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