CREATING AN HONORS CURRICULUM TO MEET THE NEEDS OF ALL STUDENTS by Natalie Youngberg A project submitted in partial fulfillment of the requirements for the degree of MASTER OF EDUCATION IN CURRICULUM AND INSTRUCTION WEBER STATE UNIVERSITY Ogden, Utah August 14, 2020 Approved ____________________________________ Stephanie Speicher, Ph.D. ____________________________________ Natalie A. Williams, Ph.D. ____________________________________ Penée W. Stewart, Ph.D. ____________________________________ Sheryl J. Rushton, Ph.D. CREATING AN HONORS CURRICULUM 2 Table of Contents Nature of the Problem .................................................................................................................7 Literature Review ........................................................................................................................9 Between-class Ability Grouping Benefits Teachers..........................................................9 Between-class Ability Grouping Disadvantages Students............................................... 10 Segregation in Between-Class Ability Grouping ............................................................ 11 Educational Studies are Being Ignored........................................................................... 12 Limited Education Leads to Limited Opportunities ........................................................ 13 Support Social Justice .................................................................................................... 14 Precedence for Change .................................................................................................. 15 New Curriculum is Needed ............................................................................................ 16 Purpose .................................................................................................................................... 18 Method .................................................................................................................................... 20 Curriculum Design ........................................................................................................ 20 Evaluators ..................................................................................................................... 21 Evaluation Tool ............................................................................................................. 22 Procedure ...................................................................................................................... 23 Results ..................................................................................................................................... 24 Discussion ................................................................................................................................ 27 Future Practice............................................................................................................... 29 Future Research ............................................................................................................. 30 Limitations .................................................................................................................... 31 Conclusion .................................................................................................................... 31 CREATING AN HONORS CURRICULUM 3 References ............................................................................................................................... 32 Appendices ............................................................................................................................... 34 Appendix A: Curriculum ............................................................................................... 35 Unit 1 ................................................................................................................. 35 Unit 2 ................................................................................................................. 62 Unit 3 ................................................................................................................. 80 Unit 4 ............................................................................................................... 101 Appendix B: Curriculum Evaluation Tool .................................................................... 140 Appendix C: Feedback from Evaluators ....................................................................... 144 Unit 1 ............................................................................................................... 144 Unit 2 ............................................................................................................... 151 Unit 3 ............................................................................................................... 158 Unit 4 ............................................................................................................... 165 Appendix D: IRB Permission Letter ............................................................................ 172 CREATING AN HONORS CURRICULUM 4 List of Tables Table 1. Qualifications of evaluators 22 CREATING AN HONORS CURRICULUM 5 List of Figures Figure 1. Scores Given to Curriculum by Evaluators. 24CREATING AN HONORS CURRICULUM 6 Abstract Separating secondary students into different classes based on their ability is a traditional way of grouping students with the intention of teaching each student at the level they are prepared to learn. However, this practice has created barriers for many students, a disproportionately high number of which belong to minority groups. In an attempt to improve education and increase social justice, people from around the world are making an effort to move toward classes that are heterogeneous in ability. Implementing these changes has been challenging. However, positive changes in the way students view learning and diversity have been encouraging. New curriculum needs to be created that will allow secondary students to be grouped heterogeneously in order to continue the movement toward equality in secondary education. The goal of this project is to facilitate teaching mixed-ability Secondary 1 mathematics classes in a way that provides universal access to both the Secondary 1 objectives and the Secondary 1 Honors objectives while using effective teaching strategies. The tasks included in this project allow for understanding at different levels, reinforce the importance of different ways of thinking about mathematics, and contain scaffolding that leads toward conceptual understanding. CREATING AN HONORS CURRICULUM 7 Nature of the Problem In secondary education, between-class ability grouping (i.e., tracking, setting, streaming, banding, etc.) is detrimental to many students (Boaler, Wiliam & Brown, 2000; Forgasz, 2010; Francis et al., 2017; Hornby & Witte, 2014; Tomlinson, 2015; Worthy, 2010). When between-class ability grouping is used in secondary education, the teachers know that there will be less disparity between the abilities of the students in the classroom. As such, teachers do not explain the content as many times because it is assumed that all students are ready for the same instruction at the same time, and teachers spend more time teaching new content (Boaler et al., 2000). Furthermore, the knowledge of the teacher can be matched with the ability-level of the class (Hornby & Witte, 2014; Tomlinson, 2015). Teachers with the highest mathematics abilities are usually assigned to teach students with high abilities in mathematics (Boaler et al., 2000). Despite these benefits, there are drawbacks to between-class ability grouping in secondary education such as demeaning messages sent to students based on class placement, lack of access to the full curriculum, and less opportunity to work with people who think differently (Boaler, 2008; Boaler et al., 2000; Forgasz, 2010; Hornby & Witte, 2014; Tomlinson, 2015). Additionally, teachers may mistakenly believe that there is no difference between their students’ abilities, which can be frustrating for the students (Boaler et al., 2000; Francis et al., 2017; Tomlinson, 2015). Furthermore, the higher percentage of students who come from low income or minority families in the lower-level classes reinforces racism and classism (Forgasz, 2010; Francis et al., 2017; Tomlinson, 2015; Worthy, 2010). Between-class ability grouping is still being used in secondary education despite the inequality it perpetuates (Francis et al., 2017; Hornby & Witte, 2014; Tomlinson, 2015). The rationale for between-class ability grouping is that it will help teachers better teach students at CREATING AN HONORS CURRICULUM 8 the appropriate level (Francis et al., 2017). However, the benefits to achievement are insignificant and sorting students into year-long classes based on their perceived ability is demoralizing (Forgasz, 2010; Francis et al., 2017; Hornby & Witte, 2014; Tomlinson, 2015). Furthermore, for