"Insert Witty Name Here"

Nets Activity and Mathematics Hiding in the Nets Article 

How could you use a similar activity with students in the classroom?  Were you able to complete the activity without too much frustration?  What are some anticipated issues while doing this activity with students?  
This is a great hands on activity that cost little to no money to do.  To my surprise I had only made on mistake in my predictions and after realizing it I wondered how I tough it would make the topless cube in the first place. 

Em, wouldn't this be a great money saver in our classroom? 

I could visually see the shapes in my head but when it comes to listing attributes, I hesitate because I am afraid I do not exactly know what to list. Third-grader Morgan clarified my thinking.  She is quoted in the article describing the nets as:

             Each net has 6 squares, and all of them have at least 4 all together and 2 on each side.
             If there were 3 in the middle and 3 on the side, it would not work.  All nets have a
             perimeter of 14 units.  They can be arranged by flips and turns.  All nets start as 2-D
             shapes and become 3-D figures.  The area of each net is 6 square units.  The nets
             all have 6 square faces to make a cube.  (Jeon, 2009)


Euler’s formula vertices (V), edges (E), and faces (F) of a polyhedron (F + V = E + 2).

Textbook Reading  

All the questions on page 60. 

3. What can you do when the students in your classroom are at different van Hiele levels of thought?  The first thing that comes to mind is differentiation.  I could always give the students with high levels a more complex problem than the lower group.  Alternatively, I could pair those who are at lower levels with the higher ones.  

4. National Library of Virtual Manipulatives: What an awesome website.  Some classrooms are not equipped with the manipulatives necessary to help students explore and understand math. It is however now protocol for all classrooms to have computers and Wi-Fi.  (I do realize this is not everywhere)  This website would be a great alternative.  Students who have computers at home could practice using the manipulatives available.  I also can use this website to teach my students concepts on the overhead.  

Em, What did you think of this website and how would you use it in your classroom?

Spatial Readings

I too feel this is a very valuable skill.  Not everyone is able to use spatial reasoning.  I feel like you are born with skills you can just do with ease.  It is as if they see the world differently without someone pointing it out.  With practice and guidance, I believe everyone can learn this skill but for some people it just comes natural. 

What do you think Em?

Building Plans 

I found the PowerPoint very easy to follow and I accomplished every task with ease. 

Annenberg 

Did you find any of these activities challenging?  If so, what about the activity made it challenging? 

Trip on a train: It was very easy for me to compare the pictures to the map and see the order.

Plot Plans and Silhouettes: I had such a hard time visualizing this.  After several tries I finial got the answer. 

Shadows: I found it hard to see that the cube could cast a shadow that looks like a hexagon. Once the shadow moved around I saw it.

Why is it important that students become proficient at spatial visualization? At what grade level do you believe students are ready for visual/spatial activities?  How can we help students become more proficient in this area?

 I believe 2nd grade is a good time to start introducing visual/spatial activities.  By introducing these concepts at an earlier age, students are more receptive to the possibilities.

 

Annenberg Tangrams Module and Creation of Manipulative 

To be honest I started having problems as soon as I got to part B.  I just could not see where to cut the shapes to make the other ones.  I printed out the shapes even looked at the solutions to try to cut them.  I really need to practice this a lot more. 

Em, I hope you understand this more than I and can maybe meet with me? Help!!!!!!

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I printed out “The Midline Theorem” to try to do part C but it is Greek to me.  I need to get someone to physically show me how to do this.

Em, do you have any pointers you could pass along?

For further discussion 

Informal recreational geometry is an important type of geometry in many childhood games and toys. Visit a toy store (or go to an online store) and make an inventory of early childhood toys and games that use geometric concepts.  Discuss ways these materials might be used to teach the big ideas of early childhood geometry.  

Wooden Geometric Stacker 

Rings, octagons, and rectangles can be slotted onto the three rods, stacked on top of each other, or lined up to compare shapes, sizes, and colors. This first-concepts set is a manipulatives all-star!

Tangoes Jr. Magnetic Puzzle 

Tangoes Jr. brings the fun, creativity and problem-solving challenges of Tangoes to a unique toy designed specifically for kids. With a large playing surface, seven magnetic puzzle pieces and recognizable puzzles, children can create 11 classic tangram images. All seven puzzle pieces store in the side drawer and All 12 puzzle cards store beneath the playing surface Integrated carry handle Two levels allow for more challenging puzzles as the child gets older.

Kids Math Games Online

This is a screenshot of one of the games on this website. I started to play and had to make myself stop. I will definitely go back and play again just to practice. I will also share it with my son. 

Squares, triangles and rectangles......OH MY!

