Module 14

Textbook

Use the Writing to Learn questions on page 26 to focus your reading. Include at least two of your responses to the questions within your blogs.
Explain what it means to measure something. Does your explanation work equally well for length, area, weight, volume, and time?
To measure something means that you are going to give a value to describe an object. This object can be anything from a cardboard box to a sound.
I think that this definition can work with length, area, weight, volume, and time, but it may need a bit more description of each.

Four reasons wer offered for using nonstandards units instead of standrds units in intructional activities. Which of these seem most imortant to you and why?
I believe the reason "when standard units are used, estimation helps develop familiarity with the unit." is most important.

Circumference and Diameter 

Questions:
Describe Ms. Scrivner's techniques for letting students explore the relationship between circumference and diameter. What other techniques could you use?
The students drew a circle on the carpet with chalk. I find this very cool, I have not seen that before. The students place a pencil in the middle of the circle and use a string to create a circle. The string was the radius. The students demonstrated holding a pumpkin in their arms. That would be the circumference. They break down the work circumference. Circle and circumference have circ-.
They are going to measure the circumference and diameter of different items in the classroom.

In essence, students in this lesson were learning about the ratio of the circumference to the diameter. Compare how students in this class are learning with how you learned when you were in school.
In school, when studying diameter and circumference of circles, I remember measuring things given to us by a teacher and also working directly with formulas. This activity allows the students to explore more.
Do you rememeber any specific activities you got to do as a student with circumference?

How did Ms. Scrivner have students develop ownership in the mathematical task in this lesson?
The students got to choose what was being measured.

How can student's understanding be assessed with this task?
With a group discussion and with their completed data sheet.

For Further Consideration…

We have explored numerous areas throughout this semester. Pick five ideas that you will later use in your classroom.

1. Tangrams and the activities
2. Mira. I thought this was fun to explore.
3. Early data collection activities (graphs) for younger grades. Maybe a data question as morning work.
4. Types of graphs and when to use them.
5. Hands on activities and technology. we have been able to explore various topics with online activities and may be very beneficial for students.

What was your favorite area this semester?

Module 13

Measurement Misconceptions

After you’ve viewed the talk-over PowerPoint and read the vignette, respond to the following questions in your Blog. 

Why do you think the students are having difficulty? 
I think the students are having difficulty reading the ruler. They do not understand that the 0 is the start of the measurement. When counting, students start with the number one and they are trying to do the same with the ruler.

What misunderstandings are they demonstrating? 
They are misunderstanding how to compare the units with the attributes of the object being measured. Like the student who measure 41 inches. He knew that was wrong because he "thought the pencil was shorter".

Have you witnessed any students experiencing some of these same difficulties?
I have not witnessed this yet with students.

What types of activities could you implement that would help these children?
If I had students that were struggling with this concept, I would start the day with a Measuring Morning work. Give the student a chance to get familiar and perfect measuring one day at a time.

Have you seen a student struggle with measuring? If so, in what way? Were you able to work with teh child?

Angles Video and Case Studies

Discuss how the children in the video view angles. What ideas make sense and what ideas need further development?

One child described angles as "when two lines meet and it is the space between them".  The students were able to show an angle using their arms. One student mentioned how the angle is at the opening of a cross, but an angle is considered at the small end as well as the larger opening. Another student mentioned that when you put your two hands together, an angle is created. She then uses her classmate as an example. The pushes his hands closer together and explains that moving these hands together creates a smaller space and that is a smaller angle as well.
One thing the students need to have a further discussion about is what makes an angle. The student drew on a board a curved line and a straight line creating a point. She said this creates an angle. Her classmates also agreed. I think the students need to go through and further discuss why this is not considered an angle and what can create an angle.

Respond to the following questions after you read the case studies: 
1. In Nadia’s case 14 (lines 151-158), Martha talks about a triangle as having two angles. What might she be thinking? 