students to have equal opportunity for success in society, they cannot be denied equal access to an education that will help them develop problem-solving and thinking skills, and prepare them for college entrance (Boaler et al., 2000; Forgasz, 2010; Francis et al., 2017; Tomlinson, 2015; Worthy, 2010). Between-class ability grouping limits this access because the most experienced teachers are assigned the highest-level classes whereas the lowest-level classes are sometimes taught by teachers who do not have the proper qualifications to teach mathematics (Boaler et al., 2000; Tomlinson, 2015). Furthermore, students in low-level classes do not have access to the complete curriculum, and quality teaching practices, such as tasks that encourage making sense of mathematical practices, are reserved for the higher-level classes (Boaler et al., 2000; Forgasz, 2010; Francis et al., 2017; Tomlinson, 2015; Worthy, 2010). Given these disparities, between-class ability grouping in secondary education is a practice that impedes social justice and should be replaced with mixed-ability classrooms. To promote social justice and prepare students to meet the challenges of the 21st century, teachers and administrators should support practices that promote equity in the classroom (Boaler, 2008; Forgasz, 2010; Tomlinson, 2015). Replacing between-class ability grouping in secondary education with within-class ability grouping, is a step in the right direction (Boaler et al., 2000; Forgasz, 2010; Worthy, 2010) because it better reflects the inclusive groupings more commonly used in the elementary setting (Boaler et al., 2000). Within-class ability grouping in secondary education is defined as flexible grouping within a multi-ability classroom based on the needs of the students (Hornby & Witte, 2014). This kind of grouping does not limit access to CREATING AN HONORS CURRICULUM 9 information or to teachers who have the necessary certification (Boaler et al., 2000; Forgasz, 2010; Worthy, 2010). Therefore, in this type of class, teachers can teach all students in a format that naturally supports social justice (Boaler et al., 2000; Forgasz, 2010; Worthy, 2010). There is precedence for the elimination of between-class ability grouping in secondary education (Alpert & Bechar, 2008; Francis et al., 2017; Worthy, 2010), but shifting toward a within-class ability grouping model in secondary education is not without its challenges (Alpert & Bechar, 2008; Worthy, 2010). For within-class ability grouping in secondary education to be effective, curriculum needs to be differentiated enough to allow all students complete access to content, and scaffolding needs to be provided to support the needs of all learners within a mixed-ability classroom (Alpert & Bechar, 2008; Forgasz, 2010; Hornby & Witte, 2014; Tomlinson, 2015). Literature Review Between-Class Ability Grouping Benefits Teachers Between-class ability grouping in secondary education is the practice of sorting students into classes based on their ability. It is known under many names around the world such as tracking, setting, streaming, banding, and others (Boaler et al., 2000; Forgasz, 2010; Hornby & Witte, 2014; Tomlinson, 2015). This type of grouping students is popular with teachers (Forgasz, 2010). Forgasz (2010) discovered 74% of teachers in secondary schools with between-class ability grouping agreed with their school’s policy, and 75% of teachers in secondary schools without between-class ability grouping wanted their school to start using this type of grouping. It is not surprising that at least 74% of teachers at the schools in this study preferred between-class ability grouping in secondary education because it has some benefits for teachers that save them time during instruction and planning (Boaler et al., 2000; Forgasz, 2010; Hornby & Witte, 2014; Tomlinson, 2015). For example, the students have a smaller range of abilities in a CREATING AN HONORS CURRICULUM 10 class when this type of grouping is used, which leads to more opportunities for whole-class teaching, such as lecturing (Boaler et al., 2000). This results in less time needed to explain the same concept at different times to different groups, which is more common with mixed-ability classrooms in secondary education (Boaler et al., 2000). Furthermore, between-class ability grouping in secondary education allows teachers to focus on their strengths or interests (Hornby & Witte, 2014). Oftentimes some teachers are excited about offering extension activities to students with high ability who fill the upper-level classes and other teachers are especially good at teaching students who struggle the most (Hornby & Witte, 2014). Therefore, teaching in a secondary education class that has been grouped by ability makes the job of teaching all students equitably appear easier (Boaler et al., 2000; Hornby & Witte, 2014; Tomlinson, 2015). Between-Class Ability Grouping Disadvantages Students Unfortunately, the benefits for teachers lead to disadvantages for students (Hornby & Witte, 2014). First, the placement of secondary students into leveled classes is problematic because ability is not a fixed trait due to the plasticity of the brain (Francis et al., 2017; Tomlinson, 2015). Furthermore, all students have personal strengths and weaknesses within a subject (Francis et al., 2017). Boaler, Wiliam, and Brown (2000) spent two years studying how ability grouping affects the attitudes of students at six secondary schools in London. At four of these secondary schools, students moved to leveled classrooms created using between-class ability grouping from mixed-ability classrooms (Boaler et al., 2000). During the study, the researches spent 10 hours observing mathematics classes, surveyed 943 students, and interviewed 72 students (Boaler et al., 2000). The majority of students in leveled classes were frustrated with assumptions made by teachers about their ability based on their class placement (Boaler et al., 2000). The students said upper-level classes are too high-pressure and quick CREATING AN HONORS CURRICULUM 11 moving for the majority of students enrolled in them (Boaler et al., 2000; Francis et al., 2017). Conversely, students were frustrated with the pace that is often too slow and expectations that are too low in the lower-leveled classes (Boaler et al., 2000; Francis et al., 2017). In short, when teachers spend more time lecturing because they think all students are ready for the same thing, individual students are less likely to be challenged and at an appropriate level (Boaler, 2008; Boaler et al., 2000; Hornby & Witte, 2014). Second, when teachers focus on their strengths or interests, students miss out on important learning activities (Boaler et al., 2000; Forgasz, 2010; Hornby & Witte, 2014; Tomlinson, 2015). For example, the most experienced mathematics teachers generally teach the highest ability-level classes and mathematics teachers with less experience or teachers who do not have a license to teach mathematics are given the lowest-level classes (Boaler et al.; Tomlinson, 2015) even though an experienced mathematics teacher is more important to students who have learning challenges (Boaler et al., 2000; Forgasz, 2010). Furthermore, all secondary education students would benefit from the interactive and engaging extension activities that are frequently reserved for the highest-level classes (Boaler, 2008: Boaler et al., 2000; Tomlinson, 2015). Boaler (2008), after a four-year secondary education teaching practices study, found that students at all levels benefit from working together and sharing the responsibility to learn. Small group discussion and the opportunity to learn different ways of thinking about mathematics, which supports all learners, should be available to every student (Boaler, 2008; Boaler et al., 2000; Hornby & Witte, 2014; Tomlinson, 2015). Segregation in Between-Class Ability Grouping Even if these prevalent problems were avoided, between-class ability grouping in secondary education would be detrimental to students because of the message sent to students CREATING AN HONORS CURRICULUM 12 and teachers when they are assigned to a particular class level (Francis et al., 2017; Oakes, 1985; Tomlinson, 2015; Worthy, 2010). The practice of between-class ability grouping in secondary education has been compared to segregation because of the disproportionate number of students from minority backgrounds and low socioeconomic status in the lowest-leveled classes (Francis et al., 2017; Oakes, 1985). Furthermore, secondary education students become defined by the level in which they are placed (Worthy, 2010). For example, when secondary education teachers were interviewed, distinct assumptions about students and expectations for behavior were based on the level of class in which the student was placed (Worthy, 2010). Secondary education students also made assumptions about themselves because of the level of their class (Boaler et al., 2000). The negative attitude about the ability and behavior