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Demonstrating the Structure of Quadratic Functions with Desmos

I am a big fan of empowering students to look for and make use of structure in Algebra 2. This is most true for me as we work with functions, parabolas, and quadratics.  I’m writing this post about what I’m finding to be an indispensable tool for helping students quickly learn about the structure of the equations of quadratic functions. This tool is easy to use. Simply project the Desmos calculator (use the links below) and activate the sliders.

One of the many great things about Desmos is some of their built in functions on the calculator. Like this one, using vertex form of a parabola:

Link 1: Vertex Form of a Quadratic Function

In this window, you can activate the sliders* individually to demonstrate to students (and share with your math team)  how a, h, and k affect the parabola. You may want to stop the slider and manually slide a to values you want to emphasize with the class (a = 2, 1, 1/2, 1/3, 0, -1/3, -1/2, -1, -2 for instance).

*To activate the sliders, click on the arrow buttons in rows 2, 3 and 4. To stop them, click again, or manually move the slider to any spot.

Next, move to standard form, which is really interesting.

Link 2: Standard Form of a Quadratic Function

I suggest you first let c slide and have students watch as the parabola moves up and down. Ask them whether the shape is changing. Some will think it is, but it’s just an optical illusion. Tell them to look again.

Then, stop c and let a slide. Kids can see how the parabola stretches, shrinks and reflects just as it did with vertex form.

Last, the fun one. Ask them to predict and then tell their partner/group what they think will happen when b slides. Will the shape change? Will it move up, down, left, right? Then, activate the slider.

This is where the math just gets cool. Ask them, as they watch the motion, “What is the path of the vertex?” (it travels along a parabolic path); “What is happening to shape of the graph?” (nothing, it stays the same); and, “What is happening to the y-intercept?’ (the parabola travels through the point (0, c) and the intercept doesn’t change).

I found this to be so helpful to me as a teacher and to students to see quickly what the structure of these equations do. To get to them and many others, just click on the bars at the top left corner of the window for the desmos calculator. There are all kinds of great functions to work with. Here’s a picture I made in paint – screen shot, save in paint, edit with brush – to help you find the drop down menu.

desmos-menu-location-for-blog

P.S. I need to create a note sheet for this where they summarize these structures and the impact of the key components. Next week. Yep, next week. 🙂

 

Last minute quiz inspiration

Yesterday, I suddenly decided on a new quiz format. I had been writing a quiz for my Honors Algebra 2 class and I just didn’t like it. It wasn’t interesting or challenging. I really didn’t want to make a second version (my kids sit at tables), and was hoping for several days that some inspiration would hit. Our text has a set of alternate assessment questions, but they are a bit involved.

So, in the 5 minutes before class started, inspiration hit like a tons of bricks.

I let them use the alternate questions, and work in pairs. There are 6 students at a table group and we were covering two chapters. I gave them packets of the questions. There were about 7 questions for Chapter 1 and 6 questions for chapter 2. The guidelines were that they had to answer one question from each packet and couldn’t answer the same questions as the people at the table group. So, that’s a total of 6 questions for the table, three from each chapter, two for each pair of students.

To make it an actual quiz, they couldn’t use notes. They also couldn’t ask me questions. Actually, they could, but they would lose a point. Each question was worth 10 points, for 20 points total. Asking one question would still yield an A. But, no one asked any questions. The kids were engaged and worked steadily for about 35-40 minutes. Most finished, no problem.

I called time at 45 minutes. Some students hadn’t finished. I gave 5 more minutes. However, a couple of groups didn’t get to the second question or had just started it. Uh-oh.

So, as this is a group of motivated, grade-stressed students, I allowed them to come back at lunch or after school or during our tutorial time to finish. They appreciated it, so we were good.

The best part, was in the ask for feedback about the quiz format.

I gave them three prompts:

  • Partner quiz again? yes/no
  • This would have been better if…
  • This was good for…

They unanimously liked the partner quiz because they had another brain to work with. Asking for feedback is gold! Making myself vulnerable was scary. Here I had changed up the quiz in the last few minutes, kids didn’t finish, they were afraid to ask a question even when it would have gotten them to the finish.

Would I do it again? Yes! Overall, it was a positive practice for them and for me. In fact, I’m doing it again today with my other section of that class. I’m happily incorporating their feedback with what I observed to make the following changes for the next class and next partner quiz. Here’s my list:

They asked me to:

  1. Make more copies of the questions – it slowed them down to have to share.Yes. Done. Easy.
  2. Allow questions. Well, I’m thinking no on that, because I think they will ask me a million questions. So, modified practice: they can ask the question. If it’s a fair question, I will guide with no point deduction. If it’s a question about not understanding the content, then I will take a point if I answer. They can ask the question, then I will answer or respond with, “Yes, I’ll answer, but it will cost a point.” Then they can decide if they want the answer.
  3. Give more time. No, but I will advise students to read the entire question before choosing (most have multiple parts) and remind them that they can change the question if they get really stuck. Also, I will be more active in making sure everyone is employing strategies to finish on time.
  4. No one mentioned this in the feedback, but I didn’t give them a time frame before they started. I wasn’t sure how long it would take. When half the class was finished, I announced 10 more minutes. I should have circulated a bit to check on their progress at around 20 minutes to let students know they should start on their second question within 5 minutes, so they have time to finish.

