Geoboard Lesson Plan II

Geoboards Lesson Plan I

Objective:

• tessellate a variety of shapes
• learn how flips and rotations affect both a shape and its tessellation
• discover that some shapes tessellate better than others
• see that shapes can be joined to form other shapes

Procedure:

1. Display these shapes on a Geoboard. (the shapes that appear at the bottom)
   
2. Ask children to use their Geoboards to show how many of the triangles will fit inside the square if none of the triangles overlap.
3. Once the class agrees that eight triangles will fit, have volunteers display their solutions.
4. Using either the children’s work or your own, point out some different ways the triangles can be arranged to fill the square.

On Their Own
(use the figures that are shown at the top)
How many copies of a shape can you fit on your Geoboard?
• Pick one of these shapes with your
partner.
• See how many of that shape you can
fit on your Geoboard. Shapes must
fit snugly with no gaps between them.
Shapes cannot overlap each other.
• All your shapes must be exactly the
same. If you are not sure, make a
cutout of the shape and use it to
check the others.
• Record your work on geodot paper.
• Repeat this process with each of the
other shapes.
• Compare your shapes. Be prepared to
discuss what you noticed.

Questions to Ask Students:

1. Do any solutions have the same number of shapes? Why do you think that happened?
2. Why do more copies of shape _____ fit on your Geoboard than shape _____ ?
3. What shapes left the most unused space on your Geoboard? Why?
4. Can you combine some shapes and think of a pattern that will allow you make up a bigger picture?
   

Color Tiles Lesson Plan

Objective:

• build logical reasoning skills
• make conjectures based on evidence
• find the fractional parts of a whole

Material:

• 12 Color Tiles, a combination of four color
• Small paper bag
• A paper clip
• Direction page

Procedure: Students will be working with a partner.

• Display a sealed Riddle Bag containing four red, three yellow, two green, and three blue Color Tiles. Tell children that you will give clues to help them answer the riddle, “How many Color Tiles of each color are in this bag?”
• Give the first clue: There are twelve tiles in my bag. Have children follow this clue by counting out twelve tiles.
• Have children respond to the second clue: The tiles are of four different colors.
• Give the three following clues, pausing after each to allow children time to adjust their tiles to match it:

1. Three-twelfths of the tiles in my bag are blue.
2. There is one more red than blue.
3. There is one more yellow than green.

• Have children display their solutions to the riddle. Then reveal the contents of the Riddle Bag.
• Discuss what qualities make a good riddle clue. Establish that each new clue should bring the solution closer.
• Point out that a good set of clues leads to just one solution.

*The Direction Page*

• Can you write a set of clues to make a Color Tile Riddle?
• Work with a partner. Chooses up to 12 Color Tiles. Notice how many you have of each color. Discuss what fractional part each color represents.
• Decide on 3 or 4 good clues for your riddle and write them down. At least of your clues must contain a fraction.

1. .......................................................................................
2. .......................................................................................
3. .......................................................................................
4. .......................................................................................

• Now test your riddle. Do this by pretending you don't know the answer and are trying to solve it.
• Put your tiles into the paper bag, close it, and clip your riddle to the bag.
• Exchange Riddle Bags with another pair. Try to solve their riddle. Then look into the bag to check your solution.
• Be prepared to talk about how you made and solved the riddles.

Closure:
Let's double the number of tiles inside the bag by doubling the tiles of each color. Compare the fractional presentation of the colors with their doubles. What is your conclusion?
Double the number of tiles inside the bag by adding tiles from different colors -no rules. Compare the fractions all together.

Video VI

How did the use of manipulative helped the students build a preparation or a readiness to study calculus? The study of speed through time -velocity, was totally related to what exactly the students learned through their elementary, middle, and high school years: observe, discuss, analyze, organize, solve, and present the work with strong justifications.
The study of the cat's movements was completely related to four story building, pizza, game series ... projects.
"Mathematical knowledge grows from actual experience of moving around and living and thinking about it. How we are moving around?," said Prof. Robert Speiser. I do said the same in one of my video reaction paper, that the primary and most important skills of math and how to observe things and make differences between them and draw conclusions are built during our early age, Exactly at the moment we started be aware of the world around us. "Get in there and move!" said Prof. Speiser. However the students weren't exposed to calculus yet, but they were able to present their work in different format: graphs and calculation. The comparison between the cat's speed (acceleration and constant speed) allowed the student to make a prediction of the cat's next movements or velocity.
Overall, the students were able to study the cat's movement mathematically and present their findings in a very convincing way. Thus, good teaching can help students build powerful and lasting understanding to important ideas, as prof. Speiser said. Moreover, fro us teachers we must understand that students will not learn and their learning will not last unless what they are learning or the information which they are getting from us is making sense to them. Otherwise, the teaching will be like filling water in a punctured container.

