Geometry Chat is BACK! Tonight on Twitter at 9PM EST

Please join us tonight for Geometry Chat at 9PM EST.  Geometry Chat will be held the first Thursday of each month.  Just follow the hashtag, #geomchat, to see all of the action!  Also, be sure to include the #geomchat hashtag when you reply so everyone can read your tweets.  This also ensures that the entire chat will be Storified for others to read later.  

Also, if you’re tweets are private, others will not be able to see your replies, so you can make yourself unprotected just for the chat, and the re-protect yourself after the chat is over if you would like.

We hope to see you all there!  

Geometry Construction Challenges – Compass, Patty Paper, and Euclid the Game

First of all, Euclid the Game is an amazing Geogebra based construction game!  THANK YOU to @mathhombre John Golden and @mrhodotnet.  It’s fun, challenging, and teaches students the basics of construction AND Geogebra all at once for the (double) win!  I told my students that it was a problem solving game.   So I would not tell them what to do, and I would not tell them the answers, even if they BEGGED (and they did – it was awesome).  But NO, I told them that they would have to THINK.  This was great for some of my freshmen, who want me to explain everything to them, step by step.  I felt today was a giant leap forward towards becoming independent learnings and being positively frustrated problem solvers.  After a while I did help students who were still stuck in the tutorial.  But basically, they all figured it out, and begged to go on.  It was awesome.

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We do constructions by hand at my school.  We start with a compass, then throw a little patty paper into the mix.  (Fun fact – Most students couldn’t guess what patty paper was actually used for.)  I have never constructed before.  Ever.  Not only learning it, but teaching it was a huge challenge that first day.  But now I am in the groove.  Although I did seriously underestimated the skill and dexterity the smart notebook compass took to master, much less all of the nuances.  After two days of painstakingly using that damn thing, I am still not even close.  At one point I just grabbed a regular compass and (gasp!) an actual piece of paper and had students gather around me so I could demonstrate.  I also “sketched” a construction freehand on the board.  Sigh.  That smart board hates me now, but I do think it will grow to love me!  I’m persistent.  I would love to throw in some Geogebra constructions to the mix, and that is why I introduced them to Euclid the Game.  But right now I am just trying to keep my head above water.  My mantra?  “Keep swimming Dory!” 

I loved Euclid the Game because students were learning how to construct things that we haven’t done in class yet.  I did Euclid the Game after only teaching them copying a segment and an angle.  So, as I am introducing new constructions now they are saying things like, “Oh yeah, you just need to find the intersection of those two points!”  It really makes my job so much easier!  It’s also more fun for them because they feel more involved.  

Finally, I’d love to give an extra special thanks to Jen Silverman who made a Geogebra Geometry Constructions “crash course” for me (note: for teachers, not classroom use)!  She is one of the kindest and most giving people in this community!  Also thanks to all of the other amazing members of the #mtbos that sent links to my construction SOS call on Twitter.  Again, I can’t even begin to express how grateful I am for this community.  How did anyone ever teach math before Twitter?  Thank goodness I don’t have to!

Calling All Geometry and Algebra 2 Teachers

I’m teaching Geometry and Algebra 2 next year and would love to start collecting resources.  I would love to know what your favorite resources are for Algebra 2 and Geometry, including great blogs, books, websites and activities!  Please put it in the comments.  Thanks to Sam, who will also be teaching Geometry for the first time next year, and has already started collecting resources!

My school uses:

  1. Geometry Textbook:  Michael Serra: Discovering Geometry
  2. Algebra 2 Textbook:  Discovering Advanced Algebra – Kendall Hunt

In addition to putting resources in the comments, if you blog about teaching Geometry and/or Algebra 2, please put your info into the form below so I can follow your blog.  I would also love to post a Geometry and an Algebra 2 blog collection.  I need to expand my community.

Thanks in advance!

Discovering the Pythagorean Theorem with cm Cubes

I usually use have the students cut out and color graph paper squares to discover the Pythagorean Theorem. I recently read a great blog post about using centimeter cubes for this and couldn’t wait to try it. I cannot find that post now to give that person proper credit but I know it was in my Feedly or on WordPress. So please let me know if it’s you and I will link back!

The cubes were much easier than cutting, gluing, and coloring those graph paper squares! The kids had more fun too. Who doesn’t love to play with snap together cubes?

Architecture Project – Design a Multi-Purpose Building

In seventh grade we do an integrated cross curricular culminating project called “Architecture as Activism”.  The students select a region that have learned about in social studies this year.  They then design a building that will address a need in that region.  Their building must reflect the culture, geography, and ecology of the region.  There are other aspects to the project as well, including an action plan and a letter, so it spans all of the disciplines.