of students in the lower-level classes combined with the high percentage of students who are minority or are living in poverty in these classes leads to social isolation, racism, and classism (Tomlinson, 2015). Educational Studies are Being Ignored Between-class ability grouping is how students have been traditionally grouped in secondary education (Boaler, 2008; Francis et al., 2017; Turney & Hyde, 1931). The use of between-class ability grouping is increasing in secondary settings (Francis et al., 2017; Tomlinson, 2015). This is happening because policy makers and parents overwhelmingly believe that between-class ability grouping is the best way to support more rigorous study in secondary education (Boaler, 2008; Hornby & Witte, 2014; Francis et al., 2017). After all, they fear the students who have high levels of ability will not be challenged in a mixed-ability secondary education classroom (Boaler, 2008; Hornby & Witte, 2014; Francis et al., 2017). However, Francis, Archer, Hodgen, Pepper, Taylor, and Travers (2017) analyzed previous research on the effect of between-class ability grouping and discovered that the academic benefit of between-CREATING AN HONORS CURRICULUM 13 class ability grouping is statistically insignificant for secondary students who have a high ability. Furthermore, between-class ability grouping in secondary education is damaging socially and academically to the students in the mid to low ability classrooms (Boaler, 2008; Boaler et al.; Forgasz, 2010; Francis et al., 2017; Hornby & Witte, 2014). Despite these findings, between-class ability grouping in secondary education is kept because of tradition (Francis et al., 2017). Limited Education Leads to Limited Opportunities Between-class ability grouping in secondary education maintains social hierarchy because of an imbalance of access to an education that prepares students for long term success (Boaler et al., 2000; Forgasz, 2010; Francis et al., 2017; Tomlinson, 2015; Worthy, 2010). Secondary students are frequently assigned to a level based on social class and parental support rather than the ability of the student (Boaler et al., 2000; Francis et al., 2017). A class-level assignment has long term consequences because of the focus of the different classes. The top classes focus on problem solving that will prepare students for success in the future (Boaler et al., 2000; Forgasz, 2010; Tomlinson, 2015; Worthy, 2010). On the other hand, Boaler, Wiliam, and Brown (2000), in surveys and interviews, discovered the lower secondary education classes focus on copying information and memorization which denies students the opportunity to learn skills essential for higher paying jobs. Forgasz (2010) had similar findings when he sent out a survey to Australian secondary school teachers. The respondents described teaching methods used in the lower-level classes (Forgasz, 2010). The teaching methods described do not teach the critical thinking skills that are necessary for higher paying jobs (Forgasz, 2010). Furthermore, Worthy (2010) interviewed teachers at eight secondary schools in an urban school district in Texas and found that most teachers had negative attitudes toward students in lower-level classes which resulted in low expectations and reduced access to activities that build thinking skills in CREATING AN HONORS CURRICULUM 14 these classes. Also, Tomlinson (2015) found that students in lower-leveled classes had limited access to the learning opportunities that would teach them complex reasoning skills that prepare a student for success outside of school. Overall, students in low-level classes reported copying off of the board (Boaler et al., 2000) and teachers reported using teaching techniques that do not require students to build the problem-solving skills necessary for future financial security (Forgasz, 2010; Tomlinson, 2015; Worthy, 2010). The problem with student placement in secondary education is exasperated by the inability for students to move between the levels because less of the curriculum is available in the lower-level classes (Forgasz, 2010; Francis et al., 2017; Tomlinson, 2015). By the time students discovered they were not being taught information that is necessary for college entrance exams in the lower-leveled classes, it was too late for them to do anything about it (Boaler et al., 2000; Worthy, 2010). Therefore, initial placement in an ability group may change the long-term opportunities available to a student. Support Social Justice The student population in schools and the world outside of school is more diverse now than it has ever been (Tomlinson, 2015). Therefore, schools need to make changes that will meet the needs of students with a variety of backgrounds and abilities (Forgasz, 2010). When the flexibility of student placement and high-level instruction are in place for all students, achievement is maximized (Steenbergern, Makel & Olszemski-Kubulius, 2016). Within-class ability grouping in secondary education provides the opportunity for a high level of instruction and full access to the curriculum for all students (Boaler, 2008; Boaler et al., 2000; Forgasz, 2010; Worthy, 2010). Furthermore, within-class ability grouping in secondary education allows for the flexibility to meet the needs of all students equally within a mixed-ability classroom CREATING AN HONORS CURRICULUM 15 (Boaler et al., 2000; Forgasz, 2010). Finally, mixed-ability secondary education classrooms should reflect the world outside of the school walls (Boaler, 2008; Tomlinson, 2015). Therefore, restructuring secondary schools to increase the number of mixed-ability classes will best prepare all students to become respectful, contributing members of a diverse society when they leave (Boaler, 2008; Tomlinson, 2015). Precedence for Change There are people around the world trying to change how schools are structured (Alpert & Bechar, 2008; Worthy, 2010). Unfortunately, some of the leveling changes that were meant to remove the undesirable outcomes of ability grouping in secondary education have failed (Worthy, 2010). For example, Worthy (2010) found administrators and teachers believed that today’s leveling system of doing away with the lowest level of classes would be beneficial to all secondary students. Under this new system, students are grouped in regular and honors classes (Worthy, 2010). However, there was still a distinct difference between the way teachers taught different classes and the way students viewed themselves based on the level of their class (Worthy, 2010). In the new secondary education system, the students in the regular classes had the same disadvantages as the students in the lowest-level before the change (Worthy, 2010). On the other hand, some elimination of between-class ability grouping has been successful in secondary education. Alpert and Bechar (2008) researched a secondary school in Israel that used a structure the school named Open Triads. These Open Triads involved multiple teachers teaching the same content with different topics of interest and allowing students to choose which class they wanted to attend (Alpert & Bechar, 2008). The classes were regrouped regularly, leaving all students feeling included and in control of their learning (Alpert & Bechar, 2008). Furthermore, the stigma of being assigned to a lower ability class did not exist among the CREATING AN HONORS CURRICULUM 16 students and teachers at this school even when ability grouping occasionally happened (Alpert & Bechar, 2008). New Curriculum is Needed For more secondary schools to follow the pattern of student choice directing education that made this school in Israel unique, a new curriculum is needed (Forgasz, 2010; Hornby & Witte, 2014; Tomlinson, 2015). The new curriculum must teach students to construct mathematical knowledge and to respect contributions from all students (Boaler, 2008; Boaler & Dweck, 2016; NCTM, 2014; Tomlinson, 2015). This can be achieved with task centered lessons where students work together to build knowledge (Boaler, 2008; Boaler & Dweck, 2016; NCTM, 2014; Tomlinson, 2015). Boaler (2008) discovered when secondary students are taught responsibility for their own learning and the learning of their group members, they are more likely to learn the mathematics thoroughly and respect people with different backgrounds. For the tasks to be most effective, they must be open ended with multiple ways to think about and present the solution (Boaler, 2008; Boaler & Dweck, 2016; NCTM, 2014). Valuing multiple ways of thinking about mathematics helps more students to be successful and feel valued in a mathematics classroom (Boaler, 2008). When preparing these lessons, secondary teachers should start by preparing for the highest-level student and apply scaffolding and adaptations to make it work for all students in a heterogeneous classroom (Boaler & Dweck, 2016; Boaler et al., 2000; NCTM, 2014; Tomlinson, 2015). It is important to provide scaffolding and feedback that focuses on building strategies and student discovery (Boaler & Dweck, 2016; NCTM, 2014). With appropriate scaffolding provided for open-ended tasks that value multiple ways of thinking, secondary teachers can reach diverse learners without excluding or isolating students based on their ability level (Boaler, 2008; Boaler CREATING AN HONORS CURRICULUM 17 & Dweck, 2016; Forgasz, 2010; NCTM, 2014; Tomlinson, 2015). A more inclusive curriculum will benefit all secondary students (Boaler, 2008; Boaler & Dweck, 2016; Alpert & Bechar, 2008; Forgasz, 2010). CREATING AN HONORS CURRICULUM 18 Purpose Separating secondary students into different classes based on their ability, known as between-class ability grouping, has been shown to have some advantages for teachers (Boaler et al., 2000; Francis et al., 2017; Hornby & Witte, 2014; Tomlinson, 2015). However, grouping students in secondary education this way has many disadvantages for the students (Boaler et al., 2000; Francis et al., 2017; Tomlinson, 2015). It is especially harmful to students who belong to minority groups (Forgasz, 2010; Francis et al., 2017; Tomlinson, 2015; Oakes, 1985; Worthy, 2010). Unfortunately, the tradition of using between-class ability grouping in secondary education has overruled the data that support replacing it with mixed-ability classes (Boaler, 2008; Hornby & Witte, 2014; Francis et al., 2017; Turney & Hyde, 1931; Tomlinson, 2015). On the other hand, there have been secondary schools that have boldly tried to structure school as a place that provides equal access to all students using flexible within-class ability grouping in mixed-ability classrooms (Alpert & Bechar, 2008; Worthy, 2010). These changes are not implemented without their share of challenges (Alpert & Bechar, 2008; Worthy, 2010). However, the improvement in the learning and confidence students experience have made the struggle worthwhile (Alpert & Bechar, 2008; Boaler, 2008; Worthy, 2010). To replicate the success others have achieved, a new curriculum needs to be created that will allow for the teaching of students at all ability levels within the same classroom (Alpert & Bechar, 2008; Boaler, 2008; Forgasz, 2010; Hornby & Witte, 2014; Tomlinson, 2015). With the new curriculum, students should have time to work in groups with students who are at different ability-levels giving them the opportunity to learn to value the opinions and abilities of others (Alpert & Bechar, 2008; Boaler, 2008; Boaler & Dweck, 2016; Tomlinson, 2015). It is important CREATING AN HONORS CURRICULUM 19 to have small group tasks that can be solved in a variety of ways, assignments that value the multiple ways of thinking about mathematics, and scaffolding that will encourage students to think through the process being taught (Boaler, 2008; Boaler & Dweck, 2016; Boaler et al., 2000; Forgasz, 2010; NCTM, 2014; Tomlinson, 2015). The goals of the curriculum are: 1. Present the Secondary 1 Honors curriculum as an extension of the Secondary 1 curriculum to facilitate the teaching of mixed-ability classes. 2. Provide a curriculum that includes effective teaching strategies and is flexible enough to allow students who have not chosen to earn honors credit to be successful studying the honors curriculum. CREATING AN HONORS CURRICULUM 20 Method Curriculum Design In order to reach these objectives, the honors objectives were aligned with objectives of the regular curriculum that could naturally lead to the honors curriculum. This needed to be done before creating the curriculum so the student grouping could be flexible. When the honors class and regular class are taught separately, the two classes are rarely if ever on the same section of curriculum at the same time. In order to keep the class from being permanently split into regular and honors, defeating the purpose of having them in the same classroom, the pacing had to change. To solve this problem, connections were found such as the similarity between simplifying linear combinations of matrices and solving linear equations because they both heavily rely on order of operations and they both deal with linear equations. The curriculum was developed with four such units (matrix introduction, solving word problems with vectors, solving systems of equations with matrices, and linear combinations of vectors) that are aligned to be extensions of the Secondary 1 curriculum. The matrix introduction unit was made to be available along with the units involving solving linear equations. Solving word problems with vectors will be used as an extension to writing systems of equations from word problems. Solving systems of equations with matrices is an extension to solving systems of equations using substitution and elimination. The last, linear combinations of vectors, will be available during the congruence transformations unit. In order to plan for the students at the highest level and scaffold to support all learners as suggested by literature review, the objectives were unpacked into individual pieces that could be learned through small-group discovery lessons as suggested by literature review. It was important to create tasks that deliberately lead the student through past knowledge and ask them CREATING AN HONORS CURRICULUM 21 to experiment with new concepts while celebrating diverse thinking as recommended by literature review. Each unit begins with an introductory task that allows students to work in groups to build an understanding of the main concepts of the unit. Each unit contains tasks, with scaffolding that supports thinking about the mathematical reasoning, for students to practice procedures. Additional practice is available to be used as necessary. Each unit contains an assessment of learning. All worksheets and tasks are formatted to be printed for ease of use for a small group. To increase flexibility, each task and worksheet has enough information to be completed as a stand-alone activity. However, complete understanding will require the use of multiple or all tasks and worksheets for a unit. Evaluators For recruitment, the curriculum objectives were shared with three colleagues (see Table 1). The colleagues agreed to give feedback when the curriculum was developed. The first evaluator knows the Secondary 1 and Secondary 1 Honors curriculum and knows what it takes for students to learn it. This teacher is an evaluator because she has the experience to know if the new curriculum is viable. The second is an expert at accommodating and scaffolding material for students. This teacher is an evaluator because this curriculum needs the scaffolding necessary for any student at any level of ability to be successful. Her experience with scaffolding and supporting the learning of students that qualify for resource services will help her to provide valuable feedback about the scaffolding in this curriculum. She has not taught honors curriculum in the past, but will be using this curriculum because it is being created for use in every Secondary 1 classroom. The third evaluator is an expert at differentiation and has experience teaching 7th and 8th graders mathematics in a mixed-ability classroom. This evaluator will be able CREATING AN HONORS CURRICULUM 22 to evaluate the readiness of students to learn the material and how well the curriculum will work in a mixed-ability classroom. Table 1 Qualifications of evaluators Experience of colleague Area of Expertise of colleague Using the Curriculum 32 years teaching math Secondary 1 and Secondary 1 Honors curriculum Yes 12 years teaching resource math Accommodating assignments Yes 17 years teaching math Using differentiation to teach multi-ability classes for 7th and 8th grade No Evaluation Tool Once the curriculum (see Appendix A) was developed, it was given to the evaluators who graded each unit with a rubric (see Appendix B) using Google Forms. The first questions measured how well the honors curriculum (see Appendix A) is organized as an extension of the regular curriculum. This is necessary because it will allow for different groupings to be used through the year as suggested by the literature review. There will be days when the whole class is working on the same thing in mixed-ability groups and days when students choose between working on regular curriculum or honors curriculum (see Appendix A). The next few questions in the rubric address meeting the needs of the students with open-ended tasks, ways to value different thought processes, and appropriate scaffolding as suggested by the literature review. These learning opportunities will be available to all students to increase equity in the classroom. The assignments will be required for students who are seeking an honors CREATING AN HONORS CURRICULUM 23 grade and will be extra credit for students who are not. Therefore, all material needs to be prepared with the highest achieving students in mind and contain enough scaffolding for all students to feel welcome, included, and capable of working in the small group as much or as little as they want. The last few questions in the rubric measure the