video V

Starting from: observing, discussing, comparing, and analyzing then conclude.
In the classrooms, building up solid foundation of mathematical ideas: What are the possibilities?
One of the possibilities which the students learned with the different types of manipulative is how to find out different ways to solve the same given problem, also how to look at patterns and analyze them.
Prof. Arthur Powell said that it is very important to get the sutdents involved in looking for justifications. Why? Because in the heart of mathematics we can look at patterns and relationships and try to understand the underline reasons why those patterns and relationships exist.
How students think about problems or to solve problems? How do they think about that particular problem. The why and how asked questions would justify to develop an inner criteria for knowing when they're right or wrong.
Pascal's Triangle and High School Algebra
we identify the What is the connection between elemantary school and high school algebra? In pascal's triangle summitry line: when a pattern starts must reach its highetst point, then goes back to its original starting point. The symmitry line in pascal's triangle, as it is shown on the picture above, expresses the number of possibilities. the symmetry line in pascal's triangle represent the idea of going back to the starting point with the opposite.
In Jeana's classroom the students were experiencing the combination between pascal's triangle and binomials. The first row where the exponent is 1 with 2 terms, the second row the exponent is 2 with 3 terms and so on.
By 11th grade, the students gain skills that help them think mathematically and in higher level.

Video IV, Thinking Like a Mathematician

With the last three videos, we saw how children were taught how to work together to: observe, analyze, conclude, and present. We saw how they learned to listen and value the work of each other. Their engagement and openess toward each other's work, was the same picture presented in this video with the scenery where the four researchers were discussing long term problem.

"The Tower of Hanoe" activity was challenging. The students kept on analyzing their results to the point that they broke them down into pattern within a pattern, then they found the solution.



In part 2, Ms. Janett, the high school math teacher, set the students up with the practice of explaining the lessons to themselves. This was new to me as well as strange. I believe that students deserve explanations, then assign them to solve problems on their own. However, I was intrigued by her strategy.

I observed myself and my students, and note that when I assign them to explain the lesson to themselves, they are better able to focus on the topic, and get more enganged in it then when I explain it to them. I concluded a wild difference between the two situations. The main factor that affects the two situations is the "focus" factor. The main factor for teachers to succesfully relay the lesson, is focus. of any lesson is how to keep their students focused with them so they will understand what they are learning. And I concluded that Ms. Janett's strategy deserves a thumbs-up.

I teach students from fifth to eighth grade. I analyzed that whenever I assigned my students to work on their own in the computer lab, their involvement into that day lesson is always strong. All of them, without exception, were able to present and justify their work.

Video

Video III, Inventing Notation

Mathematical Notation is the use of symbols which stand for specific quantities, according to the problem, to avoid the writing of long words. It is an abbreviation that helps ease the reading of any solution. However, the most impressive thing that attracted my attention is the use of "0 and 1" as a notation to present a solution by one of the fifth graders. It was excellent. The child used a chart where he wrote the first letter of each topping and 0 for the unused topping and 1 for the used. His presentation was successful. It wasn't only that. He, also, was able to use the "four stories towers" to present the "pizza" activity. The combination he made between the two activities was mathematically correct!
When it comes to the second part, where the children were presenting and discussing their work, they were convincing. They were sure about their solutions, defending and insisting. I would say that the children gain these skills from the activities they were involved in. Thus, hands-on-learning strategy does not only inhance students' learning precess, but also their self-confidence, which I myself consider the highest goal.

Are You Convinced?
To start with, teachers must be able to prove certain formulas or rules because some of these formulas are "undigestable" without proofs. In addition, some other rules need to be worked on with the students step by step, so they will completely get them and gain the ability to use them correctly.
Myself, when our professor gave us the "Building Four Stories Towers with Two Colors without Repeating," I thought of this: 0 1 2 3 4 3 2 1 0
0= no mixture of color, the original color
1= one color interapting the original color
2= two of the same color interapting the original
3= three of the same color interapting the original
4= the interapting color gets over the original color
Then, repeat the same thing from the beginning.
And here appears the idea of "Patterns." Some problems in math need the application of "Backward" thinking to be solved, and this is another strategy shown in mathematic textbooks for Problems Solving.
After watching the video, I became convinced that the best way to deliver a proof to young students is with the use of Hands-On-Learning strategy. It is a method that must be included in teachers' daily lesson plan as much as possible. When a child gets in touch with an activity, he/she will learn and understand at the same time much easier than only reading from the text book. Logically, the brain will process freely when it is not framed with rigid instructions. The children in the video were exchanging ideas in a very relaxed environment. More over, there is a big difference between the outcomes of individual work and team or group work, as part II of video II showed.
In the second part, I saw how the students were able to organize and discuss their solutions between each other in a clear platform. They were comparing their conclusions and justifying them. Each one of them was convinced that his way was the correct path to be followed and applyed as a solution to that activity.

