Tuesday, January 31, 2017

Similar Figures Academic Vocabulary @von_Oy

This post was inspired by Suzanne blogging about the precise language she wants by not naming a method "difference of squares" when it could pigeonhole students into a procedure without knowing really why it works. I can also empathize with teaching students the definition of a function through examples. It can be disheartening when the only answer you get is it passes the vertical line test. I like when students on their assessments would draw the vertical line and the points it passes through, labeling those as the input that has 2 different outputs.

I was concerned about how my students were processing the attributes of similar figures. CPM has a great lesson where there is an original quadrilateral with a bunch of other shapes that are similar except for 2 of them. One is horizontally stretched, while the other is vertically stretched. As students compare the shapes, they can see that the corresponding angles of the non-similar shapes are clearly different measurements. Also, corresponding sides that should be parallel are intersecting. They also should be able to count how many times a corresponding side can fit into another similar shapes corresponding side to figure out the scale factor.

Since I was concerned with students understanding, I gave students a post it note, 5 or 6 minutes, and the following prompt:

How do you know if two shapes are similar? When does a dilation make the shape bigger? Smaller?

A complete answer would mention all of the following words:

size, scale factor, angles, congruent, corresponding, parallel

Feel free to add a labeled diagram to show your thinking.

I got a variety of responses. Some students included the words, but not in the correct context.

Surprisingly, many students said that similar figures are congruent. To this I commented, "always?" It was hard for students to use the word corresponding and angles in the correct way. 

We don't take notes that often, but synthesized their thinking. Students realized a similar figure that was enlarged would have a scale factor greater than 1, a shape that shrunk would be a fraction between 0 and 1, and a congruent shape would have a scale factor of 1.

This work was after dilating on the coordinate plane, and this was nailing down all the academic language.

@cpmmath 6.1.2 Rigid Transformations

If you haven't read my blog post about using Google drive, look for the one titled 5 practices. The great benefit of taking photos of students work throughout the period is you can then sequence how they are presented during the closure. Also, if 2nd period doesn't have an alternative method to a problem, but 1st period did, I can show their work and credit the student from the previous class.

It also makes it much easier to blog about because they are all stored on my Google drive or still in my camera roll.

As you can see below, students had to follow a set of instructions and transform a triangle using each of the 3 rigid transformations. A common mistake is translating the bottom right vertex and then drawing it as a different vertex such as the bottom left. Students realize they need to process that it's that corresponding point and draw the rest of the figure or translate the remaining points.

Students can visually reflect, but have a hard time describing what operation they are completing on which coordinate, and which coordinate remains the same.

Finally, I used cut up parchment paper and gave each student a piece to use on their first 2 attempts on their skill assessment and anytime during classwork. The final attempt I'm not sure if I'll allow students to use the wax or patty paper.

@cpmmath 6.1.1 Transformations Intro: Key Lock Puzzles

CPM offers a pretty cool interface to informal language about rigid transformations. It's worth a look. Students worked in partners so that they could switch off. That way they could get feedback from their partner. It made students much more engaged. Then they copied down their solving steps after solving each puzzle.

Above, a student used 2 reflections for each to solve it quickly. I was impressed.
Above are the posters I made a few years ago and had a student color and decorate. It helps students see the relationship between the informal language of sliding, turning and flipping to translating, rotating, and reflecting.

As you can see, we reviewed 3 homework problems from the night before where they practicing solving a system using substitution (CPM calls it Equal Values method), solving for y, and a word problem that could be solved using a system of equations.

y=mx+b Learning Log as Exit Ticket

I gave students about 8 or 9 minutes to graph the linear equation y=-2x+3 and answer questions in the learning log prompt linked below. I looked at all of them from each of my 3 classes, and calculated how many students successfully graphed a linear equation. Since it was an exit ticket formative assessment, I didn't grade it. I only highlighted directions they didn't follow or mistake points I saw. Then a check mark with successfully graphing and checkmark for ideas I agreed with. The statistics, which I discussed with all my classes, are below:

1*: 24/27 or 89%
2*: 19/24 or 79%
5*: 20/32 or 63%

I discussed with my classes that they're all made up of different levels of students but I think that the biggest class has the most frequency of off topic conversations. Although it's a high energy level coming back from lunch, I talk to all classes that the time of day shouldn't be an excuse and it's up to every individual to be responsible for their own choices.

