Segment Addition Exploration with Cuisenaire Rods

This weekend, I was working on a special project, and I was pushing myself to develop a conceptually rigorous activity for the segment addition postulate. When you search for segment addition postulate activities, you find a lot of problem sets, a lab or two on paper and Geogebra, and a few other odds and ends. Don't get me wrong - some of these are great. However, I really like for kids to build conceptual understanding through hands-on exploration. So, I wrote the following activity:

First, students are given a set of Cuisenaire Rods. They will need 2 lime rods, 2 orange rods, 2 red rods, 2 fuchsia rods, and 1 of every other color. This exploration will likely work best in groups, so you could easily give one group of students a set of rods, rather than each student a set. Note: If you don't have Cuisenaire Rods, You could probably cut down some popsicle stick or something. It'll just be a lot of cutting.

First, explain to students that this is a complete set of Cuisenaire Rods:

It is their job to find the length of each rod without using a ruler. Instead, they will have the actual rods to manipulate at their desks as well as corresponding algebraic expressions for the length of many rods.

Next, students will need access to the problems. I have the problems in a PowerPoint file, but I'm thinking each group probably needs a handout at their seat so they can work at their own pace. Regardless, here are the questions as a converted PDF file. If you want the PowerPoint, click HERE.

Ask the kids to work through the problems and solve. You will need to circulate around the room and prompt students when they are stuck. This activity will likely come easily to some and less easily to others.

After students have successfully solved for the lengths of these bars, the should check their solutions against the whole set to see if their solutions make sense. They could also double check themselves with a ruler( Note: measurements are all in cm).

Then, the last four slides of the PowerPoint is hopefully where the geometric representations and algebraic thinking gels. What similarities did students notice? Differences?

The last three slides takes you through the three different types of problems encountered in the activity and asks students to discuss how they viewed and solve that particular type of problem.

I am excited about trying this lesson in a classroom soon. Special thanks to @pamjwilson and @mrdardy for their help in developing this activity.

Teaching Segment Addition with Types and Socrative Task Cards

If you are interested in this post have haven't read Teaching Segment Addition with Types, I strongly recommend you skim that first before reading this post.

Now, moving on to the fourth day of segment addition and betweeness postulates, the students just needed to practice.To do this, students completed some Socrative Practice problems using the three types. Basically, each problem becomes two questions. In the first question, students must classify the type of Segment Addition problem. Then, in the second accompanying question, students must apply the strategy needed to correctly solve that particular type of segment addition problem.  Here is an example:

If you would like to use this Socrative practice assignment, just use the following code: SOC-17443219

Teaching kids to first identify the type, recall the strategy, and then solve in this very systematic way proved so beneficial for my kids and for further applications later on in the year.

Teaching Segment Addition with Types

In the past, as I had planned for segment and angle addition topics, I would almost always expect a bumpy ride. I always started the unit off with a lab/discovery activity but found my students could not transfer what they had learned in the lab to different types of segment addition problems. Therefore, I chose to give them additional guidance so as to smooth the seemingly difficult transition for them between algebraic reasoning alone and the marriage of algebraic and geometric reasoning.

I came up with three primary types of Betweeness problems. While not every problems fits nearly into one of these three categories, most do. And, by teaching my students and classifying the different kinds of geometric relationships that could exist helped them to build a firmer geometric reasoning foundation almost immediately. The difference between this lesson and lessons in prior years was quite large.

The three primary types were differentiated in the slides I gave in class. A PDF of those slides is included below:

I introduced these types over the course of two days. On the first day, we only did Type I and then quite a bit of practice. On the second day, we did Type II and III problems and practice. Then, on day three, the types were combined. Students had to classify the problems by type and then solve them. The practice sheet for their notebooks was the following:


I will definitely continue to teach the Betweeness postulates like this, simply because it helps kids catch onto the geometric reasoning layer so quickly.

Multiple Representations: Midpoint and Endpoint Socrative Practice

One value I push and is a common thread throughout my lessons is the idea of multiple representations. I think, far too much in education, we teach kids "the easiest way", which may be the easiest way for you, or for most kids, but it could very well be the most difficult method for others. I make every attempt to analyze math topics from multiple perspectives.

If I am a strong algebra student/equation solver, how would I best understand this concept?

If I am a strong visual student/grapher, how would I best understand this concept?

If I am a strong tactile student/manipulative-user, how would I best understand this concept?