I think Nadia is showing one angle which uses the bottom and one side. Instead of doing the same for the top angle, the student split that angle to distinguish the making of the second angles.

Have you come across a student that had a similar understanding of angles?

2. Also in Nadia’s case (lines 159-161), Alana talks about slanted lines as being “at an angle.” What is the connection between Alana’s comments and the mathematical idea of angle? 

This one is a little difficult. A line represents 180 degrees, but I think the student is more referring to the slant line as one of the lines that form an angle.

3. In Lucy’s case 15 (line 251), Ron suggests that a certain angle “can be both less than 90˚ and more than 90˚.” Explain what he is thinking.

I think that Ron is showing an understanding that during the action, the skater started at one angle, then moves to another. On paper, if we showed the action on paper, the skater could show both 120 and 60 degrees next to each other on a line.

4. In case 13, Dolores has included the journal writing of Chad, Cindy, Nancy, Crissy, and Chelsea. Consider the children one at a time, explaining what you see in their writing about angles. Determine both what each child understands about angle and what ideas you would want that child to consider next. 

Chad mentions how you have a small angle with two pencils together and a larger angle when you put your arms together. I think the thing that chad needs to consider is that the lines that are making the angle are different sizes, but the angles could be the same.
Cindy mentions how angles can be in any direction, but doesn't explain what an angle is. I would want Cindy needs to learn more about what an angle is, and how it they are formed. She doesn't show much understanding what an angle is.
Nancy shows a great understanding with little words. She understands that there are a variety of angles, obtuse and acute. I would move on with the student and show the student various angles that looks like the are different in size.
Crissy also has a great understanding of angles. She mentions acute, right and obtuse angles. She doesn't quite have a great description of an acute angle.

5. In Sandra’s case 16 (line 318), Casey says of the pattern blocks, “They all look the same to me.” What is he thinking? What is it that Casey figures out as the case continues? 

Based on the images provided in the case study, I am wondering if the student is confusing the linear line as a 90 degree angle since that's whats in common of the three shapes.
Casey made the connection that a 90 degree angle is still a 90 degree angle not matter the size of the object creating it.

What do you think causes students to have confusion about angles when comparing the same sized angles from different objects or shapes?

How Wedge you Teach?

What ideas will you take from this article into your classroom? Was there anything surprising about what you read that made you change your thinking about children’s understanding of calculating an angle? What possible misconceptions might children have about angles and what misconceptions did you have about angles?

When the students began performing the activity, it amazed me that the students were able to use prior knowledge of angle measurement to help determine an angle. For example, one student said “The whole circle is 360 degrees; so if you cut it in half, that straight line would be 180 degrees. If you cut it into fourths, … each angle would be 90 degrees.” This is great use of their prior knowledge. I remember as a child, one thing that I did not understand was that a circle was an angle (360 degrees). That was a hard concept for me to grasp because I assumed angles were created by two straight lines that made a point. This could be something stdents struggle with when learning about angles.

Exploring Angles with Pattern Blocks

Post your responses to the questions posed within this task. 

Was this activity difficult for you? If so, which part?

For Further Discussion

As adults we use standard measurements almost without thought. Nonetheless, nonstandard measurements also play a role in speech and behavior. We ask for a “pinch” of salt or a small slice of dessert. We promise to be somewhere shortly or to serve a “dollop” of whipped cream. In some situations we might even prefer nonstandard measurements – for example, in cooking, building, or decorating. Do you or someone you know use nonstandard measurements on a regular basis or even instead of a standard system? Give some examples and discuss why a nonstandard measurement might be preferred in some cases. 

My grandmother to this day uses nonstandard measurements when cooking. My grandmother prefers this because she always said they never made appropriate measuring cups for the amounts she needed. So she always said measurements like spoonful, handful, a pinch, a tad bit, or even a "good" amount of something. It was measurements that she was comfortable and familiar with using.

