Author: schloemerj

Rational Expressions A.APR.D and A.REI.A

schloemerj CCSS-Math Learning Progressions, HS Algebra

This learning progression is for an Algebra II class. The vast majority of the students are juniors but there are students of all grade levels. The lessons most generally use worksheets from Kuta Software to help facilitate the learning of the students; however, the class does have the Holt McDougal, Common Core Edition from 2012 that they use every now and again as the math department usually collaborates and creates their lesson plans with math tasks and activities to help engage the students. For this learning progression the students will be learning how to simplify rational expressions, multiply and divide rational expression, add and subtract rational expressions, solve rational equations, and solve radical equations. The standards that are applied to this unit plan are:

  • MATH.CONTENT.HSA.APR.D.6

Rewrite simple rational expressions in different forms

  • MATH.CONTENT.HSA.REI.A.2

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise

The Mathematical Practice Standards that are applied to this unit are as follows:

  • MATH.PRACTICE.MP1: Make sense of problems and persevere in solving them
  • MATH.PRACTICE.MP3: Construct viable arguments and critique the reasoning of others.
  • MATH.PRACTICE.MP6: Attend to precision.
  • MATH.PRACTICE.MP7: Look for and make use of structure.

A teaching strategy that will be used during this learning progression is group work and collaboration. Students will be able to work in groups in order to work in groups of three or four. In a classroom set up such as this, the teacher will be able to facilitate to more students throughout the class period and guide the lesson rather than use direct instruction. The teacher will continuously monitor the students and listen in on the group conversations to hear where the students tend to be struggling and to answer questions that the students might have. The only thing that the students will be doing separately will be the five-minute exit slip at the end of class; these will not be used every class period as students might be doing an engaging activity or they may be having a lot of group conversation which I will be able to listen in on an gauge the students’ understanding.

To view the full edTPA Learning Progression please view it here.

Graphing and Evaluating Functions: HSF.IF.A.1, HSF.IF.A.2, HSF.IF.B.5, HSF.BF.A.1.A & HSF.LE.B.5.

schloemerj CCSS-Math Learning Progressions, HS Functions

This learning progression is for an Algebra I class; the vast majority of the class contains freshman but there are representations of all grade levels The class uses the Holt McDougal, Common Core Edition from 2012 every now and then throughout their lessons but usually create their lesson plans with math tasks and activities to help engage the students. For this learning progression the class will be learning how to graph relationships and how to evaluate values using x-y tables and graphs. The standards that are applied to this lesson are the following:

CCSS.MATH.CONTENT.HSF.IF.A.1

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of corresponding to the input x. The graph of f is the graph of the equation y = f(x).

CCSS.MATH.CONTENT.HSF.IF.A.2

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

CCSS.MATH.CONTENT.HSF.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

CCSS.MATH.CONTENT.HSF.BF.A.1.A

Determine an explicit expression, a recursive process, or steps for calculation from a context.

CCSS.MATH.CONTENT.HSF.LE.B.5

Interpret the parameters in a linear or exponential function in terms of a context.

 

The CCSS Mathematical Practices that are applied to this lesson are the following:

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP4 Model with mathematics.

CCSS.MATH.PRACTICE.MP6 Attend to precision.

CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

A teaching strategy that will be used during this learning progression is group work and collaboration. Students will be able to work in groups in order to work in groups of no more than four. In a classroom set up such as this, the teacher will be able to facilitate to more students throughout the class period and guide the lesson rather than direct instruction. Every 5-10 minutes the teacher will bring the class together for a brief fist-to-five on the students understanding and go over any misconceptions that the teacher had noticed or seen while monitoring the students. The only things that the students will be doing separately on a standard day-to-day basis will be the formative assessments during class: the warm-up problems in the beginning of class, hinge questions in the middle of class, and the exit slips at the end of class.

View the full Learning Progression here.

Creating A Tangent To A Circle: Dare To Tangent G.C.A

schloemerj HS Geometry, Technology Teaching Issues

Lesson Title: Dare to Tangent

Unit Title: Circles and Their Properties

Teacher Candidate: Jenell Sellers

Subject, Grade Level, and Date: Geometry, 10th Grade, December 2nd, 2016

 

Placement of Lesson in Sequence and Lesson Rationale

This lesson is the fourth lesson of a four part unit where students will be learning about the properties and of circles. The lessons will progress from 1) Learning about the similarities between all circles, 2) Identifying the relationship between angles, radii and chords, 3) Constructing inscribed and circumscribed about a triangle, and 4) Creating tangents to a circle from a point outside of the circle.

