Tag Archives: geometry

Think Puzzle Explore #makethinkvis


I guess one can tell I taught geometry the first time I read this book.

I am grabbing this information from an Evidence File submitted for our Program Reviews…  formative assessment, student led questions, problem solving CTE, design-Art, communication, writing and exploration.

Triangle Centers:

This task was presented as an introduction to the unit for discussion, then revisited after student investigations.  I actually used the Notice/Wonder routine, however, it could easily be modified to fit TPE.

Evidence:  After constructing special points of intersections in triangles with patty paper, students were asked to share what they noticed and wondered about the geometric figures.  A list of questions generated by students.  They were given the task of choosing a number of questions to explore using Geogebra software.  Following the investigations, students shared their findings and then used the software and what they had learned to answer a problem about location of an amusement park.  See list of questions below.

triangle centers

Triangle Centers Amusement Park modified from Georgia Department of Education.

What I love about the Amusement Park task is that there is no single correct answer.  There are multiple solutions, students were simply asked to share evidence of why they chose their particular location.  Students could either write a memo and/or present their findings to their classmates, which offer led to more questions of why? what? how?

8 Videos to Engage Students in SMPs


I started off my Friday at NCTM with Ed Dickey from South Carolina.  It was a wonderful session-lots of laughter but some great opportunities to build SMP into instruction as opposed to trying tomaddress them as stand alones.

A series of tweets reminded me of one of the videos in his session…

a comedian, Brian Regan speaking about Girth as he prepares for a move.  It was a great way to add laughter to the classroom, yet introduce a topic related to SA or Volume or anything related to 3D figures/containers.

Here is a link to the Louisville NCTM handout but the PPT can also be found in the link above to Ed Dickey’s site.

Enticing Students to Think with Food


What better way to end our semester than a few tasks involving food?  Sometimes the last weeks of school can be filled with multiple distractractions.  In hopes of holding my students’ attention while they’re in class, I am bribing them to think with food.  Yes, I have fallen to enticing them with external rewards.








With the Oreo Mega Stuff,  A Recursive Process offers some research by Chris & Chris.  My plan is to follow the QFT model outlined here.  I just recently became aware of the Question Formulation Technique which I shared in this post.  The Q-Focus is simply to display my package of Mega Stuf Oreos, wondering what questions they have – recording all of their comments as questions …and follow the process allowing them to determine their own questions, lead their own learning.  Though I would hope they would approach this from a volume stand-point – letting them design their own questions may lead to other ideas and I am fine, so long as they are thinking and talking math, yes they may eat their research tools once they’ve answered their chosen question.  The final product will be a 30-second pro/con commercial Mega vs. Original supported by their mathematical findings.

Offering several stations to review surface area and volume formulas utilizing various candies as they are packaged as well as the infamous pouring water from a pyramid to a cube / cylinder to a cone will be modeled as one of the station activities.

Finally, using the  Ice Cream Cone  found at Illustrative Mathematics.

ICE CREAM prompt and file

As a “reward” for successfully completing this task, I think a class Ice Cream Party would be appropriate.  I just need to know how much ice cream I should purchase to ensure everyone has plenty to enjoy without too many leftovers.  Assuming the cones are filled with ice cream with a “spherical” scoop atop – sounds like a great homework practice problem to me…

Geometric Measurement and Dimension (GMD)  Explain volume formulas and use them to solve problems
  • G-GMD.1 – Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
  • G-GMD.3 – Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
  • G-MG.A.3 : Modeling with Geometry- Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Several times this year, I’vfe gotten the GMD (geometric measureme and dimension) and MG (modeling with geometry)domains mixed up, I am slowly beginning to internalize the new notations. 🙂

I also like this prompt: Doctor’s Appointment for GMD-A.3.

On a side note – Reading an article in MT the other night – I wondered, “Was I supposed to know that?”

The derivative of area of a circle is the circumference?  The derivative of volume of a sphere is surface area?  Similarly…derivative of area of square is half the perimeter, derivative of volume of cube is half surface area…  How/Why did I miss that? Or did I know it at some point but just pushed it aside years ago?  Interesting…made me wonder and I started looking at other figures – will share more later.