Now, I just have to grade them. That’s nice too, because I only half the number of quizzes to grade. Another teacher benefit from the partner quiz.

Please comment below with questions or ideas or practices you have tried. If you want to know more about the course or text, send me an email.

 

Tough grading moments….

One of the toughest things about grading is when the students with 79% or 89% ask/plead/argue for the B- or the A-. I do round an 89.5% or higher, to the 90%. I think that’s just doing proper rounding, as I like to teach in my classes, as opposed to truncating the grades. [Don’t know what truncating is? You can find out here] . But then, the 89.2% kid asks for the A-, too. I would be inclined if their test scores were in the A range, but they weren’t completing all the assignments, and so homework was dragging the grade down. But, if the test scores are in the B range, and homework completion is bringing the grade up to B+, I think that’s good enough.

I have several students who’ve missed a lot of school, or have ADHD and just don’t complete every assignment, or just never are there or aren’t organized enough to present the assignments for credit. If they have high test scores, I’m inclined to round their grades towards those test scores. However, high homework scores with lower test scores are not a compelling argument for me to round the grades higher, even though that’s the request I get a lot.

We just had final exams, another tough grading challenge. I think it’s normal for students to score about one grade lower on the final exam than their unit test scores. And, when that happens, I usually let them keep the grade they earned prior to the exam. An example would be a student who had a B in the course, earned a C on the final, bringing their grade to a B-. I would be inclined to let them keep the B. But, if they score low on the final (a D or an F), I do let the grade drop, but not by more that a half a grade. And, if that same student with the B earned a D on the final, they would end up with a B-. They see the B part and are still feeling content, I think. However, if a student had a B- to begin with, scored a D on the final, and ended up with a C+, they will see the C and possibly (probably) be upset about the outcome. The difference in the GPA would be the same (0.3 points) but, suddenly, the letter B to the letter C is very noticeable. That’s when I get the email with the ask/plead/argue message. Sometimes the parents get involved, too. But, I have to stick with my convictions on the grading in these situations.

My grading policies and decisions around tests versus homework and semester grade versus final exam grade are pretty generous in my opinion. Many teachers let the computer calculate the grade based on the settings for the weights they decided at the start of the semester. Many others make exceptions, too.

In addition to the above rules of thumb around my grading decisions at the end of the semester, during the semester I’ve been known to drop some low scores when the class doesn’t do well on a quiz. I think that I didn’t teach them very well when that happens, and we revisit the material.

Algebra 2 is a hard class and not everyone will get an A, even if they usually get As in other classes or in prior math classes. This is one of the tougher lessons for high school students to learn. They are hitting a level of math that really requires studying, critical thinking and perseverance for the longer, more involved problems. They aren’t all ready for that level of problem solving. Even if they are, the course is content rich, meaning there is a lot to learn and a set amount of time in which to learn it.

Students are busy with tough course loads, sports, hobbies or jobs, and social and family activities. Many students don’t have adequate time outside of school to study as much as they need to in order to get the grade they want. Others make sacrifices and get every assignment done every day. They come in and ask questions after they’ve tried to figure things out on their own. Some ask questions immediately without giving themselves time to try a solution, because they are used to the quick answer or they feel pressed to get the questions answered quickly, without a deeper understanding for when the next question comes. In learning math, you learn so much from making mistakes and trying new approaches. Especially at this level. But, I think that requires a level of calm and concentration that many teens aren’t used to. Trial and error are involved. I try to talk abut this to my students when I can.

Some people may wonder about the purpose of the final. Well, I think it’s important to review what they learned over the year. I think it’s important to have a idea of what they’ve retained and to remind students what they need to know for the next course. I think it’s good for them to have an idea of what they remember and what they may need to re-study. And, I don’t let the final exam kill their grade. I think that’s the bad part about finals, which is why I have some of the policies listed above. A final exam can bring a student’s semester grade down much more than it can raise it.

I plan to include these grading philosophies and practices, and study tips and techniques for retention and deeper understanding in my beginning of the year mini-unit next year. I introduced the idea in my blog post  Summer reading, relaxing and revamping…. and will post it when it’s done.

Comments, experiences, input welcome…

Summer reading, relaxing and revamping….