I know lots of philosophies put emphasis on catering to the different types of learners (visual, kinesthetic, auditory), but that's not exactly what I'm trying to do here. You see, I am an auditory learner - very auditory. In fact, a specialist "tested" me and said I was the highest auditory result he'd ever seen. Evidently I'm weird - nobody that knows me well is shocked.  But you see, it's a beautiful thing. In a way, we are all weird. We all learn differently. Some of us are about the minutiae - detail people. Some of us are big picture visionaries. How do these types best learn? I don't think we necessarily can have a perfectly ironed-out answer here. However, we should give students choices in how they can approach a problem

When I taught midpoint and endpoint this year, I stressed using reasoning and these multiple approaches to arrive at an answer. Students were taught how to find midpoint and endpoint (1) algebraically, (2) graphically, and (3) on a number line with marker manipulatives. Some kids heavily preferred the algebraic approach. Some initially avoided the algebra like the plague. The great thing is - by the end of the lesson set - students saw how all the methods were interwoven. Because of students' strength in one approach and the interconnectivity, these initial strengths eventually translated into a gradual strengthening of their approaches in other, more weak, methods.

To facilitate this type of practice, I used Socrative (I'm a huge Socrative fan) and a strategically assembled worksheet.

Socrative Code (if you'd like to use this activity): SOC-18350734

Here is the handout. For each problem kids had to (1) specify if the problem was an endpoint or midpoint problem (I know this is obvious, but this question seems to help the kids focus), (2) prove the calculation algebraically, (3) prove the solution on number lines, and (4) prove the solution on the coordinate plane.

Handout Download: HERE

Enjoy! Please let me know if you have any feedback on how to improve this activity!

Google Maps & Midpoint/Distance Formulas Project

As has been the theme of many of my posts this year - my geometry students are all pretty much 1:1. So, I keep trying to reimagine and reinvent what technology makes possible for my kids this year.

Last year I did a project based learning unit for midpoint and distance. Kids had to design and make water parks. That was a lot of fun, but project based learning is hard work, and frankly, the way that project is designed - it took a lot of cash to fund the purchase of a bajillion glue gun sticks, foam, felt, cardstock, popsicle sticks, etc. My school just doesn't have the cash this year, but - having a positive mindset - what do we have? Technology.

I'm sure most of everyone has read/heard about the optimized Road Trip Map produced by some data scientists. If not, you can (and should) read about it HERE.

Maybe it's just me, but I think there is something inherently interesting about a great American Road Trip. I started wondering how I could capture this for my midpoint/distance formula project.

First, there are way too many locations, so I narrowed my project down to the Northeast. It starts at my high school and returns there. I used Google Maps to record the address, latitude, and longitude (in decimal degrees) of each destination.

Here is my file. Feel free to use it, but you'll want to change the start/finish from my high school. To make it easier, tell your kids you are flying into the Lexington,KY (LEX) airport and reroute accordingly.

Map: HERE

I debated having kids plot these locations themselves. However, we don't have Google Classroom or GAFE. The lack of these resources makes having students collaboratively working in Google much, much harder. If you have these resources, let them plot the destinations themselves. It takes a little work. You need to enter all points of interest in a spreadsheet (I did address, latitude, and longitude) and then upload as a layer in My Maps. Then, you can create driving directions as additional layers. If kids had these files and could interactively work with them, I'd also have them upload a photo of each destination within the map. This year, I'm trying to not only teach math but also teach students how to use digital tools well. Very few of mine know much about technology except for how to use their phones.

Okay. So, the premise is that we are road tripping and obviously driving to each destination. Students will click on the location marker and record the important info about each place (street, city, state, zip, latitude, and longitude). Then, for almost all locations, they have to Google and determine where they are. I only gave names for strange places to search for by address - like historic districts. They'll record all this information on their handout.

Handout: HERE

Then, they have to do a little research and find out what's special about this place. For each destination, I've created some fill-in sentence-type web quests for students to complete. Each destination also has a virtual tour (except I'm still looking for something about the historic district in Annapolis Maryland) for students to take.

Here is my list of virtual tours:

#1 - Start @ High School

#2 - Mammoth Cave - https://youtu.be/fTNlZl7-s4w

#3 - Spring Grove Cemetery - https://youtu.be/mbaANqveN90

#4 - Fox Theatre (Detroit) - https://youtu.be/QSwhq7ABUMU

#5 - The Rock & Roll Hall of Fame - https://youtu.be/LvMSfrwbhyE

#6 - Shelburne Farms - http://www.shelburnefarms.org/visit/virtual-tour

#7 - Mount Washington Hotel - https://youtu.be/NxyPPLoEE24  AND http://www.historic-hotels-lodges.com/new-hampshire/mount-washington-hotel/mount-washington-hotel.htm