Central Focus and Purpose

The central focus of this lesson is to assist students in understanding the function of creating a tangent to a circle. The students will be given directions on how to create a tangent by hand and also in Geogebra. Each student will be provided with a compass and a straight-edge and also a laptop. Students will use the class period to complete both activities

CCSS.MATH Content and Practice Standards

The Common Core State Standards that are covered throughout this four lesson unit are as follows;;

CCSS.MATH.CONTENT.HSG.C.A.1

Prove that all circles are similar.

CCSS.MATH.CONTENT.HSG.C.A.2

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

CCSS.MATH.CONTENT.HSG.C.A.3

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

CCSS.MATH.CONTENT.HSG.C.A.4

Construct a tangent line from a point outside a given circle to the circle.

Prior Content Knowledge and Pre-Assessment

The prior content knowledge that the students will be building off of will be that of understanding of right triangles and the basic trigonometry.

Learning Target(s) and Plan for Clarifying Intending Learning

The learning target for this lesson will be “I can create a tangent to a circle using technology and by hand”.

 

Strategy for using assessments to guide student learning
The goal for this lesson is to allow students to make the connection between completing a task by hand and also doing the same task via computer/using technology. This will help them also make more of a connection to how they might have to complete such a task in a job.

 
Success Criteria (criteria for interpreting student success of the learning target) Plan for providing feedback and students’ monitoring of their own learning
The success criteria for this lesson will be an exit slip that they will have to complete in the last 5-7 minutes of the class period. The exit slip will consist of questions that will allow the students to express how well they understand the concept of creating a tangent line to a circle from a point outside of the circle and what help they might need in doing so.

 

 The plan for feedback for the students during this lesson is to attend to each group during the lesson to see how they are doing on their graphs and to ask them open ended questions in order to get them to think a little bit more critically about the task at hand. I will also address all the misconceptions that are written down on the exit slips in the beginning of class the next day so that all of the students can anonymously get their questions answered.

 

Academic Language Demands
Language Function Vocabulary & Symbols Secondary Language Demand
·         In this lesson, students will use the vocabulary to:

-Correctly address different pieces of the circle

-Construct a tangent to a circle both by hand and using Geogebra.

 

 

 

 

 ·         Circle

·         Tangent

·         Point

·         Center

·         Radius

·         Diameter

·         Bisect

 

 Mathematical Precision:

Students will have to pay close attention to the directions that they are given. Students will also have to understand the notation and the vocabulary in order to correctly follow the directions.

Syntax:

Students will have to

Discourse:

Students will be working on their own, however since the desks in the classroom are arranged in groups of four the students can easily collaborate with each other. They may discuss the activities during the class period but they also must each turn in their own work.

 

Language Support (instructional and assessment strategies)
Language Instruction Guided Practice Independent Practice
The teacher will begin the class period by explaining what a tangent line is; a drawing on the whiteboard in front of the class maybe necessary. To create a better visual for the students, the teacher may explain a tangent in a real-world scenario; the tires of a car on the road, for example.

The teacher will also show the students how to bisect a line and how to create a circle using their compass and straight-edge. These will be the two things that they will have to know how to do while they are creating the tangent by hand.

 

 The teacher will guide the students by providing hints and tips on how to create the tangent to a circle when they are stuck either creating it by hand or when they are using Geogebra. Their handout will have instructions but it still might be possible that they get confused or lost.  If the students need answers to questions the teacher will be monitoring the class throughout the period so that they are easily accessible to the students. After the teacher has introduced the lesson and handed out the worksheets and the laptops and the students are all signed in with Geogebra open and the materials needed are on their desk students will be able to get started. They may collaborate with those who are sitting in their group as long as each student turns in their individual work.

 

 

Differentiation, Cultural Responsiveness, and Accommodation for Individual Differences

Accommodations have been made for several students. One student with autism gets up and walks around a lot during class so in order to help him focus he sits on an exercise ball so that he can bounce in his seat so that he’s not disrupting the class. This actually works extremely well; he can still tell you exactly what his thought process is even though he’s still bouncing up and down.

Another student with autism needs to sit closer to the front of the room in order to help him visually because he has fairly poor eye sight. Every now and again he will have some outbursts out of frustration but he only needs reassurance that he’s doing okay. There is a para-educator for the classroom that helps out with his frustrations and calms him down as well.

The ELL students in my class have a hard time with the language barrier, so in order to help them out with the language barrier and to also, hopefully, improve their English reading skills I have worked with the Spanish teacher and he helped me translate the worksheet done in English into Spanish. The ELL students will receive both sheets so that they are able to see the Spanish and English translation and make a connection between the Spanish words and the English words.