practicality of the use of the units and give the evaluators the opportunity to make suggestions. Procedure After completing the IRB process, the curriculum (see Appendix A) was developed. Each unit begins with an open-ended task to connect past learning to new concepts. Next, each unit has assignments that include scaffolding to support individual concept building. Finally, each unit contains an assessment of learning. The assignments are designed as stand-alone activities to increase the flexibility of the curriculum. Everything is formatted to be easily printed for use in small groups. After completing the curriculum (see Appendix A), the evaluators were emailed along with a link to the evaluation tool in Google Forms (see Appendix B). They had two weeks to read through the curriculum (see Appendix A) and respond to the questions in the evaluation tool for each of the four units. After receiving feedback from the evaluators (see Appendix C), the comments were reviewed and follow-up questions were asked, as needed, to improve understanding of how the curriculum (see Appendix A) could be improved. For the open-ended questions, a general consensus was identified. Again, the evaluators were contacted for clarification, as needed. The improvements identified during this step were implemented in the curriculum (see Appendix A). CREATING AN HONORS CURRICULUM 24 Results The scores given to each unit of the curriculum (see Appendix A) by each evaluator (see Appendix C) were added together to compare the different aspects of the curriculum (see Figure 1). This helped to identify the improvements that needed to be made. Figure 1. Scores Given to Curriculum by Evaluators. This figure displays the points given to the curriculum (see Appendix A) by the evaluators in each area (see Appendix C). The curriculum (see Appendix A) received a perfect rating for the statement “Honors curriculum is an extension of the regular curriculum.” across all units (see Figure 1). The general consensus about this statement was reported by evaluator 1 as, “This shows a relationship to the regular curriculum, as opposed to being its own curriculum. This allows students to choose ‘honors’ activities without the risk that comes from declaring to be an honor student.” The feedback (see Appendix C) from this question showed me that the curriculum (see Appendix A) is approachable by any student even though it is focused on honors objectives. This is important because one of the values of a mixed-ability classroom is to have students at different ability levels working together to find connections between different mathematical concepts (Boaler, Scores Given to Curriculum by Evaluators CREATING AN HONORS CURRICULUM 25 2008; Forgasz, 2010; Tomlinson, 2015). Therefore, having a curriculum that is based on previous learning for all students is important for this project. The next statement, “Curriculum is flexible enough to allow students to join the honors group for any number of activities.” received the highest marks possible (see Figure 1). Evaluator 3 stated, “This is broken down into steps well enough for all students to participate, especially with support of group members/partners.” This showed the curriculum (see Appendix A) reached the goal of an honors curriculum that any student could be successful studying for any amount of time. When students have the opportunity to make choices about their learning, they are more likely to feel empowered and take ownership of their learning (Alpert & Bechar, 2008; Boaler & Dweck, 2016; Tomlinson, 2015). The focus of this project is to support social justice by making the complete curriculum available to all students. Therefore, it is important to make the honors curriculum (see Appendix A) something that any student could choose participate in. The statement “Tasks can be understood at different levels and encourage multiple ways of thinking.” received full points (see Figure 1). The reason for this rating was summed up well be evaluator 2 when she said, “I liked the multiple opportunities for students to explain their thinking. The concepts could be understood on multiple levels.” This statement shows that the curriculum (see Appendix A) values multiple ways of thinking about mathematics and different levels of understanding. It is necessary to value different ways of thinking to help all students feel included (Boaler, 2008; Boaler & Dweck, 2016; Forgasz, 2010; NCTM, 2014; Tomlinson, 2015). In the mixed-ability classrooms that are being created because of this curriculum (see Appendix A) there will be students with many different backgrounds, experiences, and abilities. Therefore, this project can only be successful if diversity is expected and valued. CREATING AN HONORS CURRICULUM 26 Next, the statement “Scaffolding is present and encourages the construction of strategies.” received less than full points (see Figure 1). Overall, evaluator 3 agreed with her colleagues when she said, “The problems begin with more guided instruction and move on to problems with less guidance and more challenge.” However, in unit 2, evaluator 3 stated, “Additional scaffold-ed notes could be used to support students with learning difficulties if they were expected to participate.” Similar problems were pointed out regarding unit 4. This showed a weakness in the goal to provide adequate scaffolding for any student to have the opportunity to be successful. When planning curriculum for a mixed-ability class, it is vital to first plan for the top student and provide scaffolding to support all learners (Boaler & Dweck, 2016; Boaler et al., 2000; NCTM, 2014; Tomlinson, 2015). Therefore, guided notes were added to unit 2 and the “Put it together!” activity was added to unit 4 to support student learning. With these additional activities, any students will have the resources necessary to be successful studying this curriculum (see Appendix A). The curriculum (see Appendix A) received a perfect score on the statement, “Students are encouraged to think about math in a variety of ways (see Figure 1).” Evaluator 1 stated the general consensus when she said, “Comparing answers with other students and explaining math thinking provide an opportunity to explore the concepts from different perspectives.” The score on this statement and the responses from evaluators shows this curriculum (see Appendix A) reaches the goal of encouraging students to investigate multiple ways of completing a task and thinking about mathematical processes. In order for a curriculum to be successful, it must value and encourage diversity (Boaler, 2008; Boaler & Dweck, 2016; Boaler et al., 2000; Forgasz, 2010; NCTM, 2014; Tomlinson, 2015). This curriculum (see Appendix A) shows that diversity CREATING AN HONORS CURRICULUM 27 is valued and encouraged by containing multiple questions that ask students compare and discuss mathematical thinking. The last section received the lowest score (see Figure 1). In this section, the evaluators were asked, “As a math educator, what is your level of comfort in teaching with this curriculum?” The problem was summed up by evaluator 3 who mentioned, “Brief teacher notes to assist teachers with limited understanding i.e. resource teachers who are co-teaching may not have the in depth mathematics background.” In creating the curriculum (see Appendix A), a lot of time went into considering student needs when thinking about mathematics in a variety of ways and having a variety of abilities. However, teachers come from a variety of backgrounds and have a variety of abilities as well. The evaluators pointed out that some teachers would need more support to teach the curriculum (see Appendix A). This step cannot be left out because when between-class ability grouping is removed, every teacher needs to have the necessary resources to teach every level of student (Hornby & Witte, 2014; Tomlinson, 2015). Therefore, more inclusive answer keys and teacher notes were added to each unit (see Appendix A). Discussion In the beginning, the topic of ability-grouping was researched because of the number of students who said they did not think they belonged in an honors class even though the questions they asked showed that they craved more of a challenge. They could get more challenge within the regular class with supplemental activities, but it was concerning that they perceived such strict limitations for themselves. Research showed the problem to be larger than one school or one school district. The benefits of between-class ability grouping to teachers were not a surprise. The current arrangement of separating students into honors and regular classes gives more time to CREATING AN HONORS CURRICULUM 28 teach new content (Boaler et al., 2000) and the knowledge of the teacher could be matched with the ability level of the class (Hornby & Witte, 2014; Tomlinson, 2015). Furthermore, the message sent to students by between-class ability grouping and how it could limit students’ options for the future (Boaler et al., 2000; Forgasz, 2010; Hornby & Witte, 2014), sounded familiar. However, the assumption about ability affecting the majority of the students in a class (Boaler et al., 2000) was a surprise. It seemed like more of a localized problem that affected a few students. Furthermore, the segregation and the extent of misplacement of students that the current system supports (Forgasz, 2010; Francis et al., 2017; Tomlinson, 2015; Worthy, 2010) was not fully understood before research. It was obvious, at times, that there were students who were misplaced. However, research showed that students from minority groups were routinely placed in the lower-level classes despite their abilities (Forgasz, 2010; Francis et al., 2017; Tomlinson, 2015; Worthy, 2010). These realizations motivated me to rethink how mathematics was being taught, how kids were grouped at school, and the unintended lessons that were being taught by separating students into different ability levels. I decided to find a way to teach Secondary 1 Honors and Secondary 1 side by side in a mixed-ability class using research supported teaching techniques. The three techniques that were broadly recommended and universally applicable are task centered lessons where students work together to build knowledge, assignments that value multiple ways of thinking, and scaffolding that encourages students to think about the process being taught (Boaler, 2008; Boaler & Dweck, 2016; Boaler et al., 2000; NCTM, 2014; Tomlinson, 2015). The curriculum (see Appendix A) was created around the use of activities that focused on the above mentioned techniques. In addition to this, the curriculum (see Appendix A) was created with the goal of planning for students with the highest abilities and scaffolding for CREATING AN HONORS CURRICULUM 29 students with lower-abilities as recommended by research (Boaler & Dweck, 2016; Boaler et al., 2000; NCTM, 2014; Tomlinson, 2015). Furthermore, scaffolding was created by reflecting on tips routinely given to students that guide them toward the solution without removing the need for them to think through the process. The feedback (see Appendix C) that was the most surprising was the reminder that all teachers are not comfortable teaching the honors objectives. This highlighted the fact that removing between-class ability grouping requires every teacher to have the necessary resources to teach every level of student (Hornby & Witte, 2014; Tomlinson, 2015). This feedback (see Appendix C) let to looking at the curriculum (see Appendix A) from the viewpoint of a teacher who was not familiar with teaching it. This process let to more thorough answer keys and teacher notes that make the curriculum (see Appendix A) usable by any teacher. Future Practice I recommend every teacher look at their curriculum and find ways to increase the social justice in their classroom. For example, teach students to learn in heterogeneous groups, teach students to take responsibility for their learning and the learning of their group members, make it clear to students that different ways of thinking are expected and valued, and provide scaffolding that will help students to develop understanding (Boaler, 2008; Boaler & Dweck, 2016; Boaler et al., 2000; NCTM, 2014; Tomlinson, 2015). All of these things are supported by the literature review as ways to improve the education of each individual student and give students the opportunity to value diversity within themselves and in others (Boaler, 2008; Boaler & Dweck, 2016; Boaler et al., 2000; Tomlinson, 2015). These techniques can be used in some way by any teacher in any classroom to help students to be prepared to be contributing members of a diverse society upon graduation (Boaler, 2008; Tomlinson, 2015). CREATING AN HONORS CURRICULUM 30 I challenge every secondary teacher to work with their school to eliminate between-class ability grouping. Putting the new curriculum (see Appendix A) created for this project to use has taken cooperation from all stakeholders. The whole mathematics department looked at the evidence and made a group decision to change the way mathematics is taught. The administration listened to the united voice of the mathematics department and provided the opportunity to present the idea to parents. The barrier between regular and honors mathematics has been removed for Math 7 and Math 8 after the original pilot program. The final piece to eliminate between-class ability grouping in all mathematics classes at the school is the completion of this project. The honors curriculum for Secondary 1 is more intense than the honors curriculum for Math 7 and Math 8. However, evidence from literature and the personal experiences I have had at a school that is working toward removing the barrier between regular and honors curriculum gave me the motivation to find a way. Equality in education is going to take a group effort. However, it can make a real difference to students when teachers that have not taught honors classes in the past have the support they need to teach a curriculum that focuses on top students and contains scaffolding to support the learning of all students. Future Research The intent in creating this curriculum (see Appendix A) is to improve social justice by making all Secondary 1 curriculums available to every Secondary 1 student. However, this project does not measure student reactions to the curriculum (see Appendix A). In the future, a study that compares students’ learning in a mixed-ability class using curriculum similar to the curriculum (see Appendix A) that was created in this project to students’ learning in a class that has been created using between-class ability grouping, would further answer questions that were not able to be answered by this project. CREATING AN HONORS CURRICULUM 31 Limitations One evaluator for this project knew the Secondary 1 and Secondary 1 Honors curriculum well, one had experience working with students who qualify for resource services at the Secondary 1 level, and one is an expert at teaching mixed-ability classes. However, none of the evaluators had all three categories of the expertise. This may have altered the feedback (see Appendix C) that was received. Conclusion Between-class ability grouping in secondary education has created barriers for many students. Those students who belong to minority groups have been affected the most. Many schools around the world have attempted to eliminate between-class ability grouping to improve the appreciation of diversity and to give full access to the curriculum to all students. These changes have been difficult, but mixed-ability classes have shown to make positive changes in student learning and attitudes toward diversity (Alpert & Bechar, 2008; Boaler, 2008). To follow these examples, new curriculum needs to be created that uses proven teaching techniques to provide quality education and full access to all students. When creating this curriculum, a teacher must focus on the students at the highest level. Then, accommodation and scaffolding that support the building of concepts must be added to support the learning of all students. This curriculum should value understanding at different levels and different ways of thinking about mathematical processes. Following these guidelines and creating teacher resources for teachers who may not have the background necessary to teach the advanced curriculum, can provide the foundation for change that can make a positive difference to students in any class. CREATING AN HONORS CURRICULUM 32 References Alpert, B., & Bechar, S. (2008). School organizational efforts in search for alternatives to ability grouping. Teaching & Teacher Education, 24(6), 1599-1612. doi:10.1016/j.tate.2008.02.023 Boaler, J. (2008). Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed‐ability approach. British Educational Research Journal, 34(2), 167-194. doi:10.1080/01411920701532145 Boaler, J., & Dweck, C. Chapter 5, 6, 7 and 9. In Mathematical Mindsets: Unleashing students’ potential through creative math, inspiring messages, and innovative teaching (pp. 57-140). JB Jossey-Bass, a Wiley Brand, 2016 Boaler, J., Wiliam, D., & Brown, M. (2000). Students' experiences of ability grouping - disaffection, polarization and the construction of failure. British Educational Research Journal, 26(5), 631-648. doi:10.1080/01411920020007832 Forgasz, H. (2010). Streaming for mathematics in Victorian secondary schools. Australian Mathematics Teacher, 66(1), 31-40. Francis, B., Archer, L., Hodgen, J., Pepper, D., Taylor, B., & Travers, M. (2017). Exploring the relative lack of impact of research on ‘ability grouping’ in England: A discourse analytic account. Cambridge Journal of Education, 47(1), 1-17. doi:10.1080/0305764X.2015.1093095 Hornby, G., & Witte, C. (2014). Ability grouping in New Zealand high schools: Are practices evidence-based? Preventing School Failure, 58(2), 90-95. doi:10.1080/1045988X.2013.782531 CREATING AN HONORS CURRICULUM 33 Isecke, Harriet. Chapter 3 and 4. In Backwards planning: Building enduring understanding through instructional design (pp. 43-86). Shell Education, 2011. NCTM, National Council of Teachers of Mathematics. (2014). Effective teaching and learning. In Principles to actions: Ensuring mathematical success for all (pp. 7-58). Reston, VA. Oakes, J. (1985). Keeping track: How schools structure inequality. New Haven, CT: Yale University Press. Steenbergen-Hu, S., Makel, M. C., & Olszewski-Kubilius, P. (2016). What one hundred years of research says about the effects of ability grouping and acceleration on K–12 students’ academic achievement. Review of Educational Research, 86(4), 849–899. https://doi-org.hal.weber.edu/10.3102/0034654316675417 Tomlinson, C. (2015). Teaching for excellence in academically diverse classrooms. Society, 52(3), 203–209. https://doi.org/10.1007/s12115-015-9888-0 Turney, A. H., & Hyde, M. F. (1931). What teachers think of ability grouping. Education, 52, 39–42. Retrieved from https://search-ebscohost com.hal.weber.edu/login.aspx?direct=true&db=rgr&AN=522268854&site=ehost-live Worthy, J. (2010). Only the names have been changed: Ability grouping revisited. Urban Review, 42(4), 271-295. doi:10.1007/s11256-009-0134-1 CREATING AN HONORS CURRICULUM 34 Appendices Appendix A: Curriculum Appendix B: Curriculum Evaluation Tool Appendix C: Feedback from Evaluators Appendix D: IRB Permission Letter CREATING AN HONORS CURRICULUM 35 Appendix A: Curriculum Unit 1 Teacher Notes To multiply or divide a matrix by a scalar multiply each element in the matrix by the scalar Example: 4[−1532]= [−420128] To add or subtract matrices add or subtract the numbers in coordinating positions. Example: [−4589]+[7−35−2]=[−4+75+(−3)8+59+(−2)]=[32137] Only matrices that are the same size, same number of rows and columns, can be added or subtracted. To multiply matrices find the sum of the products of corresponding components from each row in the first matrix and each column in the second matrix. Example 1: [1234]×[5678]=[1(5)+2(7)1(6)+2(8)3(5)+4(7)3(6)+4(8)]=[19224350] Example 2: [123456]×[789101112]=[1(7)+2(9)+3(11)1(8)+2(10)+3(12)4(7)+5(9)+6(11)4(8)+5(10)+6(12)]=[5864139154] Example 3: [789101112]×[123456]= [7(1)+8(4)7(2)+8(5)7(3)+8(6)9(1)+10(4)9(2)+10(5)9(3)+10(6)11(1)+12(4)11(2)+12(5)11(3)+12(6)]=[3954694968875982105] Example 4: [1234]×[789101112] is undefined Matrices can only be multiplied if the number of columns in the first is equal to the number of rows in the second. The number of rows in the solution is equal to the number of rows in the first matrix. The number of columns in the solution is equal to the number of columns in the second matrix. CREATING AN HONORS CURRICULUM 36 Unit 1 Vocabulary Matrix Definition Example Non-Example Scalar Definition Example Non-Example CREATING AN HONORS CURRICULUM 37 Commutative Property Definition Example Non-Example Associative Property Definition Example Non-Example CREATING AN HONORS CURRICULUM 38 Distributive Property Definition Example Non-Example CREATING AN HONORS CURRICULUM 39 Unit 1 Vocabulary Key Matrix Definition An array of numbers in rows and columns. [42−10] Example Non-Example 14 Scalar Definition A real number. Example 14 [42−10] Non-Example CREATING AN HONORS CURRICULUM 40 Commutative Property Definition 𝑎+𝑏=𝑏+𝑎 Or 𝑎×𝑏=𝑏×𝑎 Example 1+2=2+1 Or 5×6=6×5 Non-Example 2+(3 + 4) = (2+3) + 4 Associative Property Definition a + (b+c)=(a+b) + c or a(bc)=(ab)c Example 2+(3 + 4) = (2+3) + 4 Non-Example 1+2=2+1 CREATING AN HONORS CURRICULUM 41 Distributive Property Definition a(b+c) = ab+ac Example 4(5+1)=4(5) + 4(1) Non-Example 4+(5+1)=(4+5)+(4+1) CREATING AN HONORS CURRICULUM 42 Let’s Party The school decided to simplify ordering supplies for a party. For every 5 people the school will order 5 drinks, 2 pizzas, and 1 order of breadsticks. In order to keep track of the party orders, matrices are used: 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒5 [ 5 2 1] 1. The Spanish club has 30 members. Write the number of party supplies the Spanish club would need in matrix form (simplify as much as possible). 2. Describe what you did to find how much food and drink was needed for the Spanish club party. 3. How was your method the same as your neighbor’s? How was it different? Do both ways work? 4. Darin, the Spanish club president is also the captain of the football team. He is planning a party for both groups. The Spanish club has 30 members and the football team is expecting to have 50 people at the party. Write an equation in matrix form to show the total amount of food and drink needed for the parties. Simplify the solution as much as possible. Drinks Pizzas Breadsticks CREATING AN HONORS CURRICULUM 43 5. Describe the process you used to answer question number 4. 6. Compare your process to your neighbor’s. How was your method different? How was it the same? Did you both get the same answer? 7. How would you use your method of adding the supplies needed for both parties to a question that looks like this? Compare to your neighbor. 2[2−145]+3[41−2−3] 8. What would you do differently if you were subtracting like this problem? Compare to your neighbor. 2[2−145]−3[41−2−3] 9. Would you be able to add these matrices [82−35]+ [7 5 2] Why or Why not? Simplify the following expressions as much as possible. 10. 5[3−410]+[−17210] 11. −1[25−27]+4[86−12−86] CREATING AN HONORS CURRICULUM 44 12. 5[1218−6]−5[2−475] 13. −1[52−810]−[124−6−7] 14. 7[−9456]+2[219−43−2] 15. 3[5−259]+3[−2−609] 16. 6[410−7−1]−(−2)[53−21] 17. −6[13−60]−[−10−83−2] 18. 9[7053]+(−5)[4−2] 19. −5[−73−80]+2[3632] 20. 8[−77−47]−(−3)[−23−4−8] 21. 9[−3−69−5]−[55−6−6] CREATING AN HONORS CURRICULUM 45 Let’s Party Key The school decided to simplify ordering supplies for a party. For every 5 people the school will order 5 drinks, 2 pizzas, and 1 order of breadsticks. In order to keep track of the party orders, matrices are used: 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒5 [ 5 2 1] 1. The Spanish club has 30 members. Write the number of party supplies the Spanish club would need in matrix form (simplify as much as possible). [30 12 6] 2. Describe what you did to find how much food and drink was needed for the Spanish club party. 3. How was your method the same as your neighbor’s? How was it different? Do both ways work? 4. Darin, the Spanish club president is also the captain of the football team. He is planning a party for both groups. The Spanish club has 30 members and the football team is expecting to have 50 people at the party. Write an equation in matrix form to show the total amount of food and drink needed for the parties. Simplify the solution as much as possible. [30 12 6]+ [50 20 10]= [80 32 16] Drinks Pizzas Breadsticks CREATING AN HONORS CURRICULUM 46 5. Describe the process you used to answer question number 4. 6. Compare your process to your neighbor’s. How was your method different? How was it the same? Did you both get the same answer? 7. How would you use your method of adding the supplies needed for both parties to a question that looks like this? Compare to your neighbor. 2[2−145]+3[41−2−3] 8. What would you do differently if you were subtracting like this problem? Compare to your neighbor. 2[2−145]−3[41−2−3] 9. Would you be able to add these matrices [82−35]+ [7 5 2] Why or Why not? Simplify the following expressions as much as possible. 10.5[3−410]+[−17210]=[14−13710] 11. −1[25−27]+4[86−12−86] [−2−5−27]+[3224−48−3224] CREATING AN HONORS CURRICULUM 47 12. 5[1218−6]−5[2−475] 13. −1[52−810]−[124−6−7] [50255−55] [−17−614−3] 14. 7[−9456]+2[219−43−2] 15. 3[5−259]+3[−2−609] [−63283542]+ [4218−86−4] [9−241554] 16. 6[410−7−1]−(−2)[53−21] 17. −6[13−60]−[−10−83−2] [3466388] [4−10332] 18. 9[7053]+(−5)[4−2] 19. −5[−73−80]+2[3632] [6304527]+[−2010] [41−3464] 20. 8[−77−47]−(−3)[−23−4−8] 21. 9[−3−69−5]−[55−6−6] [−6265−4432] [−32−5987−39] CREATING AN HONORS CURRICULUM 48 Where’s the money? 1. On the last assignment, Darrin discovered that the Spanish club needs [30 12 6] and [50 20 10] for the football team. Since both clubs have separate budgets, Darrin needs to find the prices separately. He finds the prices at two different places and uses matrices to find the price of the Spanish club party and the football team party from each restaurant to decide who to order the food from with the following equation. (Labels added for explanation.) 2. Explain in detail how you found the prices of the parties at the two restaurants? 3. How was your strategy the same as your neighbor’s? How was it different? 4. What do the four numbers in your solution represent? Drinks Pizzas Breadsticks Drinks Pizzas Breadsticks CREATING AN HONORS CURRICULUM 49 5. Where should Darrin order the food from? 6. Could Darrin multiply these matrices to find the price at the restaurants if he decided to add salads to the order? Why or why not? 7. How can you decide by the size of matrices if you are able to multiply them or not? 8. If labels were not present in Darrin’s first equation (from problem #1), how could you keep track of the process of multiplying the matrices? Use the same matrix multiplication you used for the story problem to multiply the following matrices. If matrices cannot be multiplied, mark as undefined. 