Sunday, February 19, 2017

#clotheslinemath Linear Equations

I am going to teach this lesson on Tuesday to all 3 of my Common Core 8 classes. Any feedback or suggestions prior to that are appreciated. A reflection of how it unfolded to follow. I wanted all the images in one place for my colleagues to use and for anyone else to try. All credit goes to Andrew Stadel and Chris Shores for the lesson, I just added a small extension.

UPDATE: (2/21/17)

I noticed students drew the number line cards really big after looking at the google slides. For each period after I made sure students knew they needed room on their paper for 2 clothesline and 2 graphs to be pasted on.

I had students think for 3 minutes how the 6 parameter cards could be placed on the lower clothesline. I demonstrated how if a card lined up with a number, they would be equal or equivalent. I also demonstrated how you could clip a card underneath another using the clothespin.

I then asked students to place the cards individually with 3 minutes of silent independent work time. Then I timed them for another 3 minutes of taking turns sharing out their ideas to their group, to gain confidence in sharing their idea with the class.

With 6 parameter cards, I said that we would need 6 different students put up 6 different cards and give a statement to the class about what they did and why they chose to put it there. I had students vote using their thumbs whether they agreed or disagreed with a person's statement. I asked students put their thumb sideways if they didn't whether the person was right or wrong. In first period students immediately went for the growth of line 2, or m2, was 0.

Students seemed to see that the blue line's growth had to be 2, because it was the steepest, and it was increasing from left to right so it had to be a positive number, so they said it couldn't be 1 and picked 2. Some students also reasoned that the red line must have a smaller slope but negative because it was decreasing slowly.

One student talked about how the y-intercept of the red line had to be 2, because the blue line was also the same distance from zero being at -2. I asked the students what we call distance from zero, and some remembered that was the absolute value.

After students had placed cards m1 to m3 and b1 to b3, I handed out the desmos graph. I asked them to paste it in their notebook and then see if they agreed with how the clothesline was from the last image. Students noticed that they appeared the same, and the y-intercepts were definitely the same.

I pointed out when students were drawing slope triangles between lattice points, but am not sure if all students correctly did so as I walked around the room.

Students had a bit of trouble with the scale of the graph, with each square being 0.5. This lead them to see that the vertical growth was 1.5, and the horizontal growth was 1. I asked them what the slope triangle was in grid squares, which gave 3/2. Students reasoned that must be between 1 and 2.

I did not get to the part of putting m4 under -1, and b4 under 1. If I had more time I would have. My colleague said his class also had trouble substituting the new m values into the equation y=mx+b to try to confirm that the lines would intersect algebraically.

It was nice to see that after some students noticed the lines would intersect, that they could prove it graphically. In each class a student hung x and y under the respective coordinates for the intersection point.

Some students thought there might need to be 3 sets of x and y coordinates since there were 3 different lines.

Overall I loved the activity. It had a sense of mystery. Also, having a graph with grid lines achieved my goal of practicing calculating slope. I am still concerned how many students left the class with a solid handle on it. I liked how I made sure they had some independent think time before sharing with their group because if it's straight to group it tends to be the same people sharing. Students also liked the chance of going up to the board with the whole class watching, while others were nudged towards giving their reasons for agreeing with someone from their seat.

Wednesday, February 1, 2017

Hot Seat Study team Strategy (@cpmmath Transformations)

This past summer I had the pleasure of applying for and being accepted to CPM's TRC research group in Sacramento. After brainstorming ideas, my group was formed with Aristotle and Erica. The great part was that we all taught 8th grade and were concerned about students doing their homework, specifically the Review and Preview. We decided we would focus on taking breaks from the lessons once in awhile to focus class time on individual practice as well as group work on review preview problems. We are also trying to show students the benefits of mixed spaced practice compared to a bunch of practice problems of the same type. Another concern was if students don't do the homework, they are getting very little individual practice to self-assess. It's also important to make homework worth part of the grade, but not a big part. My colleague and I settled on 15%.

