I cannot believe how quickly this semester has come and gone. Throughout the class we have reflected after each lesson that we taught.  It’s interesting to see where we started at the beginning of the semester compared to how far we have come. In this lesson specifically I felt more comfortable teaching and I think this is because I’ve been given several opportunities to teach throughout the semester.  I don’t necessarily think that teaching became easier, but I definitely felt more comfortable in a classroom setting and for this I thank my classmates and Dr. Nagle.

In this lesson we began by reviewing angle bisectors.  I had students complete a worksheet that required them to find several missing angle bisectors.  After the class finished the worksheet and we reviewed the answers we moved on to the activity.  The activity was more of an exploration, it had students create a triangle, incenter, and incircle.  Students were required to look up the definition of an incenter, which states that if all three angles in a triangle are bisected and those lines are extended, the intersection of those three bisecting angles is called the incenter.  I think I should have stopped students at this point in the lesson to make sure that they had all found the same answer/definition online to make the lesson more clear and make sure students were not mislead.  If I were to re-teach the lesson I would also have students label the measurements of the bisecting angles within the triangle.  This would be so when students are moving any vertex of the triangle they can still see that the angles are still bisecting angles.  However, I thought the directions for the activity were clear and easy to read.  I also think the students were engaged throughout the lesson.  I was able to hear from every student and for the closing problem a student was able to present her work with the class using the document camera.  We concluded the class by restating what an incenter and incircle are.  I collected the students activities and closing problem to look over their work.  To improve the lesson I would add a more formal form of assessment, maybe by using string and have the class (multiple students) construct an incenter/incircle within a triangle on the bulletin board in the classroom.

I’m surprised to see the difference from where I began to where I’ve come. When I taught my first lesson I was more focused on presenting the content and using appropriate academic language (mathematical terms).  However, at this point I think more about how I will ask the students questions and how little information I can provide so the students can discover more for themselves.  It’s also important to think about when I will ask the questions and what questions I will ask students.  I still need to consider what adaptions need to be made throughout the lesson, i.e. when students aren’t at the same level and some complete the task much earlier than others, I should have something ready to ask or for those students to complete while other students are still finishing the problem. The feedback I’ve received throughout class has helped me tremendously, not only from my professor but also from my peers.  Thank you!

This activity uses Fathom to simulate random data, in this case simulating roulette. Using Fathom, we create a “wheel” and account the different cases/outcomes. By the end of the activity the probability of a certain color or number can be calculated. More data can be collected, or we can simulate more trials by selecting “Sample More Cases”.

Using Fathom allows us to create simulations which can be useful tools for finding approximate probabilities for situations in which the actual probabilities are too difficult to calculate.  We can also model real world situations which can make the lesson more interesting.  Simulations are not just a pedagogical tool, but are used by statisticians and scientists.

Data Analysis Activity

1. Transferring files and screen captures.

Teachers are able to share files from the calculator to computer or computer to calculator. They can also capture screen images and share them with the class. This can be beneficial in the classroom because students can see/share the right/wrong answers which can lead to further class discussions or clarification.

2. Using function tables.

When graphing functions on the Nspire students are also able to see a function table.  The table shows the value of x and also the value of the function corresponding to the x value.  I think this is helpful because students are visually able to see the function represented graphically but are also able to see the numerical values as x increases/decreases.

3. 3D graphing.

The Nspire allows students to graph 3-dimensionally but also has a feature that allows students to see the graph from each axis (x-axis, y-axis, z-axis).  I personally am a visual learning and I struggled with imagining what an expression would look like in 3-d.  I think this feature will help students because they can visually see the expression graphed in 3-dimensions.

4. Manually entering a list of data.

Numbers in the list can be entered or expressed as functions.  If you have used excel before you should be familiar with this. In excel, say you are entering numbers increasing by one.  Instead of entering numbers in manually, 1, 2, 3,… excel allows you to enter =A1+1.  The Nspire has the same feature, this allows students/teachers to save time because manually entering in data takes away from the exploration part of the lesson or activity.

5. Defining & linking variables to a coefficient in a graph.

Students can define and link a variable to a coefficient in a graphed expression. I like how variables in one problem have no relationships with variables in another.  This just means that if students had two different graphs but the same variable, those variables would not have the same definition unless they were defined to be the same by the student. I think it’s helpful for students to see different representations when the variable is changed.

In this activity (Classifying Quadrilaterals) students drag edges and vertices of various Sketchpad quadrilaterals to discover which are constructed to have specific characteristics. As students make these distinctions between the figures they are able to deepen their understanding of the definitions, relationships, and properties between/among them.

Using this technology students are able to themselves construct different properties of quadrilaterals.  I believe this gives them a deeper understanding and a better experience.  Students can visually see various quadrilaterals and their properties.  They can even manipulate the quadrilateral into what may appear to be another type. Using Sketchpad rather than having pictures on a worksheet will keep the students visually engaged and will expand their knowledge on quadrilaterals.