Quick Images

I decided to play along with the class.  When first show the image it was easy to see a crescent moon with a small circle or hole in the middle.  The first child said he saw the moon and I thought wow that is what I saw. However as the video went on and she asked other students I felt as if they were answering how they saw the shape now not the strategy used to try to remember to draw it. I feel as if the children were trying to give different answers to her question so that they can be heard. I do feel like all the input the children gave were all great strategies to help remember the shape but I feel she could have worded her questioning different such as, “Did anyone use the moon as a reference to remember the image.

Case Studies

I was taken back by the misconceptions that most of these children had about what makes a geometric shape a shape.  They were could not see the difference between a mathematical terms like triangles or square and familiar objects like diamond or hearts. (line 47) They seem to be able to list certain attribute that make a shape a shape but have a hard time seeing those attributes in any other form.  They could not clearly defined what made a square a square as they had the same definition for a rectangle and seamed to be totally fine with that. 

Evan was one of the exceptions.  He seemed to get the concept that it doesn’t matter if we skew the shape of a triangle by making the sides longer, it is still a triangle. 

Em, I feel that this type of thinking will help him with recognition of other shapes. After all if you stretch out a square into a rectangle it still has any attributes of a square. What do you think?

Annenburg – Polygons

This module is a little scary for me. It reminds me so much of trigonometry, yet another math topic I struggle with. 

Problem A4

How many polygons can you find in the following figure? 

Polygon	Names	Score
Triangle	RUV, RST, VWT, RQT	4*3=12
Quadrilateral	RSTQ, RUWQ, RVWQ, USTW, USTV,	5*4=20
Pentagon	RUVTQ, RSTWV	2*5=10
Hexagon	RQTSUV, RQWVTS	2*6=12
TOTAL		54

It really helped but was very time consuming to draw out each shape.  I had a hard time seeing them all but with help from the solution I was able to find them all.

Em, did you draw yours our as well?

 Problem B2

As a warm-up for the game, put each of the labels Regular, Concave, and Triangle next to one of the circles on the diagram. Place all the polygons in the correct regions of the diagram.

Working this problem instills in me the importance of manipulatives.  It was so much easier to solve the problem once I had it in front of me.

Problem C2

A figure is convex if, for every pair of points within the figure, the segment connecting the two points lies entirely within the figure.

Which of these definitions work for convex polygons? A polygon is convex if and only if...

a. all diagonals lie in the interior of the polygon. True

b. the perimeter is larger than the length of the longest diagonal. True

c. every diagonal is longer than every side.  False

d. the perimeter of the polygon is the shortest path that encloses the entire shape. True

e. the largest interior angle is adjacent to the longest side. False

f. none of the lines that contain the sides of the polygon pass through its interior. True 

g. every interior angle is less than 180°. True

h. the polygon is not concave.  True

EM, listed above are the correct answers given on the site but I am still unsure of how they got most of these answers. I can work out the problems when it comes to shapes but sometimes I have no idea when it comes to expressing in words. Any thoughts to help me?

Color Tiles

This was a fun little activity that I enjoyed doing.  I had no problem finding the two and four figures.  When it came to five, I had about 6 different shapes found and paused.  I knew there must be more but I just couldn't see any more.  Maybe I gave up too early.

The real shocker came when we had to name the shapes.  I feel so dumb, when I think of an octagon I picture a stop sign. I believe this to be what everyone thinks of. I had no idea that it was any 8-sided polygon (a flat shape with straight sides)

It make total sense to me now. This confirms what I talked about above. If we don't show our students when they are young it could take them until they are in their 40's to figure it out!

Ok Em, did this come as a surprise to you?

Geometry

What are the key ideas of geometry that you want your students to work through during the school year?

I would want my students to become familiar with vocabulary needed to understand geometry. I would like them to understand that there is so much more to a shape than its outer appearance. For example: A square is a square because that is what we were told. 

A Square is a flat shape with 4 equal sides and every angle is a right angle (90°).

Van Hiele levels and Polygon Properties Article

The van Hiele levels are built sequentially upon one another.  I know we all start out at the visualization level.  That is we see a shape for what it is.  A square is a square because it looks like one. One masters each level depending on the quality of instruction they have.  The van Hieles mention 5 phases of learning.  I can remember geometry in school and I always felt I was good at it until college. I do not remember my instruction getting much further than the information stage. 
I do like how the article and the power point stressed that if the teacher is not thinking at the child’s level little to no learning will occur.  I think this to be true in many cases.  “The teacher needs to remember that although the teacher and the student may both use the same word; they may interpret it quite differently”.  We need to be mindful of our students and ask questions to make sure our students understand.  They may be just memorizing what they think they want them to say and retain little knowledge of the topic.

