byc.html

ALGEBRA II - BIG YEAR CURRICULUM THE RELATIONSHIP

EXTRAVAGANZA

BY LISSA STEFFEY, ANGELA SHINOS AND JOHNNETTA RICKS

What's this page all about?

This page is the home page of our curriculum for a second year Algebra class. In the following document, you will see our theme for the course, our assumptions about what the students are bringing to the classroom, a flow chart containing our natural progression of ideas for the course, and our rationale for each of these. Our curriculum is based on the relationships between two variables with an emphasis placed on the exploration of connections between those relationships. The learning environment we feel this course should be taught in is such that each student is encouraged to find his or her own real life applications for each type of relationship. Each of the curriculum developers were responsible for two topic plans, totaling six of the seven listed in the natural progression of ideas. Our rationale for choosing each of these topics is contained in each individual topic plan.

Mail the authors: Lissa Angie Johnnetta

Course Theme - Relationships of two variables in the Cartesian coordinate system and understanding the relationships between these relationships. (Click on Course Theme to see our rationale for choosing this theme.)

Course Assumptions We have two main types of assumptions about our school: assumptions about the students and assumptions about the available technology.
To be able to carry out this curriculum, we will need to have access to graphing calculators for each student and to a computer lab for our students with the following capabilities:
                                ---Netscape Gold
                                ---GeoSketch Pad
                                ---Mathematica
                                ---MatLab

This list consists of the concepts we hope that our students are bringing knowledge about with them with them from previous courses and names the courses we think that each concept should have been covered in. (click on Course Assumptions to see our rationale)
                               --- The real number line (introduced in 6th grade)
                                    -properties of real numbers
                                    -negative numbers
                                    -inequalities
                                ---Fractions (3rd and 4th grade)
                                ---Absolute values (Pre-Algebra and Algebra I)
                                ---Manipulating rational expressions, simplifying (Algebra I)
                                ---Plotting in the Cartesian coordinate plane (Algebra I)
                                ---Linear functions (Algebra I)
                                    -slope and intercepts

Here is the course 'flow chart'. For a more in depth explanation of how to read the flow chart, click on 'Natural Progression of Ideas'. For an explanation about anyone of the elements in the table, click on that item. Clicking on the items in the bottom row of the tier will lead you to our rationale for including each item and a set of goals and rationale for teaching each.

NATURAL PROGRESSION OF IDEAS
Pre-Assessment Activity About Relationships Web Explorations

Trigonometric

Exponential/
Logarithmic

Located below is the rationale and deeper explanation for our course:

_Rationale for Course Theme_
This theme includes our thoughts about themes in Algebra I continued in Algebra II with new emphasis upon the study of relationships between relationships born of the Algebra I concepts as well as those new concepts introduced in Algebra II. The theme in Algebra I, we believe, is giving the students an introduction to relationships, specifically linear relationships, of two variables. This Algebra II theme over any other theme, allows our students to grasp the 'bigger' picture: recognizing there are several types of two variable relationships and that all of these relationships are interwoven. This theme is broad and basic, but certainly not set in stone. It can evolve to something much larger depending on the resources of the class and the individual students. We believe that our students' input (especially applications discovered in the Activity about Life Relationships Projects) can enlighten our thinking about the Algebra II course and consequently make contributions to the course's theme.

back to Course Theme

_Rationale for Course Assumptions_
We feel it is appropriate to design a curriculum that requires technology such as computers with Netscape capabilities for three reasons. First, we feel that the world is ever changing and using technology is a large part of what students will need to know to function in the world. Preparing students to use mathematics means making them familiar with the tools used to solve mathematical problems or at least making them recognize that their are technologies out there that can be used for solving these problems. Secondly, the NCTM clearly states that technology should be incorporated into mathematics coursework. Finally, the use of technology in the classroom like the graphing calculator or GeoSketch Pad can help students to visualize what is happening more quickly than hand-sketching. This visual teaching will help reach a broader portion of students when combined with symbolic teaching. The quick calculations and sketches promote exploratory learning. Students can quickly check many conjectures they might make concerning equations of relationships. For these reasons, we feel technology is an asset and a necessity in the classroom.
As for assumptions about the students, we had to decide where Algebra I ended and Algebra II started or we could not move forward with the development of the curriculum. You have to know where your kids are at to meet them there. Notice next to each assumption topic we listed what grade we expected each to have been at least introduced. We are certainly not entirely assuming that each topic has been mastered (see pre-assessment activity).

