1. Proposal Summary This project based unit will incorporate a creative way for students to learn and apply using systems of linear equations. Students will be required to solve systems of equations through the use of elimination method, substitution method, and graphing. They will be able to incorporate what they learn in the classroom to an interactive, real-world application. Students will assume the role of a graphic designer and be required to not only design a creative banner for the organization of their choice, but also use systems of linear equations to create each letter in the banner. This project will benefit all the students in Mr. Hinchman’s 8th grade advanced math class at Mebane Middle School as well as any class who repeats the project. Creating a banner for their favorite organization or team will engage the students as well as keep them involved throughout the project as they construct their banners by creating text using linear equations as well as incorporate design and appeal. They will be using their banners to advertise for the organization of their choice and at the end of the unit, there will be a showing of all of the banners made; peers, administrators, and a select few graphic designers will together vote on the winning banner which will be displayed in the school. 2. Rationale and Potential Impact/Needs As stated in NCTM Standards, during the middle grades "work with patterns should emphasize concrete situations and be informal and relatively unburdened by symbolism." However, teaching sometimes tends to focus on the skill set necessary to solve formal representations of functions before students have a secure foundation in the use of those variables. Witnessing it first hand, the concept of a variable is not well-defined for many students. When introducing systems of linear equations, teachers are often unknowingly building on a shaky foundation. Though this unit is not meant to fill in gaps of knowledge that should have been addressed early in middle school, by providing multi-faceted opportunities for students to explore systems of linear equations teachers create authentic experiences that can potentially correct misconceptions and build a stronger foundation. Specifically for this unit, students will be creating a banner for an organization meaningful to them, and this significance should motivate the whole project. Ultimately, students will understand the usefulness of linear equations, and appreciate the astounding amount of work that goes into creating a design, but that will not mean very much if not connected and rooted in something that interests them. This experience is intended to provide students with a chance to show their understanding, and furthermore is cemented in the requirements of the final product – that is the write-up, draft, and presentation. Furthermore, this project would mesh very fittingly to other disciplines such as art, language, social studies, and more. Clearly, the larger adult community can benefit from this as well considering some banners will be sent to the printer. 3. Description of Project This project –based unit is planned to span roughly two weeks. As the calendar lays it out, completion is expected after 14 school days. However, considering this creative endeavor may take longer, the 15th day is built in as a back-up work day whenever needed within those three weeks. Once students have had experience solving systems of linear equations, an investigative lesson will introduce students to graphing/creating text using linear equations. After deciding on a charity/school/sports team etc., students will design a banner to advertise their chosen organization. One week into the lesson, students will take a field trip to a printing store to discuss costs, materials, dimensions, and any other questions or concerns they may have, and complete a cost-analysis. Students will transpose their designs (including text) to a large piece of graphing paper (1-inch grid), and complete a draft using linear equations to describe the text and graphics. Along with a write-up, the final product will be the completed banner draft and a presentation to the class and some visiting judges/sponsors. Students must use and identify at least 45 linear equations to map out their design. Additionally, no more than 1/3 of those lines should be horizontal or vertical lines. Obviously, it would be a bit tedious for students to write out every letter using linear equation or to make an intricate drawing using linear equations. Students will be allowed to copy graphics or special text fonts. However, they must map out the placement, making their grid paper into a coordinate plane, and there has to be some additional background design, font, or picture that is described by 45 different linear equations as explained. Along with this draft, students will be required to write out a project overview that includes a summary of the cost analysis, explanation of linear equations used, and a persuasive argument for why their banner should be printed. Finally, students will present their banner to the classroom as well as a few sponsors (judges). The top three banners will be made and given to their chosen organization after first being displayed on campus. This product is an authentic representation of real-world issues in that students can connect and rally for something that interests them in the real-world. That could be something serious and thought-provoking, such as the Invisible Children organization, or something more recreational that students enjoy, such as a community baseball team or a local restaurant. Students will first present their banner drafts to the classroom and the judges, and then the winning banners will be displayed around the school.
Performance Objectives Students will be able to… 1. Graph systems of linear equations by hand and find their solution or point of intersection from the graph. 2. Determine if a system of linear equations has only one solution, infinitely many solutions, or no solutions by referencing a graph. 3. Demonstrate that parallel lines have no point of intersection using a graph. 4. Graph systems of linear equations using a graphing calculator and approximate points of intersection using the graphing calculator. 5. Verify the solution of a linear system by substitution of the solution into both equations. 6. To re-write an equation explicitly by isolating the dependent variable. 7. Find the solution to a system of linear equations using substitution. 8. Find the solution to a system of linear equations using elimination. 9. Analyze costs associated with printing the banner. 10. Create a design using linear equations. 11. Relate solutions of systems of linear equations to authentic contexts. 12. Collaborate with other students to produce a final product.
