To use this structure correctly, you need the right technology and a quick and easy way to prepare the lesson. A discovery lesson should only replace a typical lecture when it is just as effective, does not waste extra time, and is just as smoothly implemented.
My favorite tool for teaching with technology is GeoGebra. With the GeoGebra software, you can set up a lesson for your students in only a minute or two, and they can see relationships in figures that are impossible to view and manipulate in a typical math lesson or lecture. This software is free for teachers and students to use. I have put together a quick-start guide with a few sample lesson ideas. This will show you specifically how you can use the software to set up a smooth discovery lesson for your Geometry students. Basics
Getting into the Software & Setting Up:
1. Go to Geogebra.org and click “Start GeoGebra” (There are also materials and lesson guides available to browse.) 2. Set up a “Geometry” working screen. 3. In the menu (three horizontal bars in the top right corner), select “options.” In options, select “labeling.” choose to “display all new objects.” This helps your students to save time and communicate in pairs more easily by saying “Move point A”, etc.
Play around the first time you get in. Practice making a triangle by selecting a “polygon” from the top menu. Click three times to make the three points, then click again on the first point to close/finish the figure. Then select the arrow from the top menu to drag a vertex and manipulate your triangle. Try displaying an angle measure by selecting the angle tool in the top menu, and then clicking on the two lines/rays that form the angle. Try displaying a side length as well.
These are the basic tips you will want to show your students. Before your first investigation lesson, work together at the front of the room (if possible) on a smart board or using a projector. Allow students to practice manipulating a figure, creating new diagrams, and displaying measures. Sample: Diagonals in a Parallelogram
Depending on the level of your students, you can do the “Prep” part ahead of time as the teacher, or you can have students follow these directions and start from scratch.
Lesson:
1. Give students (in pairs) the prepared screen with a parallelogram (or directions to construct their own) and ask the following essential question: “What properties do diagonals of a parallelogram always have?” (Explain a “diagonal” if necessary.) 2. Allow students to work in pairs to add the diagonals, manipulate the diagram, and make observations. Encourage them to display angle and side measures to try to determine relationships. Have them discuss what happens as they drag a vertex. Which measurements change? Which stay the same? 3. Try to avoid giving too many hints. This learning process is great for their brains! 4. Require students to formally write up their observations. 5. Encourage students to extend the properties to other quadrilaterals. Next, have them construct a rectangle and its diagonals and observe what properties are maintained as vertices are dragged. What about a trapezoid? 6. If your students would benefit from reflection or additional structure, try using the half sheet that works for any hands-on digital discovery lesson in Geometry. Sample: Angle Inscribed in a Semicircle
Depending on the level of your students, you can do the “Prep” part ahead of time as the teacher, or you can have students follow these directions and start from scratch.
Lesson:
1. Give students (in pairs) the prepared screen with a circle and an angle inscribed on the diameter (or directions to construct their own) and ask the following essential question: “What properties do angles inscribed in a semicircle always have?” 2. Allow students to work in pairs to manipulate the diagram, test cases, and make observations. Encourage them to display angle measures and lengths to try to determine relationships. Have them discuss what happens as they drag the point on the circumference and when they make the circle larger or smaller. Which measurements change? Which stay the same? 3. Try to avoid giving too many hints. This learning process is great for their brains! 4. Require students to formally write up their observations. 5. Encourage students to extend the properties to other inscribed angles. Next, have them construct different angles in the circle that do not subtend the diameter. What about angles that are not inscribed? 6. If your students would benefit from reflection or additional structure, try using the half sheet that works for any hands-on digital discovery lesson in Geometry. Resources, Links, & Tips
• Use digital discoveries only when a physical hands-on idea will not cut it. It’s important for kids to physically touch and move items whenever possible. This just cannot always be achieved in a tangible format without technology. Try physical activities too, like this hands-on lesson for discovering Triangle Inequality Theorem with markers with snap-on caps.
• Allow students to spend a little time exploring with the technology at the beginning, or during the first lesson that you introduce this type of activity. The more comfortable they are, the more options they will have for adding to their diagrams, changing things, manipulating what is there, and exploring new ideas and properties. Your students will likely catch on to the software faster than you can, so rather than trying to spend a long time teaching them the toolbar, just let them have at it! • Take time to set up your expectations and formal structure for any period that they are working this way. Decide how you want groupings to work (pairs or teams of three are great for this) and try to keep consistent partners throughout at least a quarter. Determine how much structure you will give, and whether you will kick off each property/lesson with a question, or only a figure. Explicitly teach students what you want them to turn in at the end. Do you want only the properties they discovered or a full reflection as they work? It may be nice to have them submit a diagram or two representing what was on their screen. They can always use extra practice drawing and labeling diagrams, so this approach can serve as both purposes. • Take a look at some of the built-in lessons in GeoGebra, as well as these in SketchUp. The SketchUp software is another option that allows you to do easy tessellations, let your students explore pyramids, prisms, and more. It’s great with 3d representations. There are a set of handy videos to get you started.