#8 - Acadia National Park - https://youtu.be/MQTA8HU07zc

#9 - USS Constitution - https://www.google.com/maps/@42.3724797,-71.0566936,3a,75y,121.69h,75.55t/data=!3m7!1e1!3m5!1sJjoh75kl7doAAAQZN_qRbA!2e0!3e2!7i13312!8i6656

#10 - The Breakers - https://youtu.be/uxhsTmzqheQ

#11 - Mark Twain House & Museum - https://www.marktwainhouse.org/house/floor_plans.php

#12 - Statue of Liberty - http://www.nps.gov/featurecontent/stli/eTour.htm

#13 - Liberty Bell - https://youtu.be/bWVQS7hpr34

#14 - Cape May Historic District - https://youtu.be/491RI34usvw

#15 - New Castle Historic District - https://youtu.be/AHX_EORCfkA

#16 - Colonial Annapolis

#17 - The White House - https://www.whitehouse.gov/about/inside-white-house/interactive-tour

#18 - Mt. Vernon - http://www.mountvernon.org/the-estate-gardens/the-mansion/mansion-virtual-tour/

#19 - Lost World Caverns - https://youtu.be/jJHRkiyNDqo

The last page of the packet has some summary information the kids have to compile.  For example, for each leg of the trip, the kids have to calculate the straight line distance using the distance formula (then use a conversion to get the value to miles) and the time it would take to travel such distance at a constant rate of 60 MPH. They have to compare this to the actual distance and figure out why these straight line distances don't make sense.... i.e. the world is round and thus we should really calculate these distances using great circle calculations.

This reasoning is the best way I could figure out to get kids practicing the distance formula with latitude/longitude and still have them reason/make sense of why the numbers are so off. If anybody else has a better way to integrate the distance formula into this, please let me know!

Using Google Slides as Group Whiteboards - Angle Puzzles

If you've read my other posts, you've probably gathered that my geometry classes are all 1:1 now. I had a few kids with laptops last year, but now - virtually all of my geometry students have them.

This new opportunity pushes me to rethink the way I structure and deliver all content. In previous years, lots of practice time was somewhat traditional - student would do a problem, instructor would provide feedback. The problem here is the delay inherent between the pencil/paper feedback cycle. Don't get me wrong - I'm usually all over the classroom helping, prompting, asking questions, etc., but there are still delays with this traditional method (I mean there are 30 kids and 1 of me).

Enter our 1:1 devices. I automatically ask myself how I can restructure the lesson so it's more effective, students get better/more immediate feedback, and it's truly more personalized. To do this, I analyze the takeaways I want students to have, the structure and necessary supports needed for a particular lesson, and the way in which I need students to practice.

One such lesson is my annual day of "angle puzzles". Basically, these are complex drawings composed of sets of parallel lines and transversals that ask students to reason and apply their knowledge of angle pairs. HERE are the puzzles I use. Yes. This is a link to Teachers Pay Teachers. No. I'm not a fan, but I couldn't find them anywhere else when I was looking a couple years ago.

Having used these angles puzzles for a while, I knew the following:

1. Low to low middle kids struggle with angle puzzles and need lots of peer/instructor support. Groups are a good thing.
2. On these puzzles in particular, students get "stuck" and either disengage or wait on instructor help.
3. These kind of puzzles need lots and lots of instructor feedback.
4. When kids work in groups on these puzzles, one student tends to dominate the group.

To address these problems, I decided I needed the following features from a technology solution:

1. Collaborative work from multiple devices on same file
2. Instructor ability to see student work on own screen at all times
3. Instructor ability to provide continuous feedback to groups/teams
4. Groups/teams need opportunity to respond to feedback
5. Ability to include multiple puzzles to groups in one file

Having analyzed these needs, I determined that Google Slides could offer all these features. So, I took the snipping tool and created an image file for each angle puzzle. Then, I inserted these files as the backgrounds of individual slides. This is important. Since the image is the background, students can't move the puzzle around or delete it. Since we don't have Google Classroom, I took the slides link and shorted it with bit.ly before providing it to the groups.  Here are some samples of my students' work:

Takeways: My students (for the most part) loved this activity. Google Slides worked just how I'd hoped. The one issue I found though, was when the students inserted the text boxes. In image #2, the text boxes are large (standard size when you click to create the text box instead of drag to create them). If students do not resize the text boxes, they have trouble clicking on the correct text, since the text boxes begin to overlap. The group in image #1 have resized their text boxes appropriately and had few issues. Image #3 shows the back and forth comments/feedback between students and me.

It was a great day!

Pictionary with Socrative

I meant to write this post earlier in the year (say - oh - August?), but that obviously didn't happen, so here we go...

This is my fifth year teaching geometry, and even though I love teaching the course, the basic naming and definitions portion at the front of it can often be boring. I mean, sometimes, it's really hard to spice up how to name a plane.

