Monthly Archives: April 2013

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.

Evaluating Statements About Length and Area


This lesson can be found same as title of the post.


This is one of six cards students discussed within small groups today. A student stated, “this is going to be a thinking day,” as they began removing the clips to start reviewing their cards. Most students would quickly come up with an always, sometimes or never true. However, to create their own examples or counterexamples to either justify or refute the statements was a struggle for some of them. Several groups had similar statements for this particular card. It was when a student asked, “do they have to be triangles?” that a turning point came for some.


Within our share out as a whole group, a student shared examples of reducing area, same perimeter and less perimeter. A question they wondered…can you reduce the area but increase the perimeter?

I really enjoy days like this, students are giving me the information, I am their scribe and I am slowly learning to let them determine if they agree or disagree with each others’ claims. I’m not even sure where the key is, that way I am actively having to listen to their arguments to determine if I agree or not. (Shout out to Max @Math Forum, I am listening to my students, not listening for the answer!) I go through the cards myself prior to the day of the lesson, just like I require them to do. But I am still closed minded in my own thinking at times. Why would you limit the example above to only triangles? Because that is what shape was presented on the card. However, does it state triangles only? Nope.

A task like this may drive some teachers crazy. Once you start considering different shapes, you begin to see what works for one, may not work for another. I had students cutting scrap paper, tracing patty paper, measuring side lengths…without me telling them to do it.

The classic question, a square and circle have equal perimeters, which has the larger area? I will do my best to share more reflections as we wind up tomorrow, if we wind up tomorrow…depending on their questions, discussions, claims and supporting evidence.

Quadrilateral Diagonals Properties


Over spring break, I was surfing online resources, searching for ideas and suggestions on how to plan and be more purposeful with the Mathematical Standards, which I have realized this year just how key these are to the success of CCSS. As I looked through Inside Mathematics , I ran across some PD training materials. I watched clips from Cathy Humphrey’s class. The Kite Task, an investigation of quadrilateral properties from seemed like a great activity to ease back on day 1 when we returned.

The task in short is for a kite company, who wishes to launch a new line of kites consisting of all types of qudrilaterals. The students are asked to devise a plan for how to cut/assemble the braces for each type of kite. They are only working with the diagonals in the investigation.

Rather than running copies and cutting out, I used my paper cutter to cut 1″ strips one color card-stock lengthwise and 1″strips width wise of a different collor (I didn’t realize how helpful this would be until later on). I created a strip to use as a guide on each strip, placed 7 holes equally spaced. Odd amount is best since they will be looking at bisectors some.
Each student would receive 2 of one color and 1 of another color.

Here are some snapshots of possible braces built.

For anyone who is having trouble visualizing, I’ve added some “sides” to the diagonals:

As we began the 2nd day of class, a few groups needed just a bit more time to wrap up their investigation. Using fist to five, I asked how many they still needed to determine. Most groups only 2 or 3, so I set the timer to keep us on track. I love days like this to walk around and just listen.

As I was questioning one of the groups, trying to ensure an absent student was on track, I asked the group’s members to “fill an order” – pick 2 sticks and construct the diagonals needed to brace…kite that was a rhombus, then another shape, etc to quiz them for understanding. AHA! Why couldn’t I use this as a formative assessment for the entire class?!?! Perfect.

When all groups had completed and debriefed a bit, I placed orders for kites and the students had to build the braces and pop up to show me for a quick assessment.


These pics were actually a geometrically defined kite. If you look closely, you can see a few wrong repsonses. To address these, I used extra sets of sticks to build a correct example and an incorrect example. To ask for suggestions why one was and the other was not correct. Why was one example actually a rhombus, allowing them to really compare/contrast the two figures.

Another great mistake I saw…when asked to create a rectangle, the top sketch is what I saw from about 6 students. Of course, my initial thought was, they dont understand the diagonals must be congruent.

Then I saw a student trace their shape in the air…second sketch. I literally saw their thinking. They had not used the sticks as diagonals. Clarified and corrected!

A post-it note quiz today, I built the braces, they had to tell me the quadrilateral name. A stop-light self assess, revealed most were confident, of the 10 yellows, 7 got all parts correct. The others missed 1, 2 or 3. All green students had each part correct.

We did a little speed dating to use properties to solve problems. As I listened to their approaches, most everyone seemed on track. Overall, I was very pleased with the results of the lesson.