My Vision: 

To make all students participators in the math classroom.


What does it mean to make students participators in the mathematics classroom?

• To make students participators in the mathematics classroom means that students are actively engaged in the learning process, taking part in instructional activities instead of exclusively listening to a lecture.

How do students become participators in the mathematics classroom?

• Through communication: every student should learn to communicate about mathematics. This can happen through classroom discussions and small group conversations. Technology can assist in communication through texting and social networking sites such as Edmodo.
• Through exploration: students gain a much deeper understanding of mathematics if they are given opportunities to discover mathematical concepts for themselves. This can be achieved through technological applications that let students visualize mathematical concepts, through inquiry based-learning projects where students are asked to investigate a mathematical question or problem, or any other means of student-driven discovery.
• Through critical thinking and reasoning: students should analyze mathematics by recognizing patterns that they observe or discover and logically developing mathematical arguments based on their observations. These reasoning skills are a critical part of the mathematical skills they will need for their future careers.
•  Through collaboration: cooperative learning activities enhance all of the above activities, by allowing students to learn from each other and sharpen each other’s ideas.

Why should students become participators in the mathematics classroom?

•  Participation increases student motivation. Students are more interested in mathematics when they are involved in the learning process!
• Participation engages students in the classroom. When students are exploring math through hands-on representations or technological tools, or when they are reasoning about math and discussing ideas with their peers and teacher, they must use all of their senses in the learning process. This makes mathematics more concrete and easier to understand.
• Participation provides real opportunities for differentiation. When students are exploring math, or developing mathematical arguments, a teacher can provide different levels of exploration and critical thinking for students who are at different academic stages. More advanced students can be driven to dig deeper than other students. The teacher can provide scaffolding interventions for students who need them.
• The Common Core Standards demand student participation in the math classroom. Mathematics Practice #3 requires students to “construct viable arguments” (critical thinking) and “critique the reasoning of others” (communication and collaboration). Practice #4 requires students to “model with mathematics” (explore real world scenarios with math). Students are also required to “make sense of problems” (#1), “reason abstractly and quantitatively” (#2), and “look for and express regularity in repeated reasoning” (#8), which require critical thinking and are enhanced by exploration, communication, and collaboration.

Algebra: A Mathematical Poem

Algebra

When I met the unknown

I stopped,

And cried, familiar only

With the metronomic

Repetition of numbers.

What was this letter doing

In the midst of my math book?

From one moment to the next,

Life went from simple

To impossible,

When I could no longer

Plug questions in a calculator

To get an answer.

But when I started treating the unknown

As a number

(Shaking it like a gift

That I might find a clue),

The pieces fell in place,

And I found the answers

Right beneath my nose.

2013 NCTM Conference: "Exploring the Common Core Practices in Secondary Classrroms"

2013 NCTM Conference (4/19/13 - 3:30 pm - Kristen Bieda and Samuel Otten):

My intention in this session was to learn more about the Common Core Mathematical Practices. Kristen and Samuel provided details on these practices embedded in the standards, but even more importantly, they suggested ways of implementing the standards in a high school mathematics classroom.

The Practices are defined as follows (the links were not provided in the session):

Standard 1: Make sense of problems and persevere in solving them 

Standard 2: Reason abstractly and quantitatively 

Standard 3: Construct viable arguments and critique the reasoning of others 

Standard 4: Model with mathematics

Standard 5: Use appropriate tools strategically 

Standard 6: Attend to precision

Standard 7: Look for and make use of structure

Standard 8: Look for and express regularity in repeated reasoning

Kristen and Samuel noted that these practices require students to "do" mathematics rather than just following mathematical procedures laid out by the teacher. They suggested that "doing" mathematics requires the following:

• Students participate in complex and non-algorithmic thinking.
• Students explore mathematical concepts.
• Students self-monitor their problem-solving processes.
• Students access relevent knowledge.
• Students examine task restraints.
• Often, students will experience a level of anxiety as they wrestle with mathematics in a more real way.

Kristen and Samuel focused on the implementation of the 3rd standard (construct viable arguments and critique the reasoning of others) which may be one of the most difficult to implement. They stressed the importance of reasoning and proving for students to meet this standard. To participate in reasoning activities, students must generalize mathematical relationships, evaluate arguments, and discuss assumptions and what they are based on.

Kristen and Samuel promoted the importance of modifying tasks to promote mathematical reasoning. They provided several ideas for doing so:

• Ask students what happens in the nth case of a pattern.
• Ask students to justify their conjectures.
• Reduce scaffolding, so that students are prompted to seek out strategies for problem-solving.
• Ask students to identify prior knowledge that they assume is true and justify the use of this knowledge.
• Ask students to define key mathematical objects.
• Ask students to model a problem through visual representations.

Kristen and Samuel suggested that if students are actively "doing" mathematics, they will cover all the Common Core Practices. The Practices are interconnected and naturally work with each other.