Following Children's Ideas in Mathematic
How children develop mathematically? This question was the main purpose observed in video I.
The activity which the children paticipated in was absolutely different from direct addition, subtraction, multiplication, or division operations. It pulled them toward unusual experience where their thinking as well as their hands were both involved in finding the solution. This combination let them use two different strategies to solve the given activity: "Drawing a
Picture" or "Making an Organized List." The two startegies are found in mathematic text books in different levels and grades.
During the activity, the students were engaged in an open discussion, revision, comparison, and presentation of their work. Also, they were able to control and extend their solution. Thus, the algebraic thinking was present. It does exist.
How the algebraic thinking exists in students' mind who were never exposed to it in a direct way? The answer is simple. Let's go back in time and try to look at this students at the age of one and half to four years old. During this stage, children devolop their algebraic thinking -observ, analyze, and conclude when they are: playing -either with toy(s) or in the playground, eating, taking baths... on their own or with partner(s), guided or unguided. They are always looking for answers to their why and how questions. How is it my plate, full with food, is now empty? How is it the bucket was filled many times with sand, and then emptied, and yet the castle is getting larger? Brain development not only allows advancement of supporting yourself, but also how to solve problems in different fields, logically.
Back to our video. The question discussed within was, how children develop mathematically? I would ask: How can teachers guide students to applying algebraic thinking (which is already developed in a child, but simply unused) on their own?

The Lesson Plan / Fifth Grade / Chapter 7

Objective: Adding and Subtracting Fraction with Alike Denominators Correctly
Motivation: Remind the students about the following definitions:
Fraction= division, dividing the numerator by the denominator= decimal
Denominator= the number, which is called denominator, tells us into how many pieces
1 whole has been divided into.
Numerator= the number, which is called numerator, tells us how many parts we will
be using from the pieces of the denominator.
Equivalent= equal value
Material: Pattern Blocks
Procedure: the students will be working in groups of three.

• I will demonstrate to my students how to build three congruent regular hexagon. One is divided into 8 congruent trapezoids, thus the denominator is 8. The second is divided into 12 congruent rhombus, thus the denominator is 12. The third is divided into 24 congruent and equilateral triangles, and here the denominator is 24.
• I will let each student within each group to choose his/her denominator, start writing fraction, ex: 1/8; 3/8; 5/8; 5/12; 7/12; 9/24; 11/24 and so on.
• Add any fraction, either with the same or different denominators. Write the your observations by answering the following question: does adding unlike denominator fractions give you the same part of the same hexagon, knowing that each one was divided into specific number of parts? Was adding fractions of the same denominator gave you the same hexagon? Explain.

• Subtract fractions following the same steps.

Questions to ask students: How many equivalent fractions can you obtain from these three different divisions?
Ending Activity: Students can divide a map, for example into congruent squares, so they can draw the same map easily.
Homework: practice book

The Four Story Building
At the beginning, I gave the block to my husband and I explained to him the rules and how he must create as many 4-story building as he can, with the use of 50 orange and 50 black blocks, with out repeating the same order of colors. He tried for about half an hour and all he could accomplish was 12 buildings and he didn't want to go any further. As for me, I built 18 buildings in an hour and half. I thought of a pattern, which is starting by entering the orange color on the black building one, by one, then two by two, then three. When I reached the orange building, I did the opposite. I entered the black blocks one by one, two by two, then three -of course without repeating the same stories. Observe the picture from right to left.


The Changing Colors Rectangle
I took the "brain teaser" with me to school, where I teach and I gave it to two of my students to try it. One is an eighth grader with a B+ level, who did it in less than 3 minutes. The second one was an ADHD fifth grader. He has a good observation, quick answer to tricky questions, and gave up easily. He tried it for about 7 minutes. At first he moved correctly four steps, but then he lost it and he couldn't go back and start from the beginning. He had hard time remembering his own steps. At home, I gave it to my two sons. Anas, who is in third grade he did it in less than 5 minutes and he was able to remember his steps without any mistake. I was impressed. Aassem, who is in 9th grade, in honor classes, he tried for about an hour, giving up sometime and go back to it, then he was able to switch the colors with respect to the rules. But, he didn't want to do it again.