Proofs in the Coordinate Plane

My Geometry students are petrified of proofs; if I even mention the word they start to freak out.  First semester we worked with various formal, two-column proofs, which my students really struggled with.  They wanted to make huge leaps in logic and being required to state every piece of reasoning, along with an appropriate mathematical justification, really challenged them.  We have circled back to proofs again, but this time we are working in the coordinate plane so we are using distances, midpoints, and slopes to prove things about various figures.  In order to alleviate some of my students' stress over having to do "proofs" again I decided to give them some guidelines they could use when proving things in the coordinate plane.

Here is a list of the main things we will be proving in this unit: 

• Triangles: right, isosceles, equilateral, right isosceles
• Quadrilaterals: parallelograms, rectangles, rhombi, squares
• That a segment bisects a side of a figure
• That triangles are congruent according to the SSS Congruence Theorem or the HL Congruence Theorem

We will do a few other things, but these are the big ones that I wanted my kids comfortable with.  The added benefit of this is that in the process of doing these proofs we are reviewing a lot of material from first semester, which is great because the EOC will be here before we know it.  

I had each of my students get a piece of construction paper and two 1/4 pieces of graph paper.  We began by listing 4 major guidelines for our proofs.  

1) To prove segments congruent use the distance formula.

2) To prove sides/segments parallel show they have the same slope. 

3) To prove right angles exist prove the segments are perpendicular by showing they have slopes that are opposite reciprocals.

4) To prove segments have the same midpoint use the midpoint formula.

We then listed each of the figures we would be working with and some ways to prove that particular figure existed.  I did not give them every possible combination of ways to prove a figure is a parallelogram, but two or three options that they could work with.  The idea was to give my students who were intimidated by proofs a structure to follow, and still leave some things unsaid so that they were not overwhelmed by information and I have some options left to challenge my high achievers to figure out.  Finally, we used the two pieces of graph paper to do two 'sample proofs' as a class.  This gave them notes and examples on one piece of paper.  I gave my students a suggested format for their notes, but some deviated from my suggestions and came up with their own format.  In fact, I'm going to steal the format of one of my student's notes for next year.  Several of my students were gracious enough to allow me to take pictures of their notes.  Here are just a few of them.

Student #1 - This was my recommended setup.  Notes on the front side divided into 3 sections, then examples on the back.

Front - Notes

Examples

Student Sample #2  - This one is set up differently.  This student did her second example as a flip up section over the triangle strategies.

Notes with examples on top

Ex. 2 flips up to show triangle strategies

Student Sample #3 - This one is my favorite and the setup I will steal to use next year.  This student did all of his notes on the bottom, then taped his graph paper with examples over his notes to create flip-up sections.

Notes 

Examples that flip up to show strategies underneath

 After finishing these notes my students did practice on their own proving figures in the coordinate plane.  Based on an informal survey of my students, the notes served their purpose.  My students used them, and reported that they were extremely helpful.  This is definitely something I'll use again next year.

Volume of 3-D Figures

Volume of 3-D figures is not a new concept for Geometry students (at least it shouldn't be), but in middle school they focused on finding volume of prisms (square, rectangular and triangular).  The high school standards focus on calculating volume of cylinders, cones, spheres and pyramids, as well as composite figures.  Students also need to be able to identify the cross-sections that are created by slicing these 3-D figures, as well as identify the 3-D shape created by rotating specific 2-D figures about an axis.

For the last week we have been focusing on these concepts, and as a part of this section I had my students create 3-D 'juice containers' that met certain requirements.  I grouped my students based on their performance on the last test (high performer with high performer and so on), and assigned each pair of students a shape with specific requirements.  Some groups were assigned a cylinder and given the total volume and diameter while others were given the total volume and height of the cylinder.  My highest performing students were assigned a cone and given the volume and its height.  Students had to take their information and construct a 'juice container' that met the given specifications.  They were also asked to come up with a brand name and to decorate their containers.

I went into this activity expecting my students to sail through it with no issues, however I quickly realized that was not going to be the case.  They quickly moved through the calculations required to determine the missing dimension, but then slowed down significantly.  Physically constructing a 3-dimensional object is something that my technology focused students do not have much experience with.  While they may be able to manipulate objects using technology, physically constructing them requires an entirely different thought process.  It took many of my students lots of discussion and a few false starts to figure out the best way to build their figure and to make sure they had all of the information necessary in order to do so.  With a few exceptions their final results looked good, and they were able to take from it valuable experience in creating something that fit within specific constraints.  This is definitely a mini-project that I will hang onto for next year.