 

Materials – Instructional and Technological Needs (attach worksheets used)

The materials needed for this activity are:

-Handouts with the instructions

-Compass

-Straight-edge

-Laptop with the Geogebra Application

-Pencil and Eraser

Instructional Plan (detailed explanation for thing the lesson)
Pacing Teacher Activities Student Activities
Before Before the class period starts the teacher will need to ensure that each student will be able to have their own laptop and that each laptop also will have Geogebra downloaded already; this is to minimize wasted time and distractions by the students being online. The teacher will also need to ensure that each student will have a compass and a straightedge. Extra paper will also be necessary in case the students will want to practice or make a mistake. The student activity will be to create a tangent line to a circle using the Geogebra application on a provided laptop and also creating the same construction by hand.
During The teacher will introduce the topic of tangents and explain what a tangent is. In order to make a real-world connection for the students to relate to, the teacher might be able to explain that a tangent can also be thought of a tire of a car and the road that it’s driving on; a circle with a line touch the edge but not crossing through. The teacher will then explain that the students will be constructing a tangent to a circle using the Geogebra application on the laptops that are provided and also by hand with the provided tools. The teacher will monitor each table group and help students who have any questions and those who need assistance. If there seems to be a common misconception then the teacher may address the whole class and correct the common error that is occurring throughout the class. The students will have to turn in their worksheets at the end of class. They will also be able to print out their work that was done in Geogebra so that the teacher will be able to assess whether they understand the Geogebra application enough to operate it for another lesson. After students receive all of the necessary materials they will begin working on their handouts. Students may collaborate with their tablemates if need be. The students will be given instructions on how to create the construction but then they must be able to also do it on their own.
After Ten minutes before class is to end the teacher will instruct the students to put away all of the materials and to clear their desks so that they can fill out the exit slip. The students will be able to complete the exit slip within the last 5-7 minutes of class. Afterwards the teacher will make sure that all of the laptops have been returned and properly stored in the laptop cart and that all of the compasses and straight-edges are back in their appropriate boxes. The teacher will collect all of the exit slips and all of the handouts and prints from the students and review what the students have been able to accomplish throughout the lesson. The students will need to swiftly put all of the materials away and turn in their handouts with the print out of their work in Geogebra. The last 5-7 minutes of class they will be working on their exit slip until the bell rings and they are dismissed.

 

 

Dare to Tangent

Learning Target: I can create a tangent to a circle using technology and by hand.

Ready for a challenge?! Your mission today is to construct a tangent to a circle. This will be done by hand and also using Geogebra, the master of all things Geometry!

You may begin with either activity. Manage your time wisely, students of mine, for everything must be turned in in its entirety at the end of class!

 

Geogebra Instructions:

Construct, in Geogebra, a tangent line to a circle through a point not on the circle.

Directions:

  1. Using the ‘Graphics’ tab, get rid of the axes that are visible when you open the application.
  2. Using the Circle tool, create a circle of a size that is convenient to work with.
  • Create a point that is outside of the circle, call it P.
  1. Now, using the line tool to create the line AP
  2. Using the other line tool, bisect AP.
  3. Using the point of intersection, create a circle whose center is the intersection and whose radius diameter is AP.
  • Notice how this creates two intersections between the two circles you have created? Plot a point on each of these intersections using the Point tool.
  • Now, create a line that connects the original point that is not on the circle and the two points that you just created. Voila!
  1. Clap your hands once, raise a fist in the air and proclaim “Boomjam!” You have just created tangents to the circle using a point not on the circle! Print out your masterpiece as it will have to be turned in with the rest of the handout.

 

 

Manual Instructions:

Construct, using a compass and a straight-edge, on a separate piece of paper the following using the scheme that is provided below:

Given a circle C and a point P not on C, construct a line through P and tangent to C.

Given:

by-hand-construction

Scheme:

  1. Draw AP
  2. Bisect AP; giving point D
  • Create circle with center D and radius AD; giving points T and S
  1. Draw PT and PS
  2. Clap your hands twice, raise your fist in the area and yell “Boomjam Again!” You just created a tangent to a circle by hand!

 

 

 

Exit Slip

 

What was the most difficult/challenging aspect to the activity today?

 

 

 

Did you receive any help during this challenge? From whom (peers, friends, teacher, etc.)?

 

 

 

On a scale from Basic, Proficient to Mastery, how comfortable are you in performing this task in a stressful setting (quiz, exam)?