After we've discussed basic nomenclature, I always ask students to flip the process, meaning I give them a description, and they give me a drawing. This part is usually a little more fun. In the past, I've made index cards with the situation on the front and a possible solution on the back. One partner reads the text to the other partner, who does his/her best to draw the accompanying image on a whiteboard. I then run around like a chicken with her head cut off trying to look out for misconceptions and errors. Hey - it worked, but you never really knew when you had missed out on that one special moment... you know, the one where something was slightly incorrect, and you, the instructor, could have posed a question or a clarifying statement that could have really helped the kids build a solid understanding of the content.

This year, my students are 1:1 with Dell laptops. No. They are not touch screen. And yes - I'm bitter (but don't tell anyone). Think of what else my kids could do if they were touch capacitive. Anyway - that's not important right now. I'm off track. Here's what we did...

I used @sandramiller_tx 's graphics, since hers were digital and mine were hand-written. Go visit Sandra at https://tothemathlimit.wordpress.com. Anyway, I took these images and put them in Socrative. I used a standard multiple choice quiz format - with only choices of A or B.  Choice A was to be selected when student answers matched the exemplar or when they didn't but partners were able to discuss and remedy the mistake. Choice B was to be chosen when the partners disagreed or did not understand the proposed solution. The way the activity worked - one student had the laptop and read the written explanation. This student was also looking at the proposed image solution. The second student in the pair had a whiteboard and was drawing as the first partner was giving the directions. When partner 2 was finished, they both compared the whiteboard to the computer solution and discussed then appropriately chose either option A or option B. On my teacher dashboard, A answers showed up as green (I marked them the correct answers), and B answers showed up red. Even from a distance, I could immediately see when a student was struggling and who to assist. It's almost the same idea as the Red/Yellow/Green cups, but less troublesome.

Here are some screenshots from the activity:

If you'd like to try this activity, the share code for this Socrative quiz is  SOC-17217490.

Using Google Slides For Interactive Card Sorts

Edit: Desmos to the rescue! Access the Desmos version of this card sort HERE.

Name something. There's an app for it, right? Evidently, I've discovered the one classroom need that no programmer has addressed. Card sorts. Lots of teachers have started using card sorts in classrooms, since they are a higher-level activity and require more intellectual might than a standard question/answer kind of practice.

I looked and looked and looked. I couldn't find an app that would allow me to design a card sort for my own content. I will, however, say that Quia has a
card match program but doesn't offer a true card sort. So, I decided to try and redesign the use of a pre-existing tech tool. I wanted this card sort to be collaborative between two partners, but I didn't want them to share a device. I wanted partners to have equal control of the sort - not the primary device user to dominate the activity. I also wanted to be able to label the left side of the "sort board" vertical angles and the right linear pairs.

Really, if Padlet would allow other users to move posts, it would probably be the best choice here. But, alas, it won't. I did tweet the Padlet folks to recommend this feature (get behind me here MTBoS)!

Anyway, I decided to use Google Slides. Technically, it ticks off all my specifications. Multiple students can access the sort; I can label the left and right sides of the sort board using a slide master; and students can drag images to the left and right to sort them. I designed the original file and made 12 or so copies of the file for each group I anticipated on having. Then, I took those links and converted them to Bit.ly links for ease of access. I made group tents with their group numbers and individual Bit.ly links.

Here is a picture of the first sort students completed. It was quick and I used it as a formative check that students had mastered/learned the highlights of the previous day's notes.

After students completed this sort, I had them raise their hand, and then I either physically went over and checked their sort or I did the same thing virtually (have I mentioned how much I LOVE Google?)

When students had this correct, they clicked on slide 3, which was really the heart of the activity. They needed to sort various examples of vertical angles and linear pairs into the two distinct piles. I actually uploaded my graphics into Padlet first (I forgot non-owners in Padlet can't move the posts). Then, I took screenshots of my cards from Padlet and pasted them into Google Slides. This was nice because I had automatic card numbers. These card numbers are how I always check card sorts. For instance, I knew that cards 2, 4, 10, 12, 13, and 15 (I think - going from memory) belonged under linear pairs - making my formative check really fast.

All in all, this activity accomplished exactly what I wanted - it just wasn't perfect.

Things I liked about making card sorts in Google Slides:

• Students can easily collaborate using separate devices
• Enabled me to make multiple card sorts within each document, and students could easily progress between them
• Allowed students to freely move the cards from place to place
• I could check the sorts pretty easily given I'd pre-recorded the appropriate cards for the vertical angle pile and the linear pair pile.