Trig Ratios – #made4math


Through the years, I’ve seen students struggling trying to remember which Trig Ratio is which.  I have a colleague who draws a big bucket with a toe dipped into the water.  She says she tells the students “Soak-a-Toe” to help them recall SOH-CAH-TOA.  Another has described the “Native American”  SOH-CAH-TOA tribe as the one who constructs their teepees using Right Triangles.  The most entertaining though is the rap from WCHS Math Department “Gettin’ Triggy Wit It” on youtube.

I wanted to use an inquiry activity to help them develop the definitions of the Trig Ratios.  Basically, they constructed 4 similar triangles, found the side measures, then recorded ratios of specific side lengths.  Next, I had them measure the acute angles, then we used the calculator to evaluate the sin, cos and tan for each angle measure.  Students were asked to compare each value to the ratios they had recorded in the table and determine which ratio was closest to their value.  Here’s the file https://www.dropbox.com/s/gfvhnictujfj2ik/similar%20triangles%20intro%20trig.docx?dl=0 Similar Triangles Trig Ratios.  Anyway, its not a perfect lesson, but a starting point.  If you use it, please comment to let me know how you modified it to make it a better learning experience for students.

In the past, students sometimes struggle trying to decide which ratio they need to use when solving a problem. I put together an activity adapted from a strategy called  Mix-Pair-Freeze I’ve used from my KaganCooperative Learning and Geometry book.  This book offers numerous, quality activities for engaging your students.

You can make copies of this file, Trig Ratio Cards File, then cut cards apart to use.

Trig Ratio Cards

Each student gets a card.  They figure out which Trig Ratio is illustrated on their card (& why).  They mix around the room (with some fun music would make it better), then pair up with someone.  Each person tells which Trig Ratio and why (can be peer assessment, if one is mistaken).  They swap cards, mix and pair with another classmate.  This continues for several minutes, allowing students to pair with several different people.

When I call “Freeze!” Students are to go to a corner of the room which is designated Sin, Cos or Tan.  Within the group in each corner, students double check one-another’s card to determine if they are at the right location.  Again, peer assessment, if someone is wrong, they coach to explain why, then help them determine where they belong.

Students swap cards, mix-pair-freeze again.

I like this activity for several reasons:

  • 1. Students are out of their seats and active.
  • 2.  Students are talking about math.
  • 3.  It allows them to both self-peer assess in a low-stress situation.
  • 4.  I can listen to their descriptions and address any misconceptions as a whole-class as a follow-up.


To clarify, the intent of this activity is for students to determine what information they are given in relation to a given angle, then decide which ratio it illustrates. It is meant to help students who struggle deciphering what information is given.

Chalk Talk part 2 #makthinkvis


Another task I presented students in the form of a Chalk Talk

We had previously used a patty paper lesson to construct our kites. image

Simply enough, we constructed the kite by first creating an obtuse angle, with different side lengths. Folding along AC, tracing original obtuse angle using a straightedge to form the kite. Immediately students made comments about the line of symmetry. They were given time to investigate side lengths, angles, diagonals, etc. forming ideas and testing them to prove properties.

Their Chalk Talk task was to devise a plan to calculate the area of a kite.







Most every group approached the problem by dissecting the kite into right triangles, then combining areas. Several approached dissection as top triangle/bottom triangle, but would have to adjust their thinking when I asked them test their idea with specific total diagonal lengths. Some even extended the kite to create a rectangle. In the end, our discussion centered around 3 statements/procedures for finding area of a kite.

1/2(d1*d2) (d1*d2)/2 d1*d2

Allow them to determine which will /will not work and share evidence as to their conclusions. (Hello! MP3 critique reasoning of others.)

Sure, it would have been quicker to say here’s the formula, here’s a worksheet, practice, learn it. But its so much more fun “listening” to their Chalk Talk. Again, the end discussion is key-allowing them to think / work through each group’s findings, address any misconceptions and finally coming to a concensus as a class.

