An Actual Application of the Law of Sines

I have the privilege of teaching an algebra and trigonometry class for engineering tech students.  I am always looking for ways to help my students experience the relevance of a concept.  My parents' DIY patio project this past weekend provided an awesome practical application of the Law of Sines.

My mom wanted some shade around the pool so she found the photo below online and tasked my dad and brother-in-law with making it happen.

They were cruising along until I received the following picture from my mom:

They were wanting to figure out the exact angle to cut the front diagonal boards (near the peak) without a few hours of guess and check.  They did know that they wanted the length of the diagonal board to be 30 inches.  She also sent me a quick sketch:

I observed that it would be close to an isosceles triangle, but my brother-in-law is a perfectionist so I used the Law of Sines to give him 4 decimal places.  Here was my work:

The angles worked great and I did a happy dance because I am so excited to use this scenario with my students.  Would you have solved it differently?

Every View Has Value

I have been constantly modifying a low tech activity that emphasizes the connections between the the 4 views of a function.  Here is how the latest version works:

• Students work in small groups
• They are given a group of mixed up cards that represent the words, equation, table, and graph views of 8 scenarios.
• The task is to put together the corresponding views for each situation and then use the cards to answer a set of related questions without additional access to technology.

At first I had students stop at this point, but upon reflected I discovered that while they were matching up the views, students were not understanding the value of each view. So, I extended the activity to include a question set that had students analyze the scenarios and answer specific questions about them while at the same time emphasizing the view that students chose to use to answer each question.  This question set has been crucial in helping students understand how the views are connected to each other and I have observed that students are much more willing and likely to explore and use different views when working on problems after this activity.

The following are my set-up and facilitation recommendations:

• Use a variety of functions, but include a couple that are familiar to students
• Let students struggle (mine usually take 20-30 minutes on this)
• Add Independent/Dependent variable cards or Domain and Range cards
• On the handout as specific questions as well as observation and comparison questions
• Adapt this activity to focus on a specific type of function
• Update the scenarios as needed to they are current and relevant

Check out the files below if you would like to try out this activity:

• 4 Views Solution Grid
• 4 Views Activity Cards
• 4 Views Question Set

The Metamorphosis of an Idea

I have been making an attempt to make my existing lessons and activities more mathematically meaningful.  I think I made some progress last week when we were working on fraction/decimal/percent connections in two of my classes.  In the past I had students go to the gym and shoot a particular number of baskets and collect the data for makes and attempts and turn that data into a fraction, a decimal, and a percent.  It was fun, but didn't generate too much reasoning or meaning for the amount of time it took.

Last week, I had each of the eight students in an occupational math class create an activity that involved successful and unsuccessful attempts.  We landed on the playground doing creative things such as throwing a ball at a bowl on a spinning merry-go-round.  The students collected the data for their own events and then were to return to the classroom to determine the fraction, decimal, and percent that represented the successes of each participant.  In theory, this activity should have worked, but I was getting little math out of it until we started bouncing a tennis ball at the basketball hoop.  I told each student to see how many makes they could get in 9 tries.  The first student start and missed, but a voice said 0 for 1.  I responded, "What percent is that?" The other students replied zero.  The student tried again and made it.  He said, "1 for 2, hey that's 50%"  The activity finally morphed into something helpful as the students determined the percent after each attempt and made the connection about how one fraction/percent related to the next.  One student even stated, "It makes a lot of sense once you calculate the percent after each try."

The next hour I decided to try a similar activity with my 7th graders, but I knew it would be chaos for them to create their own challenges so I set up 8 activities throughout the room (one for each student).  I had each student create a grid to collect their attempts and successes.  I initially told them how many attempts they should have for each activity, assigned them to their first station, and told them to go.  After about 15 seconds I realized that my students needed a little more structure because some did their trials very quickly and were sitting down staring at me and others were leisurely rolling a 20-sided dice.  I decided to change the activity and told them that they need to run as many trials as possible in one minute.  After one minute they wrote down their data and rotated to the next station.  This was a crucial adaptation as it lit their competitive fire a bit and removed the down time from the activity.  After each student completed the eight stations, they returned to their desks to write the fraction of their successes to their attempts.  Then I had the students see what fractions they could easily convert to a percent.  They discovered that the 1/2s, 1/4s, 1/5s, and 1/10s were the easiest.  This lead to a discussion on how we could figure out the percent equivalents for more complicated denominators.

One activity +  two different classes = two mathematically meaningful experiences, not bad for one day...if only it worked this well everyday.