Things I disliked about making card sorts in Google Slides:

• When kids move images in Google Slides, they often accidentally delete them, make them really tiny, and crop them 
• Kids had to use the "undo" button a million times so I could reset the sort for the next class/group
• If kids used two devices, it turns out that the undo feature only partially works. I think what happens is that kids can only undo the actions they have personally taken on their device, but I didn't have time to further investigate. I just had to go in and copy/paste the original pile back in. This was safer than just having kids move them back to the middle because they often "lost cards". See bullet point #1 above.
• The usual beloved layering of images in presentation software became a problem. The cards would layer and students couldn't figure out how to display the card number so I could easily grade them. Sometimes I'd have to go in and manipulate the cards so I could do a quick formative check.

So... would I do it again? Yes... I think so. It is definitely not ideal, but it is better than cutting out 5,280 squares of paper. If you're reading this, and you know a programmer, get them on this. I would LOVE to chat with someone about the necessary specifications an app of this kind should have.

Volumes of Prisms Play-Doh

Friday was a fun day (mostly). For one, it signaled the end of the last full week of school. Yay! Let's face it - we all need a break. It was also a fun day because use used Play-Doh in geometry. The lesson was an introduction to volume of prisms (we focused on rectangular and triangular) as well as volume of cylinders. This lesson was modeled after Julie's (@jreulbach)  lesson at ispeakmath - HERE.

The kids had a regular-size cup of Play-Doh for every 2-3 people. They split the play-doh and had plenty. First, we crafted rectangular prisms. Then we sliced them into 1 cm cross-sections. Students analyzed the shape of one "slice" and found its area. We decided, then, that we could simply multiply this area by the number of slices we had (it's height) to find the volume. We repeated this process for triangular prisms and cylinders. Students derived the formulas for the volumes of all three. One student informed me that "[Volume] was the easiest thing we've done all year". Good. It's easy because they really understand the formula. Great day.

Some kids discovered some other geometric properties... Hello octagon!

...And we still managed to have a little fun!

Finishing Up Surface Area - Spheres

Today was our last day of surface area. If I had time, I'd actually do one more day, since we really need a day of practice. I just can't take one more day, though, because I have to get in volume before finals. I only have two days for volume as it is (and then two days of review).

Today, we did a hands-on lesson on the surface area of a sphere. The idea was guided by Jennifer Wilson 's (@jwilson828) blog post on the Surface Area of a Sphere.

To do this activity, I knew I needed to get my hands on lots of oranges, since I have 110 kids in geometry. I called our local Food City, and -bless them- they donated 55 oranges. 1 orange per group of two. Thanks Food City!

The kids really enjoyed this activity. It was messy, but i think - sometimes - learning should be messy. The janitor didn't agree. Oh well. You can't win 'em all. The biggest problem I kept running into was that kids were skimping on filling their drawn circles. There would be lots of white space, and the kids would think they filled 6 circles. I would go around and have the groups start disassembling one circle, using those pieces to fill in the gaps.

At the end of the day, the kids had fun, and they really KNEW the formula for surface area of a sphere. Oh - AND my room smelled strongly of oranges instead of smelly high school kids. It was lovely.

Area and Perimeter Culminating Project

School is nearing its end for this year. I'm not mad about it. The kids are basically already hanging from the ceiling. My last unit is always area/perimeter and volume/surface area. Last week, to finish our area/perimeter portion, I assigned two geometry sections to complete the Apartment Remodel Project from Sarah at Everybody is a Genius. I modified her assignment slightly in order to incorporate the perimeter piece. All I really did was add a perimeter bullet and blank to the answer sheet.

My kids loved it. They mostly worked in groups of three to complete the assignment. I told them they were roommates (There are three bedrooms in the blueprint I used. It's the same one from Sarah's blog). I also redrew the 3D model on floorplanner.com. (Link to mine: HERE) The way the kids got into the flooring selection process and argued over it was hilarious. When one group finally calculated their total flooring costs with tax, the boy in the group screamed and said he was going to have to work overtime and buy on credit. He also noted that he was doing it "for the kids", asking his female partner where he wanted her to lay the floor when discussing the sample selections. I swear we all had too much fun with this project. It's probably my favorite project I've ever assigned. The kids learned SO much and had lots of fun with it. I think it was a good reality check for some of them, too. DO THIS PROJECT!

Composite Area Assembly Lines Activity

Since it is the end of the year and my kids are extra disengaged in class (read: way hyper and ready to get out of here), I've tried to really make this last unit fun. In some ways I've succeeded, and in some ways, I've failed. However... this activity was a big success.