2013 NCTM Conference: "Take Time to Question the Questions"

2013 NCTM Conference (4/19/13 - 12:30 pm - Mark Howell):

I've come to believe that asking the right questions is one of the most important parts of building student understanding of new mathematical concepts. One of my goals in coming to the NCTM conference was wanting to continue to sharpen my questioning skills. I was therefore eager to learn from Mark's session about "questioning the questions".

Mark started the session with a bold statement: he suggested that as beneficial as multiple representations of mathematical concepts are, the use of multiple representations is not sufficient. He argued that linking between and among the representations is where learning takes place, when student understand how the representations are connected together.

Mark proposed that good questions are those that come from powerful connections between representations. These questions prompt students to explore these connections and examine what the representations actually mean.

Mark noted that it is possible to ask a lot of questions in the classroom without building a deeper understanding of mathematics. He noted that we've all seen procedural questions that ask students to provide the next step in a familiar mathematical routine. These questions tend to be very repetitive and removed from context. But students can memorize procedures without knowing how to think about math or understanding what the procedures really mean.

Teachers often provide shortcuts for students that allow them to solve problems without thinking about how math really works. This can be detrimental for students in the long run. For example, students may have memorized the "vertical line test" without knowing the properties of a function.

Mark noted that questions are essential for revealing deeper mathematical realities. But it is difficult to craft good questions that prod students into these deeper realms. There are four different levels of questions:
1. Questions that elicit feedback from students.
2. Questions that ask for factual responses.
3. Questions that ask students to explain a procedure.
4: Questions that ask a student to explore math, make conjectures, explain and verify their results, or talk about connections between representations.

Most teachers' questions remain in the first three levels, but these questions are all very shallow compared to the fourth level of questions. It is important that we consider how to craft questions that take students to this deeper level.

2013 NCTM Conference: "Technology: A Portal to Exploration and Discovery"

2013 NCTM Conference (4/19/13 - 8:00 am - Kenn Pendleton):

A big theme through the NCTM conference was using questions to prompt students to explore mathematics. I am a big proponent of inquiry-based learning, so I was excited to find other teachers with similar passions. I was able to collect a lot of fresh ideas.

Kenn suggested that we can use technology as a tool to enable students to explore mathematics. He suggested that we as teachers should ask questions which require students to use technological tools (such as graphing calculators) to explore and answer.

Kenn strongly encouraged the use of technology to allow students to explore the connection between graphs and their symbolic representations. This connection is very important for helping students to visualize math and to build a deeper understanding of what mathematical symbols actually represent.

Kenn encouraged teachers to challenge students to create conjectures based on their observations using technology. They can then test these conjectures by contrasting different scenarios and evaluating whether their conjectures are true or not. This can given them a fuller understanding of mathematical principles than the teacher given a list of facts that go in one ear and out the other.

Technology can be a powerful tool for comparing the graphs of different functions. This can help students contrast functions with their inverse, tranlations, and other properties of functions.

2013 NCTM Conference: "Discover New Ways to Make High School Math Meaningful"

2013 NCTM Conference (4/18/13 - 2:30 pm - Tim Pope):

Tim Pope is the curriculum manager at Kendall Hunt Publishing Company. While this made his session a little more of an advertising campaign, he had some good ideas to share as well.

One statement Tim made really stuck with me, perhaps more than any other at the conference. He said, "If we can remind students of a story that kids can remember, we'll always have something to build on." He elaborated on this idea further, suggesting that if we make math real to students and get them involved in the classroom, they'll remember it much better than they would if they memorized a bunch of formulas. They'll remember even better if a teacher makes math fun as well, or allows students to get up and move around, physically participating as well as mentally participating.

Tim got the audience to participate in an activity identical to one he would use with students. We calculated the amount of time it takes for a specified number of people to complete the "wave", and plotted this data in a scatterplot. We then used this data to create a line of best fit and discussed the properties of this line.

Tim advocated the Van Heile structure for building an understanding of mathematical concepts, and gave an example of each.
Level 0: Visual - students are able to visually identify shapes, patterns, etc., but can't explain why. "It is a rectangle because it looks like a door."
Level 1: Descriptive - students are able to describe the properties of shapes and patterns, but are unfamiliar with the ways different properties interact with each other. "It is a rectangle because it is a quadrilateral with four right angles."
Level 2: Abstract - students can more fully describe the ways the properties of shapes and patterns interact with each other and understand what makes each individual group unique. "I know it’s a rectangle if it’s a parallelogram with four right angles."
Level 3: Proof - students can prove why the properties of shapes and patterns are true. "I can prove it is a rectangle if it’s a parallelogram with one right angle."
Level 4: Rigor - students understand a concept so fully they don't need to consider the basic properties of the concept. "I know that is a rectangle - what would happen if I tried to put it on a sphere?"
Tim suggested that we should challenge students to wrestle with higher-level mathematical questions, that they may gain a fuller understanding of the mathematical concepts involved.