After passing it out I had students contribute to a whole class google doc of notes on agreed upon answers to the learning log questions. That file is available here. The biggest mistake seemed to be students ignoring the negative sign on the growth, or not having an idea about where the y-intercept was or how to find a second point.

Monday, January 23, 2017

Part 1: My first @desmos 3D print

Our education technology specialist took a job at a local charter school and asked me if I wanted to keep the 3D printer in my classroom. Of course I said yes, and it has been collecting dust in my closet. After seeing this tweet from John Stevens, co-author of the Classroom Chef, I was inspired to do some research and try setting it up.
Mind you, this tweet came after he showed some amazing student work samples. If you follow the link there's a Google doc that is a sample of the student instructions, which I followed.

Instead of a keychain, I figured students could create a name plate using linear equations and anything else they cared to figure out. I came up with my sample graph:
I exported the file after prepping it (John describes the instructions, but it's basically making all equations black, hiding axes and gridlines, and exporting the file as .png file and then converting that to another format to be edited in Tinkercad.

I read all the directions to setup the printer, and started leveling the build plate. To my dismay, I kept running into "Fatal temperature error" and couldn't get past that screen. I need to do some more research to troubleshoot it, and vow to persevere.

In the meantime, I shared my troubles with a class, and one student volunteered that the local library had a 3D printer anyone could use.

I called them and found out that on Mondays and Thursdays I could make a 2 hour appointment from 4 to 6 to get an introduction to 3D printing and use their Ultimaker 3D printer. I went today and learned a lot.

I downloaded my .stl file from my Google Drive and opened it with Cura. After adjusting the letters of my name to be 4 millimeters rather than 4, printing commenced.

One drawback, is that the print is not on a rectangular name plate background. But, the library employee told me next time I could try an embossed look where I could take a rectangular prism and make the name be hollow or an indentation in the prism. That seems like it would look cool too!

Here is the first layer...

And after carefully scraping it off the build plate, I had my name. Any ideas on what type of material to mount it to? probably use some nice glue.

Next steps:

  1. Troubleshoot and fix the school's 3D printer.
  2. Plan time before the end of the year to have students make a rough draft of their name on graph paper, then recreate in desmos.
  3. Final product would be an .stl file. If I was super awesome, I'd print them all. That is a possibility. Worst case scenario: students take the file and do what I did and get it printed at the library.
Any feedback is greatly appreciated. Big thanks to John Stevens for the superb instructions and inspiration.

Part 2 is continued here...

Tuesday, January 10, 2017

NCTM Blog Post #4: Replacing a Textbook Lesson

Sunday, January 1, 2017

Border problem (@youcubed @joboaler)

I also saw a video of Cathy Humphreys demonstrating this task.

It all starts with the image below (I found it by googling "Math border problems," make sure you don't forget the plural form and math or you'll get a bunch of articles on immigration policy). This is a picture of a square with a side length of 10. The border has been shaded orange. How many orange squares do you see without counting one by one? 
And then the magic happens...

In the particular class pictured below, students got answers of 37, 36, or no answer at all. I put 40 up their because I think some thought that but were too afraid to share it.

As you can see, there are 5 distinct methods that students in this cass used. The most common I believe is the one in the bottom left where the multiply 10 by 4 and subtract 4 because the corners were overlapping. I made sure students understood that it was subtracted because each corner was counted twice. This leads to 4x-4.

One student worked around that by saying there's a spiral of 9 square lengths around the border. It's hard to draw, but you can see it in the upper left. Therefore they got 9*4, which leads to 4(x-1).

Another said there were 10 squares on the top and 10 on the bottom. Therefore the left and right sides had 8 each, so 10*2+8*2. We didn't write that one as an algebraic expression.

Karin said she saw the square in each corner for a total of 4 corner squares, and 2 rows of 8 and 2 columns of 8.

Finally, Sebastian saw it similar to the visual pattern from yesterday, as a square with a total of 10^2 or 100 squares subtracted by the inside white square which is 8^2 or 64. 100-64 gets you 36. This leads to x^2-(x-2)^2.

I think I'm going to revisit this warm-up again, but slightly different as a Contemplate then Calculate instructional routine that I saw David Wees present at a Global Math department webinar. I can't find it right now but I'll post a link once I ask him.