For Summer 2016, I plan to do a lot of relaxing, enjoying time my family, cleaning out closets and thinking about next year. I also plan to revamp my grading and record keeping. And overhaul my unit plans. I want to think about making things easy for students in terms of being able to understand what’s expected of them and what their grade depends on.

For next year, I’d like to make unit packets for students that consist of

  • Pre-requisite skills questions
  • Sample test problems
  • Practice Sets
  • A portfolio project
  • Open questions
  • Note organizers
  • Formulas
  • Reflection prompts
  • Spiraled review questions

I know, that’s an ambitious list. Well, I’m an ambitious gal. But, I’ll leave myself some wiggle room. I may not include everything on every plan. Or, it may turn out that some things aren’t needed while other things are, depending on the topics. Some spaces will be blank, allowing students to fill-in as we go. After all, I don’t want to toss all the cool open questions out in a lump. I want to use those to set a low entry point, an opportunity to engage students and an opportunity to talk to the class about options. Those will likely be saved for openers during block periods.

I also think, it will get easier as I get through the first couple of units. I’d like to have the first two units ready to go before school starts. That’s my summer goal.

I want to start the year with a mini-unit on study skills for math. I want to talk with them about passive versus active learning and how that makes wanting success and high grades either a wish or a goal. If you are passive, then your goals are more like wishes. If you are active, your goals are more likely to be achieved. I want them to know what working towards a goal is versus doing the minimum and hoping for a high grade.

Another part of that mini-unit will include how to handle absences. We have a lot of absences at my school. I kept a record in May/June (see photo) because it was really getting to be too much to compensate for. We have block periods, so when kids are absent, they miss a lot. However, I think these unit packets will help them when they are absent too.

20160607_165821.jpg

The top display is for my Tuesday/Thursday classes, the bottom is for my Wed/Fri classes. Everyone meets on Monday. Some of those percents are pretty high and there are perhaps too many days with more than 10% absent. I didn’t even include senior ditch day.

So, the absences are driving my desire to make things really clear at the start of the unit and have a packet so that kids can know what’s coming and when. I don’t plan to give them a 50 page booklet, but at least an outline, a schedule, a grade tracker, etc. I think I may handout homework packets on Mondays and collect them the following Monday, too.

Part of my planning will be doing some reading and research. I’m looking forward to re-reading some books and finding some new ones. Teaching Struggling Students in Math is my newest acquisition and should be a good one. Check out my new and ever growing list of favorite books for teaching, especially teaching high school math.  My Favorite Books on Teaching Math

Enjoy your summer!!

How 2 questions took 1 hour to solve

I was responding to a post by Dan Meyer about his awesome recent talk at NCTM. Here’s what I wrote in my comment:

I am working hard to keep engagement high. This week, we actually started with two blank triangles, one right and one obtuse. I asked kids to solve the triangle, and ask me questions. There were no measurements on the triangles. The triangles were to scale. They could ask me for one value.

From there, they had to measure another part of the triangle and let the solving begin. They could only measure one other part, no more. So, depending on what they chose, they would use different tools to solve the rest of the triangles. They used rulers or protractors, ratios, triangle sum rule, right triangle trigonometry and Laws of Sines/Cosines. They worked in groups to try to figure out what information they needed, they worked together to try different strategies. In the end, the only way they knew the answers were correct was by rationalizing whether or not they made sense.

We spent 1 hour on two problems. Engagement was high, completion rates were nearly 100%, participation was 100%. It was a really great day for me and I was able to coach them. They did the lifting. It was great. Great conversations.

So much of Dan’s work inspires mine. I love his 3-act-math ideas, though I don’t use them much. More so, I respond to the idea of opening up questions. I love the idea of putting out a skeleton and asking kids what they need. They actually answer, they actually engage in the problem and start to think and ask questions. Once they are invested like that, they don’t like to give up. Giving up has been like an epidemic in mid level high school math classes like geometry and Algebra 2.

Today’s geometry warm up looked like this:2016-04-22-13.44.41.jpg.jpg

We didn’t even get to the third problem. Yes, it was hand drawn. Like, free hand. That actually made the problem kind of interesting because I don’t really have a right triangle, do I? Just almost one. So, depending on what side or angle students measured (by the way I told them the base side of the triangles in both problems one and two was 15 units) the answers they got might be a bit different.

It was also a great talking point about how real life problems are rarely perfect and we rarely get to check the back of the book to see if we did it right. We have to trust our methods and our team.

Great conversations. Such a great day. It was hard for some of them to keep going to finish the second problem. They were a bit tapped out. We have block schedules, which means 90 minute periods. This took way longer than I expected. But, totally worth it. We’ll do the third problem next week. They actually asked for a book assignment when we were done. I guess their brains were tired. But, they did it! And, they did really well! Go team!

More about problem three in another post. Maybe.