Part of the work we did was work on how to check it, and with some collaboration with Erica and some ideas from Dylan Wiliam's Embedded Formative Assessment, students have a quarter sheet of paper to check 5 assignments. I can share this if anyone wants it. Most days students self check. Others they peer check. Once in awhile, a whole group switches their work with another group so they get a different group's opinion of their work. Also, if and when students lose the checking sheet, if they want credit, they have to show it to me personally, and of course I look it over a lot more closely!

Participation quizzes have been helpful, as well as pair checks, which I detailed in an NCTM blog post recently. After a recent Skype meeting, Erica reminded me and suggested I use the Hot Seat strategy. The rules are detailed below in the Google slides, as well as the problems students focused on. I had never used this strategy before as I was concerned about the competitive feel, students feeling inferior, and the timing of the activity. I will address those concerns in this post.




So, as described in the first slide above, 1 student brings their seat to the back of the room, and works individually with no help. To make this quick, I said resource manager (assigned team roles based on the alphabetical order of their first name in their group). If they get it right, they earn their team 2 points. If everyone in their group gets it right with all work shown, the team gets 1 point. I kept track of the points, and students were very honest. It was instant formative assessment.

As you can see, we are finishing up Chapter 6 which has closure problems. Instead of saying students, work on these problems. I mixed up the order and focused it on transformations.

For the first problem, students only needed 3 minutes to figure out the scale factor. Some asked which is the original and which is the new figure? I told them they could decide. I also said to not use calculators.
Students in the back row are on the hot seat.
After keeping the score, facilitators were up next. I gave them 7 to 8 minutes for this because they had to start with one transformation, and then finish the steps. I would circulate asking questions and giving advice to students who needed it.

I asked students for feedback informally in each class. A student that doodles and doesn't engage with her group much liked that it was more individual based and the timing made her focus more on finishing the problem. A struggling student said she didn't like it because she couldn't get help from her group when on the hot seat. Students were much more quickly to adjust their paper and body to point to their work while verbally explaining a concept.
From my classroom door, you can see everyone focusing on the problem, book, or glancing at the board for the problem.
Also, when creating the graph, the class got dead silent in concentration, and then you could start hearing whispers in the groups when they were comparing how they rotated or reflected a shape if their triangles looked the same.

I was also able to implement the 5 practices using my Google Drive app by taking photos of student work to share when going over the answer and awarding the points.

For the second problem for list 1, it asks for a 180 degree rotation. A student asked which direction clockwise or counterclockwise? Another student answered, "it doesn't matter it will look the same!"

Here you can see that same triangle successfully reflected over the y-axis. Then student volunteers said you could then reflect it over the x axis and then translate it up 1 and over 2.

This photo shows the same problem, but also the second one where after graphing a triangle students were asked to reflect it over the y axis. Separately they were asked to translate it, and describe what happened to the coordinates.

Here is 2nd period's scoreboard.

Numbered steps that were easy to read.

Here you can see the horizontal, then vertical translation. You can also see the correct reflection over the y-axis. I asked students that made mistakes reflecting, "how far away is the shape from the axis you are trying to reflect it over?" This moved them in the correct direction.

Here you can see how students were able to show expressions for what happens to the coordinates when moving 4 to the right and down 6. They would have to add 4 to the x coordinate, and subtract 6 from the y coordinate.

Also, for reflecting, they had to multiply the x coordinate by -1, and leave the y coordinate alone.

My colleague and his students really enjoyed the hot seat activity as well. I picked a wednesday to try it since it was a shorter period and I knew they wouldn't be able to finish the 6.2.6 lesson during a short period.

Unfortunately, we only got through 3 rounds, but a LOT of learning happened, and these were dense problems. I had a short warm-up, reviewed 2 homework problems briefly, passed back a test that I went over, and then did the Hot Seat. The resource managers didn't get a chance to be on the hot seat, but I'm definitely going to do this again with a goal of students being on the hot seat more than once.

At the beginning, I told students that the goal of this activity was to see where you are at individually with their understanding, and to also appreciate the PRIVILEGE of working cooperatively in a group. I am very confident that students will be working much harder together tomorrow when working on a problem solving task about scale factors between models and real life objects.

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

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.