2013 NCTM Conference: "Using Teacher- and Student-Made Videos in the Mathematics Classroom"

2013 NCTM Conference (4/18/13 - 3:30 pm - Janet Andreasen, Deborah McGinley, and Zyad Bawatneh):

Janet, Deborah, and Zyad gave a discussion advocating the use of videos in the classroom - created both by teachers and students - for instruction and assessment. This is a concept I had not personally explored, so I was intrigued by the ideas they shared.

Janet suggested that student-created videos can give teachers the opportunity to view students' problem-solving process instead of just analyzing the end result. The common core mathematical practices require students to draw geometric shapes with given conditions, find solutions using technology, and many other "practices" that are as focused on the journey as much as the destination. Student-made videos allow teachers to assess the practices more fully than they may be able to do through informal observations.

There are many different types of software that can be used for screen capturing and voice recording. Jing and Educreations are free software packages that students can use to create videos. Camtasia and Explain Everything offer more features but come with a price tag. The TI-nsspire also comes with video recording features.

Janet and Deborah suggested many uses for videos in the classroom: First, as mentioned above, student-created videos can be used for authentic assessments to help teachers evaluate the process as well as the final product. These videos, if presented appropriately, can provide insight into students' thinking in a way that a test cannot.
Second, teachers can use videos in a "flipped" classroom to provide instruction for students to view at home or to provide extra support. Teachers can use videos to not only speak to students, but they can also use software such as GeoGebra or visual manipulatives to give students visul representations of mathematical concepts.
Third, videos can be used for student-to-student support. Students can share their ideas with other students through videos. Deborah explained that their district uses Edmodo to encourage students to collaborate together and share student-made videos with each other.

2013 NCTM Conference: "Keeping It Real: Teaching Math through Real-World Topics"

2013 NCTM Conference (4/18/13 - 12:30 pm - Karim Kai Ani):
Karim from Mathalicious.com gave an awesome presentation on the value of "real-world" math to engage students in the mathematics classroom. Karim suggested that math has become very distasteful in our society, "a recipe to something no one wants to eat". He suggested that we need to make math "delicious" again by engaging students in math through topics that they find interesting.

Karim suggested that students today look at math as a bunch of arbitrary rules that somebody made up to torture people for thousands of years. On the contrary, students who explore math based in real life will be more interested in how math works and will be more likely to participate and ask questions.

Karim showed attendees some example lessons from Mathalicious.com that teachers can use in their classroom. The first example he demonstrated was using the Domino's web site to graph the cost of each topping on a medium Domino's pizza. The site doesn't describe the cost of a single topping, but through mathematical evaluation, students can find the cost of each topping and predict the cost of a five topping pizza. The teacher can build on these findings to develop the concepts of slope and the slope-intercept equation of a line (where the y-intercept is the price of a cheese pizza, and the slope is the cost per topping). Students can also compare the graphs of medium, large, and small pizzas.

Karim suggested that teachers can use videos, movies, internet sites, and hands-on applications of science, as well as many other applications of math, to make mathematics more real to students. The world is an interesting place, and there are a plethora of interesting questions to ask. Math is the tool we use to explore these questions.

2013 NCTM Conference: "Using Problem-Based Learning Tasks to Foster Reasoning and Proof"

2013 NCTM Conference (4/18/13 - 11:00 am - Enrique Galindo, Julie Evans, and Jason Walton):
In this session several teachers from Indiana shared their journey of using problem-based learning in their classroom. These teachers worked together to create units based on problems that prompted students to respond to a question in such a way that they are actively engaged in the learning process and driven to learn appropriate skills and concepts.

The first example that one of the teachers gave was a PrBL which challenged students to consider how long they would have to grow their hair in order to donate it to Locks of Love. The students were required to research the necessary data, plot a graph of hair growth, and answer the question using mathematics to defend their opinions.
The students had to research the following questions:
What is the "starting point" of their hair now? Where would this point be on a graph?
What is the rate of growth for their hair? How does this affect the graph plotting hair growth?
The students made predictions of the rate of hair growth based on a video depicting the hair growth of a young man who grew out his hair for "Locks of Love". Students measured their own hair and researched how long hair must be in order to donate it.
Students then used the data to create a graph and to model it using an algebraic equation. They then gave a presentation on their findings and defend their predictions and mathematical calculations.

Another teacher provided an example PrBL which challenged students to create a new snack using chocolate and peanuts which could not exceed specified conditions regarding weight and fat content. This project required students to apply systems of inequalities to a real life scenario.

A third teacher provided an example PrBL which asked students to use trigonometry to analyze the "biorhythms" of the two quarterbacks in the superbowl to predict which quarterback would win the superbowl.