I start the culminating project off in math.  We first learn about floor symbols and floor plan scale.  Before they start their culminating project, they do a mini-project so they can learn about and practice scale and area.  For their mini-project they get to design a new multi-purpose building for our school.  It has to be under 1,000 square feet and must include a bathroom, a storage closet, and some furniture.  Since 1,000 square feet is pretty small many of my students are adding outdoor features and courtyard.  Interesting building configurations make excellent area problems.  Also, students do not realize how small 1,000 square feet really is.  They have to make tough choices or be creative enough to make it all fit.

Design a Multi-Purpose Building Project Sheet

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Volume of Cones Discovery Lesson

I spend two days with my students discovering volume.  I do this because the volume formulas LOOK horrendous, however they make perfect sense.  I like for my students to discover these formulas, because then they do not think of them as formulas.  It just makes sense.

IMG_0554The first day I do volume of prisms, rectangular, triangular, and a cylinder using play-doh.  The next day I do cones and pyramids.  A few years ago I had a project where students made cones that fit into cylinders perfectly.  I still use these models as they are such a great visual for students.

 

I start by showing them a cylinder, and having a student write the volume formula on the board.  I have not asked them to memorize the volume formula of a cylinder, but after day 1, they all know it.  It’s magical.  I then pull a cone out of the cylinder and ask them what they notice and wonder.  They notice that the circle has the same circumference and the height is the same.  Excellent.  I then ask them what they wonder about the volume of the two objects.  Most groups decide that it is about 1/2, except for this group.

IMG_0550I then let them pour cheerios (or marshmallows) from the cone into the cylinder.  They notice it takes THREE cones to fill up the cylinder.  I have them talk with their table to adjust the volume of a cylinder.  Today, ALL of my students said, “Divide the volume of a cylinder by 3!” and one group even told me that was the same as multiplying by 1/3.

 

After I write their observations on the board, I show them the actual formulas from their Geometry Booklets and ask them to write down what they learned today on that page.  I think this is the most important part because I don’t want them to forget HOW they discovered the formula.  Today a student said, “We just invented already invented math!”  It was so awesome I had to film it, of course.


Now that is a magical day in math.  Please, please, please do discovery learning with your students.  It does take more time but will actually will save you more time in the end because you won’t have to re-teach it!

Student Created Math Hunt of Area of Irregular Shapes: CCSS 6.G.1

I racked my brain for at least a week trying to come up with something interesting for this.

CCSS.Math.Content.6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

I had a few worksheets in my files on finding areas of irregular shapes.  Boring.  I even had a Math Hunt that I created last year (from a similar worksheet.  But, that was pretty dry as well, the only difference is they get to walk around the room.  It’s been a tough week, but these activities just weren’t inspiring me.

My students do love a good math hunt, but they also love being creative.  Also, I didn’t just want them to find areas, I wanted to have them do the higher level skill of finding unknown measurements if they are given the area.

I decided to have my students create their own math hunt.  The instructions were:

  1. Fold a sheet of paper in half, creating a half sheet card.
  2. Draw either a regular or standard shape on the bottom half of the inside of the card.  Decorate this shape however you want (make it something), and then color it with colored pencils.
  3. Give your card a title on the outside cover.
  4. Find the area of your shape on the back of the card (show all steps).

After the pictures were drawn and decorated and the areas found, they switched cards with a partner to make sure they were correct.  I ended up with cards in several different categories.

  • Find an unknown measure given the area.  These were for standard shapes like (circle, triangles, quadrilaterals)
  • Find the area of irregular objects
  • Find the area of an object where you have to subtract out another object.

After all of the cards were finished, I had students partner up and check each other’s work.  I created a partner check sheet for this to ensure that they were thorough.

Create A Shape Partner Check Sheet

I then taped them up all around the room.  Students moved around the room, and got to pick one card of each type to work.  After they worked one of each type, they got to try to pick one of each shape.  I also created a foldable worksheet for this so that they would not forget to pick different kinds of shapes.  Create A Shape Search WS

They enjoyed creating the cards and working with the other students creations as well.  Plus, they got a variety of practice on different aspects of area.

One drawback to this activity was that I did not get enough triangles and trapezoids.  If you do this activity, I would randomly assign students a shape so that you are sure to get a good variety.

Geometry – Google Forms To Pre-Assess

Screen Shot 2013-04-22 at 8.26.20 AMIt’s time for Geometry in 6th grade once again.  I love this time of year because Geometry is so much fun!  Each year, I have my student create Geometry Booklets.  They are half sized booklets that we make from folding a small stack of 8.5″x11″ copy paper in half.  The booklets are a quick way to review past topics without spending days reteaching the topics or taking notes.  Each Geometry lesson I teach goes into the booklet and then they make a wonderful Cubism cover in Art Class.  Taking notes in the booklets are a change from their usual math notebooks.  Change is good in the last weeks of the year to keep the student’s interest up during “Spring fever”.

