Rorschach Inkblot Test

While studying reflections and reflectional symmetry in 8th grade math, I showed the class images of the Rorschach Inkblot Test. I also showed the test to one of my geometry classes, and they had a blast with it! They were bursting to share their answers. A couple of times students saw the same image without talking about it first. Some of the images they saw:

 

two pigs

two pigs carrying a beetle

Patty Cake explosion

two Asian mean giving high-fives

two men pushing a fat woman down a well

Miley Cyrus looking in the mirror

Two people twerking

 a bat

a moth

 an iguana breathing fire

 two lion fish (in blue)

My First Favorite Proof

I have been struggling with finding good proofs for my geometry classes - ones that aren't obvious and that are within reach of my students' abilities. Then I found this problem on Don Steward's blog.



It fit well with the work we were doing in class about parallel lines and transversals, so I presented it to the students. The directions I gave were:

• Draw two parallel lines (maybe using both sides of your ruler).
• Draw a line that cuts the two parallel lines (called a transversal). Measure the two consecutive interior angles.
• Bisect the angles that the line makes with the two parallel lines. Measure one of each of the bisected angles.

Every student created the same construction, though their transversals all met the parallel lines at different angles.

Then I asked: What is the angle at their intersection? Why? 

I had them think about the problem silently for about 5 minutes. Then we talked. Not many students got it the first time around. I explained it, as well as had other students explain it. Then I gave them the assignment to write 5 sentences explaining why the two bisectors intersect at a right angle. 

The next day, very few people had completed the assignment, presumably because they didn't understand why it happens. Or they didn't know how to communicate it in written word. Or some other reason. So we spent about 15 minutes the second day talking over the problem. I had as many students as I could in that time give an explanation. That way, students could hear explanations from multiple perspectives and different wordings, and they could practice communicating verbally. I have found that just because students understand a problem doesn't mean they can explain it. I tested the waters by asking how many understand it, how many can explain it to someone else, and how many have no clue. There were students in all categories. So I had students keep explaining until they got tired of it or no one else was willing to explain. I also showed them an applet I made in GeoGebra of the problem to make it more dynamic.

This is the very first proof that I felt comfortable presenting my students. I wish I could find more!

Test Taking Environment

Just wanted to share something I've noticed in the classroom. When I turn off a set of lights during a test-taking situation, the room feels calmer. Students seem to prefer a set of lights off when they're taking a test or quiz, and I get the sense that it relaxes them. I originally only did it in one of my eighth grade classes when they asked, but then I started doing it in all of my classes. I ask them if it's okay to turn off a set of lights, and they all say Yes! Some want all the lights out, but I think that would put them to sleep. So I turn a set off, and it instantly feels calmer in there, which is great for taking a test. Of course I wonder if it enables easier cheating, but I don't think it does. I also give multiple versions which I hope prevents a lot of cheating.

I'll let the pictures speak for themselves

 

"Adjacent" angles

 

Camo Day (that's a Ghillie suit)

Heart Fractions

Which Television to Buy?

I talked to my brother about two weeks ago and he told me about this problem he was working on. I used it in two of my geometry classes today, with mixed results. My students still aren't very good with solving problems without knowing the steps in advance, so I had some difficulty getting them to continue to stay interested when they didn't know quite what to do. I still like the problem. Here it is:

TV Problem

There are two types of TV’s: the old tube TV and the new widescreen LCD TV. When you buy a TV, they quote the size by the number of inches from one corner to the opposite corner. The actual height and width of the TVs aren’t quoted at all.

However, you can figure out the length and width of a TV by using the length of the diagonal and the aspect ratio. The aspect ratio is the ratio between the length and width of the TV. For an old tube TV, the aspect ratio is 4:3 (length : width). For a new widescreen LCD TV, the aspect ratio is 16:9 (length : width). So no matter how big the TV is, the ratio will always be the same.

QUESTION: My brother Matt wants to buy a new TV for his arcade cabinet he’s putting in his house. He wants to buy a 19” TV, but isn’t sure whether the old tube TV or the new widescreen TV is going to give him more surface area and therefore more pixels. He’s a video gamer, so more this matters to him. Which type of TV would you recommend and why?

EXTENSION: What percent larger (surface area) is the one screen than the other?

How Many Stars in the Universe?

I found this lesson by Robert Kaplinsky and was excited to use it to teach scientific notation to my eighth graders. I have a class of 13 students and a class of 38 students (with two aides). I wasn't sure if I could pull off the same lesson in both classes, but thought I'd give it a try. I also found this lesson from the Shell Center that I thought would tie in nicely, and an online app called Scale of the Universe.

We do Andrew Stadel's Estimation 180 every day in class, and I thought that the question for today's lesson would substitute nicely for our usual routine. I had students make a high and low estimate and their best estimate. We had some great conversations around astronomy, including whether the size of the universe, how light travels, and how stars are born and die. Some of my students had a lot more knowledge about these things than I knew. I wasn't expecting students to ask how we take into account the fact that stars are being born and dying all the time, or whether we should count the stars that have died, but we still see them because their light is still traveling to us. Or the stars that are exist, but we don't see them because their light hasn't made it to us yet. I also didn't really know how to answer the question about the size of the universe. Many students said it is infinite, but I thought that the universe has a boundary but is expanding. Correct me if I'm wrong.

I had to do some work to get students to think about how we could estimate the number of stars in the universe by multiplying the number of stars in or galaxy by the number of galaxies in our universe. Many students said that galaxies are different sizes and contains different amount of stars, so we wouldn't get a very good estimate. I asked them to consider whether we could come up with an average of the number of stars in each galaxy and use that in our estimate. I don't think every student agreed with this strategy of estimating the number of stars in the universe, and many still thought the task is impossible.

However, once I showed the video on the number of stars in a galaxy and the number of galaxies in the universe and had students work on multiplying the two numbers, they did some great work. We had to review how to write 100 billion and 400 billion in decimal notation. Many students used their phones and came up with 4e+22. Some got 2x10^22 on their calculators. Very few thought to add the zeros on 100 billion and 400 billion. I had students write the different representations on the board, including 4 followed by 22 zeros. We talked about what 4e+22 means because I think it's important for students to be able to interpret what their calculator is telling them. At this point, a student suddenly raised her and said, This is sci--- sci--- scientific notation. I was so excited that someone recognized it!

After talking about the solution, we practiced writing numbers in scientific notation. Then students worked on a modified activity from the Shell Center. I turned their activity that has students matching cards into a worksheet because I didn't think my classes could handle the matching activity, especially the class of 38 students. In the smaller class we had time for students to match the objects to the measurements, but in the larger class we're saving it for Monday. 

After doing the worksheet, I showed the Scale of Our Universe. It's really great for this lesson because it shows the scale in scientific notation. It also peaks kids' curiosity. I started by zooming way out to see galaxies and other objects in the universe, and then I zoomed way in. Kids had a lot of questions and thoughts, which was great to see. 

I was surprised that this lesson worked well for both of my classes, even with how different they are. Some students wanted to know why we were learning about science in math class. I hope they see that math is not a self-contained subject, but it touches on many other subjects. That's something I need to work on expressing and demonstrating to students.

Here's the powerpoint I created to present the lesson, as well as the worksheet I created from the Shell Center activity, in DropBox.