The teachers provided the following suggestions about implementing PrBLs:
First, student grouping and collaboration is important for driving discussion and investigation. They suggested allowing students some amount of choice in who they work with, so they are enthusiastic about working together. Students must be willing to discuss ideas in order to work together to achieve the expected results.
Second, the teacher should provide students with a rubric so that they have a goal to aim for. Students find it frustrating to work on a project if they are confused about what the teacher desires.
Third, the teacher should ask many questions to guide students' thinking. Instead of telling students "that's wrong", they should challenge the students to defend their ideas and use mistakes as a base for learning.
Finally, authenticity is extremely important. Students have to be "hooked" to get passionate about what they are doing.

PrBLs are excellent for developing 21st century skills, helping students understand how to really use math in their lives. Memorizing mathematical facts and formulas is not enough.

They provided the following site for people interested in PrBLs: www.pbl-online.org

2013 NCTM Conference: "I See It: The Power of Visualization"

2013 NCTM Conference (4/18/13 - 9:30 am - Marc Garneau):
The second session I attended at the 2013 NCTM Conference was Marc Garneau's "I See It: The Power of Visualization". As a visual learner, I know how helpful it can be to provide students with more concrete representation of mathematical concepts.

Marc suggested that most students have been taught symbolically their whole lives, and the lose touch with what the symbols really represent. Students who do not understand what the symbols really mean have no idea how to visualize math or even to apply it in real life applications. Visualizing math can help students reason about math and give them a fuller understanding of the abstract concepts.

A typical example of visualization is creating diagrams to depict fractions. These can be very helpful building students' understanding of fractions, which can be a very challenging concept for students. Comparing fractions can be difficult symbolically, but is much simpler if students are able to depict fractions visually and comparing the concrete representations of these numbers.

A more exciting example of visualization for high school teachers is the use of patterns to develop algebraic reasoning. Teachers can use visual patterns to first help students develop recursive reasoning (i.e. "look at these two first figures in a pattern: how many squares will the third figure have?") , and then to build to a more abstract understanding of algebraic patterns. Teachers can use patterns to prod students to reason about what the 10th or 100th item in the pattern will look like, without recursively counting the items in each step. This helps them understand algebraic modeling without even realizing what they are doing.

These patterns can be very simple or quite complex. Marc showed us a quadratic pattern that can be used to help students begin to explore quadratic functions, which could be a much more tangible introduction to this family of functions.

2013 NCTM Conference: "Communicating Performance for Common Core State Standards"

2013 NCTM Conference (4/18/13 - 8:30 am - Forrest and Elizabeth Clark):
I was first introduced to standards-based grading (SBG) by Shawn Cornally at Think Thank Thunk. I thought the idea was a fascinating one, but I was interested in learning more. When I heard that Forrest and Elizabeth Clark were giving a session on their experiences with SBG, I knew I wanted to come collect the fruits of their experiences.

Forrest pointed out that assessments should predict what students actually know. He suggested that a gradebook should not be used to direct students' behavior, but should be used only as a measurement of academic progress. Therefore, the gradebook should not be used to score effort, participation, attitude, behavior, or, more controversially, group work (except individualized portions) and homework.

In defense of this stance, he noted that he can predict how well his students will perform on state assessment tests, simply by looking at his gradebook. A good grade indicates that a student understands the mathematical concepts explored in his class, while a bad grade suggests that students do not yet understand these concepts.

The first key to SBG is that grading is based on standards instead of a specific test or quiz. These standards are assessed and graded individually. Students' performance on each standard is easily communicated to students and parents, so all parties know what each student needs to work on. This is very useful for creating a plan to help students receive extra help on the specific skills defined by each standard.

Another key component to SBG is allowing students to be re-assessed if they performed poorly. This allows students a second chance to relearn concepts for the natural reward of a better grade.

The process for implementing SBG is as follows:
1. Identify learning targets
2. Align curriculm with the learning targets
3. Create standards-based assessments
4. Create the assessment process
5. Inform parents and students

1. Identify learning targets:
The teacher must identify what students should know, do and understand. These concepts should be drawn from the common core standards, but may need to be simplified as needed. It is easily possible to draw 30 concepts from the standards, but Forrest suggested whittling these down to 15-20 concepts that will be measured in the grade book.

2. Align curriculm with the learning targets:
The teacher needs to identify where in the curriculum each of these concepts will be addressed. What order will he or she teach these concepts? Instruction should be modified to flow with the learning targets, instead of meandering aimlessly.

3. Create assessments:
The teacher must create assessments that identify how well a student understands the learning targets, so that each student can receive an accurate score. The teacher can use existing assessments that already address the targets, edit existing tests, or create new assessments, depending on his or her needs.

4. Create assessment process:
Each teacher must ask himself/herself the following questions:
a) When should assessments be given?
b) When will re-tests be given?
c) What is required of students before a re-test?
d) When will extra help be provided?
Forrest provided plenty of ideas from his own classroom. He requires a mandatory re-test when students score below 80% on an assessment. Students who score 80% or above have the option to re-test as well. He provides the first re-test during normal classroom time (students who are not re-testing work on homework or other assignments during this time), but any other re-tests must be done before or after school. Students are only tested on specific targets that were not mastered, so the re-test may not be as long as the original test. He provides extra help as he can, during his planning time or before or after school as students have time. The re-test score replaces the original score, for better or worse.