Arc Length & Sector Area

We have been working with circles in Geometry the past several weeks and just finished up the standards on finding arc length and area.  I didn't want to do worksheet practice, but this is a topic where my kids need a little bit of practice using the formulas. I decided to have them draw their own circles and create sectors within them, then find the arc length and area of the sectors they created.  We then extended the activity and I had them 'discover' that circles meet the definition of similar figures since the ratios of corresponding parts are equal.  Finally, they worked backwards given specific information about an arc length or sector area to determine the radius or central angle of the sector.  I had butcher paper in various places around the room and had them post their circles by class period.  This gave them an ok to get up and move, which for some of my boys is necessary.  This activity took about 1 1/2 class days (we have 50 minute periods) because we took a few minutes to review how to use the protractor to measure angles.  My kids actually did well with the drawing and measuring parts of the activity, which to be honest I was a little concerned about.

Here is the assignment and some pictures of student work.

Circles Created by 6th pd

Close up of circles

Similarity Discussion

Working Backwards -- Sample 1

Working Backwards Sample 2

Arc & Angles in Circles Activity

So we are currently in our circles unit in Geometry.  We started this semester with the basic definitions related to circles, and for the past week and a half we have been working with the relationships between the arcs and angles created by intersecting diameters, chords, secants, and tangents.  Getting the kids to recognize the relationship and figure an arc or angle measure when there is only one thing going on isn't too bad, but when the figures get more complicated and there are multiple segments and lines intersecting in various places things get a whole lot more difficult.  We spent some time working on these types of figures this week and I found that being able to mark on a figure, erase, then mark again was helpful for many of my students, so I designed an activity that allowed them to do this.

I separated my students into groups of four and each group got a supply basket with dry erase markers and erasers (washcloths), as well as four copies of each of three figures.  The copies of these figures were each in a page protector.   Also in their supply baskets they got a page protector with multiple copies of the answer document.

Each student got a copy of the answer document, and they could choose which of the three figures they wanted to start with.  The figures were all in page protectors so students could mark on them with the dry erase markers as they figured various angle and arc measurements about the circle and their answers for specific arc and angle measures were recorded on the answer documents.  My students seemed to like being able to draw on the figures and erase things easily if they made a mistake or no longer needed a specific piece of information.  I also heard a lot of great math conversations as I circulated about the room checking on progress and answering questions.  Any activity that sparks good math discussion is a winner in my book.

Here are the figures that I had my students working with.

FIGURE 1

FIGURE 2

FIGURE 3

Answer Document
Note:  I had students add in the arc markings on the questions asking for arc measures.  

 When attempting an activity like this I highly recommend having group supply baskets pre-made.  I invested in a half a dozen shower caddies from Dollar Tree a couple years ago and they have been great to have for any kind of group project that requires supplies.  Sorting into the baskets is easy and they keep the kids from wandering the classroom in search of things they need. Clean up at the end of each period is also easy as the students just need to put everything back into the basket at each group of desks.

Real-World Applications in the High School Math Classroom

Why is it so hard to integrate real-world applications of math into the classroom?  So much of what I've been able to find in textbooks and other resources is so contrived it's ridiculous, and the kids know it.  It's hard enough to get get kids excited about investigating mathematical ideas, but when the scenarios we pose are so far outside the realm of what anyone would ever do it just makes it harder.

I try to incorporate legitimate (or as legitimate as I can make them) application scenarios in every unit in my Honors Advanced Algebra class.  Of course my kids whine and complain; how many teenagers actually want to do word problems in math class that require them to think?  I continue to persist though, and eventually the whining fades and my students become resigned to the fact that they are a required part of every unit.  Every year I get a little bit better about incorporating applications into the curriculum and it becomes a little bit easier to design scenarios that contain a little bit of realism.  One thing I have discovered is that anything that allows the students to build or experiment with something that is hands-on makes things much more interesting and grabs students' interest.

Here are a few examples of things I have done with my students in the last couple of years.

• Measuring angles of elevation with hypsometers in a repeater Geometry class.  

I don't know what standardized testing days look like everywhere, but there are days it means I have a class of kids for 3+ hours because we don't transition while major tests like the Milestones are given, and at the high school level these tests are only given in certain classes.  Let me just say that getting 3+ hours of math out of a group of kids who don't like math is an 'interesting' challenge.  My solution with a group of Geometry students I had last year was to make homemade hypsometers and use them to calculate the height of the flagpole out in front of the school, the height of the awning in front of our main entrance, the distance from the ground the science hall windows, etc.  We took the plastic protractors we use in class, taped straws to the straightedge side, tied fishing line through the hole in the middle of the protractor, and put a fishing weight on the bottom of the fishing line. You hold the protractor upside down, look through the straw and you can roughly estimate the angle of elevation.  Not an incredibly accurate device, however it suited my purpose and cost me about $5 since I already had a roll of fishing line in my classroom (it's amazing how often it comes in handy).  While I won't say every kid absolutely loved the activity, one of my boys made a point to tell me on the way back to the classroom that the hypsometers were "really cool and he never knew you could do stuff like that".  I called that a teacher win.

