Private Universe Project 4: PROBLEMS AND POSSIBILITIES

One of the key goals of mathematics education is building problem solving skills. How does a mathematician approach problem solving? What does it mean to think like a mathematician?

In this video we viewed a mathematicians work.
We observed students in a variety of problem solving situations to uncover parallels in the way that students and mathematicians solve problems.

For mathematicians solving difficult mathematical problems like this is a little like working on a puzzle or a game.

In this video the mathematician used the Towers of Hanoi to explain mathematical problems and how they are like solving a puzzle.

When a mathematician is working on a problem, they don't always know the answer.
And if you do know the answer you are not going to give a mathematician the answer, they are going to try to solve it on their own.

First thing a mathematician does is simplify the situation.
Second mathematicians look for patterns

Private Universe Project 2: TEACHERS BUILDING PROOFS

This video takes place in Englewood NJ. The principal has invited Arthur Powell, Associate Professor at Rutgers University - Newark, to work with Englewood teachers, grades K through eight. She hires Arthur Powell to implement a professional development initiative that encourages teachers and students to think deeply about problems and to justify and convince others about their solutions.

The goal of the workshop:
1) Introducing teachers to a new way of working in the classroom; having them pay careful and close attention to what students actually do and say.
2) To have teachers rethink how they view themselves in front of mathematics.

During the workshop, Arthur asked the teachers to work on combinations problems such as - How many different towers four blocks high can you make by selecting from blocks of two colors? For each solution, teachers explained and attempted to convince the others that they had found all possible towers, four high.

The main focuse of this video is getting to the idea of teaching mathematics; the improtantce to get involved in looking for justifications. The basic idea of mathematics is the idea that we can look at patterns and relationships and try to understand the underlying reasons why those patterns and relationships exist, given the particular mathematical objects. And in reasoning - in understanding why they exist - one is developing ideas of proof.

Often teachers teach to teach the lesson. It is important to teach the skills and the thinking process to help understand why things happen. Students may know how to solve a problem but it is also important to understand and be able to explain why their solution is correct.

Virtual Math Manipulative

The virtual math manipulative I picked this week is:

Fraction Bars

I felt this virtual math manipulative game would be a great way to reinforce the lessons taught using Cuisenaire Rods

Here is an example of it being used to cover the giraffe

How to use:
1) Students pick the length; then new
2) They can click on the bar they created and choose a color for that bar
3) Repeat to make other fraction bars

Lesson Plan 2: Cuisenaire Rods

I took my lesson from the cd; it its called: Cover the Giraffe

Materials:
Cuisenaire Rods; 1 rod of each color per child
Cover the Giraffe worksheet; several pages per child
Crayons
Overhead/Elmo Cuisenaire Rods and/or Cover the Giraffe transparency

Objective:
Children will cover the outline of a giraffe using a specified set of Cuisenaire Rods. They compare their work and try to identify as many different solutions as possible.

In this lesson children will develope spatial reasoning

• visualize a region as the sum of component parts


• find multiple solutions

Procedure

• Outline the shape shown on 1 centimeter grid paper. Then copy and distribute it.


• Challenge children to cover the outline using only 1 white rod, 2 red rods, and 1 light green rod.


• Establish that several different ways of covering the shape are possible.

ON THEIR OWN
Work with a partner. Each of you make your own cuisenaire rod staircase.
Now use the rods from your staircase to completely cover a giraffe that looks like this (model)
Compare your work
Record both solutions if they are different; Record just 1 of them if they are the same.
On other worksheets, find more ways to cover the giraffe. Record each way.
When done we will discuss and share as a group.

Lesson Plan 1: Cuisenaire Rods

The lesson plan I choose was from the CD.

Build a Boat

Materials:
Cuisenaire Rods 1 set per pair
1 centimeter grid paper
Overhead/Elmo to display Cuisenaire Rods
1 centimeter gride paper transparency if using overhead

Objective:
Children will create "boats" using Cuisenaire Rods. They will then make an estimate how many white rods are needed to cover the shape of their boat.
In this lesson children will:

1. Discover numerical relationships among the rods
2. Develop strategies for adding

Procedure:

Review: Ask how many white rods would be needed to exactly cover a yellow rod.

Model 5 white rods would be needed to cover the yellow rod.

