Module Fourteen

Textbook

Question 2. A general instructional plan for measurement has three steps. Explain how the type of activity used at each step accomplishes the instructional goal.

Step One-Making Comparisons; by making comparisons based on the attribute, longer/shorter, heavier/lighter the students will be able to see that the objects or lengths are different based on the same attribute.

Step Two-Using Models of Measuring Units; using models, for example, index cards to cover the table, allows the students to see that the unit of measurement, the index card, uses how many to fill it. This will enable students to make comparisons of the attributes. Using nonstandard units of measure to start with will help the students understand the use of standard units.

Step Three-Using Measuring Instruments-using common measurement tools with understanding. Allowing the students to construct simple measuring instruments will allow them to better understand how an instrument measures. 

Each of these steps builds upon the students' comprehension of measurement, by enabling them to work with making comparisons and then moving on to units of non-standard measurement they are able to better grasp the concept.  The steps of the instructional goals are to first get familiar with the unit, can select an appropriate unit, and know how the units relate to each other. By using the instructional plan, we are paving the pathway for the students to build the needed understanding to master units of measurements. 

Question 3. Four reasons were offered for using nonstandard units instead of standard units in instructional activities. Which of these seems most important to you, and why?

While all the reasons are important, I think that the one that sticks out as most important to me is that nonstandard units provide a sound rationale for using standard units. As students begin using the nonstandard units and find different results, it goes to show that the measurements that they are using are different. For example, they use their feet to measure the size of the classroom, each student is going to come up with a slightly different answer. When the students are at this point of understanding, they will be able to grasp using a standard measurement tools usefulness a little more. 

Circumference and Diameter

• Describe Ms. Scrivner's techniques for letting students explore the relationship between circumference and diameter. What other techniques could you use?

Ms. Scrivner has the students use hand gestures to show the meaning of the word circumference and diameter. She then has the students work together to measure circular items around the room to find both the circumference and diameter. She also makes connections in the vocabulary words of "circle' and "circumference." I think the hands-on activity is a great way to allow the students to understand the measurements that they are collecting. Maybe having the students use nonstandard units of measurement would help, but I am not sure what point in their understanding of measurements they are at. Aubrey, what do you think?

• 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.

I don't remember doing a task like this in my elementary grade levels, but I would hope that my teacher would have done a fun, hands-on task like this one in my class. 

• How did Ms. Scrivner have students develop ownership in the mathematical task in this lesson?

By allowing the students to do this activity hands-on, they are able to do the measuring themselves and be accountable for the results they came up with. 

• How can student's understanding be assessed with this task?

By getting group responses to the activity to share with the class. The teacher was also able to walk around the room and monitor what the students were doing and the measurement they did. It shows her questioning the one group on the diameter of the trash can, and then explaining how the shape of the head is not a circle. 

For Further Consideration...

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

1. The use of math manipulatives inside the classroom, I think having objects available for the students help them work and figure out what we are asking of them.

2. Incorporating the use of geometric shapes from other countries and tessellations. 

3. Online math games that provide useful learning instructions that will enhance the learning that happened in the classroom. 

4. Using the Mira tool, we've had so much fun with this little tool, I will learn more about how to incorporate it into the classroom.

5. The Annenberg website and activities, I have learned so much more from the activities that were found on this website.

Indeed the list goes on and on from last semester to this semester, I almost cannot believe this is the last module!!! What about you Aubrey, do we have similar ideas to take with us? 

Module Thirteen

Measurement Misconceptions 

Why do you think the students are having difficulty? 

While the students have seemed to grasp the idea of how to use the ruler regarding putting the end of the object at the end of the ruler they are not sure as to which unit they measure with and the attributes that differ on the ruler itself; inches versus centimeters.

What misunderstandings are they demonstrating? 

One of the most significant misunderstandings they are having is counting the zero as a one on the ruler. Henry is counting every mark on the ruler as an inch, instead of counting the larger lines.

Have you witnessed any students experiencing some of these same difficulties? 

I have not during this FE placement or my last placement. This placement I am in currently is working on adding, subtracting and base-ten facts now. I'm not sure if it has been covered already, but I don't think it has. Even in my placement the last semester we never had measurement as part of our math lessons. 