 

 

 

 

 

 

 

 

 

 

 

Exit Slip

 

What was the most difficult/challenging aspect to the activity today?

 

 

 

Did you receive any help during this challenge? From whom (peers, friends, teacher, etc.)?

 

 

 

On a scale from Basic, Proficient to Mastery, how comfortable are you in performing this task in a stressful setting (quiz, exam)?

 

 

 

tires-on-the-road

Passcode Picture Problem SP.B

schloemerj HS Statistics & Probability, Picture Problems

You are on a mission. You’re mission, should you choose to accept it, is to unlock your best friends phone so that you can take a selfie and set it as their wallpaper. If the passcode to their phone is four digits long and you know that the first number is 7, and that 0 and 2 are never used; it is possible that numbers may be repeated. How many combinations are possible in order to unlock their phone?

CCSS.MATH.CONTENT.HSS.CP.B.9

(+) Use permutatphone-lock-screenions and combinations to compute probabilities of compound events and solve problems.

CCSS.Math Makes Cents?

schloemerj Uncategorized

One thing that is important to remember when working with mathematics is that it, in fact, does relate to real life. An extremely drastic example would be the show Numbers; an FBI agent’s little brother is a professor of mathematics of a local university and uses his knowledge of probability, statistics, physics, etc. to help catch ‘the bad guys’. Like I said, although this is a very drastic scenario it really does help one see that math can actually be used in everyday life. Another thing to remember, especially when working with middle school and high school students is to remind them that math is really figuring out relationships between elements; how fast you go in a car and the time it takes you to break, the more interest that is charged to a credit card and the minimum payment that you might have to pay, investments, etc. Teenagers and adolescents hardest concept to come by while struggling through their math classes is that they don’t think that math is applied to them directly and they don’t see the relationships within math itself.

One of the Common Core State Standard sections for mathematics is all about modeling. Modeling means to create a picture, or graph, that represents a function, or relationship, to better understand what the overall effect is. We do this with exponential growth and decay, with parabolic functions showing the movement of something being throw into the air, with investments of businesses and companies, even with the world population. This modeling standard is stated, by the CCSS.Math website, “Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (*).” Other CCSS for Math that is included are:

CCSS.MATH.CONTENT.HSF.BF.A.1.B
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

CCSS.MATH.CONTENT.HSF.BF.A.1.C
(+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

CCSS.MATH.CONTENT.HSF.LE.A.2
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

An experiment called ‘Making Cents of Math’ will be how we will tie everything together with a pretty mathematical bow. This experiment uses a Vernier sensor called ‘Dual-Range Force Sensor’. This sensor is used to measure pulling and pushing forces. It could be used to help study the impact of car crashes and it could also be used to study trucks and the amount of force that they are able to pull and it could lead to some findings on how to increase the strength of the truck in order to pull more weight. However, today we will be using it to create a relationship between the amount of pennies and how much they weigh. The sensor looks as such:

The cord you see here will plug into a laptop and it will record the data for you. Instead of the metal mass attached to the hook on the picture on the left we are going to hand a paper cup from the hook. This is going to be how we will measure the weight of all of the pennies. Using the Dual-Range Force Sensor and the appropriate software on the laptop we will be able to create an accurate table, which will then give us our coordinate points that we will graph on a graphing sheet of paper.

Starting out, each group of mathematicians will receive 32 pennies (4 sets of 8 pennies). After their paper cup is properly hanging of the sensor’s hook each group will slowly measure the first set of 8 pennies, only recording the weight after the cup has stopped swaying. This will ensure a more accurate weight. Each group will continually add the second, third, and fourth set of pennies and recording the resulting weight. Now, they can’t have the computer record everything for them! They will have to find the slope of the line that these data points created. They’ll have to use two sets of data points and the slope formula, , and write a complete linear function in the form of , where m is the slope and b is the y-intercept. Also, to assist them in their progress throughout this experiment, they will have to work through the handout which can be found here: https://www.vernier.com/files/sample_labs/RWV-02-DQ-making_cents_of_math.pdf. This will allow them to see the relationship between the pennies and the weight (as well as the picture that they’ll have to create) and it will also allow them to pick apart the experiement and each peice of the equation and determine why each part is important.

This is just a very small experiment, however it teaches the students how they might be able to find a relationship between two different elements and how they can apply math directly to a real life scenario. This is especially important for those students who struggle in realizing that math is important in the everyday life and can be used in the everyday life and that it’s not just something that they’re forced to take for a minimum of 2 years in high school.