Before I did the activity, I split my kids into groups of 3 - heterogeneously by skill level. One person in each group needed a phone for the purpose of reading the QR codes. This person was partner #3 Then the other two numbered off as partners #1 and #2. I borrowed ( I read their terms of use, and it said it was fine.) a bunch of composite figures from Math-Drills.com. However, I modified them a lot. Several of the measurements made no sense. For instance, there were several "right triangles" where all three side lengths were given, but they didn't satisfy the Pythagorean Theorem. So, in publisher, I used a lot of white rectangles to cover up nonsensical measurements. Anyway, partner #1 finds the area of shape #1; partner #2 finds the area of shape #2; and partner #3 finds the area of shape #3. Partner 3 then adds the three answers together to get the total area of the figure. Finally, the partner uses his/her phone to scan the QR code. If the answer is correct, they go to the next card. If the answer is incorrect, the card goes back to partner #1 to check his/her work and find the error.

My initial hope was to laminate these cards, so student could write their partial areas on the cards before passing them to the next partner. However, this didn't happen. I just ran out of time. That's my next step.

I'm attaching the PDF. I think I've fixed all the weird measurements. If you catch one, be sure to let me know.

Also, I know problem #1 is weird... it's really just two steps - area of a trapezoid and of a rectangle, but I wanted to split it into three to show kids you could do it with rectangle/triangles rather than the trapezoid.

Download the PDF File (HERE)

Editable Publisher File (HERE)

An Idea for A Culminating Volume/Surface Area Project

As of this week, my geometry classes have begun the final unit of the year - area/perimeter and surface area/volume. I threw out my old unit and am starting anew (let's face it - the old unit was junk). I have some neat hands-on stuff lined up for learning all these concepts. I also am using the apartment project from Sarah at Everybody Is a Genius as my culminating area/perimeter project. I have been considering what to do for the volume/surface area part of the unit as far as the assessment goes. I've done a lot of googling and have found some interesting projects but none that really got at the depth of inquiry and thinking that I want.

So... here's my idea. I'll put the kids in groups of 4. Each group will receive 12-15 nets of 3D figures. I will already have calculated the volume and surface area of each figure. Using combinations of these 3D figures, I will give the groups certain composite 3D figures to assemble with given volumes.

For Example (just making these numbers up):

Group #1 (Each group will have different figures/scenarios):

Combine your figures to make composite 3D shapes with volumes of

(1) 66 cubic in.

(2) 54 cubic in.

(3) 40 cubic in.

(4) 82 cubic in.

Here, I will assess with a rubric for correctness and calculations before students move to part II.

After student groups have successfully grouped their 3D figures, they'll then have instructions on how to assemble them according to surface area calculations.

For instance:
(1) Assemble the figures with a volume of 66 in sq. so that they have a surface area of ________.

(2) Assemble the figures with a volume of 54 in sq. so that they have a surface area of ________.

(3) Assemble the figures with a volume of 40 in sq. so that they have a surface area of ________.

(4) Assemble the figures with a volume of 82 in sq. so that they have a surface area of ________.

The students will tape their figures together appropriately to meet these specifications. I will then grade groups on the correct assembly of their composite 3D figures with a rubric.

Okay, MTBoS.... will this work? What would you change? All input is appreciated.

Right Triangle Trigonometry with Clinometers

I've been going out of my way this year to make my geometry classes more applicable. At the first of the year my kids really struggled, so I've tried to re-invent the way I teach. I'm doing a lot more application and less theory. I'm not saying that's the way it should be done, just that this is what works with my particular group of students.

Today we went outside and measured the heights of some of the taller objects around campus. We used homemade clinometers to do this. Since we only had one measuring tape, my co-teacher went outside and pre-measured distances from the building to the point where kids were supposed to measure their angles of elevation. (Note: next time, we'll spray only non-permanent surfaces like gravel. That green 10 ft mark may never come off the sidewalk.)

Anyway. The kids loved it. It didn't seem like work. Each class also discovered that their measurements didn't "make sense". They were too short. They decided that they measured from eye level up, so we needed to add on their heights to determine final measurements. This went great. Even my students who usually really struggle enjoyed it.

Clinometers and Trigonometry Handout

Some Pictures...

Following Up Pythagoras (It Went Great!)

I started off my geometry classes' reintroduction to the Pythagorean Theorem with a simple bell ringer. It was a 3-4-5 triangle, with a missing hypotenuse. With some figuring, they remembered the Pythagorean Theorem and applied it. A student said "The answer is 5." I agreed and asked why. Why was it 5? The response I received was that a^2+b^2=c^2. Yep. That's true - when you have a right triangle. Why does a^2+b^2=c^2? My students decided they didn't know.

