Day 58-60 - Cumulative Check-in

Scholars completed a cumulative check-in that is based on all of the material that has been covered so far this year. The review took a lot longer for the scholars to complete as previously thought. Due to the fact that the final will involve the material from the beginning of the year, I will be allowing the scholars to use a help sheet for the final.

I was gone for two of these days in order to present at a national conference.

Day 57 - Review from the Year So Far

Students worked on the material from the beginning of the year up until now. The problems involved the following:

• Midpoint, distance formula
• Graphing a line
• Finding an equation of a line
• Functions
• Evaluating functions
• Composite functions
• Proportionality constants
• Quadratic functions
• Long division
• Synthetic division
• Intercepts and asymptotes of rational functions
• Logarithmic/Exponential functions

Continue for pictures...

Day 56 - Logarithms

Students worked on their assessment for logarithms and exponential functions. This will be the last day that we will be working this material. After this, we are going to do a quick review of all of the material in the year so far, then move onto to trigonometry.

Yay! Trig!

Day 55 - Exponential and Logarithmic Functions

Students worked on the finishing their understanding of exponential and logarithmic functions.

In an effort to save time, I allowed the students to use the following equation sheet to help them with the assessment.

The main idea that students need to understand with using logarithms in algebraic problems is that they are the inverse of exponential functions.

Day 54 - Logarithms

Teaching logarithms was a bit of a challenge, but once students found out that they were nothing more than the opposite of exponents, it helped with the process.

Continue for pictures...

Day 53 - Exponential Functions

Students went over the basics of exponential functions in class today and broke down the characteristics of them.

This included such things as analyzing the domain and range, finding any intercepts, and analyzing the graphs for asymptotes.

They seemed to be pretty comfortable with the material and noted that as a increases in the function f(x) = a^x, the graph makes certain changes.

With the understanding of exponential functions going smoothly, I anticipate that logarithms will not end being too bad. Once we are through with this material, we will start to focus on trig.

Fingers crossed!

Continue for pictures...

Day 51-52 - Rational Functions

Students worked on reviewing the ideas behind rational functions. The most important parts that we looked at included:

1. Finding the roots of rational functions.
• The biggest issues that students had with finding roots was checking that their answer was really a root. This relates strongly with the Habits of Mind, namely persistence.
2. Finding the vertical asymptote
• To find this, we first discussed how and why you cannot divide by zero. The students blogged about this.
• From there, we discussed the definition of what a rational function is and I asked what would happen if the denominator of a rational function were to equal zero at a certain input. This is where things got really confusing for the students.
• Graphing the rational functions helped explain what happens at these points. The ideas behind infinity and negative infinity seem to be difficult for some.
3. Finding the horizontal asymptote
• Because the definition of a horizontal asymptote involves a value that f(x) never reaches but gets infinitely closer to, I decided to teach horizontal asymptotes using limits. This may have been a bad idea since it ended up eating a whole week.
• I am happy that the scholars got to see an brief intro to limits at least.

Continue for pictures...

Day 49-50 - Rational Functions

Students began work on analyzing the components of rational functions. Being that it is the week before spring break, the students seemed a little off task, but nothing too major.

Day 47 - Rational Roots Theorem (cont'd)

Students continued their work on Rational Roots Theorem and prepped for the next section on Rational Functions.

Day 46 - Rational Roots Theorem (cont'd)

Students continued their work through the Rational Roots Theorem. Luckily for me, one of my students mentioned to me that they want to become a math teacher when he gets older and asked that I be his mentor. It took some time for me to figure out something to best help this student, but I think that I came up with a great idea.

The student will be taught the material after school and try to keep ahead of the students. The students will be taught by me the same way that they have always been, but this time, I will also have an assistant. Throughout the time spent after school, I will teach my mentee how to use best practices when helping the students learn the material.

Day 45 - Rational Roots Theorem

Most of the class started the basics behind Rational Roots Theorem. The issue that many of them were having came with the proof that I used for the lesson. I did not intend for them to learn the proof or even understand it for that matter. I simply aimed to give them an idea of where Rational Roots Theorem came from. I liken it to Pythagorean Theorem. Most students do not know how the PT was derived, but know that it works and use it. It's safe that to say that only 1 out of my 21 students actually understood the proof for the theorem. It does not matter as long as they understood my next video on how to apply the theorem. This video got much better results and students were better able to understand the point of using Rational Roots Theorem.

News:

• Due to the fact that I have a student that wants to be a math teacher, I am training him to become my assistant in the class to get a firsthand view of what it is like to teach mathematics. He stayed after school to work on getting ahead of other students on the material so that he can better help them in class.

Continue for pictures:

Day 43-44 - Rational Roots Theorem

Some students started their work on framing the ideas behind Rational Roots Theorem in their mind. None of the students were ready to take the exam on the theorem and thus worked through the material from the website.

Some needed to do further work on Remainder Theorem and Factor Theorem. Hopefully, as a class, we will done with Rational Roots Theorem by the end of next week.

Day 42 - Remainder Theorem and Factor Theorem

Things went well in my first class. Mostly due to the fact that the Internet was working. During the second class when I had to get observed was not so good. The Internet went down so my video lectures were not able to be watched by the students. I had to think quick in order to figure out how to deal with the situation. I had students watch the videos on my iPad. The only issue is that the videos were unedited and contained all of my mistakes. It was almost like a blooper reel. It ended up all right in the end.

I had backup plans that I could have used. They were the following:

1. Students could have been given the videos by way of flash drive.
2. Students could have been given the book to use. We have yet to have used the book all year.
3. Students could have been lectured to by me.

All of those would have been viable options. The students were great of course. A lot of learning (and struggling) went on in the class and the students really stepped up to the plate. I was really proud of them.