5. Inform parents & students:
The teacher should let parents know the grading policy as often as possible, until they are used to the "new" style of assessing students' understanding. Forrest and Elizabeth suggested including the grading policy in the syllabus, in school newsletters, etc., and should explain the process the process to parents during open houses and parent nights.
Parents should be told that grades are determined by assessments only, but good behavior is still important and will be enforced in other ways. Parents need to understand that if a student scores below the standard on a learning target, the student will be re-tested and the most recent score will replace the earlier score.
Finally, it is important for administrators to be on the same page, or the SBG process will not work. If the principal does not have the teacher's back, the revolutionary style to grading will only cause strife.

The Taste of Success

Some of our students have a problem with motivation. Even though they know how important school is, they don't have the drive to succeed. Getting them over this hill has been one of the most significant challenges I've faced.

I've found a solution that seems to work better than anything else I've tried. If I give them a small and very attainable goal, this will give them a chance to succeed - even in the tiniest of ways.

Once they know they can succeed, motivation increases a hundredfold. That taste of success builds their confidence and self-esteem and helps them to believe in themselves. From here, I am able to help them form bigger goals at which to aim. With their newfound confidence, they are much more eager to pursue bigger and better goals as success builds upon success.

There are still moments when they forget the taste of success, and we have to start over again. But once they taste it again, I can see them lift their eyes to the stars, and I smile.

Inquiry-Based Learning

I believe that often learning is more about asking the right questions than providing the right answers.

I am a huge fan of inquiry-based learning, where students learn by seeking out the answers to questions, whether they are prompted by the teacher or the student. With the student population I work among, this usually means that I'm the one asking the questions to propel learning. Rather than telling my students the "answer" when they have a question, I try to ask a question to make them think. If they aren't able to answer my question, I ask another question, trying to build on what they already know.

Sometimes I have to give raw facts or the "right answers". But it is my goal to ask questions first, to help my students think and to challenge them to explore math for themselves. I would a hundred times rather that my students leave my class having learned to think critically than that they leave having memorized a few formulas that they'll forget in a matter of months.

Math lab: discovering the length of line segments needed to create triangles

I've been working on creating math "labs" that allow students to explore math concepts in a hands-on way. My first lab was in geometry; my goal was to help students understand the relationship between the lengths of three line segments that is necessary for creating a triangle.

First I had students try to make a triangle with three line segments which could NOT create a triangle. They concluded that it was impossible because two of the line segments were not long enough. Then I had students make a triangle with three line segments that were sufficient for creating a triangle.

I then wanted students to record a hypothesis for evaluating any three line segments' ability to form a triangle. This was really tough for students. They were able to describe why three segments could or could not form a triangle, but they had trouble generalizing their findings.

I want to keep working at making my students think about math in a broader sense. They are used to solving problems (as long as the problems are set up in a familiar format), but are not able to apply what they know to scenarios beyond what they are familiar with. My hope is that if I continue to challenge them to think beyond the box, they will begin to learn to see the big picture of what math is really about.

Making a difference

In the fall I had a student (Tony) who had serious issues with anger. He often butted heads with anyone in a position of authority, which lead to many suspensions, and even lead to expulsions from many other schools.

Tony responded well to me, because he could see that I cared for him and I wasn't just here to tell him what to do. In time, I saw him begin to change. First, he didn't explode as often. Then I saw him start to recognize when he was getting angry; when this happened, he would talk to me about needing to leave, so that he didn't blow up. In time he got better at first recognizing he was getting angry and then dealing with his anger.

After Tony had significantly improved, he transferred out to give another school a try. I didn't see anything of him for months; I just hoped that I had made an impact on his life in a way that would stay with him for the rest of his life.

I saw Tony again this morning, for the first time since he left. Tony was really excited to see me, and I could tell that he changed. He is holding a good job making decent money. I know that I made a difference in his life - which is the most satisfying feeling in the world.

How many above the mean?

Robert was challenged by a question regarding the following scenario: consider a skating rink with 200 skaters. How many are skating above the average speed? He first assumed that about half would be skating the average speed, and therefor a quarter (50) would be skating faster than the average speed.

I asked Robert to consider the batting average of a baseball player. If a player had an average of .285, how many games does he actually bat .285? Robert had to think this one through for a minute, but then suggested that the baseball player will not have any days of batting exactly .285; this is just the average of all his days.

After this, he had a better understanding of what the mean measures. He came to the conclusion that half of the skaters would be skating above the average speed (assuming an even distribution of speeds above and below the mean).