So, we proved it. ( I didn't call it that because my kids hate proofs.) I used this awesome 3-4-5 template from Jennie at mathfoldables.blogspot.com (HERE). I didn't use the two layers like she recommends. One worked just fine. We discussed the right triangle formed in the center, its side lengths, finding the area of the squares, and how this all relates to the Pythagorean Theorem. Students really understood and were able to draw connections between what they had colored (see picture below) and the concept behind the Pythagorean Theorem.

It turns out that this introductory coloring activity was a great transition to the Converse of the Pythagorean discovery activity. I modeled completing the INB page for a 3-4-5 triangle. We also completed a 5-7-9 triangle as an example. From there, students were instructed to use any of their squares to form additional triangles and complete the form. This was a great way to spiral in the Triangle Inequality Theorem we'd just learned in the last chapter. Students would ask if x, y, and z would work, and I'd ask them... how could we figure this out? They use those skills and really started to see patterns in their Pythagorean Theorem inequalities and the triangle types.

Tomorrow we'll finish up our tables and discuss takeaways. This was a great activity, and after a tiny bit of productive struggle, the students caught on well.

Teaching the "WHY" of the Pythagorean Theorem

Every year, there are two things I can reliably predict about my new batch of Geometry kids: (1) They know enough about transformations and (2) They know the Pythagorean Theorem. Now, let's expand on the second. They "know" the Pythagorean Theorem - well kinda. What they really know is that a^2+b^2=c^2. They have no idea where it came from, why it works, who Pythagoras is... none of the good stuff.

So, this year, instead of just reviewing how to plug and chug with the Pythagorean Theorem, I decided I want them to re-discover (and thus re-learn) the Pythagorean Theorem. I set out on a virtual journey to find the perfect Pythagorean Theorem activity - one that shows kids WHAT they are doing when they use the Pythagorean Theorem, not just HOW to do it. I searched high. I searched low. The #MTBoS came to the rescue, as usual. I had happened upon Dan Meyer's Pythagorean Theorem discovery activity during a google search, but Lisa Bejarano's blog posts about it (HERE and HERE) helped me to re-think how it might be successfully implemented in my classroom. Jacqueline (@_Cuddlefish_) also sent along some really awesome links, especially an informal proof of the Pythagorean Theorem based on Perigal's proof of the Pythagorean Theorem (HERE). 

I decided to re-work Dan Meyer's Pythagorean Theorem idea into an INB page that would provide a little more support for my students. It covers both the Pythagorean Theorem and its converse. Here is what I came up with:

Pythagorean Theorem and its Converse INB page (Click for Google Drive Download Link):

FRONT:

INSIDE:

I plan to still use the square blocks included in Dan Meyer's Pythagorean Theorem Discovery (HERE - Week 25), but I'm going to white out the area calculations and have students calculate those themselves.

Triangle Midsegments & Exterior Angle Speed Dating

I've learned so much from the #MTBoS. So, so much. One of the best things I've ever learned, though, has to be Kate Nowak's Speed Dating (See post HERE). It's just amazing and does so much for kids on so many levels. With our three thousand (perhaps two thousand ninety-nine) snow days, we've needed to review a lot of topics. This was the perfect way to do it. I primarily wanted my kids to review the triangle midsegment theorem and the exterior angle of a triangle theorem. I also threw in some isosceles and equilateral triangle questions from the pre-snowpocalypse era.

I got comments like "Ms. Boles, I really learned a lot today. I really did."  "I like this game."  "When can we do this again?" "We should do this everyday!" I said no to the last one. However, I really loved all of the positive feedback I was hearing.

Since I've done speed dating for three years, I've learned quite a bit about how best to facilitate it. First, count the number of students in your smallest class and half it. My smallest class was 22, so I wrote 11 questions that really hit the main ideas I wanted reviewed. I obviously have to write more questions, but the rest are usually very similar to the first 11. This way, students don't have to make it all the way around the room to review all necessary topics. In my class of 34, speed dating all around the room would take far too long.

Midsegment & Triangle Exterior Angle Speed Dating Index Cards (Click for Google Drive DL)

Breakthrough!

I've been having a rough year. Things haven't necessarily gone the way I would have hoped in many of my classes this year. There is a disconnect between my normal teaching strategies and the way my sophomores learn. It's been a struggle, but I'm still fighting to figure it out. Today, however, was a great day.

