Logarithms - UGH

I think of myself more as a geometry and trigonometry teacher and less as an Algebra 2 or Statistics teacher.  We are on an integrated math system and I teach third and fourth year which amounts to 1/3 geometry, 1/2 algebra 2, 1/4 statistics, and some precalculus for integrated math 3 (which I know is >100%) and integrated math 4 is mostly trigonometry and precalculus with a bit of function modeling and vectors.
But truth be told, I do not like logarithms.  I remember hating them in my Algebra 2 class growing up.  I knew that I was not doing justice to them as a teacher, so I hit the web looking for a better approach!  I have to give all the credit to some great bloggers that I found!  Sarah Hagan is amazing: if you are reading this blog, then please check out Sarah's blog it is the best at Math = Love.  Thank you to Sarah Hagan I used your log notes with the loop method.  So credit for the loop must be given to Amy Gruen with square root of negative one teach math.  We also did log "war"!  Logarithm war is like the card game war you played as a kid, but this is with evaluating logarithms.  There is a link to the cards that I used on the previous blog.  I also added cards that had solutions to the logarithms that were fractions or negative numbers.  This was awesome. I asked my principal to come in and observe.  The students were excited about logarithms.  (I wanted to take a mental picture of this!)  They were engaged and trying to beat each other as they raced against their partner to answer the logarithm quickly.  This activity got in about 80 practice problems per person within the time in class.  I don't think an 80 question worksheet would have fared as well.  The opportunity for differentiation presented itself easily.  I paired some of the lower students with higher students for round one.  In round two, I let similarly abled students pair.  Some students took longer to get through the 30 cards - this was fine because they were thinking about it and really trying their best.  Some higher functioning students were able to quickly process logarithm answers.  We stepped it up and played "speed log war."  In this version you had to give the response the quickest (and be correct) to get the cards.  Excitement and energy were great!  I even jumped in for a round.  If we had more time, I think we could have designed a basketball type bracket for the logarithm championships.
Even as students come to me in integrated 4, I remind them of the loop method and they immediately remember.
A suggestion to make is that every card was written in the format log_2_8 where a student could give the answer 3.  When we worked on practice later, some students struggled with a problem written as x=log_2_8.  They were trying to do the loop and the variable was not on the right side.  So, next year, I will provide variety in the format of presentation so that the lower students understand (regardless of the format) the "loop" method.

Motivating the Unmotivated

A challenge that we all face is motivating the unmotivated learner.  It seems that high school education culture has grades and college as a high motivating factor for most students.  This is a problem as high school and college no longer seems to be about educating and learning for the sake of learning and to be informed about a body of general ideas, but rather a training ground where students are motivated to "make money."  This bothers me, but is not the point of the post.
How to motivate the unmotivated...
I want to devote my thoughts to the unmotivated or underachieving students as the focus for this post.  I teach the regular-level classes (non-honors) and have a range of students from those who really enjoy math and routinely master the material, to those who loathe coming to class.  One of the first things that I do in class is make intentional relationships with students.  I try and get to know them and see them for more than who they are in my class.  I am thinking about trying to reach the lowest 10% of kids here; those who dislike school; those who hate math.  Often, I can develop a relationship such that when I request a student pay extra attention to homework, they will comply.  I can request that a student come in for additional help, and usually they will.  What about the kids who won't?
Here is my struggle: motivating the students who hate math and do poorly.  This past year I had students in study hall that I could help who were lower achieving.  This took daily work of my looking over their shoulder to ensure that homework was complete and correct before they even came to class.  I also used this time as remediation with them so that after poor performance on assessments, we could review and remediate weak areas.  Then they could reassess.
Two specific examples of student interactions this year.  Student A struggled with a D all year and with some increased work in the trigonometry unit was able to pull his grade close to a C.  As final exam time came close, I made a contract with the student.  I highlighted 40% of the 10 page review guide.  The students had to complete all highlighted portions of the review with correct work shown for each part of the problem before I would let him take the final.  If on the final exam day, he came to school without the highlighted portions complete, I would make him work on those questions while his peers took the exam.  When he finished he was eligible to take the exam during the make up exam period.  This strategy worked for student A.  He came to the period with excellent work on the review packet and was able to take the exam and keep a D in the class for the semester.
Student B dislikes school more than student A.  During the trigonometry unit I gave a 20 point formative on finding exact values, then a 20 point formative on solving for angles.  (I think of this as working forward then working backwards.)   After the two twenty points, we put them together for an assessment working on both skills worth 40 points.  Any student who earned less than 15/20 was told to come in to remediate and retest on either/both of the first parts.  This would ensure a basic understanding of the material as every trig assessment for the next two years builds on these concepts.  Almost all students eligible came in to work toward mastery, such that the median score for the combined assessment was 37/40.  Student B did not take advantage of this opportunity.  So, instead of taking the 40 point quiz, I had him remediate and retest his previous scores of 3/20 and 11/20.  After he retook these, then I had him take the combined assessment, earning 32/40.  What a success.  Unfortunately, student B still earned a D in the class such that the final exam was the determining factor of whether or not he passed the class.  He did not complete the review, nor did he come to review sessions, thus earning a 35% on the final and failing the class.
Why didn't I make the contract with student B too?  At what point does the responsibility for wanting to learn and succeed take over?  These are juniors in high school.
Some colleagues have instituted a two strikes homework policy.  Students who miss two homeworks must come in during lunch or after school to complete the next day's work.
Some colleagues have adopted A, B, not yet.  This is interesting, but oh the retesting and remediation.  With 130 students I think that I would be exhausted and not physically be able to remediate all of these students and make enough new assessments.
I wonder if you used standards based grading.  Would this be an improvement?
What about if students completed units of study as projects of the body of knowledge.
Education is changing so fast.  I struggle with having students learn math because it is important and the amount of real application involved.  Truthfully, there are some lessons where the application seems so much harder, I don't know if there is even benefit to say "with 10 more levels of difficulty you could use ___ in the real world."
I would love some thoughts and feedback here.