back to Course Assumptions

_Natural Progression of Ideas Explanation_
The natural progression of ideas is a 'flow chart' that represents the order that the topics will be covered in the Algebra II course. The first tier represents the first topic covered in the course: Pre-assessment Activity. Again, we feel this must be the first item on our agenda because you must have an idea about where a student is to be able to reach them, in mathematics or otherwise. The second tier contains two of the essentials of the course. These are the Activity About Life Relationships and the Web Exploration. We suggest covering the Activity about Life Relationships first because it is something that you can refer to throughout the year. It's where the students have made sense of some relationships around them mathematically. We believe this activity's results should be referred to throughout the year to continue to show connections. The Web Exploration is something that you can cover within each of the topics discussed (the 'topics' are the third tier of the flow chart) in the course. The Web Exploration is a way for students to further see how to relate what they are learning about in class to the world around them. Each topic in the curriculum should have some kind of Web Exploration included. The third tier of the Natural Progression chart contains the main topics of the big year curriculum. We have not defined in what order the topics should be covered (other than the fact that the Activity About Life Relationships should go first and the Web Exploration should be included in every topic plan) because we feel that should be defined by the questions that the students are asking in the course. Student-defined analysis of relationships will certainly be more meaningful than teacher-defined analysis from what we have seen in the field and through our own experiences. You will see within each of the topic plans the rationale for covering the topic plans and the relationships between each of the topics of the course (hence, the name of the BYC, "The Relationship Extravaganza.") If you would like to see one progression through these topics and the rationale for this order, click here.

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_Pre-Assessment Activities_
The following are example of activities/questions which will help us to assess what the students already know and what may need to refresh their memories on. These are activities that cover the course assumptions. The sample activities/questions are as follows:

        a.) Real number line:
                Put the numbers -10, 5, 7, -2, 0, 25 in order from smallest to largest.
                What does the minus sign (-) in front of a number mean?

        b.) Fractions:
                Compare the fractions with greater than, less than or equal to sign:
                    5/10 __ 1/2        3/4 __ 3/5        -1/3 __ -1        2 __ 5/2
                    2/5 + 3/5 = __        7/8 + 2/4 = __
                    (And any multiplying, dividing, adding and subtracting examples).

c.) Absolute values:
What does absolute value mean to you?

        d.) Manipulating rational expressions:
                Simplify the following:
                    2x + 3x = __
                    3y²+ 2x + 3x² + (-5)y² = __
                    x²yz/y³ = __
                    x/(x - 1) + (3x²)/(x + 2) = __

e.) Plotting in the plane:
Plot the following points: (1, 2), (4, 6), (-2, 9)

        f.) Linear functions:
                Find the x,y intercepts:
                    3x + 2y = 7
                    y = (5/6) x + 4
             Which of the following are straight lines?
                3xx -7 = 4y
                x + y = 0
                x = 2x + y -4
We suggest reviewing for up to two weeks any material this assessment reveals the students are unfamiliar with. If much of the unfamiliarity involves linear relationships, a formal review is not necessarily because this topic will be explored to the extent that students unfamiliar as well as those familiar will feel included. If what needs reviewing is quite varied from student to student, we suggest that covering the review when necessary (when the time comes that the students need that information which they have forgotten).

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_Activity About Life Relationships_
This activity is for the students. Its purpose is to get the students thinking about dependence relationships and eventually, the relationships between those relationships. We suggest that you continue throughout the course to refer back to these situations when they are applicable. It will help you to show the students how valuable their are to the course and how they contribute to their own learning. The following is an example of this type of activity. We suggest allowing your students to spend a few days at the library and that you should possibly present an example of your own to the class so the students have an idea of the type of relationship you're interested in for the course.

Activity About Life Relationships Find something that is dependent on another thing. Describe the things and what makes the one thing dependent on the other. Try to mathematically represent this dependence.