Make a train of 2 yellow rods. Ask how many white rods would exactly cover this 2-car train.

Now explain today's activity/model

• Work with a partner, build a boat according to these rues:
• use at least 5 rods; but no more than 10
• make the boat lie flat
• use any colors except white
• estimate how many white rods would cover your boat
• check your estimate by figuring out the exact number of white rods that would be needed. Find 2 ways to do this
• Record how many of each color rod you used in your boat. Record the total number of white rods.
• Leave your boat in place. We will discuss as a group when everyone has finished.

Math Man

Here is another great game to try with your students

http://www.knowledgeadventure.com/games/math-man.htm

Math Games

This week I googled math games to see what I would find.

I came across this website; http://www.knowledgeadventure.com/math-games.htm

I decided to play: Picture Math
http://www.knowledgeadventure.com/games/picture-math.htm

You have a choice: addition, subtraction, multiplication or division.
This game is nice because if gives students a choice. You work on different math facts.
Students solve math facts. They are given an equation ___ + ____ = 9.
Students must fill in the two missing numbers. If they are correct the numbers they chose disappear and a picture appears. There is a time frame they need to complete this in.
This is a great review of math facts; and keeps the students engaged and motivated.
Below is an example of what the screen will look like.

Geoboard: Inside Outside

Does anyone know how to post a power point?

Lesson Plan #2: Inside Outside (also taken from CD)
Grade Level: K

Objective: Students will develop an understanding of area as a number of square units needed to cover a region.

Materials:

• Geoboards (1 per a child)
• Rubber bands (several different colors)
• Geodot paper
• Color Tiles
• Crayons
• Pencil
• Record Sheet
• Smartboard/ Overhead or Elmo

Procedure:

*Allow student the day before the lesson to have 20 minutes to explore with Geoboards.

1. Model to students how to use one rubber band on the geoboard to create the largest possible square. Explain this is the outside region.
2. Keep the outer region and this time create a smaller region. (Refer to Teacher made smartboard lesson)
3. Pass out geoboards and have chidren duplicate what you made using two different color rubber bands.
4. Review that they have created two different regions (the inside "smaller shape" the other region is the outside region)
5. Ask children which region they think covers more space.
6. Disscuss/brainstorm - how we can find out they are correct (ex: count the pegs; use geodot paper; record data)
7. Model to students using color tiles how they can measure the outer region or inner region.
8. Model on the smartboard the same method.
9. Allow children to explore on own for a few minutes.
10. Call children attention to brainstorm what they have discovered. Call on children/pairs to post their drawings to see if anyone had a drawing with one square inside. If so ask how many small squares were outside that shape (record) Ask which shapes had more squares outside/inside.

Follow up:

• If your guess was close to your count, what helped you make such a close guess?
• After you found the number of squares inside your shape did you ever know the number of the outside squares without counting? Explain.
• What do you see when you look closey at the chart (record/data)?
• What do you notice about shapes that can be covered by the same number of squares?

Geoboard Lesson : Picture This

Lesson Plan (taken from CD) - Picture This
Level K

Objective: Students will create Geoboard shapes to represent objects.
Keeping the objects hidden; children will try to guess them. Last; they will
copy a representation of the object.

Materials:

• Geoboards (1 per a child)
• Rubber bands
• Geodot paper to create flashcards of objects or take pictures of objects you created using the geoboard.
• Smartboard/ Over head or Elmo

Prodecure:

1. Explain to students the objective for todays lesson
2. Model an object to make (ex: kite, chair, flower, rocket, house and or fish) state different characteristics of the object. Have children guess the object.
3. Using flashcards - allow one student to go to back table to pick a flashcard.
4. The student will try to give some hints what object they picked (teacher may need to assit).
5. Students will guess what the object is; then recreate on their geoboard (or using the geodot paper).
6. The child that guesses the object will have a turn to pick a flashcard.
7. Repeat steps 3 - 6

Take pictures of each object and create a book. Write down questions or characteristics. Create a flap to cover the picture of the object/geoboard. Print out book and have students share with parents of family memebers.

Posts and Trees - Area

As discussed in class we found out the area of triangles.

Peg = post outside touching rubberband
Tree is a peg inside the rubberband not toughing a rubberband.