4. Evaluation The final product will be graded based on four criteria: overall organization/details, banner mathematics, write-up/proposal, and presentation. At most, students can earn 24 points on their final project. The first category, overall organization/details, involves the completion of a group role worksheet and group member evaluations, as well as having the slogan/banner wording pre-approved by the teacher. Since it does not specifically require any particular math knowledge, this criteria is only worth 4 points (17% of total). The banner mathematics criteria addresses the specific expectations of the benchmarks to be learned from this PBI Unit. Students are expected to accurately impose their banner design on a 1” grid graphing paper with x- and y-axis clearly identified. Each design should be the combination of at least 45 lines. No more than 2/3rds should be horizontal or vertical lines. The linear equations should be written directly behind the lines on the back of the banner paper. Students are also to complete a smaller scale finished model. Clearly, this criteria contains the heavy mathematics and thus is worth 8 points (33% of total). Thirdly, students will be evaluated on their project write-up. They will have to complete a full cost-analysis as a group, and additionally, each member has to write two or more paragraphs explaining how they found the linear equations that map out the design and how it relates to solving systems of linear equations. Since this criteria focuses on the reasoning and conceptually ability of the student, it too is worth 8 points (33% of total). Finally, students will present their banner designs and explain their linear equations and most of what was in the write-up. Again, though the verbal explanation is important to reflection and essentially metacognition, this criteria is only worth 4 points (17% of total) because some students may have better presentation skills than others. Aside from the final project rubric, there are multiple forms of formative assessment built into the project. In particular, many worksheets given throughout will be graded with feedback for students to progress in their understanding (see Appendix B). Again, specifically for the final product, the group role forms, wording draft worksheet, and cost-analysis write-up are all types of formative assessments to allow me to evaluate where students are in their project. Furthermore, many in-class formative assessments are anticipated. For instance, during the first lesson, students will work in pairs to complete a T/F table with their reasoning to describe possible solutions to linear equations. As a teacher, this will allow me to understand student thinking in a way that a computational problem might short-change that process. Every day has some type of formative assessment planned by the close of the lesson (see Appendix B). Some days that evaluation is as simple as an exit slip prompt or one word problem, and other days there is time set aside for individual meetings with the teacher.
5. Budget
If we are awarded this grant, we will be in need of $4,281.06. This seemingly high budget is more fully explained by the table above. The majority of the cost is in the purchase of a class set of TI-Nspire graphing calculators. Students will be able to quickly and accurately graph linear equations and manipulate those graphs to better understand their properties. These versatile graphing calculators have many features that make them not only ideal for the linear equations involved in this unit, but to many other applications outside of this unit. Additionally, these are recyclable items - meaning, this class set may potentially last for 10 or more years if well taken care of. These calculators are an investment for hundreds of students to come. The smaller, non-recyclable items are well-worth the investment in terms of their application within this project. Yardsticks and crayons are invaluable for making the banner drafts both exact and colorful. The roll of 1" grid paper will also simplify and organize the drafting process. Though the field trip site costs nothing, student transportation is most likely only achievable using Public School Bus Transportation, hence the additional $400 fee.
6. Calendar Summary
For full calendar, please see Appendix B.
Week 1: First two Benchmark lessons will jump-start the unit. Students will complete a matching cards activity involving solving systems of linear equations followed by a T/F table. Students will gain practice solving systems of linear equations by using a graphing calculator on the second day. The third and fourth days will be an investigatory lesson on graphing letters followed by a lesson on substitution. Friday will be the project introduction and engagement. By the end of the week, students ought to have formed the groups and begin their first word draft.
Week 2: The week will begin by a project lab day where they will spend time working on their banner draft. We will cover the elimination method on the second day, and then do some 1-1 group sessions with the teacher Wednesday. By mid-week, students should have their banner wording finalized and their design draft begun. Thursday will be the field trip to Target Copy to complete the cost-analysis. Finally on Friday, students will complete a warm-up/formative assessment, and continue working on their banner design.