 

Hungry To Learn

schloemerj Technology Teaching Issues

Hungry to Learn

Using Geogebra to Bring Out the Inner Mathematician.

Written by Jenell Sellers


INTRODUCTION

Times have changed, can anyone deny that? Gone are the days in which primary and secondary students sit row by row and take notes to a lecture. The image of school and education is changing and morphing into an environment where adolescents are learning how to collaborate with their colleagues and learn as a group instead of in solitude.

Now, instead of rows we have desks put into groups. Instead of the students only holding pencils to paper, they are holding manipulatives (algebra tiles, fractal cards, hand-made unit circles, etc.). This is all to help students become effective citizens. Yes, times are changing indeed.

What we are seeing in our students is that they want to learn; they want to understand what is going on within the math class, it’s just that they don’t think that they can. They’ve seen math taught may a couple of different way, but it just hasn’t really sunk in yet and they start to think that maybe math is really only supposed to be taught a certain way. This is where we start to break the boundaries that they, or maybe other teachers, have set up around them; around their mind and ability to learn.

One change that is also being implemented is trying to introduce technology into the classroom. Our students, being part of the Millennial generation, are growing up with technology at their fingertips. It is our way of connecting with them and bringing their world into our classroom instead of bringing the students into our world of pure mathematics and computations.

One piece of technology that is immensely helpful within the classroom to help students see a better visual of what they are learning within the classroom is Geogebra. Geogebra can be used for Algebra I and II, Geometry, and even Trigonometry. Geogebra is a free application that can be downloaded from the Geogebra website (www.geogebgra.org) to the aspiring mathematician’s home computer or laptop. For the schools that have laptops in the classroom it can be especially effective so that every student can explore the program. Throughout this article we will be examining how Geogebra may be implemented within the Algebra II classroom.

STEP BY STEP, WE PROCEED…

The beauty about Geogebra is that it is amazingly simple to use. When the application is first opened we see the x and y axes, just waiting for an equation to be plotted. Only by going under the Graphics tab, we can make the grid show so that we can see each individual intersection of the x and y values; this makes it significantly easier for the students to follow along with the values that are shown when you plug in a specific graph, especially if it is being projected in front of them.

Let’s say that we are in an Algebra II class setting where we are learning how to translate graphs. This can be an especially hard lesson for students because it can be so hard to visualize the moves of a graph when the teacher is explaining; “If the value is inside the parentheses you move the vertex left or right in the opposite direction of what the value is. And when it’s outside the parentheses you move the vertex up or down the number of units according to the value.” Umm, what? Let us see a visual, shall we?

Take the equation, . When we type this into the input bar at the bottom of the screen within Geogebra we get the following visual:

This function alone is not that hard for students to plot on a graph. However, when it comes to the translations the students feel that it’s a lot trickier that what it really is.

Now, let’s take the equation         . The students are told within their class that the horizontal shift is going to be ‘the opposite sign of what is in the parentheses’. So instead of shifting over to the right 2 units, we go to the left 2 units. Like this:

Plugging this into the input bar, we can see that the graph does indeed shift over to the left when we add 2 within the parantheses. After showing the students the parent equation, the students can see how the graph physically moves.

Now let’s look at the final transition of this graph. Typing  into the input bar we see the following:

During this final transition, the students are able to see that when the value (minus 4) is outside of the parentheses then we actually move the graph down 4 units.

Not only might it help for the students to see each individual shift by itself, but it would also be well worth it to have them see each individual shift put altogether. Like so:

It is of upmost importance that the students are able to see each of these translations, rather than just hear about them and the instructions. The lecture piece is important, but without seeing what is physically going on with the graphs, then it is easier for the students to get lost.

ACHIEVING THOSE STANDARDS

This lesson would most definitely prepare the way for students to achieve a couple of common core state standards: “factor a quadratic expression to reveal the zeros of the function it defines” (CSS.MATH.CONTENT.HSA.SSE.B.3.A) and “understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output” (CCSS.MATH.CONTENT.8.F.A.1). Not only will they be able to be prepared to achieve these standards and more, but they will also be able to apply the learning targets to go along with the lessons. The learning targets that could go along with the lesson mentioned here could be something along the lines of “I can write the equation for a translated parabolic function” or “I can show that a parent function is still one-to-one.”

CONLCUSION

As time moves along, change comes with it. We have made several innovations for cars, computers, phones, and so much more. The future generations are continually learning, but they are learning at a far different level than what students used to. It is about time that we change our teaching methods as well to match the changes that have over the years in the education system. Geogebra, among many other applications, is one way that we can further assist our students in visualizing the mathematics of the world.