What types of activities could you implement that would help these children?

I'm not sure what activities I could implement that would help the children, I guess worksheets or more hands-on lessons would be beneficial to building the knowledge for the students understanding. 

TCM Article – Rulers 

The first idea would be to make sure I have a set of those different color rulers, I think that using those tools helped the students start to grasp the concept of measurement. I cannot say that anything changed my thinking because I haven't had much experience with measurement aside from my children's homework on occasion. I was surprised at how quick students made the connection to the color of the ruler to each of the objects that needed to be measured. Using spatial concepts helped the students understand that using a ruler not only measured quantity but also space. Using these types of rulers will allow the students to grasp the concept of units without the label, and create an avenue to deeper understandings. Many children struggle with the number of lines on a ruler and knowing which unit is which and how to accurately count on the ruler itself. 

Aubrey, would you use this lesson if your students needed more help in measurements?

Angles Video and Case Studies 

The children in the video understand that creating two straight lines that have a point of connection make an angle in the space in-between. Many of them were able to understand how to demonstrate an angle with their hands or arms, and some were given a chance to write their thinking down on paper. One of the ideas that they need to learn more about is whether or not an angle can be formed with curvy lines. The one girl in the video thought not but then when the teacher questioned her thought process she retracted her statement and gave a new example. 

1. In Nadia’s case 14 (lines 151-158), Martha talks about a triangle as having two angles. What might she be thinking? 

Martha is missing the third angle because when we draw representations of angles, we commonly draw them in this fashion "< or >" so to Martha she does realize that the ABC angle is also an angle. It goes back to our previous modules ideas on misconceptions of shapes. 

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? 

Alana realizes that the lines are at an angle and not necessary that the line formed an angle. I feel like I am missing the connection that you are asking for, but all I can think of is the fact that she recognized the angle of a line versus the angle of two rays. Aubrey, what do you think?

3. In Lucy’s case 15 (line251), Ron suggests that a certain angle “can be both less than 90˚ and more than 90˚.” Explain what he is thinking. 

When we look at an angle, we often look at it from the viewpoint that it is given to us in, with the skater most people would see the angle as starting at 90 degrees and then getting smaller as the skater bends down more. However, as Sarah points out the angle could also be from the standing position to the bend over position. Within many angles, there is an angle on the flip side (or turn) that helps create the other part of the angle. 

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 angles and what ideas you would want that child to consider next. 

Chad- Might have an understanding that the lengths of the rays can be different sizes, but when reading his comparison it makes me think that he is talking about the size of angles and that they can be big or small. I think the next thing to consider would be the names of the angles and how they are formed. 

Cindy- knows the different ways an angle can be facing, whether it opens to the left, right, bottom or top it is still an angle. I think an important next step is introducing the sizes of angles and the names the angles go by. 

Nancy- has the idea of size in her angle thinking, but we no idea if she has any other understanding other than angles being pointed. Having her describe the difference between acute and obtuse would be the next step and have her demonstrate her understanding of the orientation of the angle. 

Crissy- gets that all angles have a degree to them, I think it is important to mention that she understands that obtuse angles are more than 90 degrees. Just like the teacher said though I would question her idea of 90 degrees for the right angle to make sure she knows what she is referring too. Chelsea- thinks she knows what constitutes a right angle. However, she has started to create right angles in most all of the angles she diagramed. I would have her show me how the shape can have the right angles she has given it. Giving her the opportunity to explain her reasoning first is crucial to her understanding. 

5. In Sandra’s case16 (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? 

Casey only sees the square shape, it is like tunnel vision, once you see it it is hard to see anything else. But as the articles continue, he begins to grasp the concept that you can turn the shape to get it to match the sides. Tracing the shapes and rereading the worksheet helped him overcome his "tunnel" vision. 

How Wedge you Teach? 

I will take the knowledge that using inquiry-based approach using a hands-on activity before using a traditional approach is helpful to the students. They are often able to make deeper connections with the inquiry-based learning than traditional learning methods. I was surprised that taking a simple circle and creating a "wedge" by folding it prompted such great responses to the question of how would you name the angle without using the terms 90 degrees, 180 degrees and so forth. Students can have an issue naming the angles between the common angles, for example, 35 degrees is still an angle just not one they would think would be correct. I remember a lot from my math classes at CCCC so I am not sure of any misconceptions I have now, ask me about 2 years ago and I would have had some. :) Aubrey, what misconceptions do you have?