Module 12

Coordinate Grids

List the websites that you explored and discuss which ones you would use in your own classroom? Discuss the advantages and disadvantages of using online programs in the classroom? 

http://www.shodor.org/interactivate/activities/GeneralCoordinates/
http://www.beaconlearningcenter.com/weblessons/GridGraph/default.htm
http://www.shodor.org/interactivate/activities/MazeGame/

I liked the maze game as an activity to do in the classroom. All the websites were fun and incorporated points on a grid different ways. The disadvantage of the maze game is you can select any points on the grid to make a maze. You are not having to locate specific points. The beacon learning center game does require you to figure out points 1-5. The sight however is not very kid friendly.

Did you find any websites that you like? What were they?

Miras, Reflections, and the Kaleidoscopes

After you’ve completed all the activities, take a picture of the boy on the swing set showing how you were able to successfully use the mira. Have you ever used a Mira before? Did you find any part of this problematic? How did this build on your understanding of transformations? Discuss the ideas from the article that you will take with you into the classroom.

This was my first mira activity. I had never used one before. I had trouble getting the mira just right to make an accurate reflection. These activities were very fun. It helped me better understand transformations by having to perform them. In the article they mention that for students to have a better understanding they must learn more than the definitions. I think that the activities discussed in the power point and article can give the students a visual representation of the word reflection.

Was the mira hard for you to do? Have you done anything like this?

Case Studies

Respond to the following questions in your Blog:

1. What ideas about measurement do the children in Barbara’s class (case 12) bring to school before they are taught about it?

The students were discussing the size of the box and emphasized on the box being big. They then started comparing themselves to the box. The pointed out the box was bigger than certain students.  The students were asked how they could measure the box at school, and the students began using objects in the class to measure the box. They tried bins, a chair, and books.

2. Many children struggle with the idea that the larger the unit, the fewer the number of units needed to cover a length. Go through the cases by Rosemarie (case 13) and Dolores (case 14) to identify how different children are making sense of this issue.

In case 14 Dolores, the students were given the scenario of two boys measuring a room. One got 30 steps and one got 43 steps. Students had varying conclusions like one miscounted, one took longer steps, and one had bigger feet. All the students had different understandings. on case 13 Rosemarie, the students wanted to measure the distance using hand and feet. They made the connection that they could not use hand and feet. They must use things the same size to get the right measurement.


3. In Dolores’s case, line 245, Chelsea notices that Tyler and Crissy both measure the width of the basketball court as 62 “kid feet.” Why didn’t everybody measure the width as 62 kid feet? What discrepancy is Chelsea noticing? What is Henry noticing? How are their observations related to the issue that arises in Sandra’s seventh-grade class (case 17)?

Not everyone had 62 because of the varying shoe size and mistakes made. One student mentioned when comparing the measurements that some of the measurements were the same or close together. Later Chelsea made the conclusion that people with bug feet take fewer step.

How would you demonstrate to the students that the different lengths mean the unit they are using are different sizes?

4. The children in cases by Mabel (case 15) and Josie (case 16) are working out the use of standard tools for measuring length. Specifically, the children in both classes discuss how to place the tool and how to read the number of units. What do the students have to say about these two issues? What do they understand about measuring with accuracy and precision?

They discuss using a ruler to measure objects. If the object is longer than the ruler, you have to use several rulers. If you do not have a more rulers, then you can out your finger and move the ruler. The students also discuss that when using your finger as a place holder, it can alter your length a little because of your finger. If you have to do this several time, it may make a big difference in length.

5. By comparing the cases from second, third, fourth, and seventh grades to Barbara’s kindergarten (case 12), can we identify ideas that, by the older grades, are understood by the children and no longer warrant discussion. What are some issues that still lie ahead for Barbara’s students to sort out?

The older group discussed measurements using various terms and were able to understand how to measure objects with different types of units. The younger group needs to work on their vocabulary and how to describe objects when comparing sizes.