Suggested Sources:   library
                                internet
                                life examples

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_Web Exploration Explanation_
What we mean by Web Exploration is students exploring the web to find concrete examples of where the relationships we discuss in this course occur. This is a research option for students as well as a resource for clarifying the topics discussed in class. We expect it to be used during the teaching of each topic in the curriculum, but that after a few topics are covered, in the course that the students will be self-motivated to do so.

Here is a list of sites that could be used for this course:
Parabolic Relationships
    http://www.krellinst.org/uces/archive/resources/conics/node68.html
    http://dept.physics.upenn.edu/courses/gladney/phys150/lectures/lecture_sept_29_1997.html
Polynomial/Rational Relationships
    http://www.mcs.kent.edu/docs/maxima/maximahints/section3.15.html
Other Related Sites
    http://forum.swarthmore.edu
    http://www.wnet.org/nttidb/lessons/as/pathas.html
    http://library.advanced.org/10470/netscape/alg2

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_One Ordering of Coverage for The Relationship Extravaganza and Rationale_
    As stated in the Natural Progression of Ideas Rationale, covering the Pre-assessment Activity is our first priority. We need to know who our students are mathematically speaking before we can attempt to meet them at where they are. After this activity, we have suggested either a few weeks of review or a 'review as necessary' throughout the year. The next coverage will be a project for the students to complete called Activity About Life Relationships. This activity will call for the students to look in the world around them and come up with examples about how things depend on other things. This activity, again, will provide us (the teachers) with examples that are student generated, making them more meaningful than examples that we would come up with for the students to study.
    Our next progression is linear relationships. At this point, we're not calling them functions. Function is a big word that mathematicians, in our opinion, haven't really seemed to be able to define very clearly. We feel like this because we never had a very good idea about what a function was until upper level mathematics courses. Functions is a topic in and of itself. We would like to cover linears first because it will be a relationship that the students are familiar with from Pre-Algebra and Algebra I. It is the simplest kind of relationship because it is defined mostly by proportionality. We expect that many of the Activity About Life Relationship ideas will revolve around linearity. From here, we saw several possibilities .... you'll understand as we progress through the chosen progression.
    Functional Relationships coverage is a possible next move. We chose this topic next because we feel that most of the time texts and teachers choose to call linear relationships 'linear functions.' This naming can be very meaningless to students unless they know what a function is. This topic will help the students to define what a function is (beyond those text book definitions!) and will allow them to recognize which relationships are functions in further class discussions. This topic will allow us to introduce functional notation (very useful notation!) that can be used throughout the course.
    Parabolic Relationships is a possible next move. Students can think about parabolic relationships as those generated by multiplying two linear equations together and as functional relationships (minus sideways parabolas...). Certainly, not all parabolas are generated by multiplying two lines together, but this connection can provide flow from linear relationships to parabolic relationships. There are also connections between the y-intercepts of the linear equations and parabolic equations that the students may come across. Rate of change also comes to mind as a connection because linear equations have a constant rate of change while parabolic equations have a constant 'change in the change.' The Activity About Life Relationships might not have produced as many examples, but the Web Exploration should provide the students with several concrete examples of parabolic relationships.
    From parabolic relationships, we felt a next step would be to Rational and Polynomial Relationships. We feel this progression might make sense to students because we have generated parabolic relationships by talking about multiplying linear equations together. Students might be wondering what happens if you multiply more than two linear equations together or multiply a linear equation and a parabolic equation or divide any one of these items. These relationships are functional relationships as well. Again, the Activity About Life Relationships might not have produced as many examples, but the Web Exploration here should provide the students with several concrete examples of rational and polynomial relationships.
    From rational and polynomial relationships, covering Systems of Equations seemed to be the next move. Systems Relationships are ones that use some of the types of relationships we've talked about, but use more than one at a time. Thus, the students will be able to recognize the connections to previous topics quite easily. Sub-topics like optimal solutions will be discussed that students may relate to maximums and minimums of parabolic relationships and the rational and polynomial relationships. Many economic and business situations involve using systems of equations, so some of the things that the students mentioned in the Activity About Life Relationships might apply very nicely here. The students may have more difficulty finding examples on the Web, but your searches may be plentiful and we suggest sharing search words with your students and consulting Dr. Math for ideas (Dr. Math is a interactive web page.)
    Exponential and Logarithmic Relationships may be found in the students' Activity About Life Relationships since so much about growth and decay are modeled by these types of relationships. Certainly, systems of equations could have contained exponential and logarithmic relationships. Students can contrast the rate of change of something growing linearly (or even something growing parabolically) with how things grow exponentially. The definition of e is given by taking the limit of a polynomial expression raised to a variable power. These functional relationships may be more difficult for the students to understand because of the different ways the variables appear (in superscript or following log or ln.) The Web Exploration should lead to a wealth of information for the students.
    Finally, Trigonometric Relationships are the last topic of choice. These relationships are functional as well and connections can be made to the previously covered topics. Linear equations can be related to the tangents of trigonometric equations at certain points. The symmetry of parabolic graphs and rational and polynomial graphs can be related to trigonometric graphs. Trigonometric functions can also be approximated by polynomial equations. Things like sound waves may be a popular topic in the Activity About Life Relationships and in the results of the students' Web Exploration.