Exploring Angles with Pattern Blocks  

Green Triangle- Each angle is 60 degrees, I know this because the inside of the triangle adds up to the sum of 180 degrees, since this triangle has no right angle it is safe to assume that it is an equalateral triangle. 

Blue Rhombus- We know that 2 of the angles are acute and 2 are obtuse. The 2 acute angles are 60 degrees each, and the 2 obtuse angles are 120 degrees each. The total degree of the angles that are adjacent to each other can only total 180 degrees.  

Red Trapezoid- Same as the blue rhombus, the 2 acute angles are 60 degrees, and the 2 obtuse angles are 120 degrees. By drawing a line to create 2 right angles within the trapezoid, you can make smaller triangles which have a total of 180 degrees inside. 

Tan Rhombus-This one got me at first, but the same concept is applied create a smaller triangle within the shape. The 2 acute angles are 30 degrees, and the 2 obtuse angles are 150 degrees, which makes the angles on one side equal 180 degrees.

Yellow Hexagon- Each angle is 120 degrees, to be honest, I remembered from the readings :) 

For Further Discussion 

My father doesn't measure when he is cooking, growing up in a family restaurant business I would watch him cook in the kitchen, and he would always seem to know just the right amount of spices and seasonings to add to the dish he was cooking. Even though he is a "chef" at home for just the family he still cooks like he did back in the day, if I ask him for a recipe he has a hard time telling me the measured amount of ingredients to use and will have to refer me to a website that has a similar recipe to what he knows by heart. 

Another nonstandard measurement is "just a minute" which could truly mean a minute or less or when I get a second. Sometimes I am busy but my children don't always understand that and they will ask for me to do something over and over again, so my "just a minute" give me a little more time to finish the task that I am working on at the moment. 

Module Twelve

Coordinate Grids

http://mathsfirst.massey.ac.nz/Algebra/CoordSystems/Coordinates2D.htm

http://www.beaconlearningcenter.com/weblessons/GridGraph/default.htm

http://www.shodor.org/interactivate/activities/MazeGame/

http://www.learnalberta.ca/content/mesg/html/math6web/index.html?page=lessons&lesson=m6lessonshell18.swf

https://www.funbrain.com/games/whats-the-point

http://mrnussbaum.com/stockshelves

http://www.math.com/school/subject2/S2U4Quiz.html

http://www.math.com/school/subject3/practice/S3U1L2/S3U1L2Pract.html

I decided to visit all of the websites. However, I was not able to access three of them. Because I want to teach kindergarten, I don't plan on using any of these in my future classroom, but if I am not lucky to get a kindergarten job and I teach an older age, I would use websites from shodor.com and math.com. The activities would be engaging for the students for a certain amount of time, and I think they could reinforce the teaching that the teacher has done. Some of the issues of using these types of technologies are that they are the same unless the programmer makes changes the games will be the same most of the time. But like I mentioned before I think these could be used as a reinforcement to the teachings of the teacher. Aubrey, did you have luck on getting the first two links to work or the graphing applet? I couldn't access those.

Miras, Reflections, and the Kaleidoscopes

I have not used a mira before, but I have to say that it is a pretty cool device. I did find the task difficult if the shapes or letters were too close together. I was having a tough time drawing the lines of symmetry across the middle of the letter instead of just above or below. This is what I came up with for the lines of symmetry in the letters given, I think if a different font were used, H would have 2 lines of symmetry and E would have 1 line. I found the words SAY, WOW, TOMB, and ACT for the challenge of finding a word that has symmetry. Aubrey, how do you think font would change the results? Did this hurt your eyes at all, my astigmatism was going a little crazy.

No Lines of Symmetry	1 Line of Symmetry	2 Lines of Symmetry	3+ Lines of Symmetry
E, F, G, J, K, L, N, P, Q, R, S, Z	A, B, C, D, H, I, M, T, U, V, W, Y	X	O

I have a better understanding of the differences between reflections and transformations along with rotations. I enjoyed the article and the learning and hands-on activity the students did. I will definitely bring the idea of hands-on learning into my future classroom, and I think it would be fun to introduce the students to the ideas of reflections and mirror images. 

