# Final Project for Algebra 2

As we head into finals season, students have asked to do another project! This blog post will be updated as we get underway, but so far, here is the updated rubric draft for Spring Semester! I thought I’d put it out there after the last post on this type of project has had so many views and downloads lately. đź™‚

# Unit 1 Planning: Get them engaged on Day One!

(8/21) I will have my first day with my Algebra 2 students on Thursday (8/24). Here I sit, thinking about what to start with. Last year, we learned how to write numbers in different bases. Kids enjoyed it and asked to do more with it. But, we didn’t have a lot of time for it and it did not show up on the first test. It’s not part of the Algebra 2 curriculum. However, I remember learning about different bases somewhere in my high school math classes. It’s disappeared from the curriculum. Or, maybe it’s reappearing somewhere else.

Having kids learn to write the number 25 in base 5 Â (looks like 100), 7 in base 7 (looks like 10), 12 in base 4 (looks like 30), really is interesting for them. Their little neurons start firing. They say things like “Whoa!” and “That’s so cool!”

We also spent time issuing texts, getting kids signed up on RemindÂ , and doing getting to know you activities. We went over some expectations, etc. But, I knew I had won them over. Then I started teaching.

This year, I want to do that again. And get them on DesmosÂ quickly – download the app, etc. Play Marbleslides: Lines, etc.

(8/23) I make a plan often based on all of the great ideas and inspiration I get from Twitter math teachers (#mtbos #iteachmath), Jo Boaler, Dan Meyer and many others. I recently read a great article about not grading students in the first month (or some time frame like that) and I thought, “Wow, what a great way to build culture and address equity issues.”

So, I’m thinking about that right now.

I feel that the most important things I can do as a teacher is invite students to be curious, let them know they are an important part of the class, and teach them mathematical concepts. After meeting with my excellent colleagues, I’ve come up the following plan:

We are going to see if those iPads work. If they do (fingers crossed) I want to get students to do a card sort activity, and hopefully play some marbleslides and then get into some vocab around linear functions and translations. We will get signed up on Remind and learn about a few classroom expectations. Seriously kids, no phones and limit your bathroom breaks. Â Be nice.Â That’s really it. However, when working with teens, there is always a need to discuss these things, come to an agreement and then move forward. They will test these agreements. We will need community building. I look forward to that.

I’m really excited to get going. I’ll be working some great activities and mathematical ideas into my lessons. We’ll explore some history, look at different bases, play games on Desmos, be creative, and have another great year!

# Excellent Summer PD: Desmos Summer Institute

This summer I went to a two day Desmos training in sunny San Diego that was completely dedicated to improving student learning through activity builder.   Dan Meyer, their CEO, kicked off the training with his typical charismatic, humorous and collaborative way of learning about us and generating the goals and plans for the next two days. There were about 25 teachers and a group of Desmos staff and Desmos Teacher Fellows. So, we were surrounded with amazing support, creativity and collaboration.

The Desmos staff said we can share the materials from the workshop. I love that attitude of sharing. In that spirit, I’d like to share with you one of the great shares of Day 1, the Desmos scavenger hunt. The Hunt provides tasks and solutions so you can test your skills and learn new ones. We had great fun with that.

Using the activities or making your own requires you go to teacher.desmos.com. I recommend you get started by watching the one minute video and then create an account. You can login and start searching existing activities.

I recommend starting with searching the existing activities and activity bundles. There are individual activities or bundles of activities by topic (there is a list on the sidebar to the left of the screen). For instance, in Algebra 2, there is a bundle for exponential functions. There are currently seven activities. By doing a quick preview of each activity, you can decide which ones you want to use and when to use them during your unit. As you preview the activity, you’ll see a green pop-up that gives teacher tips. They are really helpful.

I love the activities in Desmos and have had great success with them. Students work and I am freed up to circulate and help as needed. At a glance, I can see where every student is from the teacher dashboard. I can use the teacher tools to anonymize student names and project individual student work or entire class screen overlays so students can see multiple ways of solving problems. Students get instant feedback on their work. Pacing is individualized and many activities get more difficult as you progress, which challenges every student. Students naturally start to ask each other questions. I can partner students as needed. I can even pause an activity – which always leads to groans and the question, “Why did you stop it?!?” You can read about my marbleslides experience Whatâ€™s great about marbleslides, if you want more details on a specific activity.

My takeaways:

1. There are amazing educators out there. Find them and stick with them. Then, find more. Share your ideas. Share your lessons. Build better lessons together.
2. There are great activities for Algebra 2 already built in Desmos. I’m not sure I want to build any myself. It’s not easy to do and more are being added all the time. There are teachers all over the country adding to the bank of activities.
3. The activities are varied and can be used in many ways – introducing a topic, vocabulary builders, practice, and formative assessment. So far, the ones I’ve used have all provided differentiation for students.
4. I will build some activities for my AP Economics classes, which I will be teaching for the first time and am very excited about. I’m picturing supply and demand shocks as a starting place. I built my story board using post-its, as recommended, and am ready to start building!