Posts Trees Area
0 0 .5
1 1 1.5
3 2 2.5
3 3 3.5

From these observations as a class we came up with a formula:
A (P,T) = .5(P) + 1(T) - 1

Professor Flint asked we go home and see if this formula will work with quadrilaterals.

Quadrilaterals: means "four sides"
(quad means four, lateral means side).
Any four-sided shape is a Quadrilateral. But the sides have to be straight, and it has to be 2-dimensional.

There are special types of quadrilateral:
Some types are also included in the definition of other types! For example a square, rhombus and rectangle are also parallelograms.


Posts Trees Area
4 0 1 .5(4) + 1(0) - 1
4 1 2 .5(4) + 1(1) - 2
4 2 3 .5(4) + 1(2) - 3
4 3 4 .5(4) + 1(3) - 4

In conclusion I found the formula to work the same for quadrilaterals.

NLVM Fraction Bars

This week I focused on the NLVM fraction bars.
Fractions are something we cover briefly in Kindergarten.
I felt this Virtual Math Manipulative is a good game for reinforcement of new material.

How to play:
1) First set the desired length with the up and down arrows.
2) Click the new button. You may add multiple bars by repeatedly clicking the new button.

In this game you can choose any colored bars a variety of lengths from 1 to 10.

This game can be modeled to your students first.
Students can be assigned to computers. Teacher can create a table using word.
For example if you are working on fractions (halves, thirds or forths) have 4 columns; two rows.
Ask students to show a half, by coloring the table. They may use the fraction bar games to help.

Pictures from Lesson Plan

Below are pictures from my lesson on finding area using GeoBoards

Setting the border

Teacher modeled the shape the children will make to find the area of.

Teacher also modeled how to use color tile squres to find the area of that space.

Students then worked at their desks; while one student came up to the smartboard to solve the same problem.

When completed children checked their answers to see how many squares took the space of the space.

After that they used the same shape but calculated the area outside the shape. Once again students worked at their desk and one study was working the same problem on the smartboard.

This continued for 4 more different shapes.

We tried to solve the pattern for the area. (This was a difficult concept for my students but the kind of got the idea. I had to model to them the correct problem sovling techniques).

This lesson was taken from the CD. It is called Inside Outside. I modified it by using the color tiles and the smartboard.

Diffy

I was looking at the NLVM site and came across the game: Diffy.

The name stood out and I thought to give it a try.

Diffy is an interesting puzzle involving the differences of given numbers using whole numbers, fractions, intergers, decimals and money. This is a great problem sovling activity for all.

I tried the whole numbers. You first start with 4 problems on the outside of the puzzle and work you way in. Its a great way to challenge your students in addition and subtraction.

Below is the problem I was presented to solve: (the numbers in the middle was blank and I had to solve the missing number.

Here is an example of a problem. What do you think are the answers?

Virtual Math Manipulatives - Base Blocks

Base blocks was a perfect way to review this weeks Math objective: count/use objects to show 6, 7, 8 or 9.

First I modeled to my students how to use the virtual manipulatives. I explained the value of the ones and tens columns (because this is a new concept). The teacher can dictate different numbers for students to represent using the base blocks or students can work independently on computers by clicking "Show a problem".

For a kindergarten level if you clicked Show a problem the direction stated: Use blocks to show 3. Use blocks to show 5. This was a nice way for me to review our topic of numbers 6 - 10. I asked my students to use blocks to show me numbers 6, 7, 8, 9 or 10. When I asked a student to show 10; they went up to the smartboard and clicked ten ones. The virtual manipulatives did not correct to show one 1 tens = 10 ones. When the child I was done I said that is one way and then modeled again the tens column.

This is also a great way to introduce addition or subtraction. If you add one more block how many are there? Now take away two how many are left? I plan to use the base blocks again when we get to addition and subtraction.

Virtual Manipulative Game - Geoboard

My students LOVED playing Geoboard!

This was a great game to play with my class to review shapes and colors.

Directions consisted of the following steps: Coloring Triangles and Squares
Color the inside of each triangle green. Color the squares yellow.

To play the game a child had to click on the shape then click the appropriate color.

This was a great way to practice two step directions, waiting in a group, using the smart board, identifying shapes and colors. Along with motor skills.

This was a very basic and easy game for my students to use. However unlike other games there were no directions or feedback "nice job" "wow" or any kind of reinforcement. This is something I would like to see added to this game.