Week 3: On Monday, students will be expected to complete their design. Additionally, the cost analysis part of the final project will be due. On Tuesday, students will spend time doing informal peer presentations. Groups will be paired off with other groups and will practice presenting their final banner. Wednesday is the last chance students will have to do any preparations before they present. Thursday will be the times for formal presentations.
7. Appendices
Appendix A: Teaching Philosophy For much of my naïve youth, I was under the impression that teaching was heavily formulaic. Simply be a strong disciplinarian with eyes in the back of your head, good handwriting for the blackboard, and enough knowledge to intimidate any student into respecting you, and success will surely follow. This seemed especially true in the realm of mathematics, a subject seemingly so clear-cut and objective. Frustrated with such common norms, it took some time for me to appreciate the effort that continues to be put forth by educational institutions and teaching communities to educate future teachers so as not to resort to the aforementioned highly unsuccessful teaching approach. Considering the material we have covered and the small amount of experience in the classroom I have been fortunate to enjoy this past year, I found five principles central to effective classroom instruction: careful lesson planning, culturally-responsive teaching, accommodating lessons, providing opportunities for mathematical discourse through cooperative learning, and finally, teaching through problem-solving. In order for teaching to be effective it must facilitate student understanding. This measure of student understanding of mathematics is commonly defined by Kilpatrick’s et al. five strands of mathematical proficiency. As outlined in Adding it Up (2001), the five strands of mathematics are conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. There is in no strict delineation between each strand, but rather they are each interdependent on one another. Conceptual understanding implies that the learner has a strong comprehensive and useful handle of mathematical ideas. Part of that understanding means recognizing the logic of common operations, relationships, and underlying theorems, which bleeds into the strand of adaptive reasoning. At its core, procedural fluency equates to having the necessary skill set to perform one operation or to carry out multiple steps of a problem. This strongly relates to strategic competence, which is the ability to solve problems using mathematical ideas. Therefore, both are inherently linked to conceptual understanding. If a student can conceptually understand a mathematical idea, he or she will see the procedural application. Sometimes it works the other way as well – it can take multiple examples of the application for the concept to click. Adaptive reasoning refers to the use of logic in problem solving in many different situations, which entails careful reasoning and justification. This strand requires the first three, but also tightly binds the first three together. Finally, productive disposition lends personality to all the strands. It requires motivation on the part of the learner to make sense of math and utilize the tools of math to recognize useful applications. These five strands imply much for the role of the teacher. Limiting valid problem-solving methods, representing problems in very few contexts, or not properly explaining the rules behind foundational concepts may seriously handicap students’ progress in mathematical competency. Such poor teaching methods models very little of the five strands of proficiency, which teaches students to follow suit by example. Rather, teachers ought to model the five strands and structure lessons accordingly in order for student understanding to take place.
Obviously this implies some serious forethought in regards to the first principle, lesson planning. While in UFTeach, we have adopted the 5E lesson planning format – engagement, exploration, explanation, elaboration, and evaluation. The lesson is usually planned to flow from an exploration activity that is student-centered into more technical explanations. This enables a shift from teacher-centered direction to learner-centered direction through the guided discovery process in the exploration. As we read in chapter 2 of our textbook, guided discovery “tends to result in better long-term retention and transfer than expository instruction” (pg. 30). Students have more ownership over their own understanding if they are playing an active role in constructing it, which helps support the strand of a productive disposition. Careful lesson planning can save time and distractions, which would certainly detract from the learning process.
Moreover, I think I have yet to write a lesson that is less than 10 or so pages long. Most of that length comes from writing out specific directions, probing questions, and possible student answers including misconceptions. We are also required to write a detailed explanation of the concept, propose objectives, align the lesson with Sunshine State Standards, create a knowledge package, and include accommodations for students with a learning disability. Writing a lesson plan can be a bit tedious, but the preparation goes a long way. As Liping Ma proposes and as we discussed in class, content knowledge should not be lacking in a teacher. By creating knowledge packages and anticipating possible student misconceptions, we are challenging and enriching our own understanding of the content, which in turn offers more resources to the student.
However well-written a lesson plan may be, disaster can still strike if it is not implemented well. Such is the lot in which we are cast as fallible creatures. From my own experience, “guided-discovery” explorations have the potential to confuse and frustrate students, waste time, and create more misunderstandings. Many of the above problems can be remedied by the effective execution of the remaining principles. Concerning cultural responsiveness, I believe teachers have a responsibility to get to know their students in order to understand how to motivate them both individually and collectively. Clearly, what goes on outside of the classroom is beyond our control, but every effort must be made to remedy issues related to culture or socio-economic status of students in order to create a fair learning environment for the class. In our class a few weeks we were asked to create a concept map representing important factors related to teaching mathematics effectively. One group drew a centralized concept map enclosed in a large circle, which represented equity. That ideal setting requires getting all students involved in the mathematics lesson by students gaining confidence and recognizing the value in their work.