Curbing Learned Helplessness with Questions

I am nearing the completion of my Master of Arts in Teaching Mathematics and in the midst of my action research project.  My problem/purpose/question is to determine if a focus on reasoning and sense making tasks in the classroom will help my students become independent learners.  Since my classes range from 3 to 20 students, I fear that my students' access to me has created a culture of learned helplessness.  For example, if a student has a question, I am able to respond immediately and the student has little time to problem solve independently.  I have been working on giving better responses (more questions) to questions and stalling a bit before I acknowledge the raised hand.  I plan to use my Algebra and Geometry classes as the research subjects during the third quarter, but I have been practicing my questioning strategies with my 7th, 8th, Applied Math, and PreCalc students.  I have discovered that it is becoming more natural for me to ask a question in response to a question.  Some of my favorites include:

• Why?
• How can you start?
• What have you tried?
• Does it (the solution) make sense?
• Is there another solution?
• Walk me through what you did.
• Try drawing a picture.

While the questions themselves are not particularly profound, they have lead to a dialogue between my student and myself.  These dialogues have enriched my practice and I believe are helping my students understand and retain the concepts rather than just getting through a task or an assignment.

Musings on the First Week

Favorite quote:  "Will you please just tell me the answer and not ask me another question when I ask you one?"
Reason:  That's my focus this year - to be less helpful and ask questions

Best first day activity ever:  A Bingo/Jeopardy board that the students worked on to earn up to five free homework or grade booster passes.
Reason:  It was awesome to hear mathematical conversations and problem solving on the first day.

Best new idea:  Taking the time to make a Notebook file for every hour for every course.
Reason:  Even though I teach six different courses and this takes a lot of time, this has helped the lessons to be focused and engaging and has significantly reduced the down time.

Engage Immediately

I remember sitting through many first days of school, moving from one class to the next, bored to tears as the teacher went over the syllabus.  These un-engaging hours quickly zapped my excitement for learning.  As a teacher I have learned that it is important to clearly explain my expectations, but I have struggled with how to do this in an engaging manner - especially when I have the same students year after year and they basically know my expectations and just have to adjust to the two or three bright improvements I come up with over the summer.

One of my expectations is that class time will be used for learning thus this year I have created a Jeopardy/Bingo/Find a Person Who...conglomeration of a game board to give to the students on the first day.  They are typically given one free homework pass per quarter, but this year they will earn their free homework passes based on their completion of the game board.  The questions are based on the previous year's standards and my expectations so this task will help me assess retention over the summer and help the students refresh their skills.

An Attempt at RtI

Our school has decided to implement a Response to Intervention (RtI) program this year.  We are targeting math and reading...the challenge is that our school is tiny and we only have one teacher per subject for grades 7-12.  So, most of the 7-12 teachers have gathered this summer and we have split into two teams - math and reading.  I will have 4-5 teachers working on the math team with me and below is our plan:


Math will focus on the following standards
First Quarter – Numbers and Operations and Measurement
Second Quarter – Algebra and Patterns and Measurement
Third Quarter – Geometry and Measurement
Fourth Quarter – Data, Statistics, and Probability and Measurement

*We have a partial block schedule so grades 7-9 will have Math RtI on Purple Days and grades 10-11 will have Math RtI on White Days.  I have identified essential standards within each major standard and each teacher will focus on one of the essential standards each quarter.  Students will rotate from one teacher to the next after 4-5 days of instruction on a particular essential standard.  Since we all prep 6 different core classes every day, this keeps our prep time for RtI manageable and allows us to do one concept really well.

Student Placement for Math

Students will initially be identified as in need of intervention based on the most current MAP score (Spring 2011).  Students who score below the 66^th percentile on a particular standard will take a pre-test on the standard to confirm their need of an intervention.  If a student scores above an 80% on the pre-test that student will be exempt from the intervention on that standard.  Teachers will also make decisions to include or exempt a student based on observations, classroom performance, and state standardized test scores.

Leaving RtI

Students will be given a post-test at the end of each quarter.  Teachers will discuss individual students who are not demonstrating progress and modify the intervention as appropriate.  Students who demonstrate proficiency based on formative assessments and an 80% or higher on the post-test may no longer qualify for the intervention.  Students will have the opportunity to move from the intervention to service learning at midterm if they score an 87% or higher on the midterm assessment and have earned at least 90% of the engagements points.

2012 Seniors      

This years seniors will not be part of the RtI program. They will have the opportunity to earn a senior privilege based on class work completion and attendance. Power school will be checked every Monday morning. If a senior does not have any missing assignments and was not tardy or absent for any reason the previous week then that senior may leave the school building after 7^th hour. If the student does not meet the criteria then he or she will report to the library for 8^th hour. Failure to comply will result in the loss of senior privileges for the remainder of the semester and the student will be referred to the discipline matrix.

Students Exempt from RtI

Students that are exempt from intervention will participate in service learning projects (SALT). 

Teacher collaboration

All High School teachers will collaborate every other Wednesday.  On those days students will participate in service learning projects, career presentations or other speakers/presentations.

Our plan seems to be working on paper and all of us on the math team are working on our essential standard for the first quarter.  I am hopeful that this intervention will have a positive impact on our students and I'll keep you posted on modifications and lessons that we develop.