Case Studies

1. The students have an understanding that people are bigger or smaller than each other. They believe that the box is big, mostly because it is bigger than most of them. They also understand that stacking things can be a way to measure something when some of the children recommended using a "measuring tape" to measure they already knew that there is something already made that will tell home tall or big something is. 
2. In Rosemarie's case the teacher first thought that the students had the correct idea on the measurement of feet concerning the size of the foot, but once the children performed the task on their own, they had a difficult time answering the question of whose foot was the biggest. At the beginning of Dolores's case, the children had the same struggle of measurement and the idea that a bigger foot meant fewer steps. With the guidance and homework assignment that the teacher gave the students, it seems that most of the students were able to finally make the connection.
3. Chelsea is understanding the relation to the foot size of Tyler and Chrissy but takes notice that the other measurements should also be the same measurement. Henry has also noticed the difference that the children have given in the measuring of the different lines. They both seem to have an understanding that if the results are similar in measurement on one line, they should be the same for the other lines. These observations relate to Sandra's class because they have the same discrepancy, the students believe that the bigger feet mean bigger numbers, and that is the opposite thought process. I was a bit surprised by the 7th-grade responses, to be honest, I thought that they would have a better understanding of size. What about you Aubrey, did this particular case study surprise you like it did me?
4. In both of the case's the students had discovered that using different tools for measurement is important depending on the size of the item. For the 2nd-grade class, some of the students thought that placing a finger on the mark would be okay because "nothing is perfect" however other students disagreed with this and gave other ways to accurately measure the length by "flipping" the ruler if it was not long enough. Some of these students also understood that the smaller marks between the inch marks meant "1/2". The 4th-grade class seemed to understand the precision of measurement, but a few students struggled with the very beginning space on the rulers, but they all seemed to have grasped the idea of the ruler starting at "0" and moving up. 
5. The unit of measurement and geometry should be introduced in the early school-aged years because as the students progress through the grades what they learn will continue to grow, expand and change in understanding. The kindergarten students are well on their way of building knowledge of measurements, but they still have a lot to learn. They will need to learn that the size of the unit used for measuring gives way to the number of units used, they need to learn measuring tools, and much more. By reviewing the case's we can see that this concept is still needing to be learned in the older grades, so most of these topics will still warrant discussion even if just for a refresher. 

For Further Discussion...

Teaching students what the definitions and terms within the realm of geometry are important, but we must introduce and show our students learning through doing. If we wait until the students know all the terms of geometry, we would never be able to teach them the intricate details that geometry holds. This misconception of the students knowing the definitions is what is holding our students back. Engaging the students in activities that allow them to learn and explore for themselves will help the students learn how the definition fits the word, it creates a long-lasting "bond" to the knowledge that they can learn. Waiting to introduce things will just hurt the students as they progress in school, open their minds to the concepts now so they can build and make connections. 

Geometry is much more than I had originally posted about, the key ideas I spoke about included shapes, perimeter, area, and knowing the different angles. I hadn't realized that spatial knowledge was such an integral part of geometry, or that coordinate grids where included. After discussing all the information we have the last few weeks, I have learned much more about geometry, and I am more aware of the content knowledge, and a little less scared of the word geometry. 

Module Eleven

Pentomino Activities

Being that I want to teach kindergarten, I am not really sure that I would have these types of activities within my classroom, but I did find some interesting websites that students can go to and play with pentominoes on their own. 

https://teachbesideme.com/pentomino-blocks-math-game/

http://www.mathsphere.co.uk/fun/pents/pents.html

http://www.transum.org/Maths/Activity/Jigsaw/Pentominoes.asp

http://pbskids.org/cyberchase/math-games/cant-wait-tessellate/

http://www.shodor.org/interactivate/activities/Tessellate/

https://www.mathsisfun.com/geometry/tessellation-artist.html

During the PowerPoint, I did just fine with learning objectives that were given. But I don't struggle with finding area or perimeter. I didn't have any frustrations until it came to creating the Pentominoes with spatial sense, last week I said I thought I had good spatial sense, this activity makes me rethink my words. How about you Aubrey, did you have any problems?