Culture impacts cognition and thus should not be ignored in the mathematics classroom. As Ladson-Billings proposes, math is not sterilized from culture but integral to it. As teachers we can envelop and take in as much as we can about our student’s respective cultures. We can attempt to bridge the operations of mathematics to what students value and appreciate. To encourage achievement, teachers ought to keep high expectations that are attainable along with the development of self-efficacy, build the difficulty of material from the foundation of students’ prior knowledge, and truly care about each student.
The third principle, classroom accommodations complements becoming a culturally-responsive teacher. We investigated four specific accommodations set by Margaret Williams in her article, “Strategies to Support ESL Students in Math”:
By encouraging math acquisition in the primary language, linking math to the four domains (reading, writing, speaking, listening), utilizing multiple learning modalities, and differentiating both in-class and homework assignments, teachers can support English language learners and help these young students to be successful in math (http://margaretmwilliams.suite101.com/strategies-to-support-esl-students-in-math-a116386).
Though aimed to help students with learning disabilities, many of these strategies are beneficial for any student. By differentiating instruction, teachers can engage a wider range of students thereby reducing off-task behavior and eliciting student interest. Considering the literacy achievement gap in education, visual aids can certainly benefit more students than just English Language Learners. By varying learning modalities, teachers can incorporate culturally relevant activities by developing activities using technology, manipulatives, and/or body movement to link mathematical ideas to student interests. Though speaking, listening, reading, and writing are not commonly linked to mathematics, they are students primary tools for reflection and metacognition, both of which are vital to understanding (Teaching and Learning Mathematics with Understanding, pg. 87). Furthermore, providing or developing tools such as a displayed or hand-held accessible classroom-constructed dictionary would be very beneficial to all students.
Integral to all of these issues, is the idea of developing a classroom norm of whole-group mathematical discussion. This is only initiated and achieved in a cooperative learning environment. Referring back to the 5E lesson format, usually the exploration is structured to facilitate cooperative learning. Students either work in pairs or small groups, or occasionally individually. In Johnson’s et al article, “Essential Elements of Cooperative Learning,” there are five core components of cooperative group work: positive interdependence, face-to-face promotive interaction, individual accountability and personal responsibility, interpersonal and small-group skills, and finally, group processing. It may seem a bit forced or unnatural for all of these to be carried out through role-playing as the article suggests, but teachers can still build from small successes and perhaps develop cooperative group skills within the classroom. In order to facilitate whole group discussion eventually, group work provides ample opportunities to ease students into discussion because they can consult and reason openly without the gaze or ears of an entire classroom.
In order to keep group work coherent and focused, students must be presented with a challenging mathematics task or problem. As outlined by Lester and Schroeder in the article “Developing Understanding in Mathematics Via Problem Solving,” teaching through problem solving is an inquiry-based approach to lesson designs. This requires a de-emphasis on procedural skills over conceptual skills and the tendency of teaching a limited set of problems with pre-defined solutions after introducing concepts. Rather, problem-solving becomes a “means for acquiring new mathematical knowledge and a process for applying what has been learned previously” (Lester & Schroeder, pg. 39) by covering formal definitions and concepts only after the discovery process. The class will then have to work toward a common goal, which is essentially to construct their own understanding. However, such a process could never take place without students’ articulating their thoughts throughout the problem-solving venture. Teachers need to incorporate effective questioning and facilitate student interactions to achieve such necessary discussions, and thus the remaining two principles facilitate each other.
In total, cooperative learning and problem-solving contribute to the development of the first four mathematical strands not only on an individual but a cooperative level. Cultural-responsiveness and flexibility with lesson plans to accommodate students, greatly benefits the final strand of a productive disposition. Lesson planning is the reflection and articulation necessary for a teacher to prepare himself or herself for the daunting but more-so rewarding task of promoting all five mathematical strands within the classroom. Ultimately, exemplifying active learning as the teacher, developing a relationship with your students, recognizing and building off of prior knowledge, and maintaining openness to variation can lead to effective classroom instruction.