Chalk Talk part 1 #makthinkvis


I have wanted to try Chalk Talk, a strategy from our #makthinkvis bookchat, for several weeks.  However, I wanted it to be an authentic learning experience rather than a contrived activity just to say we did it.  This past 2 weeks, I found myself able to use it in 2 very different contexts.  Chalk Talk requires students to communicate written dialogue, no verbal.

The first was at the end of a unit of study.  I used the “2 Minute Assessment Grid” discussed here,


as a reflection tool for my students a couple of days before the unit assessment.  At the end of the previous post, I wondered how to address student questions/misconceptions.  I chose to recopy the questions onto a post it, placed in the middle of a dry erase poster.  Students were curious as they entered the room that afternoon and saw the posters hanging around.

Students took a dry erase marker and were instructed to respond without verbally talking, to suggest, explain, give examples or ask questions on the posters. 







Notice 2 posters were red.  I explained to students that red flags went up for me as I read the statements from their classmates post-it note reflection on the 2MAG. 

After students had opportunities to respond on each poster, we carouselled around to read responses.  I’ll be honest, I was hoping for more guidance, in depth statements from them.  There were some good examples, but majority were point-blank, straight forward surface statements without in depth explanations.  However, as we discussed the posters, I felt the thoughtful ideas came through.  “Here’s how I remember this…”, “If you can think of it this way…”

Which shows most of them can verbally give ideas, explanations but written is not as strong.  How do we assess them? High stakes testing is almost always written.  Another reason I am not am not a fan.  It just seems unfair we judge students and even teachers based on written, mc tests that don’t allow opportunity to showcase strengths of all students.

Overall, I feel like this task gave students a chance to address those ideas they were still fuzzy on, gaining suggestions from classmates, whether written in the Chalk Talk or our wrap up discussion.  On our unit assessment, questions that targeted the concepts from Chalk Talk, students performed very well on.  I do feel the opportunity to discuss/process verbally as the follow-up is key. A wrap upmdiscussion gave me opportunity to address any unclear / incorrect comments as well.

I look forward to finding more opportunities to use Chalk Talk to move learning forward and make thinking visible.

See, Think, Wonder #makthinkvis


For our next Making Thinking Visible chat, we were asked to read Chapter 3 and implement the first routine presented – See Think Wonder (STW) pg 55.  I realized late Wednesday evening students were scheduled off for a staff PD day on Friday.  I scrambled wondering how I could incorporate this strategy in a meaningful way.  We had worked with parallel lines / transversals and the angle relationships created.  My goal was for students to look for ways to prove lines parallel.  How could I use STW to get this accomplished?

When I searched for images of parallel lines in architecture, I ran across a picture of a building in Australia and a picture of the Illusion as well.  You can find more here Cafe Wall Illusion.

My plan was to use the optical illusion – the placement of the black and white blocks causes one to think the lines are getting closer / farther apart.  However, as I flipped through my book, I saw the routine of Zoom In and wondered if I could combine the 2 somehow.  And here is what I did:

Zoom In – Ask Students what they see, pretty standard – black rectangle.  So many ideas (some silly) of what this could actually be part of…


Slide 2 was a little more interesting, alternating black / white rectangles with several things they thought it could be a part of – keyboard, referee’s shirt, prisoner uniform, zebra…


Slide 3 eliminated some of their predictions…I did have a person actually state a building. Hmm.  I think they must have seen it before.


When I revealed the final picture – it was fun listening to their comments.  One was very perplexed “Why would anyone want their building to look that way?”  It is found in Melbourne, Australia.


After a few moments of sharing / discussion – students were comfortable.  As I shared with students that we were going to do a thinking routine called See Think Wonder – I tried to explain each step.  This is the slide I shared with students:


I went through each step, allowing time for students to record what they saw, what they thought and anything they wonder (a question they could investigate/answer).    We then shared our responses.  After the first 2 statements, I paused and revisited what we were to do for each step.  We agreed the statements would actually go to “Think.”  Here were responses:


After sharing, students were given a copy of the Cafe Wall Illusion – but not allowed to use rulers/protractors to measure anything.  You can see from the snapshots, several chose to use patty paper.

