Private Universe - Workshop 4 - Thinking like a Mathematician

The main thing I took from this video was the importance of engaging your students in the lessons, especially in mathematics. It is easy to present information to a group of students from a textbook and in a lecture format. With this approach a majority of the students will "check out" and that is when behavioral problems will occur. In classrooms where math instruction has changed to hands-on, inquiry based instuction, students are taught that there is more to solving problems than simply getting the right answerand the students are more engaged. This approach of inquiry also allows the less confident students to realize they may more know than they originally thought they did. If this type of instruction begins early in the grade levels students would develop more confidence in their mathematical abilities. Teaching that moves from the traditional - teacher has the answers and tells the students who is right and/or wrong to an inquiry-based learning environment the students develop more confidence in their abilities to learn.

Private Universe - Workshop 3 Inventing Notations

As I watched this video all I could think about was how important it is to allow students to express themselves in their own ways. When we allow the students to explore and in turn explain their thinking they learn so much more than simply feeding them the process and the right answers. When we give them the time to develop their own processes and proof we are allowing them to become problem solvers and thinkers and these skills will only benefit them in the future. This video sparked my interest and I wonder how my fourth grade students would go about solving this pizza problem and how they would display and prove their answers. I can't help but wonder will they make any connections to the tower problem that they were presented with weeks ago. I will be presenting this problem to my students and will bring the results to class to share.

Virtual Manipulatives 4 - Function Machine

The virtual manipulative I will work with this week is Function Machine - Grades 6-8 Numbers and Operations. I was looking for something that my afterschool tutoring groups could use as a reinforcement activity for input/output tables. This was a weakness of the group based on an assessment done at the beginning of the tutoring program. This manipulative seems to be pretty good. It gives the students an opportunity to see a function machine and see the table that is created. When they put in the numbers 1, 2, 3 and 4 the computer tells them the output numbers. They have to use what they know to determine the pattern and fill in the remainder of the table, numbers 5 - 7. Once they figure out the pattern they can move on to a new function. I will use this with my group and see how they do with it. It does not seem to be as exciting and attention grabbing as some of the other virtual manipulatives we have explored.

Lesson Plan - Color Tiles

In this lesson students will use the color tiles to continue their exploration of perimeter and area. This lesson will be used with my after school tutoring group of fourth graders. The group was designed to give the students who either did not pass or just barely passed the NJ ASK 3. I gave a LEARNIA assessment test at the beginning of the tutoring sessions and found that perimeter and area were weaknesses for the group. This activity will help to reinforce these concepts.

Lesson Plan - Color Tiles
This lesson is based on: "Changing Areas" Grades 3-4

Anticipatory Set:
Complete/Review Do Now and homework

Learning Expectations:
* measure to find the perimeter of a shape
*develop the understanding that figures with the same perimeter can have different areas

Activities:
* Display this shape made from Color Tiles of one color. Have students copy it. Show students how to move the edge of another Color Tile around the shape to measure its perimeter. Have a volunteer give the perimeter. Confirm that the perimeter may be expressed as 10 units or 10 inches. Have another volunteer give the area. Confirm that the area may be expressed as 4 square units or 4 square inches. Ask students to use four more Color Tiles to make a different shape in which each tile touches at least one other tile along a complete side. Have students find the perimeter and the area of their shape and share the results.
~~How many different Color Tile shapes can you make that have the same perimeter?
• Work with a partner. Use 3 to 8 Color Tiles to make a shape. Each tile in the shape must touch at least 1 other tile along a complete side.
• Record your shape.
• Find the perimeter of your shape and write it above the recording.
• Now use Color Tiles to make as many different shapes as you can that have the same perimeter. Record each shape and its area.
Thinking and Sharing
After you made your first shape, how did you go about making different shapes with the same perimeter?
Do you see any patterns among your shapes? Explain.
Do you notice a relationship between shapes with the same perimeter and the areas of those shapes? Explain.

Assessments:
* teacher observations, student-centered discussion, color tile shapes

Materials:
* color tiles, color tile grid paper

Tech Infusion:
Promethean board - flipchart, laptops

Accomodations/Modifications:
* teacher designed groups
* differentiated assignments/activities (centers to reach all levels)
* extended time-classwork/assessments(when needed)
* restructured homework assignments for both above level and below level students
* use of manipulatives/graphic organizers
* extension activities/alternative assignments
* organizational strategies - notebooks/book boxes/folder systems
* cloze-notes/study sheets/practice sheets for further reinforcement
* modified assessments - M/C, SCR, ECR, O-E
* preferential seating
* seating near positive role model
* encouraging independence/student initiative
* includes a rich variety of resources, media ideas, methods and tasks
* proximity to teacher
* personalized monitoring tools (checklists, goal sheets, thumbs up/down level understanding monitors)


Homework: Tell students to imagine that a zoo needs help designing a play area for monkeys. Students can help by making a design with a perimeter of 36 units (or yards) of fencing. Direct students to use Color Tiles to make their designs. Then have them record the design that gives the monkeys the most room to play. Students can write a note to the zoo telling why their design would make a good play area for monkeys.