When we were done with the directions given I cleared the board and had student come up to the board and try to create their own shapes and pictures. This was great to watch!

Lesson Plan: Color Tiles

Objective: Students will use objects to count to 6 and 7; make 6 and 7 using color tiles (show two parts)

Materials:

• Crayons
• Color Tiles
• Paper
• Pencil
• Smart Board

Procedure:

• Review numbers 6 and 7.
• Hold up an index card students will use color tiles to show that number.
• Model to students on the Smart Board how to make 6 and 7 in two parts.
• Break up into 3 groups.
• Group 1: Workmats
• Group 2: On Computer watching Interactive Learning Video: Making 6 & 7
• Group 3: Hands on work with color tiles

Assessments:

• Workmats
• Teacher observation
• Data Collection

Exploration with Color Tiles

Today, Tuesday March 2nd

I decided to take out the color tiles and let my students explore. I placed them on the table after lunch. My students had a blast! They immediately began forming patterns. They showed pride in their work and were eager for me to take photos of their work! :)

Private Universe Project in Mathematics 3

Workshop 3: Inventing Notations

"In mathematics, how do we make visible an idea"... this is important for teachers to realize. Students at all ages are unique; they all have a way of preceiving a problem. As a teacher we must keep this idea in mind and try to realize how our students go about a problem/their thought process. "The teacher in a sense has to become a learner."

As a teacher of children with special needs I try to visualize the best way to break up a problem for my students. I try to visualize methods my students will use to make a connection. I often use connection to real world situations to help them apply their knowledge. My students benefit from pictures. Just like mathematics we use manipulatives and symbols.

In the video it mentions, "there are limitations on what a teacher can do, given classroom time". This is true. I have observed in the faculty room many teachers discuss standards and testing. I believe many teachers get lost in these requirements and only test to the test; they forget to make learning fun. I feel these workshop video demonstrate how learning can be educational, motivating and enjoyable. In each problem the students are communicating with eachother, learning, have a different thought process - that they can teach to their classmates and not bored!

Pattern Block Website

Below is a preview of the website I was trying to share in class.
I feel many would enjoy using this site with your class. It is a great way
to integrate technology into the classroom. You can use on the smartboard or assign
to students individually on computers and allow them to print work upon completion.

Hope this helps!
http://www.arcytech.org/java/patterns/patterns_j.shtml

Bar Chart

I search the Virtual Manipulatives site to try to find something my students could use.

It is difficult to find appropriate games to fit their needs and skills.

I came across the bar chart. This was a nice tool to use as a group.

We discussed how everyone gets to school: bus, walk, or car.

Together we create a bar chart: (see below)

This tool is nice because as a teacher or class you can decide how many columns or rows you will need to create your chart. You give your title a name and label your columns. This a great way to introduce making graphs and collecting data.

Private Universe Project in Mathematics 1

A long term study conducted by researchers at Rutgers University followed the development of mathematical thinking in a randomly selected group of students for 12 years - from 1st grade through high school. This study showed surprising results. The videos I watched show an overview of the study and was able to witness the conditions that made their math achievement possible.

In first and second grade they were presented a problem: Stephen has a white shirt, a blue shirt and a yellow shirt. He has a pair of blue jeans and a pair of white jeans. How many different outfits can he make?

When in first grade the students solved the problem and came up with 5 different combinations. They used symbols and words to solve the problem.

A few months later the same students entered third grade. They revisited the same problem with the shirts. This time when they started counting outfits a young girl Stephanie, began drawing lines to connect the different shirts and pants. In conclusion she came up with a total of six possible combinations. Just in one year students mathematical thinking process grew. Students learned from the first time and changed strategies when given the problem a second time.

I liked watching the students solve the problems. I would try to solve the problems along with the video. It was interesting to try to see their view on problems. I liked how the reserach developers presented different problems over the years with similar concepts. Children were solving high-school math problems in elementary school. In today's society there is alot of pressure to meet state standards that does not allow for exploration. In the video students were given fun math problems; over the years they had no idea of their teacher's objectives. They were learning and having fun. The best part it was real life situations and hands on!

PATTERN BLOCK LESSON PLAN - Kindergarten

Objective: Students will recognize that shapes can be combined to make other shapes.