Chalk Talk

I think this idea is awesome!

http://www.schoolreforminitiative.org/doc/chalk_talk.pdf

"Irrational" Sequences

I had an idea for a visual metaphor for irrational numbers, that I would like to use to help students understand these mysterious numbers. First, I would ask students to imagine that there are an infinite number of videos on youtube (more on this later). Then I would ask student to pick out their favorite music video. I want them to call this video "1" and write "1" on a piece of paper.

Student will then click the 1st link on the sidebar. They will call this video "2", and write it on a piece of paper. They will then click the 1st link on the sidebar of this video. This third video will be "3". Students will continue this process for 25-50 videos.

When students are finished, we will discuss whether their video sequence fell into a looping pattern, or whether all the videos of their sequence were different. I would compare a video sequence that "loops" to a rational number - all rational numbers have a pattern of decimal digits that eventually repeats itself. I would then compare a sequence that never repeated itself to an irrational number, which never contains a repeating pattern of digits.

I would then tell students that it is impossible for a video sequence on youtube to be "irrational", because there is a finite number of videos on youtube, so eventually any sequence must fall into a loop.

What do you think?

Area in real life

I want to make geometry more interesting to my students by making it feel more "real". In an attempt to do so, I created the following exercise:

Lillian wanted to reseed her lawn after installing a new gazebo. The area that needs to be reseeded is represented by the shaded part of the diagram below. Help her find the area that she needs to reseed.

Part I: What is the area of the gazebo (the octagon)? Assume that the gazebo is the shape of a regular octagon. Remember, the apothem is the distance from the center to an edge. The distance of the gazebo from edge to edge is 4.8 meters.  

 Part II: What is the area of the whole yard (the rectangle)?  

 Part III: What is the area of the lawn that needs to be reseeded? Show your work.  

In the future, I would like to develop this activity further by giving students a set yard size and allowing students to shop for their own hexagon or octagon-shaped gazebo to put in the yard (which will change the area of the lawn that needs to be reseeded). This will give them more freedom in the assignment and allow them to use their creativity.

Identifying the names of different sides of a right triangle

I have found that one reason my students have struggled with applying trigonometric ratios to mathematical scenarios is because they are unable to identify the differences between the "opposite" side, "adjacent" side, and the "hypotenuse". I decided to try to build up their vocabulary and boost their understanding of trigonometry at the same time.

First, I tackled the hypotenuse. My students were already familiar with the term, due to the Pythagorean Theorem. I reminded them that the hypotenuse is always the long side of a right triangle. I demonstrated that the long side is always the side opposite the right angle. Regardless of the way the triangle is "facing", the hypotenuse is always opposite the right angle.

I then told them that in trigonometry, they must think of themselves as looking at the triangle from the perspective of the angle they will be using in their trig function. I encouraged them to picture themselves standing at that angle and evaluating the triangle from there.

I asked my students what the word "adjacent" means. They seemed confused at first, so I asked them what it would mean if I said I was "adjacent" to my desk. One of my students suggested that I would be "next" to my desk. I encouraged this line of thought, and suggested that the "adjacent" side of the triangle is the side (other than the hypotenuse) that is touching the angle. The students were able to identify which side this is.

I then told them that the "opposite" side is the one that is opposite the angle - that is, that doesn't touch the angle at all. They were also able to identify this side.

After we discussed the names for the different sides of the triangles, it was much easier for the students to identify which trigonometric ratio they would use to find a missing side.

Like a family

My favorite part of teaching in a small school is the close friendships that have developed between the staff. My fellow teachers and staff have become like sisters and brothers; we are a team working hard to help our students, and this effort has brought us together in a very special way. I know that they care about how I'm feeling and how my day has gone. I know that they are there for me when I'm stressed during a bad day and I need someone to vent to.

Last week we participated in a team-building exercise. We all wrote something we appreciated about each of our fellow staff. It warmed my heart to read the encouraging notes that they wrote for me, and to know how much they appreciate me.

Wherever life may take us, I know that this experience has built friendships that will always leave a mark. I am grateful for each person that I have had the opportunity to work beside.

Infinite Series

Robert had trouble understanding how it is possible for an infinte series to converge. I asked him to consider an exponential expression with a base that is a fraction less than one. First I asked him to consider 1/2 squared. After he found 1/4, I asked him to consider whether this number is larger or smaller than 1/2. He wrestled with the concept for a moment, and then realized that 1/2 times 1/2 is the same as a half of 1/2, which is of course a smaller number.
I then asked him to consider 1/2 cubed. He was quickly able to understand that this number is smaller yet. I then asked him to consider 1/2 raised to the 1000th power. This number is very close to zero.
Robert was then able to understand that as x approaches infinity, (1/2)^x will approach 0. This enabled him to understand how an infinite series can converge.

Triangle problems

I have found that in large tests such as the Ohio Graduation Test, students often have trouble recognizing how to solve different problems involving right triangles. Scenarios involving right triangles are abundant in such tests, and it is vital for students to be able to approach these problems with confidence.

