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.