Category Archives: triangles

Posting Learning Targets yay or nay


Thanks to @JustinAion,  I got thinking…

It depends… on my class and the students and the activity…to determine if I actually post it.

However, when I do, I refer to it at the beginning, throughout the task – to remind students of the end goal, and again as a wrap up – whether reflection, exit ticket of discussion/summary to end class.  And I like to refer to it the following day as we begin the next lesson, just as a quick review.

I, personally, would prefer to have an overarching Essential Question for each lesson to use rather than a specifically stated target.  However, I sometimes struggle a lot with Writing EQs, would love a colleague to collaborate on these.

Here’s a section of the unit organizers I’ve used this past year (thanks @lisabej_manitou). 


And a link to this file.
Unit Organizer
Functions Overview

I give them to students toward beginning of unit, we complete the words worth knowing for vocabulary (thanks @mathequalslove). Then read through actual targets.  When quizzes are given back or practice problems checked, students have a place to reflect/record thwir level of learning as well.  Because students have this in their INBs, I can quickly refer to them if not posted on the board on any given day. 

Card Tossing & Spiraling Curriculum #tmc14


Awesome session Mary and Alex!  Thank you. Thank you. Thank you.

The session focused on their experiences with Grade 10 Applied students ( Canada).  The entire course is activity based which allows students to not miss out on big ideas as they would in a traditional unit by unit aligned course.
Students have repeated opportunities to experience big ideas. The tasks are rich  with multiple entry points and different approaches to solving.  It’s a collaborative environment with accountable talk.  There are fewer disciplinary issues with increased engagement.

Each 6 weeks a mini – exam over entire course up to that point takes place.  Questions are in context and tied to activities they have completed.

We began with beads and pennies on our desks and this task… Cole has 2 smarties and 3 juju bed for $.18 while Noah has 4 smarties and 2 juju be for $.20.  They shared that systems are presented this way – no algebraic forms- for the first several weeks of class.  I, personally, can see how effective this strategy could be.

The next activity shared was Sum of Squares (he doesn’t refer to it as Pythagoras Theorem, yet – or did he say ever?)

Students are asked to cut all squares from side length 1 to side length 26.  Each square is labeled with side length, perimeter, area.  Then they build with them.

Basically students explore and eventually they focus on triangles formed with question, are there 3 you cannot make a triangle with?   Which combinations form different types of triangles. Begin looking at 3-4-5 triangle families, similar triangles (Kate suggested dilations here), discuss opposite side and adjacent sides, then give them a TRIG table and allow them to figure it out.

Compare side lengths with perimeter, or side length with areas.  The possibilities of math concepts are endless.
We ended the day with Card Tossing by collecting data, then using rates to make some predictions.

Video of Alex & Nathan picture below is only a screenshot.


@AlexOverwijk downed by @nathankraft 75 to 72

Each person in the room completed several trials of tossing our cards for 20 seconds.  We found our average rate of success, then determined who we thought might beat King Card Tosser.

Alex asked us to predict how long they needed to toss if he gave Nathan a 35 (?) card advantage so it would be super close and exciting.  Our prediction 38 seconds about 75 cards. Many ways of making the predictions were possible. Not to shabby, huh?

This task was fun, exciting, engaging.  Definitely on the to-do list.

This approach is definitely something I would like to consider, if administration will allow it!

Triangle Centers #MTBoS30 Day 3



For triangle centers, I like to let students construct them on geogebra if the lab is open, then notice and wonder about them…and their properties.  Ideally, they would explore and investigate their questions and prove/dispute their claims. 

A few questions that arose this morning…

Do the 6 triangles created by medians have equal areas?

I wonder if you dilated the incenter for the inscribed circle, would it become the  circumcenter of the circumscribed circle? 

Another student stated, not always, unless your angle bisectors became the perpendicular bisectors.  (When would that happen?)  Without that happening, it’s a dilation and translation.

Comparing the areas of the circles and corresponding triangles, a student asked,

Is the area of the circumscribed circle twice the area of the triangle?

Is the area of the inscribed circle…one half the area of the triangle?

Now to explore the questions…

These were all follow-ups to discussing how these constructions would aid in solving various real-life contextual problems presented at the beginning of the lesson.

Hinged Mirrors & Polygons


The last session I attended on Thursday afternoon at NCtM last week was with Erin Schneider from a
Louisville, KY.   Several hands-on and open ended tasks, sharing and talking.

The hinged mirrors were fun to play with and I wondered how I could use them in my classroom.


The hinge is placed either off the edge of a sheet of paper or on the edge of a paper.






Convince me its a square.

How can you create a rhombus that is not a square.

What happens as the central angle gets smaller? Larger?  
For my students, I feel this allows them to really see a polygon diseected into several triangles from the central angle.

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 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.

Questions, Blocks & Shadows…


What a chain of events.  Last summer I created Pinterest boards to tag some amazing classroom ideas I kept running across.