From this resource I would incorporate two modules during class, one would be the row operation calculator and the other, solving a system of linear equations.  The row operation calculator would be used for the opening activity which would be a review from last class.  The majority of the class will be spent solving systems of linear equations. The row operation calculator is an interactive activity that allows the user to perform a sequence of elementary row operations on an m x n matrix.  While the solving a system of linear equations module allows the user to solve a given system of m linear equations in n unknowns.

The row operation calculator allows students to keep track of all the operations done on the current matrix. There are also buttons that will solve the matrix for the student if they get stuck, showing the step-by-step operations.  When students use the row operation calculator they are able to add, multiply, and interchange rows respectively in order to eventually obtain a matrix in row echelon form or reduced row echelon form.

The solving a system of linear equations module allows students to see the step-by-step solution for solving a system of m equations and n unknowns. The solution in separated into two parts.   Part 1) The transformation of the augmented matrix to the reduced row echelon form & Part 2) How to interpret the reduced row echelon form.

Below are both modules from the Linear Algebra Toolkit resource

Linear Algebra Toolkit – Row Operation Calculator 

Linear Algebra Toolkit – Solving a System of Linear Equations

I chose this worksheet because I’ve seen this concept before in Geometer’s Sketchpad and I thought, HOW COOL!  The worksheet has a right triangle and the right angle remains constant. However, the three sides of the triangle represent one side of a square, giving three squares total and can be manipulated.  When exploring the worksheet you notice that no matter what the side lengths of the squares are, they are equal to the lengths of the triangle, and both lengths (square & triangle) yield the same answer for a^2 + b^2 = c^2.  The app/worksheet allows students to visually see the numbers change and that no matter what they choose for the lengths, pythagoreans theorem still holds. The link to the worksheet is attached below.

Pythagoreans Theorem

Overall I think the students worked well with the technology, and using the technology helped with the lesson.  Not only were students visually able to see what the fractions represented but the students also made/created the representations themselves. I think to improve the lesson/activity I could have gone through the directions and showed the students what they were supposed to do, but not do an actual example of the problem, leaving room for students to explore the applet.  I didn’t talk about reduced fractions either, I could have gone over what it means for a fraction to be in simplest form or even created a worksheet for students involving manipulations with fractions.

Below are the PDFs for the lesson, opening activity, and closing activity

FractionsANDPercentages

Fraction_Finder_Exploration_Questions__doc_

PIZZA ACTIVITY

This article discusses whether or not students having individual laptop computers on their desks would form barriers between students, possibly hindering their communication.  Fonkert observered two classes implementing the Core Plus Mathematics curriculum, the two classes consisteted of course 2 and course 3, with tenth and eleventh graders, respectively. CPMP-Tools is a free software accessible to students via the Internet, it includes general-purpose and custom tools developed for each mathematical strand of the curriculum (algebra, geometry, statistics, & discrete mathematics).  Three key attributes were mentioned to attain successful group work; group-worth task, assignment of group roles, and development of positive norms of behavior.  The most frequently observed behavior throughout using the technology during class was students building on each others ideas.  Other behaviors observed in the classroom were asking questions, giving ideas, and answering others’ questions.

I enjoyed the dialogue included between the students/teacher in the article.  Fonkert used the dialogues several times to support her observations and give evidence to show that although students were working individually their was still communication and learning throughout the lesson among students.

The article also mentioned the advantages computers have over calculators and physical maniputlatives.  The CPMP-Tools engage students and have the power to illustrate mathematical concepts clearly.  The software is also easier to manipulate (clicking & dragging), there is better accuarcy when measuring, and the movement of objects are more precise.

PDF article:  mt2010-11-302a

This resource involves students in mathematical modeling and making predictions by measuring the height of oreos. There is an add data interactive Excel spreadsheet that allows for running two simple simulations to investigate changing the average cookie thickness and how it influences the slope of the graph.  Later the students see the random variation of individual cookie thickness and how it influences the scatter of the data and value of r-squared, a measure of goodness of fit.

Students do not need to have prior knowledge of Excel in order to complete this activity. Students only need to be able to enter the data into the cells and be able to measure using a ruler.

The goal for this activity is for students to discover whether or not there is a relationship between the height of a stack of oreos and the number of cookies in the stack. Throughout the activity students should develop a mathematical model from experimental data. Then, they will make predictions with the model and consider the variations & the impact those variations have.

This activity uses technology to ask questions from the students that demand reflection, sense making, and reasoning.  In this activity students are asked questions such as:  what does the slope represent in terms of the variables investigated, units of that slope, what the y-intercept should be for this specific mathematical model, heights of 20 & 150 cookies, verifying predictions made, conclusions about the uniformity of the thickness of the cookies, etc.  The worksheet with this activity helps students make the connection between the action and consequence.  It is helpful that when students enter the data into excel the consequences are immediate, visual, and mathematically meaningful.  From these consequences students can then reflect on their new knowledge/conclusion and understandings.

link: cookie stack activity