Enrichment: Challenge students to build two Color Tile shapes, one with a perimeter of 20 units (or inches) and having the least possible area and the other with the same perimeter and having the greatest possible area.

Virtual Manipulatives 2 - Peg Puzzle and Towers of Henoi

After the experience of trying to solve these puzzles in class and feeling frustrated I was a little concerned about sharing these with my fourth graders. Since we had just received our laptops in our classroom and we had a little extra time after a lesson I decided to show them both puzzles on the interactive board. I was surprised at the fact that some of them had already seen the peg puzzle: "I play this at Cracker Barrel!" was the response of one of the boys in my class. They were very excited to be given the opportunity to try both puzzles. I left a group by the interactive board and the rest of the students went onto the laptops. They spent about 15 minutes totally engaged. They were excited and some were experiences frustration. When I spoke to those frustrated students and told them how I had difficulty in my class solving them it seemed to ease their frustration and they got back to it. It was interesting to see which students could solve the puzzles and which ones had the difficulty. The next day I had a few students come in and tell me how they showed their parents the puzzles and how they could solve them when their parents had a hard time. This was the confidence boost some of them needed. This is definitely a site that we will continue to use throughout the year in my class.

Lesson Plan - Pattern Blocks

The lesson I chose was dealing with the pattern blocks and would be done a the day after all students completed their free explorations. We discussed the relationship of the pieces to the hexagon. Students were able to quickly see the relationship and this also served as a review of writing fractions. Once we agreed that only the parallelogram, trapezoid, and triangles could be turned into the hexagon I then took it one step further. The lesson involved assigning a value to the hexagon and then based on that value determining the value of the remaining pieces. For example: the hexagon is worth $3.00 what would the value of the remaining pieces be? Once the values were determined we created a design that had a total value of $10.00. We utilized the interactive whiteboard and the virtual manipulatives to model the process. Students were then allowed 10 minutes to create their own $10.00 design. We then discussed how to prove that our designs were actually worth $10.00. Independent practice was to create a $12.00 design and homework was to create a $15.00 design. Students could use either the hands-on pieces or the virtual manipulatives to complete both practice and homework. Lesson plans will be distributed during class.

Lesson Plan
This lesson is based on the lesson: "What's My Shape Worth?"
Anticipatory Set:
Complete/Review Do Now and homework

Learning Expectations:
* investigate relationships among the areas of different shapes
* perform computations with money
* discover that shapes with the same area may look different

Activities:
* Review the characteristics of the pattern block shapes to be used - hexagon, parallelogram, trapezoid, triangle
* Determine the value of the shapes if the value of the hexagon is given - If the value of the hexagon is $0.30 what is the value of the remaining shapes?
* Give students an opportunity to create a design that is worth $1.00. Call for volunteers to put design on Promethean Board (Virtual Manipulatives - Grades 3-5 Geometry Pattern Blocks)
* Discuss ways students went about deciding which shapes to use in their designs.
* Break into centers to create a $12.00 design. (Values are given in directions)
~ Laptops - use Virtual Manipulatives - Grades 3-5 Geometry Pattern Blocks
~ Hands-on - use pattern block pieces - use triangle paper to recreate design
^ work with students who are having trouble with monetary values and building designs

** Enrichment Activity - assign a money value to each of the shapes and create a design that is worth more than $7.00 and less than $20.00.