This post, Blocks and Shadows from Best Case Scenario intriqued me. blocksandshadow

Several weeks ago, I was reading some posts by@jgough at Experiments in Learning by Doing where she suggested the book Make Just One Change, Rothstein & Santana (2011) .  The premise is to help student ask their own questions.  This book was deinitely on my summer reading list.

A few days ago, I mentioned the same book to @druinok on Twitter, which leads one of the book’s authors to my blog.  He shares a link to The Right Question.  Last night, I take some time to check it out and read an article Teaching Students to Ask Their Own Questions which briefly outlines 6 steps of the QFT -Question Formulation Technique.

So where is this going?  After working yesterday to complete a narrative for an application I’m submitting this week, my mind is in a mode where it won’t shut down.  I woke at 5 this morning, thinking about blocks, shadows, QFT.

Here are my thoughts…

1. I share pictures from our opening discussion of our Right Triangle Similarity unit, which include snapshots from The Vietnam Veteran’s Memorial in Frankfort, Kentucky

vietnamsundial From the memorial website: The design concept is in the form of a large sundial. The stainless steel gnomon casts its shadow upon a granite plaza. There are 1,103 names of Kentuckians on the memorial, including 23 missing in action. Each name is engraved into the plaza, and placed so that the tip of the shadow touches his name on the anniversary of his death, thus giving each fallen veteran a personal Memorial Day.

The location of each name is fixed mathematically by the date of casualty, the geographic location of the memorial, the height of the gnomon and the physics of solar movement. The stones were then designed and cut to avoid dividing any individual name.

and other shadow snapshots of random objects outside my classroom.

2013-04-23 12.01.49 2013-04-23 11.55.582013-04-23 12.08.55

I am hoping this will be enough for my Q-focus, but since I have not read the book, I feel like there’s more to it.  Improvements to the lesson next time…

Next, set out blocks, flashlights, making available measuring tools such as grid paper, rulers, protractors, etc.

2. Students get time to play, explore and prodcuce questions!

Prior to beginning 2, I will explain certain steps and “rules” from the QFT model outlined here.

The 4 rules as discussed in the article: ask as many questions as you can; do not stop to discuss, judge, or answer any of the questions; write down every question exactly as it was stated; and change any statements into questions.

Here is where I need some help, I feel like I should impose a time limit to keep students focused and on task, but what is reasonable?  Even with an imposed time limit, I am one who will bend if I see my students are on task and into the mathematical discussion.  My initial thoughts are 10-20 minutes to explore and generate their questions before moving to the next step.

3.  Students improve their questions, noting difference between closed/open, etc.
4. Student prioritize questions, submit their focus to the teacher.
5. Discuss next steps.
6.  When all is said and done…reflection on their learning.

Please offer suggestions or even how you’ve used a similar activity in your classroom.  I am VERY interested in offering more lessons like this – where students guide their own learning.

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.

Triangle Centers #made4math Monday


I stumbled upon a learning task this weekend on a Georgia DOE site involving triangle centers.  The task is simply to choose a location for an amusement located between 3 cities.  Yep. Simple enough, until I sat down and started deciding how I would approach the situation.

The final task is for students to write a memo with their recommendation when cost of building new roads is taken in to account.

Here is copy…

Triangle Centers Task

I am looking forward to reading what recommendations my students give and their reasons why!

Providing Students Time to Reflect #makthinkvis


Making Thinking Visible online chat has really challenged me to think differently this semester about my questioning, looking for opportunities for students to share their ideas but most importantly, giving them time to reflect.

To begin our unit on triangles, I used the Generate-Sort-Connect-Elaborate, with plans to elaborate towards the end of the unit. As a class, I simply went around the room, each student generating an idea/concept related to triangles and I added what they shared to the list.  I placed students in groups of 3 and asked them to sort the ideas any way they wanted and to connect each set of ideas to the triangle central theme.

Most had measuring, classifying/types, etc. However, several had made some connections back to our Day 1 activity with the Chaos game, Sierpinksi’s Triangle, Midsegments and their properties.

Today, in class, I asked them to flip back to INB page 47 and take a couple of minutes to do nothing but read through their original concept maps/webs. Before I could give them further instructions, one asked if they could add to it? Of course! That’s exactly what I want you to do! I’ll see if I can manage some before/after pics.  The following few minutes were great. Listening to them think and share outloud. One even said, “Man, I’ve sure learned a lot!”

The next task is one I read about inmy reader a few weeks ago. I apologize, if you blogged about this and I’ve forgotten your name, but I really, really liked it! I gave each student 4 sticky notes, directing them to place a + sign in the corner of one, ? on another, ! on the third and finally a student asked, “you’re not going to make me draw a lightbulb are you?!?”


I explained what each note would include:

+ One Improvement – this could be either an improvement they still needed to make OR an improvement I could make in teaching the unit. A student asked if it could be something they improved on during the unit..sure!


! What NOT to forget!


? A question they still have.


Lightbulb moment during the unit…


I gave them some time to flip back through their INBs, instructed them to place their notes on the board in the back of the room.  A few asked if they could bring theirs in tomorrow. 

