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.