Ways to teach and introduce the unit circle, reference angles, and exact values

I have thought through the best ways to teach students to love trigonometry.  I tell them this is "The Best Unit Ever" because of how great it is for students to first see the relationship between a triangle and a circle.  I am hoping that you will comment on this blog post to give your version of how to begin to introduce trigonometry.
First, I think it is essential to have a few days of "back to basics."  We refresh operations with fractions, simplifying and rationalizing radicals, and basic fraction simplification of of sine, cosine, and tangent ratios.
From this point, I introduce all six trig ratios.  Students primarily work in quadrant one.  They should be able to find all six trig ratios given cosecant of theta = 2/3.
Now for the fun.  I begin by giving two homework questions asking students to find all six simplified trig ratios.  The first triangle is such that sine of theta is 5/10.  I use this so that students will find all six ratios simplified and can refer back to this triangle as they get into making values for any angle with a 30 degree reference angle.  Similarly I use a second triangle such that tangent of theta is 7/7 so that students can refer back to this with any 45 degree reference angle.

Next we begin the whole next class period with the 30 degree triangle that had sides 5, 10, 5root3.  I discuss the unit circle as having a radius of 1.  From there I use the idea of similar triangles to "scale" down the triangle by 10, thus creating a triangle of hypotenuse 1, height of 1/2, and base as root 3 over 2.  We place this into quadrant one.

 

As we started, we only had quadrant one and we took time to label the actual triangle's base and height and discussed how you might write the ordered pair at the edge of the circle.  We also discussed how by looking at the triangle's opposite and hypotenuse we still get sine as 1/2 and why sine is the height or the y value.  They struggle at first with labeling points as with ordered pairs that include square roots.  After this we continued through each of the four quadrants only drawing the triangle and labeling the triangle's sides with 1/2 and root 3 over 2.  As we ended the class period, we were able to use reference angles to label each angle from standard position in degrees (radians were drawn in later).  Students also started to label exact values in ordered pairs.  Their goal in homework was to complete x and y values as ordered pairs.  The next day about half of students had considered positive and negative values since we were in quadrant two or three, etc.  This was great to discuss at the beginning of class.

The next day, we took the 45 degree triangle above and similarly placed it in quadrant one as a whole class activity.  Then students continued working in class on correctly identifying all four quadrants with angles in standard position in degrees  and with ordered pairs. 

Students completed this day's assignment by taking the 30 degree triangle and flipping it to make a 60 degree triangle in quadrant one.  We did this part together and discussed placing x and y values at this mark.  Students were to continue by making the 60 degree bowtie diagram.

I would love to hear comments on how you introduce and teach the first week of trigonometry to your students!

End of year reflection

Today is the last day of school and that always makes me be reflective and a bit sentimental.  This is the end of year TEN!  I am still filled with energy and passion.  I think that becoming a mother has changed me over the last ten years to have more compassion for students.
As I was driving in to work today, I thought about the types of students that have grown to be my favorites.  I love the hard working student who is eager to learn and strives for success.   But more than that, I love the willing student who dislikes math.  I like the challenge of taking a student who professes to "hate" math and by the end of the year being one (small) factor for their change into a student who has to work hard but now has confidence in their mathematical abilities and enjoys mathematics.
I feel excitement that I have chosen the correct career path.  I truly love what I do.  I love forming relationships with students and helping to teach them some life lessons along the way, not just drilling them with math.  I still get refueled by the student who takes pride in their accomplishment of success and "finally gets it."  Yes, this is why I still want to be a teacher!
Happy Summer!

This week in class we are PARCC testing and using a block schedule.  We typically have an 8 period 50 minute schedule, so I am able to get in some good exploration activities.  First we practiced using construction tools to construct the perpendicular bisectors of a triangle.  We found the circumcenter and circumscribed a triangle about the circle.  It was interesting to see how students excelled at this activity (hands-on application) that do not traditionally do well in class.
To extend the idea of perpendicular bisectors and circumcenters of triangles, I designed a coordinate geometry activity.  Feel free to use this activity. Exploring with the circumcenter

I will re-post with a summary of the class activity.

Objective: Use coordinate geometry as a method for finding the intersection of perpendicular bisectors of a triangle, the circumcenter.  Write the equations of the lines of the perpendicular bisectors of sides of a triangle.  Use the circumcenter to write the equation of a circle that is circumscribed about the given triangle.