• Catapulting Gummy Bears in Advanced Algebra.

What is better than having candy in math class?  Being able to build a catapult and see how high you can launch it!  Quadratic functions model the basics of free-fall motion using the function            h(t)= -gt² + v₀t + h₀ where h = height above the ground, t = time from launch, g = acceleration due to gravity, v₀ = the initial velocity, and h₀ = the initial height, so we do an activity where we use this function to model the height of a gummy bear that has been catapulted vertically.   I break my students up into groups and have them build a 'catapult' they will use in order to launch their gummy bears.  I have supplies for them to use; normally things rulers, rubber bands, highlighters they can use a wedges, etc.  I try not to give too much guidance on building the catapult because part of the goal is to get them to think about what type of design would be effective in launching their gummy bear.  I also tell them the goal is launch their gummy bear as vertically as possible because we don't know enough physics and trig to deal with horizontal motion yet.  The more vertical they are able to make their launch, the more accurate their function will be.  After the catapults are built the groups begin their launches.  I normally have them launch from the ground and from something a couple of feet above the ground.  The groups have to video and time their launch (from release until the gummy bear hits the ground).  This gives them the initial height and a (h, t) ordered pair that they can use to determine the initial velocity of their launch.  I then have them answer a series of questions including writing the function that represents the height of the gummy bear over time, determining the maximum height reached by the gummy bear, discussing average velocity, and describing the domain and range of the scenario.  This activity takes a couple of days (we are on 50 minute class periods); it normally takes the groups a little while to settle on a catapult design and get it built successfully, but it is SO worth the time.  It is a lot of fun and hearing the kids discuss the scenario in mathematical terms and explain why the velocity of their gummy bear varies over time, as well as why the velocity as it is dropping is different depending on where the gummy bear is launched from makes my math teacher heart so proud! 

• Pendulums in Advanced Algebra

We are currently wrapping up our radical functions unit in Advanced Algebra.  I was hunting for a cool application scenario that I could use with square root functions and ran across information on a physics site about the relationship between the period of a simple pendulum and its length.  I took that and ran with it.  It's a very basic concept but a fun one because really, how hard is it to build a simple pendulum to model this scenario?  I begin by having my students do some basic research about simple pendulums (BYOD to the rescue), answer some theoretical questions about scenarios that might impact the period of a pendulum, and then we have some fun.  The kids then get to work together to construct a basic pendulum (that fishing line and weights come in handy here too).  Again, I give them very limited direction on how to construct it and where to suspend it from.  They conduct several trials in which they time the period of their pendulum and then average their results together.  They then compare the results of their experiment to the period predicted by their function and discuss reasons for any major discrepancies.  A word of caution, for this scenario I give them acceleration due to gravity in m/sec^2 and they tend to measure their pendulums in inches.  This leads to a great discussion about the importance of units and knowing what units they are working with.  

Feel free to take any ideas here and run with them.  I know that my best ideas are normally sparked by something I see another teacher using.  Also, if you have any great applications that you use in your classes I'd love to hear about them.  I'm always on the lookout for new things to do with my kids.

Desmos Activities

Thanks to one of our county curriculum coordinators, I have discovered Desmos Activities.  I have used the Desmos online graphing calculator for several years, but had never checked out the options for classroom activities.  I must say, I wish I had done so sooner (no, Desmos is not paying me to advertise).  There are quite a few pre-made activities to choose from; some made by the Desmos staff and some shared by teachers.  There is also a feature that allows you to create and share your own activities.

I took advantage of the ability to create your own activities and made one for my Geometry kids to practice with the properties of parallelograms.  Here are a few screen shots of my activity.

Multiple Select Question

Free Response Question

Card Sort

I checked out a classroom set of Chromebooks from our Media Center (we are blessed with access to such technology) and did this as a classroom activity.  My students enjoyed the variation in our routine, and I was able to see their work on my iPad and give feedback as they were working.  I will definitely use this activity again.

I have set up another activity for my Honors Advanced Algebra students to investigate the graphs of cube root functions.  We will do this after we return in January.  Hopefully it goes as well as the one that I did with my Geometry classes in November.  Here are a few screen shots from the Advanced Algebra activity.  

Free Response - Students can view each others' answers

Graph with sliders so students can investigate the impact of changing various parameters

Students create their own graph

Free Response - Answer viewable only by the teacher

Multiple Select Question

The links to my activities are below.  Check out some of the other activities available also.  There is lots out there!

Advanced Algebra - Graphing Cube Root Functions

Geometry - Properties of Parallelograms