A quick glance showed that many still are not comfortable with proofs, a few are having trouble with the ‘names’ of triangle centers. I am more concerend they know/understand each of the centers’ special properties for problem solving. There were a variety of lightbulb moments.  And even a few misconceptions are obvious in some of their responses.

My plan is to address common questions as whole class.  I had originally thought I would respond to the individual questions/misconceptions by using different color sticky notes up on the board.  However, now, I’m thinking I may recopy some of the misconceptions onto dry erase boards and use them in a chalk talk carousel activity. 

To begin, have a variety of comments, some I agree with and others I am concerned with.  Give students red, yellow, green stickers – they carousel through the statements, placing green on those they agree with, yellow or red on those they have questions about.  Would this or the chalk talk be more beneificial here? 

#75FACTS week 4 – #24 I used to think… but now I know…


This week we’ve been off from school for fall break – a road trip down south to visit Winter the Dolphin in Clearwater and a few days of warm sunshine on the beach has me somewhat re-energized.  I’ll be honest, my book is still at school.  The directions for this week were to use one of the FACTS #1-10 but I haven’t been in class to do this. (Sorry)  Before leaving school last week, I chose FACT #24 I used to think…, but now I know…  as a left-hand page assignment for my Geometry students’ INB.

#24 I used to think… Now, I know…

Eight Standards for Mathematical Practices

Practice 1 Make sense of problems and persevere in solving them.  This FACT allowed students to reflect on their learning, an opportunity to share what they used to think and what they now know after working with the concept.  Students responded to this prompt after exploring in small group investigations, pair-share processing, independent practice and finally whole class discussions/questions over Triangle Congruence.  I used to think… but now I know gave students the chance to make sense of the ideas they have been working with in class.

Facts and Teaching Goals

The goal of the lesson sequence was to allow students to recognize and determine which side-angle combos were appropriate and would guarantee triangle congruencies and finally applying those ideas with informal proofs.  By allowing them to respond in writing, I was made aware of their initial misconceptions – but also able to see they had in fact realized on their own how to prove trianlges congruent with a limited amount of given information.
I learned that the AAA and HL were the two students had struggled with most but they wrote about how the activities / discussion helped them realize specifically what was needed with each combo.  Another common error they pointed out in their reflections were that order of the included sides/angles did matter with situations of AAS and ASA.

Planning to Use and Implement Facts

One reason I chose this FACT was because I am looking to implement more literacy strategies into my instruction.  This FACT provided students with the opportunity to reflect on their learning in written format – a different type of processing that just talking/telling what they’ve learned.  By the time the prompt was given, students had explored in small groups, shared verbally with a partner, practiced individually.  The writing component seemed to complete the various types of literacy strategies.  By giving students a chance to respond to this prompt, I was able to see in-depth their full understanding of the intended concepts.

Small Steps

Were your students engaged?  Yes, I was very pleased observing students as they wrote their responses.  Most students took their time to share insightful reflections.  There were a few who tried to skim by with very vague responses, I gave them written feedback and asked they resubmit their responses.  Based on their new responses, I expect those few will give their best effort first time around next time given this prompt. 
Were you confident and excited about using the FACT?   I felt it was a good opporutnity to have students share their learning in writing.  I was not as excited about the FACT until after I actually started reading their responses…
How did use of the FACT affect the student-to-student or student-teacher dynamic?  Student to teacher – I felt they were honest in their responses – and most were insightful – I was encouraged to use this FACT again because it allowed me to see into their thinking.
Was the information gained from the FACT useful to youYes,  however, I don’t think I will change my approach to the lesson in the future – students were able to adjust their thinking because of the lesson format.  The FACT let me see this as a successful sequence – what a good formative assessment strategy should do!
Would you have gotten the same information without using the FACT?  I’m not sure I would have given students the opportunity to reflect had I not used the FACT.
What added value did the FACT bring to teaching and learning? Based on student responses – I believe most appreciated having the opportunity to think about their learning – it “tied up loose ends” for them in the end.
Did using the FACT cause you to do something differently or think differently about teaching and learning?   It made me realize I’ve failed to provide students with good opportunities to refelct on their learning between lessons / practice and before “official assessment” occurs.  This is something I plan to implement more for my students!  It was quick, little/no prep and offered me the chance to really see what students thought about their learning.
Would you use this FACT again?  Yes.
Are there modifications you could make to this FACT to improve its usefulness?   I believe next time I will plan more time for students to share out their responses – maybe within a pair-share then as a whole class, possibly using the ‘Around the Clock’ appointment cards idea from Global Math Department.

Using Data from FACTs

Most students realized that AAA could only guarantee similarity amond the triangles.  There were several misunderstandings about HL I was not aware of until after I read student reflections.  I will be more puposeful in defining the included parts in the various combos, for example I shared examples with students and asked how AAS and ASA are alike / different because this was one that a few still had struggle with.  During this discussion / sharing – it was obvious some a-ha! moments occurred.