Having the time to really delve in and learn the tools and process for building activities was a gift. There is a second training August 10-11 in San Mateo, CA. Sign up by July 21

# The kids were a bit unruly today…

I take that unruliness as a challenge to work on a more engaging experience for them. Today I was teaching polynomial expression operations, which is, admittedly, one of the more nuts and bolts type of topics and not terribly exciting math. This blog post is about how to find ways to raise the engagement on some of the dryer topics that we cover.

And, what’s ‘dry’ to me means that I can’t readily think of great activities, applications, or problems that engage.Â

To create a higher level of interest is to create a higher engagement level. This meansÂ less need for a disciplined atmosphere centered on direct instruction when the kids are just not in the mood. Which is often in my 5th period (after lunch) class.

The kids are energetic, they are social and they are comfortable enough that they interrupt, throw stuff and eat candy, throwing the wrappers on the floor, sometimes near the garbage can. God bless ’em. đź™‚ I really do love these kids and I have fun withÂ them. BUT, I do have a hard time getting through direct instruction for 20-30 minutes, so it drags out longer, which makes it even tougher for me and for them. Way too much!! Especially for the half of the class that is quietly waiting to get through a concept or problem.

Let me say, direct instruction has it’s place, but it’s not working well for me with this group. So, I need options. First stop: Desmos. What great activities already exist for us?

So many! Here’s a link to the classroom activities that come up when I search on Polynomial Functions:Â http://bit.ly/2drqtGd and a screen shot of the list. If you haven’t already, please set up a teacher account at Create Desmos Teacher AccountÂ and get inspired!

It think for polynomial function operations,Â though,Â I’m not really seeing anything that I could use. Bummer. Hmm… Let me think about a flipped approach.

At the site, you can click the Semester 2 tab, then click polynomial functions, there’s a lesson for operations. The site providesÂ a student note page that students can print and fill out while they watch the video. This way, they haveÂ guided notes, they can go at their own pace, and they can ask questions when they get to class. In class, we can quickly summarize the key concepts and ask questions. They can do that in groups, or as a whole class.

Would this really help in terms of engagement? Well, hard to say, but at least I wouldn’t be trying to hold their attention so long when it’s just physically hard for them to stay tuned. They would get a very similar experience of direct instruction, just when they are not in a group with their friends after lunch on a warm day. So, I think it’s an improvement, but it’s not exactly innovative or exciting.Â

Next, if I do the flipped math for instruction, what activity could I have this energetic group do during class? One option is some sort of matching activity. But, wouldn’t it be better to do a live matching activity where they are the variables? Like, everyone gets to be a cubed-x or a squared-x or a single-x or a constant term? Then, I could writeÂ problems on the board and they couldÂ group themselves as the equation and solution, and maybe make a video, and maybe put it on YouTube and maybe I could tweet it and blog about it. đź™‚ Wow, I’m gonna do that next time.

Another option is to create some open questions.Â Ways to do this include using some closed questions, like most of the text book questions and simply withholding some of the information and/or instructions, then ask students what are we going to try to solve and what information do you need?

If only I had thought ahead. Well, for me, next time as I look ahead in my planning, I’m going to be a bit more proactive for the sake of this particular class.

Direct instruction + Dry topic = Headache by the end of the block. Never again. đź™‚

# Demonstrating the Structure of Quadratic Functions with Desmos

I am a big fan of empowering students to look for and make use ofÂ structure in Algebra 2. This is most true for me as we workÂ with functions, parabolas, and quadratics. Â 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)Â 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 vertexÂ form of a parabola:

In this window, you can activate the sliders* individually to demonstrate to students (and share with your math team) Â how a, h, and k affectÂ 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 theÂ sliders, click on the arrowÂ buttonsÂ in rows 2, 3 and 4. To stop them, clickÂ again,Â orÂ manually move the slider to any spot.

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

I suggest you first let c slideÂ 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 anÂ 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.Â 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 forÂ theÂ 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.

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. đź™‚

# Teaching Conics in Algebra 2

I really like opening the day with an open question. They’ve been kind of easy to think of so far. But, what about conics? What’s a good group of open questions that can be used with conic section lessons? Before I could think of that, I really had to look at the new standards for Conics. In doing that, I realized I really hadn’t examined exactly what the kids are supposed to learn. So, I had to research and think about that for a while first.

I did a little research about conic section topics and standards that need to be covered in Algebra 2. I checked this publication on the California Dept. of Education website: California Common Core State Standards for Mathematics. You’ve probably all seen it, if you’ve been working in California.

Here’s the conics standard for California – yes, there’s only one, but it’s loaded (p. 83):

3.1 Given a quadratic equation of the form Â axÂ˛ + bx + cyÂ˛ + dy + e = 0,Â use the method for completing the square to put theÂ equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola andÂ graph the equation. [In Algebra II, this standard addresses only circles and parabolas.] CA

Um… okay. Let me think about that. first of all, is this the same or different for what we’ve been doing at my school for teaching conics. Do we need to address the directrix and focus or foci? Because, I talk about those whenever I talk about conics. Even when introducing them in Geometry.

Here are the Geometry standards (p. 74):

Translate between the geometric description and the equation for a conic section.