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_Resources for The Relationship Extravaganza_
Our resources for this curriculum are not merely books. We credit (Mr) Bill Rosenthal and Melissa Dennis for their input and expertise in analyzing this document and our topic plans. We found our experiences in our observations at Holt High School through the teachings of Kelly Hodges, Josh Minsely, and Dave Hildebrandt, and at East Lansing High School with Jesse Turner invaluable in writing this curriculum. We also thank the students of Holt High School, East Lansing High School, and the Math Enrichment Program's College Algebra Course at Michigan State University for sharing their experiences in mathematics with us through their course work and participation in our assessment activities. The choice of topics for this curriculum is due in part to our analysis of several algebra texts including College Algebra: A Graphing Approach by Larson, Hostetler, and Edwards, Intermediate Algebra by Phillips et. al., Algebra 2 by Larson, Kanold, and Stiff, Contemporary Algebra: Book Two by Smith, Lankford, and Payne, Algebra and Trigonometry by Dolciani, Sorgenfrey, Brown, and Kane, and Precalculus Mathematics: A Graphing Approach by Demana, Waits, and Clemens.

_Our Self Critique of The Relationship Extravaganza_
    The work for this course can never be done because no curriculum should be set in stone. We recognize our plan does not have a definite order from topic to topic. We know that students don't necessarily have the same background as the teacher and will not always recognize flow in the same way as the teacher. The curriculum should be so flexible that it can be changed on a moment's notice and should be student guided. We do not make claim to this being a perfect curriculum as it has not been tested (only some lessons have been carried out from the topic plans.) We used our own understanding of the subjects and expanded on it to make a curriculum that makes sense to us and hopefully to students.
    In an effort to gain a broader understanding of Algebra II, we examined several sources. We listened with respect to our students' learning and incorporated what we learned into this document. We observed our collaborating teachers without judgment and were able to take some ideas from them which appear in our topic plans. The text books we examined did not dictate what was included in our document, but merely gave us an overview of possible directions for the course to take.
    The rationale given has been sound. We have provided rationale for our theme and written a curriculum that follows that theme. Each of the topic plans shows not only the specific relationship but also how they relate to the other topic plans. The rationale for the course assumptions are based on our own presumptions about the material covered in courses before Algebra II. Although we assume that the students are coming in from a year of algebra, we do not assume that the students have mastered all of the topics introduced in that course. We created an assessment to analyze the issue. We incorporated some of the topics that the students may have already seen into the course not as review, but as content. We used this content as a part of the course in conjunction with the course theme.
    We recognize that times are changing. Technology is relevant in several aspects of learning and life. Incorporating technology into the course is a must and has been done in several forms. In the topic plans, we have shown how graphing calculators and mathematics software can be utilized. In the flow chart, there are two forms of student activities: Activity About Life Relationships and Web Exploration. These aspects emphasize how we have taken a realistic approach to Algebra II. This realistic approach is more beneficial than an abstract approach because it provides a way to include all students and calls on students to recognize the usefulness of mathematics in technical and non-technical applications.