Monday is my post-observation so I will see what my principal thought about everything.

Day 41 - Quadratic Review

Students completed a quadratic review that was gone over because of the lack of a school day from the blizzard, Nemo. Surprisingly, it took the students the entire period to complete. This was a nice surprise as it was originally not planned.

Day 40 - Complex Numbers Review

Students completed a complex numbers review. In order to take a quadratic formula exam involving complex roots of quadratics. This was meant to serve as a review for the students as many of them were not able to show up to school due to the blizzard, Nemo.

Day 39 - Synthetic Division/Complex Numbers

The students finished their pre-assessment on synthetic division. From here, we will be going to finding the roots of polynomials. Before we do this, I feel as though it is imperative that we revisit complex numbers and the quadratic formula. Some of the roots will be rational for polynomials, however, some might also lead to irrational roots or complex roots. In order to make this transition easier, we did a quick review of complex numbers and how they are related to the quadratic formula.

Breakdown of the order of things so far and where we are headed:

1. Long division of polynomials
2. Synthetic division (given the divisor)
3. Complex numbers
4. Quadratic formula (special emphasis on its relation to complex numbers)
5. Finding the roots of polynomials

Continue for pictures from the day...

Day 38 - Synthetic Division

Students seemed to do well with synthetic division. I feel as though they will do well on the exam covering this material. The standard that covers synthetic division will be the following:

• Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

So the plan is to have the students complete the problems on synthetic division then the following.

1. Review of complex numbers - They will need this when going over the factoring and finding the zeros of polynomials.
2. Review of quadratic formula - This will be helpful with finding the zeros of polynomials and will also help with the review of complex numbers.
3. Rational roots theorem - for finding the zeros of polynomials
4. Putting it all together!

After this, I will be covering rational functions, asymptotes, and then hopefully move on to exponential and logarithmic functions before ending the year with trigonometry.

Fingers crossed!

Continue for pictures...

Day 37 - Long Division of Polynomials Exam

Students completed the long division of polynomials exam. The exam covered the following standard:

• Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Day 36 - Long Division of Polynomials

Students continued their work on long division of polynomials and completed a pretest on them. The results were as expected. I am finding that the pretests really help to open the students' eyes to the material that we are covering and can help them gauge where they are in relation to the standard.

Continue for pictures...

Day 35 - Long Division of Polynomials

Students completed exercises related to long division of polynomials. Using the rules from long division of ordinary numbers, many students were able to complete some of the basic example problems.

Many of the students need more work on the problems. I made a video and we will go over more on long division next time the class meets.

Continue for student work...

Day 34 - Parent Functions and Power Functions

Students completed a bell ringer in which they found a piecewise function to describe y = |x|. Many of the students got  answers without much help from me. Further challenges included finding a function that is not piecewise that gets the same outputs as y = |x|.

Students then completed challenges on parent functions and power functions. They looked for patterns with the integer values of the exponents in the power function.

Continue for pictures of student work...

Day 33 - Standard 7b Assessment

Students completed the Standard 7b assessment on the following functions:

• square root
• cubic
• absolute value
• piecewise

The next challenges will involve combining the ideas piecewise functions to describe the absolute value function without using the absolute value.

Day 32 - Square Root, Cubic, Absolute Value, and Piecewise Functions

Students completed various activities involving the graphing of different types of functions as explained by CCSS. GeoGebra was used to help the students graph the functions.

The challenge was based on the last pre-assessment in which students had to complete similar problems. I think focusing on just square roots would have been the most ideal although it would have used a lot more time which I don't really have.

Day 31 - First Day Back!

Some of the students' grades have been slipping as of late. In an effort to get them back into the swing of things, I treated this two day week as a make up week. Students focused on completing any and all work that needs to be turned in.

Day 30 - Seems a lot like Day 29

The students seem to be focusing on their upcoming winter break rather than the material that we have been covering in class. Nothing much to report today!

Happy Holidays!

Day 29 - More Quadratics

I feel like things are taking a lot longer than they should be. I want to be done with quadratics, but I feel as though the students still do not grasp the understanding of basic math to move onto to harder material. Wednesday will be the last day of working on quadratics. We will then move onto square root, cubic root, and absolute value functions.

Day 28 - Last Day of Quadratics

The students covered the ideas behind -b/(2a) for finding the x coordinate of the vertex. The students struggled with the understanding of where this value came from. Unfortunately, I do not have anymore time in class to cover its derivation, but I will allowing students to come in at other times (lunch, after school, etc.) to think about this a little more.

I really need to start taking more pictures in class...

Continue for pictures of student work.

Day 27 - Key Features of Quadratic Functions

Students completed exercises involving finding the x and y-intercepts of quadratic functions. They ran into trouble when we were going over functions that did not solve for x easily. We reviewed and went into the theory of why completing the square is used. From there, we tried to use the completing of the square principle to solve for x in the standard form of a quadratic function.

Students struggled through the solving steps, but some were able to do it.

Continue for pictures...

Day 26 - Linear vs. Quadratic Intercepts

Today's lesson involved finding the intercepts of both linear and quadratic functions. The students were surprisingly not as comfortable with finding the linear intercepts as I originally thought. Being that the course is precalculus and it is supposed to be college, I am a bit surprised. Maybe students also tried to "remember" how to do certain problems rather than try to figure out how to do them based on the definitions of certain terms. A student mentioned how to solve math problems which led to a great revelation. In mathematics, the answer lies within the question. I really like thinking about math that way. I feel that having students think about challenges in mathematics that way, they are better apt to find the clues within the challenge that can be used to solve it.

Day 25 - Standard 7 Pre-Assessment

Students took the new Standard 7 pre-assessment. They will be taking the same exam over once we cover all of the challenges for the standard.