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find theÂ center and radius of a circle given by an equation.

2. Derive the equation of a parabola given a focus and directrix.

In the past, I taught about the foci of an ellipse and parabola and hyperbola. Last year, we didn’t test on graphing hyperbolas. According to the Algebra 2 standard above, it seems like it’s all four, except for that part in the brackets. Is that only for other States? I needed to find out. After all, I’m serving on a County wide committee to discuss teaching this class and we should really know what those apparently contradictory statements mean. The bold is supposed to be California.

So…. here’s what I found out… Go to this website IXL, scroll down to whatever standard in which you are interested, hold your cursor over the standard and a sample questions will pop up. Wow. Great stuff. No foci/directrix stuff until Pre-Calculus. Okay – I guess they’ll handle that in Pre-Calc. Looks like just graphing parabolas, those that open horizontally or vertically, and graphing circles. For circles, be able to find the center. I think they still need to be able to tell whether the conic is a circle, ellipse, parabola or hyperbola from the equation, though. Â Please make a comment below if you understand this all to be different than what I’m writing here. This seems much less than what I’ve taught in the past. So, maybe that’s a good thing. đź™‚

Next, what will the lessons be?Â Then, I checked the NCTM website, Teachers Pay Teachers the NRICH websites for ideas related to those standards. And, of course, Desmos. Well, on the day I had to start the topic, I didn’t have a good ‘open question’ opener. I just asked kids about the equation of a parabola, in vertex form. I asked about the equation of a circle. I put it on the board. I asked if they’d seen that before. I asked them to notice that there’s a x-squared and a y-squared term. I asked if that meant it’s a function. So, it was a weak start compared to what I would have liked. But, it was the day before break and my goal was to introduce conic sections. I had them watch thisÂ Conics video from YouTubeÂ This was pretty much a vocabulary lesson with a graphic that was pretty good for getting them to understand the basic concept of what conic sections are.

Then, the fun began…Â  as they started to use the Desmos activity, Polygraphs: Conics, foundÂ here. Now, I’m figuring out my unit plan. I have the week off. I plan to find open questions, interesting activities and relevant homework for them. Something that spirals old stuff, too. I plan to write more about it, too…. Â Ideas?

Marbleslides is a great teaching and learning game created at desmos. It’s been a great addition to my units in Algebra 2. So what’s great about it? Well, it’s tough to know…

…where to start the list…. hmmm…

How about instant feedback for students? They know immediately whether or not they ‘got it right.’ If not, they try again. Don’t you wish they would do that on their homework? Check the answer, if it’s wrong, try again. This is so interactive and quick, they are more likely to stick with it. ‘Stick-with-it-ness’ is a new term I’ve invented. You may have invented it, too, or some version of it. đź™‚

The marbleslides activities allow kids to stick with it, even if they are not as far along as other people in the room. They get to work ‘where they are’ without getting behind the rest of the class. It’s a way to differentiate seamlessly, without it being obvious, because of the high engagement level students experience. You can very easily spend more time with the kids who need you. You can make suggestions, but never give away the answer. You can remind them to read the instructions if they missed them (which happens a lot). They can then ‘reset’ the problem and give it another go with much increased success. They are feeling challenged whether they are on slide 7 or slide 17.

For the teacher: Very little planning time is involved and you get to use that time to assess and reflect. It provides instant information to be used for formative assessment. And, kids can complete the activityÂ later if desired by you or by them.

Pretty cool.

Here’s the play-by-play of how I used the Desmos marbleslides activities with my Algebra 2 students:

First, I had students review graphs of rational functions using the marbleslides activityÂ here. We had already learned this and I thought it would be good to start them with something familiar before moving on to new functions. I had hoped this would also strengthen their understanding of the transformations of that parent function. It did. Yay!

So, they were able to learn to navigate desmos and how marbleslides works. Then, a week later, after introducing exponential functions, I had them do the marbleslides activity for that function. I heard comments like, “Whoa, I understand this now.” If you are interested, go to teacher.desmos.com and create an account. Use their pre-made activities and/or the learn.desmos.com tutorials. You will be up and running in no time. You can email their team or me if you want more info about what I did.

Will this always work to get every student super in love with the topic/lesson/learning goal? I’d love to say YES!, but it may be more realistic to say that probably not everyone will fall madly in love. However, this will certainly raise engagement and increase understanding. And, time flies when they’re doing the activity. This definitelyÂ deserves space as another tool in the toolkit to increase overall engagement and mix up your activities. What’s needed? Well… devices – iPads or computers. It won’t work on the phones… yet. So, you may need to schedule some lab time. Totally worth it.Â

Should you do it everyday? No. Keep mixing it up so you capture or engage your audience. Some kids will be more engagedÂ one way, some kids another. Some like partners, some like to work alone, some like lecture, some like ‘discovery.’ But, what’s great is, by ‘capturing’ them one day, they will most likely increase their general interest level and they are more likely to be willing to go with it another day with another method. Read Styles and Strategies for Teaching High School Mathematics by Thomas, Brunsting and Warrick for great ideas on differentiating your practices.

More on that book next time.