 In my sixth period geometry class, we began with a discovery activity about triangle midsegments. It's my favorite lesson of the year. I think it's because it spirals so many of the prior ideas we've learned. Anyway... I asked "How many midpoints does a triangle have?" and thus "How many midsegments does a triangle have?". I know - I know.. this is one of those crappy questions with a definite answer. Bad teaching moment. However, one of my students turned it into a definite win. He raised his hand and said infinite. Now... he is THAT kid. You know - the one that sits in the corner and waits until he can chime in with something "cool" like infinite or no solution or Illuminati. But, this kid is brilliant - just too cool for school. He went on to say that the three midsegments of the triangle form a second triangle, which also will have three midsegments, etc. Brilliant - and yes- just yes!

Student B. Says... "So tell me again why we can't find the midpoint of a line?". Other kids respond that to take a midpoint we must know where the segment begins and ends and that we don't know this about a line. So, student B says can we just use the midpoint formula and say negative infinity + positive infinity divided by 2. We then discussed that we can't treat infinities like variables and that a negative infinity and positive infinity don't necessarily reduce to zero, unless we assume equal infinities. We started talking about how some infinities could be more infinite than other infinities... awesome conversation.

Student C then says "Doesn't a line segment have infinite midpoints?". I didn't really get where he was going, so I asked him to explain. He said "You can take the midpoint of the line segment, then take the midpoint of that point". Another student chimes in "The midpoint is a point. You can't find the midpoint of a point." So, attending to precision, he re-phrases and says "Can't you take the midpoint of a line segment then take the midpoint of one half of the line segment, then take the midpoint of that, etc". I told him that what he was doing was partitioning the segment into certain ratios - halfs, fourths, eighths, sixteenths, etc.... all while the newest midpoint was approaching the first endpoint (according to how he'd explained it).

All three of these awesome conversations from this one discovery activity about triangle midsegments. Here's the link: http://wiki.mhshs.org/images/f/f8/Midsegment_of_a_triangle_theorem.pdf

Now... if we could only have these types of discussions everyday.

Reflecting on Coordinate Geometry Project Based Learning

As a school ARI (Appalachian Renaissance Initiative) lead, I was encouraged to implement project based learning in one or more of my classrooms. I opted to implement a project based learning activity in geometry as part of the coordinate geometry unit. The project asked students to work in groups of 4 to design a waterpark. The project work flow looked something like this:

Days 1 & 2: Water Park Research

Day 3: Make Land (Oversized Coordinate Plane)

Day 4 & 5: Draw & Finalize Blueprints

Homework: Write a paper to your boss defending your blueprint design, using the research notes.

Day 6: Draw in Walking Paths & Find Lengths (Distance Formula)

Day 7,8, 9, & 10: Build Waterparks (3D)

Day 11: Find the Midpoints of Each Path on Blueprint

Day 12: Build Benches & Place at Midpoint of Each Path on Model

Homework: Write TV Advertisements for Your Park Opening

Day 13: Record TV Advertisement & Finalize Model/Proposal Components

In good news, I LOVED this project. The kids (seriously - all of them) were into it at some level. They worked steadily, many worked hard and taught themselves how to do distance and midpoint. Other teachers loved the project, the principal loved it, the superintendent loved it - it was great (well... the janitor didn't love it... we made a mess!). 

I gave my exam yesterday. Some of the results were great - the kids that put effort into the project have a wonderful understanding of distance and midpoint - and not just at the level of the standard algorithm. However, the kids that simply went along with the project, doing what they were told to do by group members but little more did not form a great understanding of the content. 

A large part of this is my fault. Reflecting back on it, I should have planned more checks for understanding to make sure the kids were "getting" what they needed to get. I failed there. I went around and formatively assessed a lot, but there was no way designed for me to hold their feet to the fire until the ending examination. A distance quiz after day 6 and a midpoint quiz after day 12 would probably have largely addressed this problem. Next year I'll do this again, but next year there will definitely be quizzes! However, I really recommend project based learning. It's so fun to teach this way! I'll post pictures of their projects in another blog entry after Thanksgiving Break.

Making Error Analysis Work

Happy planning period! I am in need of some help in reflecting on yesterday's assignment and why/how it went wrong. The kids didn't have the reaction to them that I desired, and I want to correct it for next time, because I truly feel the activity is a valuable one.

Using Steph Reilly's Error Analysis format, I came up with, what I thought, was a great activity for my geometry classes to reflect on the major errors they commit when solving equations. I should frame this by saying that I gave an exit ticket last week and saw that my kids were still severely struggling with solving equations, so they truly did have lots of things to learn by doing the Error Analysis activity.

Here is the sheet of exit ticket errors and reflection space I made:

Link: HERE

And here are they type of reflections I got from about a third of my first period class:

Now, some kids seemed to find it helpful, and the quiz scores today reflect that. However, the activity yesterday really confused a few kids, and it was also reflected in their quiz scores today.

How do I restructure this activity so it's beneficial for everyone next time?