Day 23-24: Similarity Day 2 + Trig

Busy day yesterday and wasn’t able to post – I’ll combine yesterday with today:

We finished the similarity unit by talking about perimeter, area, and volume ratios. I started class with an investigation on area and perimeter of similar rectangles. The sheet is downloadable below. I first started with some students giving guesses for how the ratio of perimeters and areas changed with the similarity ratio and If students finished early, I asked extension questions like, “Will this work for all rectangles? Will this work for all shapes? How is the volume ratio affected?”  I liked that students were able to confirm or disprove their guess by this investigation.

Then, after looking at similar solids we reviewed the similarity unit. I used another trail activity for this. I actually made this when I student taught for an honors geometry class. I still use it because, even though the questions might be more challenging than what’s going to be on the test, the students don’t know that and they still try their best to get through the trail. They all were able to complete it (some with some guidance) in about 45 minutes to an hour instead of the 30 minutes that I had given in student teaching. I still think it’s a great way to review, practice, and challenge the students. It is downloadable below.

The class ended with a test.

Today, we started with an introduction to trigonometry. I feel like it’s really hard to just give an introduction, though. I got into the ratios and how these are really functions instead of multiplying by a “sin” or something like that. It’s hard to understand the inverse trig functions without having a firm grasp on the fact that these are functions, but then some of my students have never heard of a function before. I don’t remember running into this issue before but it definitely took a little longer to introduce it than normal. I had groups in the class trying to answer some questions about trig properties – Can sine ever be greater than 1? Can cosine ever be greater than 1? Can tangent ever be greater than 1? Can any of the trig functions be less than 0? What about the inverse trig functions? Some students just started by trying to plug in a ton of numbers in their calculator. Eventually, groups realized that they really had to look at the ratios.

When we got into angle of elevation I was so excited to try Kate Nowak’s “Measuring a Really Tall Thing” activity. I had the meter/yard sticks and the students had a member of each group with a clinometer app (the iPhone actually has it automatically in it within the compass app so most of my students were able to get it). I then took them outside and they got to measuring. I only had six meter/yard sticks so I had groups of 2 and 3 and about half the groups were able to do it correctly within half an hour. They then had to help the other groups figure out what was going on. I blame this on not giving enough time for the kids to really figure out what they were doing and also test how the clinometer works.  I was so excited that I just kind of said “go”. Next time, I actually should have them mess around with the clinometer maybe even before we leave the classroom. Overall, I think they saw how this could be applied to find the height of something very tall and were excited to be applying what they learned to something outside (even in the 90 degrees). It was fun and the students were excited, and next time it will be much better because I’ll know how to introduce it better. I really wish I had taken pictures…

Je suis trop occupée maintenant. C’est difficile quelquefois d’écrire le blog.

Day 17: Area Projects and Geometric Probability

Today was very free-form. They took a quiz at the beginning, learned about geometric probability, and then had free time to work for almost two hours. During that time, I expected them to work on one of the two projects I gave and/or study for their test, which will be at the start of the day tomorrow. The two projects are attached below and involve finding the area of a lawn to be fertilized and the area of a house to be painted. I like the house painting one better because it asks some more higher-level thinking questions (but not that high-level) at the end, but many of my students have done it before if they are taking my class for grade replacement so I decided to find another one.

A note about geometric probability – I find this topic so interesting! I guess I wished I taught more about probability and statistics, but I haven’t been able to yet. It’s a short section in this unit and I don’t really have time to go deeper into it, but I had the idea today that I should bring in a dart board and actually calculate out the geometric probability of landing in certain places. It would make it much more real, but I just have to find a dart board. Good thing I have another whole year before I might be teaching it again…

Projects:

Je me suis rendue compte de la folie de mes étudiants quand je les donne le temps libre, mais c’est la folie productive.

Day 16: Wheel of Theodorus and More Area

The Wheels of Theodorus turned out great! I actually only had about half that turned them in today all complete. The other half asked for more time to make their wheels more artsy and creative. I couldn’t say no to that! I’ve realized that I maybe needed to walk the class through drawing the first few triangles so that they saw the whole point of the 1-unit and the direction it has to go, especially with the overlap. Some were thoroughly confused, but I am still seeing good products in the end. I have the pictures of the ones that were turned in and will add more tomorrow when I get the rest.

The rest of the day was pretty boring. Go over new vocab about regular polygons and circles, practice with some examples, more challenging practice, class discussion, blah blah blah. I am missing my word wall, but in the short time span that my students have with the vocab, I’m not sure it would be all that useful. Tomorrow we will do another area project before taking our unit test.

La créativité dans les maths est vraiment source de joie pour moi.

Day 15: Area and the Pythagorean Theorem

So we are now on second semester! The final exams were pretty good. Second semester actually has no proofs, but I don’t mention that to the kids.

As it is second semester, I lose a few kids that were only taking the first semester to replace 1st semester’s grade and I gain some new students who are replacing second semester’s grade. Because of this, I wanted a good group activity to get the new students acclimated. I saw Sarah’s post on the game “31-derful” and decided to try this out. I loved it and was amazed at how the kids were actually all engaged in solving it with their group for the entire time. It took most groups around 30 minutes to complete and the last group finished in 45 after listening to some strategies that were shared. About 20 minutes in, I had one student from each group come into a huddle with me and we shot off some ideas for getting the rows and columns to work. This definitely gave some groups a push. A few were still doing guess-and-check, but this gave them an idea of how they could work a little more efficiently. I loved that at the end, we had all different sets of 31 and we had groups that used wildly different strategies. Here are some pictures:

Next, we got into our first unit of the new semester – Area. This starts with the formulas for the area of a rectangle, triangle, parallelogram, trapezoid, and rhombus/kite. We talk about where these formulas come from and do a lot of examples. We then get into the Pythagorean Theorem and its converse and some special right triangles. I showed the best video about how knowledge of the Pythagorean Theorem and isosceles triangles makes you sound really smart, even if you have no idea what you’re talking about: Wizard of Oz. I then gave the class the last hour to do the Wheel of Theodorus. I have never done it before, but I saw a coworker’s results when she did it and I just couldn’t pass up the chance. I told them to finish for homework. I’m excited to see how they turn out.

L’annee est au mi-chemin!

Day 4: Bingo Review and First Test

Today started by continuing to look at basic composite area/perimeter problems. All that was used was circles, rectangles, and triangles. I gave students one of those Geo Joke worksheets with the corny jokes. The students loved it! I remember loving them too, even though the jokes are so bad. I feel like it has something to do with the answers being there so they can immediately check if their answer was correct or not.

We then started looking at special angle pairs – vertical, supplementary, complementary. We did our first proof to prove that angles supp to the same angles are congruent and I didn’t force a 2-column proof on them. I actually like the 2-column proof because it is very clear that every statement needs a reason, but I also went through four years of being a math major and if you did a two-column proof in a 400-level math class you would just get laughed at. So I accept paragraphs or bullets as long as each statement has a clear reason.