Em, I feel like I am barely a two on the van Hiele scale.  What about you? I think my 8 year old knows more about this that I do.

How did you do with the activity in the PowerPoint?  Knowing what you now know, what do you think about geometric instruction and how will this impact you’re planning in your own future classroom?

I was slightly nervous as there were many shapes and I did not know what to expect.  I did manage to get them all right and do plan to show Alexander to see how he does. I did not realize how important manipulatives were until my tutoring experience last semester.  In the classroom I am in now, I have seen many times where the use of base ten rods would have helped with comprehension.  I will work to give my students the opportunity to use hands on activities to enhance their learning. 


Em, how important is it for you to use maniplitives in your future classroom?

Annenberg Triangle and Quadrilaterals Module

Linkage strip: Problem B1

I don’t think I fully understood this problem.  I could tell if the units formed a triangle but I was unsure what they meant by “Can it be deformed?”  I did figure it out in the end by noticing the note given. 

Em, did you have a problem with this?

Building Towers: Problem C1


I did not have the necessary materials to try this activity but I have played a virtual game like it.  In order to make it to the next level you had to build a bridge across hole of some sort.  You have a limited amount of materials so you have to choose you construction carefully.  I had more success when building with triangles than any other shape. 

Thinking about Triangles: PowerPoint activity

Look around the room and find some triangles? 
 Sadly, there are no triangles in this room.  I could split a square shape in half diagonally and make a few. 

What do these shapes have in common? I
f I did have a triangle in my room, I would say that they all have 3 sides and 3 angles

Where do you think the word triangle comes from? 
Tri- means three

What other words I know that start with tri: 
Trifold, trilogy, tripod… that’s all I got

How are they similar to the word triangle and how are the different? 
The all have three of something.

What is the definition of a polygon? 
 I said they had straight lines and were closed figures. 

Is it possible to make a three-sided polygon that is not a triangle?
No, because any polygon that has three sides is some kind of a triangle.

Is it possible for a triangle to have two right angles?
No, In order for me to make a triangle I only have the option for one right angle.

How many different right triangles can be made on the geoboard? 
This was hard for me for some reason.  I could only rap my head around 10.  I wish there was an example on the power point.  I Googled it and found a picture but I guess I would have helped me if I was shown.

How many different types of triangles can you find?  
To be honest I don’t know triangle types very well and I had to look this up.  After I looked them up, I was able to make them all. Correction, all but one the Equilateral Triangle.

Em, please tell me I am not the only one who had to look this up!  So much for the future of our children!

1. How would you structure this lesson for students in an elementary classroom?  I would definitely demonstrate what was expected first.  My students will not all be on the same level and everyone will know exactly what is expected.  
2. What parts did you have issues with? This question should state, what parts did you not have trouble with.  I Felt I struggled with a lot and I am a little embarrassed.  I should have done this assignment after the Annenberg.
3. Did you need to revisit some vocabulary words to remind yourself of their meanings? If so, which ones?

• Acute triangle: A triangle for which all interior angles are acute.
• Acute Angle: An angle that has measure less than 90°.
• Obtuse triangle: A triangle that has an obtuse angle as one of its interior angles.
• Obtuse Angle: An angle that has measure more than 90° and less than 180°
• Equilateral Triangle: A triangle with three congruent or equal sides or angles.
• Scalene triangle: A triangle in which all three sides are a different length.
• Isosceles triangle: A triangle with two equal sides and two equal angles.

Em, I had to look almost every triangle up.  I am going to have to take some time to memorize this.  How did you do?

What are the chances???

Annenberg

Ah, probability. The first thing that comes to mind then I hear this word are topics like the weather and the lottery. I lived in Las Vegas, Nevada and worked in a casino. Everyday I was witness to people hoping that they would beat the odds and win the jackpot. When many people think of probability, they think of rolling dice, picking numbers at random, or playing the lottery. In fact, games of chance, which often involve dice or other random devices, rely on the principles of probability.

Working through the site I  found the problem very similar to what was covered in my statistics class a couple of years ago. 

I did come across a couple of problems that I just did not seem to get. 

Problem C8
Write a formula for determining the number of possible outcomes of n tosses of a fair coin. 

I think this confused me because I still struggle with algebraic equations.
Can you write out the steps for me so I can see it and walk through it myself? 

Problem C10
Use the binomial probability model to determine the following:
a. What is the most probable score you'll get?
b. What are the least probable scores you'll get?
c. What is the probability of getting at least two answers correct?
d. What is the probability of getting at least three answers correct?

I just could not think of how to start the problem. Is it just like the coin toss example?