Click here to download a printable PDF version of this article including all the steps. It also contains a half-page recording sheet for your students to use to accompany any digital discovery lesson.
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15 Comments
3/18/2016 16 Comments Going Over Math Homework
The time balance never seems quite right. I don't have the perfect answer, but have noticed certain methods working better at certain times, and at least thought I could share some of the strategies I have tried.
As I struggled through each of these, I found the last strategy (the split end) to be the most versatile. It's also one that takes a certain classroom culture, so it may not be right for everyone. Here are the methods that can work in different situations. I doubt I am alone in this struggle, but who knows! Maybe it's just me. Either way, I hope that trying one of these will help you feel that you are finding a better balance with homework review. Homework Partners
Calling out Names
How it works: The teacher calls on one student at a time to provide an answer, moving quickly through the rows. Students put a star beside problems that they would like to ask about at the end. After going through the answers quickly, and correcting in red pen, there is a chance to ask questions about any problem that students are not sure why they got it wrong.
When it's best:
How long it takes: Once you get into questions at the end, this can drag on longer than I like sometimes. I have been known to cut them off and transfer the rest of homework checking into a split end strategy (see below) when I start losing a few bored students, but others still have questions. Either way, keep the verbal answer portion moving quickly. It should only take 3-4 minutes, unless you assign more than 30 problems. Student Presentations
How it works: Each student gets one problem assigned to present. The teacher can assign the names along with the homework problems, or can choose randomly when it is time to present each problem. The student comes to the board, solves the problem, and explains his/her method. Points can be assigned based on the presentation (as an assessment or simply as a participation grade). This method is great for helping students understand different approaches to the same problem.
When it's best:
How long it takes: This one can be the most time-intensive, but also can have the biggest payoff, for both the presenter and the class. It's so beneficial for kids to see a different method, and a different explanation other than the teacher's. Allow about 5 minutes per challenge problem. Self Check
How it works: Students silently and quickly look over an answer sheet that is displayed or projected on a screen at the front of the room. Each self-corrects his or her own work in a different colored pen. You can leave time at the end for questions.
Even for classes that I usually do a different method, I pull this one out for special days when we have shortened classes. It is super speedy when you are in a time crunch, but want to make sure everyone at least knows how they did and what to study. When it's best:
How long it takes: Potentially under 5 minutes (plus any time at the end that you reserve for questions) Teacher Station
How it works: On a day that students are working in learning stations, one station is the "teacher station," where a small group goes over the previous night's homework with the teacher. Additional follow-up example problems should be on hand for extra practice together with any type of problem that the students in a particular group are struggling with. Students can ask questions in a small group setting and work together and with the teacher to clear up confusion on the lesson.
When it's best:
How long it takes: 10 minutes, repeated with each small group, while others work on independent learning stations around the room. This time is really maximized for big benefits because the time is intensely spent on each group's individual needs, and you have each student's full attention because of the small group setting. Split End
How it works: After spending a few minutes taking questions and going over the most needed homework problems, stop the homework review altogether. Begin the new material, and proceed with your lesson as planned. If additional homework time is needed at the end of class, allow students to split into two groups. Those who "get it" move to the back of the room to begin a teacher-specified task. Those who would like additional help with the previous homework or lesson (or need more review) move to the front of the room to work at the board with the teacher.
I started this differentiation strategy after realizing that some periods I would spend almost 30-40 minutes going over homework, and never get to the lesson I planned because only a group of 5 students needed me to go over every single problem. Once we developed a culture that worked for this, everyone was happy! As long as no one is embarrassed to be in one group or another, they all feel that they are winning. I was surprised at first to realize that the attitudes were not what I expected. Those that need help feel that it's a privilege to get extra time with me, and those who don't need it feel that it's a privilege to go work on something else instead of sitting bored and feeling held back. It made me wish that I had tried this sooner. The reasons that I had always avoided splitting this way were far outweighed by the benefits. And the challenges that I expected never came, because each student felt the freedom to decide what they needed. I've tried this in middle school and high school, with great results and positive feedback from both groups of students. I always let students self-select their own group. If I see someone consistently making a poor choice of which group to join, I address it privately with that student outside of class time. The students love this method, because everyone gets what they need. The group in the back can get deeper into a critical thinking activity, work on an enrichment activity, or sometimes work ahead. I vary it from one class to another. I keep these groups flexible, because during one lesson, a student may be ready to move on, but in another topic feel totally lost. Once students get used to the "split end" structure, they love it and gain confidence. They also take great pride in self-assessing themselves correctly for each lesson, and knowing which area of the room to head to in order to set themselves up for success. When it's best:
How long it takes: anywhere from 10 - 20 minutes (It's not really worth splitting and getting re-settled for less than 10 minutes, in my experience)
I have had some success with mixing and matching these. It helps to add variety and keep homework checking from becoming tedious. Depending on the type of work, the group of students, etc. I have changed up my methods from one day to the next. This helps to keep the kids on their toes and accountable.