Materials:

• Pattern blocks: 6 green triangles, 2 red trapezoids, 2 orange squares, 2 blue parallelograms, 2 tan parallelograms, 2 yellow hexagons.
• Smartboard/computers.
• Workmats from www.pearsonsuccess.net
• Workbook pages: 50-52.

Procedure:

• Introduce with Interactive Learning Video: 7-03 Geometry: Making Shapes from Other Shapes
• Class discussion: Pose problem: Miss Smith wants to use small shapes to make this island in her pond. Hold up a trapezoid. How can she figure out how many of this shape to use? Have children share their ideas before modeling the solution.
• Group discussion: Guided practice on Smart-board
• Break up into groups 1) group with teacher smartboard 2) computers 3) individual or group work in pairs using hands on manipulatives - pattern blocks

Assessment:

• Participation
• Work-mat
• Practice pages

PEG PUZZLE

I must admit in class when given the peg puzzle I struggled trying to solve the problem

We were given a board with 8 pegs (4 pegs in purple 4 in blue). The task was to: move the pegs on the left pass the pegs on the right, you can jump over only one color; cannot move backward only forward.

After class I tried the peg puzzle on NLVM: http://http//nlvm.usu.edu/en/nav/frames_asid_182_g_4_t_1.html?from=grade_g_4.html

I liked the way the online game was set up.

It starts off with two pegs: (shown on left)

Once I completed and won two pegs;

I was able to move to the next step which is four pegs (below)

After completing four pegs you can choose six or eight pegs. Once I have mastered playing with four and six pegs I felt confident and moved to eight pegs.

The pattern I observed was:

It does not matter which side you start with.

Move red on left into open space, jump with blue, move blue pieces forward (empty space at end of right side), make all jumps using red peg, blue can move to end (two blue pegs at end on left), move red into emty spaces, then jump using all blue pegs and puzzle completed.

As a teacher students will come up with different ways to show patterns.

Please share with me ways to describe the pattern you used to solve the puzzle.

Color Patterns - NLVM

I was eager to try the Virtual Math Manipulatives with my students. My class is labled LLD-S (learning language disabled severe). Their are 5 boys and 1 girl in the class; the majority diagnosed with Autism. They are all on a Pre-K to Kindergarten level. In October we reviewed color and shape patterns, so I decdied to try Color Patterns: http://nlvm.usu.edu/en/nav/frames_asid_184_g_1_t_1.html?from=topic_t_1.html

They were excited to try something new on the Smartboard.
The first pattern to appear was: Blue, Pink, Purple, Purple, Blue, Pink, Purple, Purple, Blue, Pink, Purple, ?, ?, ?, ?, ?. The goal was to complete the pattern using the colors to the right. I modeled first what to do, then called a student one by one to complete the patterns. My students really enjoyed this.
The next pattern was easier for them to complete: Blue, Red, Blue, Red, Blue, Red, Blue, Red, Blue, Red, Blue, ?, ?, ?, ?, ?. This followed an ABABAB pattern, and was easier for my students to solve.

Sometimes I found the task to be challenging for a preschooler or kindergartener to complete.
For example the next pattern (shown left) that appeared was:
Purple, Dark green, Light green, Dark green, Blue, Pink, Purple, Dark Green, Light Green, Blue, ?, ?, ?, ?, ?

This task was complicated and took alot of time, thinking and analyzing.

I like the idea of the Virtual Math Manioulatives, I feel anything with technology helps motivate our students want to learn. I wish the color patterns were organized from easy to hard starting off with patterns ABABA to ABBABBA to ABCABC instead of a random approach.

4-Story Tower Activity

I tested the 4 tower lesson on a friend. For this activity I gave my friend a total of 100 connecting cubes; 50 yellow and 50 green. The directions were to create as many different 4 story towers combination using two colors (no pattern can be repeated). Immediately, he stated 4X4=16; there are 16 total combination. I asked him to show me using the connecting cubes and prove he is correct. He began to make various patterns. When he said he was done I asked what his strategy was? He stated: First he took three of the yellow and one of the green and worked his way down and up (total of 8 story-tower). Next he created two solid towers of one color (total 2 story-tower). Last he mixed it up two and two of each color (total 6 story-tower). Creating a total of 16 combination.

(Picture describes method from right to left)