Student A traced the lines to show they were actually straight, then translated the copied lines over the originals to show they were parallel.


Student B traced the edges of the rectangles, then translated to different levels to show the lines were equidistant at all parts, thus parallel.


Student C over-layed tape, traced edges at 2 different levels, then peeled the tape and matched them up…IMG01093

One student used the pink line on the notebook paper and overlaid it to show the lines were actually straight and several traced the rectangles onto patty paper and translated to others to show congruence.

It felt a bit contrived – I’m not sure what level of thinking was achieved, but I will use See Think Wonder again.  It was a good start to model the 3 steps of the routine.  Following the activity, students could be asked – if I only had 2 lines – how could I prove they are parallel?

In discussion some responses:

  • to extend the lines to see if they ever intersect (student knows the definition);
  • measure the distant between the lines at different points (again, student understands they are equidistant;
  • draw a line perpendicular to one line, extend it, if its perpendicular to the other line, then the 2 lines are parallel (yep a student came up with that one)
  • and finally, cut both lines with a transversal, measure/compare the angles to see if the relationships exist (the understand converses/working backwards to prove).

What I appreciated about STW – I didn’t tell students what question to answer or even how to answer it.  They created their own question and chose a way to answer it.  The only problem with this – they may not wonder/choose “the question” I’m wanted them to investigate/answer.  In the end, if you can get students to make a connection with the content, give them opportunities to notice/wonder, allow them to come up with their own questions – they’ll be interested in finding the answer…

Midpoint – on a different day than Distance


In years past, I’ve usually taught Midpoint and Distance on the same day or at least on consecutive days.  After a reminder of some brain research last fall – how our brains store information by similarities but retrieves information by differences – I decided to try things in split them up this semester – hoping to lessen the confusion students often face (do I add or subtract with midpoint/distance formulas?).  Again, this confusion stems from teaching a procedure without paying close attention to in-depth student understanding.

I chose to introduce “Midpoint with Coordinates” the same day we were working with segments, bisectors, midpoints of segments.  No bells/whistles here – just the basics


I gave students a grid index card and the points A(2, 1) B(8, 11) and C(8, 1) to attach to their INB RPH.  Simply starting with locating the midpoint of the AC and BC.  But also asking them to compare/contrast the coordinates of ACE and BCF each time.

Finally, asking them to locate G, the midpoint of AB.  Walking around the room, it was quite fun watching the various strategies.  The great thing was asking students to share their different strategies.  One used rise/run, several “counted diagonals” from A and B until they got to the middle, one used the midpoints of AC and BC and traced up from E / over from F until he found where G was located.  After discussing methods using the graph, a student stated “I just added my x’s then divide by 2 and added my y’s then divide by 2.”  When discussing how the coordinates were alike/different, a student asked “Isn’t that, what C____ did? Just averaging the x’s and averaging the y’s?”

So, I never actually gave them the “Midpoint Formula.”  Awesome.  Of course, we went on to practice the skill a few times.  I also chose 8 questions from Key Curriculum’s Discovering Geometry (did I mention, I *LOVE* this book?!?!? And have since the mid-90’s!)  – that required a  bit more thinking beyond skill/drill.  Two questions that led to some great discussion today was:

Find two points on segment AB that divide the segment into three congruent parts.  A(0,0) and B(9,6).  Explain your method.

Describe a way to find points that divide a segment into fourths.

But in class, I offered another – what about if I need to divide it into fifths?  Students worked individually, pair-share – then class discussion.  Quite different approaches.  I loved it.

What was even better, a student asked, “But  the examples we’ve used all have an end point at the origin.  Will it still work if the endpoint is not at the origin?”  Aaahhhhhhhhhhhh! That’s music to my ears!  Wow. Wow.  I love it.  I love it.  I love it.