During any option, I also check homework for completion, and grade on a 4-point system, which you can read about here. As for myself, I am feeling like I will never be fully satisfied with any one solution to going over homework that works well for every class and every topic, but my goal here is really to share as many options as possible. These are the ones that I have tried, but please share what has worked best for you in the comments! I'd love to collect more strategies for our math teacher community. Thanks so much! To Read Next:
On my first try, I tried to embrace the fun. I had a short activity, but it was not really structured enough. All I wanted to do was avoid 45 minutes straight of Pi Party.
I had told my classes ahead of time that they could bring any round treat to share. As part of our activity, each student would answer a question one at a time and then get their turn to go serve themselves from the giant treat buffet (a long table that took up one entire wall of the room!). This turned out to be a big mistake. There were too many options of food. And too many ridiculously messy treats. Being middle schoolers, they brought tall dishes of oreo pudding with a flat spatula instead of a scoop, big crumbly cookie-pies on paper trays that did not fit, cakes that had been smashed when they fell into the aisle in the bus.... etc. One kid even brought a pizza. But only one, so everyone wanted some, and there clearly was not enough. I refused to give up on the math content portion of the day, so I was trying to run my Pi Day Fact musical-chairs style learning activity I had developed while trying to also help each student navigate the messy dessert bar at the same time. Again, it was definitely chaos. Of course I learned from my mistakes. So I'll share some of the upgrades that I made to make the day run more smoothly. TREATS
I cracked down on the treat options. It made a world of difference to limit the offerings. The following year, I decided that from now on, students could bring only pie, and only two varieties. We set up the options ahead of time. I let the kids vote, and it turned out that everyone was ok with either apple pie or chocolate pie. So those two flavors were the only ones that I allowed them to bring. That way, when it came time to serve (which I only did for one group at a time -- see the stations idea below), I only had to ask each child which flavor he/she preferred, then serve it.
I also made myself the only pie server. This was definitely worth it. I structured the class period so that the student activities ran themselves. This freed me up to serve one group at a time. Shockingly, I had much less mess to clean up at the end of the day. Another key: I required that the pies be brought in boxes. This meant that I could stack them by flavor, and eliminate the monstrous buffet line. ACTIVITY
So, how did I structure the class period so that the learning activities could run themselves?
Well, we math teachers love our Pi Day as much as the kids do. It's our special dorky thing, where we get a day to celebrate the coolness of math. I'm all about the fun and variety. However, I learned that for my own teaching style, I really need it to be more structured. I settled on the idea of stations, or centers, that would cover all the activities I wanted to do. This filled the time so nicely. No time was wasted. Plenty of learning was accomplished, but students still felt the joy of a fun celebration day.
For station 1, I laid out a bunch of different round objects, and had kids measure with string. They recorded their observations for a few and then derived an estimate for pi.
In Station 2, students receive a Pi Fact Sheet with fun tidbits about pi and its history. Using the sheet, they are asked to develop something creative to share a few of the facts. It can be a poem, song, graphic, etc....
I leave out special Pi-themed paper for drawing or writing. I also set out some blank white paper, so each student can choose.
Stations 3, 4, and 6 are fun worksheet-style stations. Students do a word search in one. In another, they apply pi to determine what size pizza is the best deal by finding price per square inch of pizza. The 3d one focuses on a spherical ball.
Even on a fun day, clear guidelines, defined expectations, and a little structure go a long way!
IDEAS & TIPS
1. Be sure to include different types of activities if you offer stations. Make sure you throw in a little something for the hands-on learners, something for visual learners, reading out loud for auditory learners, etc. I loved that each of my students had a favorite station, since I incorporated all of these.
2. Another Pi Day idea that I love is to include a charitable donation. This idea works for high school level, too! There are two great ways to do this:
RESOURCES
The stations that I use are available in my TPT store. I also have a copy of Pi doodle notes you may want to try! They offer a fun and colorful way to engage your students. Read more about doodle notes here.
Remember to set clear expectations. I announced ahead of time that if students got stumped on a station, they should wait until I had finished the pie serving, then I would come help them until the timer went off to switch tables.
I love the changes that I made, and now am even thinking of making it a 2-day event. I think it would be great to have time for both the stations AND the doodle notes in the same week. Enjoy Pi Day this year with your students! Hopefully, this post gave you a few tricks to have up your sleeve as you plan, implement and manage the day. Have fun! For more teaching ideas, free math resources, tips, articles, and updates, subscribe to the Math Giraffe email list below: You may also like...
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