A Whale of a Tale article 

Impossible	Certain	Likely	Unlikely
It is impossible for humans to breath in space.	It is certain that you will be another day older tomorrow.	It is likely that I will not get to bed on time any time soon.	It is unlikely that I will get another pet.
It is impossible to run faster than the speed of light.	It is certain that the sun will shin in the morning.	It is likely that I will go to school tomorrow.	It is unlikely that I will follow through with my New Years resolution.

Dice Toss 

Ms. Kincaid wanted the students to make predictions about their experiment on the basis of mathematical probability. Discuss preconceptions that students exhibited about tossing dice even after discussing the mathematical probability. Discuss the instructional implications of dealing with these preconceptions.

While introducing the lesson to her students Ms. Kincaid reminded her student about what they already knew about probability and if they could recall what they have learned previously about it and one student could recall an activity they did that evolved a coin toss.  Even though students seemed to know the definitions and could recall previous lessons, I still feel like they would benefit from more practice.  One student said she thought a two would come up more that 12 because in the past she had rolled a lot of ones.  However, the majority of the class felt that the number seven would come up a lot because there are more ways to make a seven using two dice than any other number.

Were these students too young to discuss mathematical probability? What evidence did you observe that leads you to believe that students did or did not grasp the difference between mathematical probability and experimental probability?  At what age should probability be discussed?

When asked students were able to recall the definitions to mathematical probability and experimental probability students were able to give appropriate responses. I do feel that they did grasp the material being taught.  I agree with Ms. Kincaid, we should introduce our younger students to probability so that they can develop their skills over time.  We should introduce our students to forms of probability in Kindergarten.  The simple basics such as, “If I drop this rock into water what is the probability it will sink or float?” to begin an introduction of vocabulary and modeled demonstration.

The teacher asked the students, “What can you say about the data we collected as a group?” and “What can you say mathematically?”  How did the phrasing of these two questions affect the students’ reasoning?

When she asked, “What can you say about the data we collected as a group?”, one student said that the data looked like a rocket.  This child told her what the data “looked” like to him.  When Ms. Kincaid asked the same child what it says mathematically it made him think about the data in a different way, he was able to tell her there were more sevens and gave her the reasoning behind his thought.

I feel that the way we word things particularly to children is very important.  Do you feel that you can get totally different answers to the same question just by asking the question in a different way?  

Why did Ms. Kincaid require each group of students to roll the dice thirty-six times?  What are the advantages and disadvantages of rolling this number of times?

6x6=36  this number is a fair amount to roll the dice because it give each number 1-6 a fair chance to be landed upon.  He down side of this is it may be hard to keep track of the number of rolls and students may over roll like the group in the video.

Comment on the collaboration among the students as they conducted the experiment.  Give evidence that students either worked together as a group or worked as individuals.

In each group, students were devising plans to implement their experiment with the dice.  One child discussed what had worked for him in a past group.  I feel the kids worked well together.  When one group realized that they had rolled to many times, they worked together to try to fix their data.

Why do you think Ms. Kincaid assigned roles to each group member?  What effect did this practice have on the students?  
How does assigning roles facilitate collaboration among the group members?

By assigning different roles to each member, it gives students equal responsibility for the work to be done in their group.  Everyone has to pay attention and work together in order for the data to be correctly gathered, recorded and analyzed.

Describe the types of questions that Ms. Kincaid asked the students in the individual groups. How did this questioning further student understanding and learning?

She asked her students if their results of the experiment were the same as their predictions.  She also asked them what turned out different than they thought.  She asks questions to gain insight into their thought process and to learn how they are arriving at their solutions.

Why did Ms. Kincaid let each group decide how to record the data rather than giving groups a recording sheet that was already organized?  When would it be appropriate to give students an organized recording sheet?  Discuss the advantages and disadvantages of allowing students to create their own recording plans.

She let her student pick their own way to organize their data to see if they as a group could chose a way to show the data clearly.  This allowed her to walk around the groups discussing their choices.  One group we saw mad a poor choice and had to reconfigure the data again so that her group could analyze it correctly. By allowing student to choose their own way of recording data, it allows them to visualize the goal and plan for the outcome of the interpretation.  This in turn gets students to critically think about what they are trying to accomplish and plan for their predicted outcome.  It would appropriate to give students an organized recording sheet during assessment of lesson.

For further consideration…

Knowing what you now know about probability concepts in the elementary school, how will you ensure that your students have the background to be successful with these concepts in the middle school?

I feel it is important to introduce young minds to probability concepts.  Students of all ages are touched by probability everyday in one way or another.  I feel it is very plausible to incorporate probability into activities in the classroom such as Science.  Connecting probability to concepts they already know and use will help with understanding.

Everyone loves Bill Nye!