Assessments:
* teacher observations, designs, student-centered discussion

Materials:
* pattern blocks, trangle paper, colored pencils, communicators

Tech Infusion:
Promethean board - flipchart, Virtual Manipulatives website, laptops

Accomodations/Modifications:
* teacher designed groups
* differentiated assignments/activities (centers to reach all levels)
* extended time-classwork/assessments(when needed)
* restructured homework assignments for both above level and below level students
* use of manipulatives/graphic organizers
* extension activities/alternative assignments
* organizational strategies - notebooks/book boxes/folder systems
* cloze-notes/study sheets/practice sheets for further reinforcement
* modified assessments - M/C, SCR, ECR, O-E
* preferential seating
* seating near positive role model
* encouraging independence/student initiative
* includes a rich variety of resources, media ideas, methods and tasks
* proximity to teacher
* personalized monitoring tools (checklists, goal sheets, thumbs up/down level understanding monitors)

Homework: Math - $15.00 design with pattern blocks - use the shapes to create your design or you may use the Virtual Manipulatives and print out your design

Standards: MA.4.4.1.4 B.4.a, MA.4.4.1.4 B.5, MA.4.4.1.4 B.6.a, MA.4.4.2.4 E.1, MA.4.4.5 C.1

Virtual Manipulatives - 3

As a preview to the lesson on pattern blocks I gave my students the opportunity to freely explore the blocks. The only direction they were given was that their designs must be complete in 2 minutes. Most of the students created symmetrical designs and used all of the shapes they were given. When the designs were completed I took a picture of each of the designs. The next period the students were given a laptop and were instructed to use the pattern block manipulative online and recreate their designs. This allowed the students the opportunity to explore the virtual manipulative and most were successful in this task. A few students were unable to recreate their design exactly, either because it was too large or they were unable to manipulate the shapes on the computer how they had done with the hands-on pieces. Designs were printed and mounted on construction paper as a Math and Technology display. I will bring a few to class to show the finished products.

Lesson Plan - Volume

Currently I am working on geometry with my 4th graders. We used the geoboards to discover what perimeter and area mean and how to calculate both. As a next step we will be exploring volume of rectangular prisms. I looking at the lesson plans on the disk I found a lesson involving base 10 blocks and calculating volume. I would typically introduce volume using connecting cubes but this year I will use the base 10 longs as an introductory lesson and then continue with prisms of different lenths.

Lesson Plan

This lesson is based on: "Building Boxes"

Anticipatory Set:
Complete/Review Do Now and homework

Learning Expectations:
* explore volume of rectangular prisms * utilize spatial reasoning to create prisms
* discover that rectangular prisms with different dimensions can have the same volume
* Develop proficiency with basic multiplication and division number facts using a variety of fact strategies

Activities:
* Review the properties of a rectangular prism- what makes it a prism?
How many different boxes can you make from the same number of Base 10 longs?
*Work with a partner. Talk about ways of making a box using Base 10 longs only.
* Now build all the different boxes you can from 8 longs. You may put the longs in 1 layer or in more than 1 layer.
• As you work, make a chart to record the length, width, and height of each box. Also record how many cubic units make up each box.
• How will you know when you have all the possible boxes? How can you prove it?
* Look for patterns in your chart.

Extension:
1. Challenge children to repeat the activity using 12 longs.
2. Ask children to use longs to build several different boxes, each with a volume of 200 cubic units.
Centers - * FasttMath on laptops
* open exploration Virtual Manipulatives - peg puzzle, towers of hanoi, etc

Assessments:
* teacher observations, various prisms, progress sheets, student-centered discussions, Base 10 longs
Materials:
* communicators, story, base 10 blocks, laptops

Tech Infusion:
Promethean board - flipchart, laptops

Accomodations/Modifications:
* teacher designed groups
* differentiated assignments/activities (centers to reach all levels)
* extended time-classwork/assessments(when needed)
* restructured homework assignments for both above level and below level students
* use of manipulatives/graphic organizers
* extension activities/alternative assignments
* organizational strategies - notebooks/book boxes/folder systems
* cloze-notes/study sheets/practice sheets for reinforcement
* modified assessments - M/C, SCR, ECR, O-E
* encouraging independence/student initiative
* includes a rich variety of resources, media ideas, methods and tasks
* personalized monitoring tools (checklists, goal sheets, thumbs up/down level understanding monitors)

Video Review - Workshop 2

In watching this video I found it interesting to see both the teachers and particularly Jeff getting frustrated by the question of how can you prove what you are saying. Like the video stated most teachers expect to attend workshops or trainings and learn some activities they can take back to their classrooms and be shown the right way of doing things. It is no surprise that there was some frustration in this particular situation - most trainings in schools weren't typically done in this manner of questioning and requiring the teachers to prove their thinking.

I feel very lucky in that our district has made a major move toward inquiry-based learning and we have been offered similar training in hands-on math through LLTeach (Paul Lawrence). This had been a slow process in getting everyone on board to this newer way of teaching math. Teachers see that although the basics are important for the students to learn, when the students are given these types of learning opportunities with manipulatives they can develop their higher order thinking while still reinforcing the basic operations.