For further discussion

A fellow teacher says that he cannot start to teach any geometry until the students know all the terms and definitions and that his fifth graders just cannot learn them. What misconceptions about teaching geometry does this teacher hold?

This teacher is relying on the fact that students need to know definitions. You shouldn't strictly rely on the students vocabulary knowledge. In the article, the author mentions "Geometry is more than definitions; it is about describing relationships and reasoning".

Did you have a different misconception? What do you recommend when teaching geometry? Definitions?

Now that you’ve had some time to explore the world of geometry, how has your view of the key ideas of geometry that you want your students to work though changed? 

My view has changed. In school, I remember learning definitions. It wasn't until later grades was I required to show what each term meant. I think that the best way to learn and understand geometry, you have to show students examples while teaching the definitions. Working with just definitions and minimal examples can be confusing, especially for lower grades.

Module 11

Pentonimo Activities

http://www.mathsphere.co.uk/fun/pents/pents.html
This is one I found. You are given all the pentonimoes and must fill in a rectangle with them. It caused a lot of frustration because you can't flip them and place them perfectly together. It was very hard. I had to hit the help button.

http://pentomino-puzzle.toogame.com/play
This game is very similar, but you can use some of the pentonimoes more than once. You can also turn the shapes and make adjustments.

For the powerpoint, I completed the activities just fine.

Did you find any interesting websites you would use in your future classroom?

Pentonimo Narrow Passage

16 spaces 

Was this difficult for you? How many spaces was your path? Did you have any strategies?

Tessellating T-Shirts

Summarize the article and then describe how the article furthered your understanding of transformational geometry?
It is expected for students k-2 should work with putting together and taking apart two- and three-dimensional shapes. Grades 3-5 should be transforming shapes. Many teachers believe these concepts should be taught at a higher level. The best way to start introducing transformational geometry is by creating translating tessellations. When creating the shirts, the students need to use problem solving strategies. Students are more successful with large shapes.

What does it mean to tessellate?
Creating a repetitive connecting pattern.

Look online for different examples of tessellations and share what you’ve found.

I love the tshirt idea to try in my future classroom. How would you allow the students to create their tessellations? Would you provide shapes or allow them to create it?

Tangram Discoveries

Post pictures of the shapes you created and then share your responses to the following questions:

triangle, square

Trapezoid, square

Parallelogram

  Which polygon has the greatest perimeter? …the least perimeter? How do you know? (You must describe how you know this to be true.)
Greatest perimeter: trapezoid, parallelogram, an triangle,
Least perimeter: square
I gave my own lengths of sides and determined a value for each.
 Which polygon has the greatest area? …the least area? How do you know?
The area is the same for each shape since they are using the same three shapes all the same size.

How did you figure these out?

Ordering Rectangles

Share your answers to questions 1-5 from this activity in your blog. Spend some time discussing your thoughts with your blog partner.
1. My first thought was C had to have the smallest perimeter, but when I really examined them, C and D seem pretty close.
AGEFBDC Largest to Smallest
2. Smallest Area is c. Largest is G.
GAEFBDC Largest to Smallest
3. (A D E) B G C F Small to Large
I did horrible on this one. I am surprised how far off I was.
4. C D B A E F G Small to large
I did a lot better on are than perimeter. I only have a few displaced.
5. I need to remember that when looking at perimeter and area, area is the amount of space the shape takes up and perimeter is the outside length. Area and perimeter are different and can have different answers when comparing the two.

Was this difficult for you? In what way?

For Further Discussion

Multicultural mathematics offer rich opportunities for studying geometry. Research the art forms of Native Americans and various ethnic groups such as Mexican or African Americans. What kinds of symmetry or geometric designs are used in their rugs, baskets, pottery, or jewelry? Discuss ways you might use your discoveries to create multicultural learning experiences.

I see a lot of triangles, squares, and circles used for design.
I would make a connection of how various cultures used geometric shapes for design and use different images as examples. Then the students can create their own masterpiece using certain geometric shapes.