I am challenging my students to recognize and differentiate between the following scenarios: (1) when they know two sides of a right triangle, (2) when they know one angle and one side of a right triangle, and (3) when they know two angles. Furthermore, I am challenging them to assess what they need to find, whether it is (1) the length of a side, (2), the area of a triangle, or (3) the measurement of an angle.

I am confident that if my students become familiar with differentiating between these scenarios, they will be able to recognize when they need to use the Pythagorean Theorem, trigonometric ratios, or other geometric properties of triangles to solve a problem.

I am a teacher

As a first year teacher, I began my journey asking a lot of questions, such as "how should I best help special-ed students?", "how should I grade this assignment?", "what rules should I enforce in the classroom?" I never felt like I was the one with the answers. I was always looking to see if I was meeting expectations, and fulfilling the role I am supposed to fill.
As I have grown as a teacher, that has changed. I am now confident in my role as a teacher. I know what my students need. I know how to help them learn. I know that there is still so much that I want to learn, but I also know that I can learn it.
I am a teacher. I care. I work with passion to meet my students' needs. I give, I guide, I motivate, I provide. I am a teacher.

Exploring Pi

One of my students was frustrated by the work involved in understanding infinite series. To spark his interest, I told him that the mysterious number that he has used most of his life, pi, is actually equivalent to an infinite series.

I told him that pi is equal to the series 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ... We decided to explore this series together. We used a calculator to find 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11. We found the this is approximately equal to 2.976. This number doesn't look very much like pi, so I suggested that we go further.

Next we used a calculator to find 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 + 4/17 - 4/19 + 4/21. We found that this is approximately equal to 3.232. This value is much closer to pi, and this time fell above pi, which is what we expected since we ended with a positive term.

As we continued to explore this sequence we found that the more terms we added, the closer our approximation got to pi. While this exercise didn't help the student solve problems involving geometric series, it was a small example of how fascinating math can be.

Losing a student

Last weekend, one of our students passed away in a car accident. While I didn't know her personally, it is impossible for an event like that to affect me as a teacher. As teachers, we pour so much of our lives into the lives of our students, giving ourselves to build the dreams of others. When one of those lives is snuffed out, we lose a part of ourselves.

I wish that it was possible for me to help my students always make wise choices in their lives. But that is not my role. There is no magical formula that we immediately mature my students. The process must occur naturally, through little steps at a time. I can't (and wouldn't if I could) force my students to change their habits, but I can believe in them. And when they know that I do, it can provide the seeds to the hope and courage that they need to change. And the opportunity to plant these little seeds is worth all of the sacrifices I make.

The Power to Work With Exponents

One of my students, Sarah, was really confused by the process of simplifying complicated expressions containing many different variables raised to different powers, such as: (a^2 * b^-4 * c^5)/(a * b^-2 * c^-3). I decided that the best way to help her was to give her example with numerals instead of variables, to provide the scoffolding necessary for her to understand the more abstract principles at work.

I asked Sarah to consider the multiplication of 3^2 and 3^3. If we write 3^2 as 3 * 3 and 3^3 as 3 * 3 * 3, then we can write 3^2 * 3^3 as 3 * 3 * 3 * 3 * 3. Sarah was quickly able to tell me that this can be written as 3^5, and this is why 3^2 * 3^3 is the same as 3^(2+3).

I then asked Sarah to consider 3^5/3^2. I wrote this out as (3 * 3 * 3 * 3 * 3)/(3 * 3). I asked her to simplify this expression. She canceled two of the 3's in the numerator and denominator, leaving 3 * 3 * 3. This allowed her to see why 3^5/3^2 = 3^(5-2). At this point, she was comfortable working with variables. She quickly picked up the ability to simplify expressions such as x^4/x^2. With a little more direction, she was able to conquer the more complicated expressions as well, and had a greater understanding of negative exponents as well as positive ones.

Fighting the fear of fractions

One of my students, Bonnie, was having trouble knowing how to deal with a fraction that needs to be distributed, such as 1/2(x + 10) = 20. She wasn't sure what to do with the fraction or how to solve the problem in the simplest way possible.

I encouraged her to treat the fraction like any other number. She could either distribute the fraction, or she could use division to immediately remove the fraction.

She wasn't sure how to do this at first, so I asked her to consider what she would do to "get rid of" a whole number such as 2 in an equation like 2x = 12. She suggested that she would divide both sides by two.

I suggested she apply this principle to the fraction in the first equation. If she divides both sides by 1/2, she will easily be able to solve for x. She seemed unsure of how to do this at first, so I encouraged her to explore the idea: to try it and see what happened. She canceled the halves on the left side of the equation, and then tried plugging 20 divided by 1/2 into her calculator. This enabled her to solve the problem she was working on.