Every year I jump into the Geometry books, but this year, I wanted to know what they remembered from 5th grade (as I have students from multiple elementary schools) before I started teaching.  After their last test I gave them a Google Survey asking them if they knew certain geometry definitions.  I had them answer yes if they were certain that they knew the definition, and no if they did not know if (or even if they weren’t sure).  The form looked like this.

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Screen Shot 2013-04-22 at 8.36.49 AMGoogle forms will display the results in pretty pie charts like this.  However, I prefer spreadsheet summaries, so I tallied up my own results.  I counted the results with a CountIf in Google Forms and then conditionally formatted with color to show me at a glance what I really needed to focus on.  The results looked like this.  The red 20 means that 20 of my 32 students did not know the formula for the circumference of a circle.  The green 0 for parallelogram means that all of my students know what a parallelogram is.

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I was surprised that so many students did not remember the area formulas (I had similar results for the areas of triangles and rectangles).  I thought many more students would have remembered the area formulas from 5th grade.  This shows me I will not only need to focus on the area formulas, but more frequently reinforce if I want my students to really learn and then remember these formulas.  Area is such an important topic throughout all “areas” of mathematics.  🙂

Creating Tin Men to Explore Surface Area Project

I read about the amazing Tin Man project on Elissa’s blog Misscalcul8.  Please go there and read all about it in more detail!   I had to duplicate it for my 6th grade students.  They love being creative and surface area is tricky.

At their request, I let them make a tin man or animal.  The students chose their materials (they had to have a cylinder, a rectangular prism, a sphere, and a cone).  They then had to measure their materials and find the surface area of each item.  (My Tin Man Project Worksheet).  After the surface areas were calculated, they added it all up to find the total surface area of their tin man and how much foil they would need to completely him.  I created a Google Spreadsheet to automatically check all of their answers according to the measurements they had recorded.  This was essential for grading ease since every student had different materials and thus different measurements.  Then, they taped their tin man together, measured and cut the foil, and covered their tin man/animal in tin!  After they finished, I had them do Elissa’s reflection as well.  To save paper (and printing issues that students always have), I uploaded the questions to Google Documents.  Students had to copy the document to answer the questions, and then share it with me for grading.

A best aspect of the project was applying the aluminum foil to their tin men.  The students had calculated how much foil they needed to cover their man using surface area.  Once cut, if the amount of foil was too little or too much, they had to meet with me to talk about why this happened.  It was math in action!

Another great piece of the project was coming up with how much foil to cut.  They had to take their total surface area and divide it by the width of the foil roll (30 cm) before cutting the foil.  I made them figure this out before cutting the foil to deepen their thinking about area.  They then had to talk about why they divided my 30 in their reflection.  It was great to see that every student did understand why they divided by 30.

This project took longer than I thought it would, but was worth it. On the unit test,  my students scored better on surface area than on volume, even with the Play-doh activity!  Their biggest challenge were actually applying the foil to the tin men.  Next year, I will have them apply the foil first, then tape their tin men together.  They also wanted more time to decorate their tin men, but I had too much I wanted to squeeze in at the end of the year to give them an extra day.  Next year I will try to build in one more day.

I can’t thank Elissa enough for this project idea OR all of the help and suggestions that she gave me via email.  I love all of the fabulous math teachers in my PLN and am such a better teacher because of them!  Reach out teachers and connect with each other on Twitter.  Read their blogs, try their ideas.  It makes teaching a blast, and the students love it.  Everybody wins!

** Edited 4/27/16:  I added the Google Spreadsheet to check the formulas.  

Volume of 3D Shapes with Play-doh

When my students are excited about what they are doing in class, they are engaged.  They will listen and best of all, learn.  The kids were ecstatic when they saw the Play-doh.  They couldn’t wait to get their hands on it!  The benefits for learning were amazing.  All I had to say was “Remember the Play-doh?” and they knew what to do for any volume problem.

Supplies per student:

  • One fun-sized Play-doh
  • Ruler
  • Plastic knife
  • Ruler

Instructions:
Many of my students had heard somewhere along the way that volume equaled length times width times height.  So, I started by having them make a rectangular prism out of their Play-Doh.

Once it was made, I had them draw a picture of it in their Geometry Booklets.  Then they measured and recorded the length, width, and height of their prism in centimeters.  Next, I had them cut their prism into 1 centimeter lengths.  They observed that the cross sections were squares.  I had them find the area of this square.  We talked about how many slices they had.  So if their area was 6 centimeters squared and they had four slices how could they find the volume of the box?  They quickly deduced that they needed to multiply the 6 by 4 to arrive at 24 cm cubed.

I repeated this for a cylinder and a triangular prism.  We observed what the cross sections were, found the areas of our cross sections, and then found the volume.  To sum it all up, we talked about how volume of a prism or cylinder is the area of the base (cross-section) times the height (number of slices).

It was so much fun!