Pentomino Narrow Passage

I was not all that successful with this activity, the longest I was able to get was 17. I will keep trying though!

Tessellating T-Shirts 

While I have not done tessellating t-shirts before I have worked with students while they were doing tessellations. This article brings to light the fun that can be had while also teaching students of many grade levels important aspects of geometry and the vocabulary associated with the learning. This article points out how preservice teachers can take the learning that they are doing and creates a fun learning for the students they will one day teach. The idea of working with a larger tessellation before bringing the tessellation to the shirt is helpful to the students so they can see how the creation works before doing it. Tessellating is creating an image that can repeat itself without any gapping or overlapping of the image. 

Tangram Discoveries

• Which polygon has the greatest perimeter? …the least perimeter? How do you know?
• The triangle, parallelogram, and trapezoid all have a perimeter of 14 while the square and the rectangle have a perimeter of 12. When you take the larger triangle piece, you can stack the two smaller triangles on it to create the larger triangle. I gave each side a number to represent it and then calculated the outside totals. The larger triangle had sides of 3 and a base of 4 while the smaller triangles had sides of 2 and a base of 3. 
•  Which polygon has the greatest area? …the least area? How do you know? 
• The area for each of these shapes should be the same, as they are each created with the same number of triangles. 

Aubrey, did you get the same results? Do you agree with me, or did I think through this wrong?

Ordering Rectangles 

1. Take the seven rectangles and lay them out in front of you. Look at their perimeters. Do not do any measuring; just look. What are your first hunches? Which rectangle do you think has the smallest perimeter? The largest perimeter? Move the rectangles around until you have ordered them from the one with the smallest perimeter to the one with the largest perimeter. Record your order. 

As I moved the pieces around I decided that D and E are the smallest, A and B are next, followed by F and G, and finally, C having the largest perimeter.

2. Now look at the rectangles and consider their areas. What are your first hunches? Which rectangle has the smallest area? The largest area? Again, without doing any measuring, order the rectangles from the one with the smallest area to the one with the largest area. Record your order. 

I think C has the smallest area, then D, B, F, E, A, and G with the largest area.

3. Now, by comparing directly or using any available materials (color tiles are always useful), order the rectangles by perimeter. How did your estimated order compare with the actual order? What strategy did you use to compare perimeters? 

I decided to use color tiles to check my estimations from the previous questions. I found that E and D have the same perimeter with 14, then A, B, and G have the same perimeter of 16 and C and F have the same perimeter of 18.  In my previous assumptions, I thought that C had the largest perimeter but was wrong. I was correct that D and E had the smallest perimeter. 

4. By comparing directly or using any available materials (again…color tiles), order the rectangles by area. How did your estimated order compare with the actual order? What strategy did you use to compare areas? 

I found that C had the smallest area of 8, which I had assumed in the previous question, D had an area of 10, B and E an area of 12,  F and area of 14, A an area of 15 and G an area of 16. I had also guessed that the largest area was with G. Using the color tiles was rather simple and helped to see the difference between the sizes. 

5. What ideas about perimeter, about area, or about measuring did these activities help you to see? What questions arose as you did this work? What have you figured out? What are you still wondering about? 

These activities helped me see that my visual thinking of the differences in the sizes of the rectangles was pretty accurate. However, using the color tiles is what really set the activity apart for me. I have done a lot of area and perimeter with my oldest daughter recently in helping her learn the area of rectangular prisms and pyramids and triangular prisms and pyramids, so I have a hand up on this learning. :) 

Aubrey, what about you, did you have any difficulty putting them in the correct order? 

For Further Discussion

I decided to look up Native American rugs first and came across a site called indians.org. Here it talks about the symmetrical balance that the rug has when it is viewed, an interesting piece of information is that not only is the rug symmetrical from left to right but also top to bottom. Many of the art forms in jewelry and pottery are repeating shapes, these shapes can include triangles, rectangles, and squares just to name a few. Introducing these artifacts and showing the geometric shapes and patterns could allowing students to create their own "works of art".