This is a nice little open question to share with your students.  It definitely allowed me to see student understanding of the task by their work / responses / discussion.

Developing Definitions


I’m back!  Nearly 2 months? Yikes. Some fellow teachers on Twitter were committing to blogging once each week.  I think  that’s reasonable – besides, usually my best reflection comes during the moments I blog.  Reflection – seems to be the first thing I push aside when I just don’t have the time.  Yet, its the most valuable use of my time.

I’m sharing some successes from Kagan Geometry (one of my favorites by the way).

I was going to be out for a number of days due to being seated on the jury for a trial (give me 100+ teenagers over the courtroom anyday!).  I wanted to leave something productive.  I did short videos (<10 minutes) filling out certain pages in the INBs in addition to other activities.

The first Kagan activity was for vocabulary.  Each strip of paper included examples and counter-examples for each term.  I modified from the round-table recording it suggested.  Students were asked to pair up (a new partner for each new term) and develop their own definitions.  I loved it simply because most were terms students had previously been exposed to in middle school.

When I returned to the classroom, I ran through all I had left during my absences to address any concerns/questions.  Several students commneted how they liked (appreciated) doing the definitions this way.  Their comments ranged from – ‘You actually had to think about the terms; Talking with someone about it really helped you process what it was before writing it down;  The pictures of examples / nonexamples really helped understand the word better.’

Yesterday, we developed more definitions about angles.  When I told them what we were doing – they were excited about the activity.  Listening to the conversations – I was very happy with their discussion / questions / specifics they included in their definitions.

I remember several times in the past doing examples / non-examples, especially when using Frayer Models.  I believe taking it out of my hands/mouth and giving them the opportunity to work in pairs really enhanced their understanding of the terms.  Even when discussing HW  today – they used appropriately terminology.  Yeah!

Another Kagan activity I used as a LHP activity

from Kagan Geometry

from Kagan Geometry

– very similar to Everybody Is a Genius’  Blind Draw.  Students were placed into small groups and given 12 cards with written directions.  Person 1 chose a card, read the directions, gave others time to think and draw a diagram with labels.  The reader confirms/coaches/praises others’ work.  A new person chose a new card and the rounds continued until all cards had been used.  One thing I appreciated about this – another card asked students to draw a ray from E through M.  This allowed students to realize differences in very similar diagrams.

Again, when I returned to the classroom, students shared how this activity was different from anything they’d done before, saying it was both challenging but helpful in that it helped to clarify certain misconceptions they had; especially with labeling the diagrams.

I have learned the Kagan strategies help students develop and process concepts.  There are “game like” activities where students must find their match and discuss.  Visual, Auditory, Kinesthetic – something for everyone.  Its not an end all – be all resource.  But the amount of HW / practice is minimal when I’ve used these strategies correctly.  I am a firm believer that they help start a strong foundation to build upon.  Hey – if students are smiling and laughing while “doing definitions” – its gotta be good.

#Made4Math Monday – Parallelogram Foldable


Its been a while since I’ve sumbitted #made4math Monday post.  I really like the idea of foldables – a kinesthetic graphic organizer…I believe they have a positive impact on student learning when used purposefully.

This one (found here parallelogram foldable) for parallelograms, rectangle, square and rhombus.  I wanted a foldable that somehow showed all were all in the parallelogram family, but still kept them separate – I chose a trifold.


When I saw an example of the tri-cut Venn Diagram, I knew I wanted to incorporate it somehow to show squares as the overlap of rectangle and rhombus.  This picture does not show the cuts between rectangle/square and square/rhombus, but I think its visible in the last picture.


The file is simply the skeleton, please feel free to make it your own (ha, just don’t go selling it as your own!)


I am still debating what should go in the center – thinking of examples / non-examples.   Possibly even giving students a couple of example problems using properties of quadrilaterals.  Istuck area formulas in at the last second – but think it may be more effective to let students discover area of a rhombus on the own.  Suggestions are always welcome!