A Tale of One School

In the story "A Tale of Two Cities", Miss Lucie Manette goes to France to care for her father, who has been in prison for decades. She finds him locked up in the garret of a friend; even though he is no longer in prison, he is not mentally able to cope with the reality of his freedom. Miss Manette is not able to speak to him of his freedom, because he is not prepared to understand the reality of what she says. Instead, she is forced to spark his emotions, doing her best to stir up the life that lost within him:

"If you hear in my voice--I don't know that it is so, but I hope it is--if you hear in my voice any resemblance to a voice that once was sweet music in your ears, weep for it, weep for it! If you touch, in touching my hair, anything that recalls a beloved head that lay on your breast when you were young and free, weep for it, weep for it! If, when I hint to you of a Home that is before us, where I will be true to you with all my duty and with all my faithful service, I bring back the remembrance of a Home long desolate, while your poor heart pined away, weep for it, weep for it!"

I feel like my students have been locked in their own prisons for most of their lives. My students have never been taught to explore math, to stretch their minds and think about the way the world works. Instead, they expect mathematical formulas to be spoonfed to them and hope they can remember the formulas long enough to pass tests.

Teaching my students is a process. I long to just tell them that they are free: free to explore the world and to discover math for themselves. But my students are not prepared for the freedom. When I fling the doors of their cells wide open, they stare at me like I'm crazy.

Instead, I have to stir their emotions one little step at a time. I have to ask probing questions that force them an inch outside of their cells. I have to give them glimpses of what could be, if they are will to step into the open.

It's a long journey, but I hope I'm making a little difference.

The properties of exponential functions

Robert was having trouble visualizing the graphs of exponential functions and the ways an exponential function can be generalized (the characteristics that every function of the form f(x) = a * b^x will share). I started by drawing a few diagrams for him:

Robert was able to understand that in a function f(x) = b^x, regardless of the base b, when x is 0, the function has a value of 1. This is because any base raised to an exponent of 0 is equal to 1.

I asked Robert to consider how the y-intercept would change if the function b^x times a constant a. First I gave him an example of f(x) = 2 * b^x. He told me that the y-intercept is (0, 2). I then asked him to extend this idea to any constant a. He was able to see that the y-intercept would be (0, a).

I then asked Robert to consider the range of a function f(x) = b^x. He was able to see that the function will never reach 0, or be less than 0. I then asked him to consider what would happen if he added five on the end, as in the function g(x) = b^x + 5. After the previous question, Robert quickly concluded that the range of the function is all numbers greater than 5.

Line segments unable to create a triangle

One of my students, Lucy, was having trouble understanding when it was not possible for three line segments to create a triangle. I decided the best way to help her was to give her a hands-on demonstration of why this is true, so that she could tell me why it was not possible to create a triangle with three given line segments.

I cut out three thin strips of paper. One of the strips was longer than both of the other strips combined. I asked Lucy if she could create a triangle with the three strips of paper I gave her. She was unable to do so, because the shorter strips of paper could not touch to form a third vertex when they were attached to the ends of the longer strip of paper. Lucy explained that the two strips were not long enough to make a triangle.

I created another three strips of paper. This time the long strip was the same length as both of the other strips combined. I showed her that in this case, thre three strips could make a straight line, but not a triangle.

Her face lit up as it all made sense to her. She told me, "Oh, so the two smaller lines have to be longer than the larger line!" That was what I had tried to explain to her - but it didn't make sense to her until she saw it.

Finding the missing angle in a pentagon

Last week I challenged several of my students with measurement problems from old Ohio Graduation Tests. We encoutered a problem that asked for a missing angle in a pentagon shaped like a rectangle with a corner cut out (see below. Source: Ohio Graduation Test 2004 Question 4 http://ogt.success-ode-state-oh-us.info/ItemRelease/Questions.aspx)




My students had no idea how to approach this problem. I suggested that the students break the shape up into triangles and rectangles, to create shapes that they are more familiar with. We broke the shape up as follows:

At this point one of my students, Sherman, was able to come up to the board and complete the problem with guidance. He broke the 120 degree angle up into two parts: the 90 degree corner and the 30 degrees left over. Sherman knew that a triangle always consists of 180 degrees, so he was able to use 180 - 30 - 90 to find the upper part of x.

Sherman thought he had his answer, but 60 degrees wasn't one of the options. I suggested that he needed to find the measurement of the whole angle x. At this point, he was able to add 60 + 90 to find the correct answer of 150 degrees.

Another approach could have been to taught students the formula for the sum of the interior angles of a pentagon. However, I knew that my students would not remember the formula or how to use it. I knew my best approach was to break the problem up into recognizable pieces.

Introduction

I teach math at a school that reaches out to dropout students. We work with a students with behavioral issues, academic challenges, and emotional instabilities.

I have found that showing students how to do advanced (or often even the most basic) mathematical procedures does not enable them to succeed at math. Something more that seeing math is required - the students have to do math, on their own.

The best way I can help my students is by asking the right questions, and letting them work through the problem on their own. This blog is about my journey in finding the right questions to allow my students to explore math for themselves.