skills covered in the game are estimating the sums of fractions and recording data on a bar graph. This game is worth the use of class time to play the game as many students don't understand fractions very well. I love this informal format and the fact that you record the sums according to 3 different categories: Sums 1/3 or less, sums between 1/3 and 2/3, and sums 2/3 or greater.

I made this game board to accompany the game, and you only use the fractions cards contained in the deck.  To make it easier for the teacher, the directions to the game are located on the game board at the bottom.

Click on the document to download it.
Deborah