9. [831411]×[686752] CREATING AN HONORS CURRICULUM 50 10. [576703]×[39415−8] 11. [−669−7]×[−3190] 12. [6042]×[230−5] 13. [902738]×[90−12] 14. [−40−8−3]×[25−82−9−2−2−2−9] 15. [607−5]×[−915−9] 16. [−929−1]×[8−31−2] CREATING AN HONORS CURRICULUM 51 Where’s the money? Key 17. On the last assignment, Darrin discovered that the Spanish club needs [30 12 6] and [50 20 10] for the football team. Since both clubs have separate budgets, Darrin needs to find the prices separately. He finds the prices at two different places and uses matrices to find the price of the Spanish club party and the football team party from each restaurant to decide who to order the food from with the following equation. (Labels added for explanation.) [123117205195] 18. Explain in detail how you found the prices of the parties at the two restaurants? 19. How was your strategy the same as your neighbor’s? How was it different? 20. What do the four numbers in your solution represent? $123 Price for Spanish club at Pizza Palace $117 Price for Spanish club at Hungry Bear Pizza $205 Price for Football team at Pizza Palace $195 Price for Football team at Hungry Bear Pizza Drinks Pizzas Breadsticks Drinks Pizzas Breadsticks CREATING AN HONORS CURRICULUM 52 21. Where should Darrin order the food from? Hungry Bear Pizza 22. Could Darrin multiply these matrices to find the price at the restaurants if he decided to add salads to the order? Why or why not? No, there is not a price for salad in the price matrix 23. How can you decide by the size of matrices if you are able to multiply them or not? The number of columns in the first matrix must be equal to the number of rows in the second. 24. If labels were not present in Darrin’s first equation (from problem #1), how could you keep track of the process of multiplying the matrices? Use the same matrix multiplication you used for the story problem to multiply the following matrices. If matrices cannot be multiplied, mark as undefined. 25. [831411]×[686752]= [71873541] CREATING AN HONORS CURRICULUM 53 26. [576703]×[39415−8]= [7343639] 27. [−669−7]×[−3190]= [72−6−909] 28. [6042]×[230−5] = [121882] 29. [902738]×[90−12] Undefined 30. [−40−8−3]×[25−82−9−2−2−2−9] Undefined 31. [607−5]×[−915−9] = [−546−8852] 32. [−929−1]×[8−31−2] = [−702371−25] CREATING AN HONORS CURRICULUM 54 Properties of Matrices Associative Property Commutative Property Distributive Property Matrix Addition (A + B) +C = A + (B + C) A + B = B + A A ×(B + C) = A×B + A×C Matrix Multiplication (A × B)× C = A ×(B × C) A × B = B × A For each property you will first be asked to guess if the property will be true for matrices. Then, you will be asked to use a graphing calculator to test your theory with the following matrices. You will NOT be marked wrong if you prove your theory to be false. A =[−9081] B=[2−265] C=[2843] 1. Do you think the Associative Property for Matrix Addition will be true? Why or why not? 2. Test your theory from question 1 with a graphing calculator. Were you right? 3. Do you think the Associative Property for Matrix Multiplication will be true? Why or why not? 4. Test your theory from question 3 with a graphing calculator. Were you right? 5. Do you think the Commutative Property for Matrix Addition will be true? Why or why not? 6. Test your theory from question 5 with a graphing calculator. Were you right? 7. Do you think the Commutative Property for Matrix Multiplication will be true? Why or why not? CREATING AN HONORS CURRICULUM 55 8. Test your theory from question 7 with a graphing calculator. Were you right? 9. Do you think the Distributive will be true for matrices? Why or why not? 10. Test your theory from question 9 with a graphing calculator. Were you right? 11. What did you learn? 12. Why do you think matrices sometimes follow different rules than real numbers? CREATING AN HONORS CURRICULUM 56 Properties of Matrices Key Associative Property Commutative Property Distributive Property Matrix Addition (A + B) +C = A + (B + C) A + B = B + A A ×(B + C) = A×B + A×C Matrix Multiplication (A × B)× C = A ×(B × C) A × B = B × A For each property you will first be asked to guess if the property will be true for matrices. Then, you will be asked to use a graphing calculator to test your theory with the following matrices. You will NOT be marked wrong if you prove your theory to be false. A =[−9081] B=[2−265] C=[2843] 1. Do you think the Associative Property for Matrix Addition will be true? Why or why not? True 2. Test your theory from question 1 with a graphing calculator. Were you right? (A+B)+C=[−56189] A+(B+C)= [−56189] (A + B) +C = A + (B + C) 3. Do you think the Associative Property for Matrix Multiplication will be true? Why or why not? True 4. Test your theory from question 3 with a graphing calculator. Were you right? (AB)C=[36−900143] A(BC)= [36−900143] (A × B)× C = A ×(B × C) 5. Do you think the Commutative Property for Matrix Addition will be true? Why or why not? True 6. Test your theory from question 5 with a graphing calculator. Were you right? A+B=[−7−2146] B+A=[−7−2146] A+B=B+A 7. Do you think the Commutative Property for Matrix Multiplication will be true? Why or why not? False CREATING AN HONORS CURRICULUM 57 8. Test your theory from question 7 with a graphing calculator. Were you right? AB=[−181822−11] BA=[−34−2−145] AB≠BA 9. Do you think the Distributive will be true for matrices? Why or why not? True 10. Test your theory from question 9 with a graphing calculator. Were you right? A(B+C)= [−36−544256] AB+AC=[−36−544256] A ×(B + C) = A×B + A×C 11. What did you learn? All of the properties that are true for real numbers are also true for matrices except the Commutative property for matrix multiplication. 12. Why do you think matrices sometimes follow different rules than real numbers? Matrix multiplication is a combination of multiplication and addition between rows and columns that is not commutative. CREATING AN HONORS CURRICULUM 58 Unit 1 Quiz Simplify the matrix expressions as much as possible. 1. −1[−56−7−9]+4 [53−53] 2. 4[1−2−91]−9[3−583−9−8] Multiply the following Matrices if possible. 3. [2−46−88−2] × [2−69079] 4. [−14−6684] × [5−54−1] Multiple Choice 5. Which property is not always true for matrices? a. Associative Property of Addition b. Commutative Property of Addition c. Associative Property of Multiplication d. Communitive Property of Multiplication CREATING AN HONORS CURRICULUM 59 Create a matrix that will represent the following information. (Feel free to include labels.) 6. Stefon needs to buy 5 boxes of crackers, 2 bags of chips, and 3 packages of cookies. Delilah needs to buy 3 boxes of crackers, 1 bag of chips, and 4 packages of cookies. CREATING AN HONORS CURRICULUM 60 Unit 1 Quiz Key Simplify the matrix expressions as much as possible. 1. −1[−56−7−9]+4 [53−53] = [256−1321] 2. 4[1−2−91]−9[3−583−9−8] = [4−8−364]− [27−457227−81−72] OR [4−8−364]+ [−2745−72−278172] Multiply the following Matrices if possible. 3. [2−46−88−2] × [2−69079] = [10424230] 4. [−14−6684] × [5−54−1] Undefined Multiple Choice 5. Which property is not always true for matrices? a. Associative Property of Addition b. Commutative Property of Addition c. Associative Property of Multiplication d. Communitive Property of Multiplication CREATING AN HONORS CURRICULUM 61 Create a matrix that will represent the following information. (Feel free to include labels.) 6. Stefon needs to buy 5 boxes of crackers, 2 bags of chips, and 3 packages of cookies. Delilah needs to buy 3 boxes of crackers, 1 bag of chips, and 4 packages of cookies. [523314] Answers may vary CREATING AN HONORS CURRICULUM 62 Unit 2 Teacher Notes Vectors are introduced in this unit with limited use. Since students have not yet studied trigonometry, the vectors will be limited to the same direction or opposite directions. Furthermore, vectors will be used to denote direction and will be labeled with a magnitude, but will not be drawn to scale. Vectors will be used more extensively in unit 4. The equation Distance = (Total Rate)Time will be used in different contexts. The Rate of the system will be the sum of two rates that are acting in the same direction or the difference of two rates that are acting in opposite directions. When the rate of the wind and plane or rate of the boat and current are represented by vectors going the same direction the following are used: Total rate = Rate of Plane + Rate of wind or Total rate = Rate of Boat + Rate of Current When the rate of the wind and plane or rate of the boat and current are represented by vectors going opposite directions the following are used: Total rate = Rate of Plane – Rate of wind or Total rate = Rate of Boat – Rate of Current CREATING AN HONORS CURRICULUM 63 How Fast? A boat traveled 300 miles downstream in 10 hours. The trip back took 15 hours. What is the speed of the boat in still water? What is the speed of the current? 1. Draw a picture with labels of: a. Boat going downstream b. Boat going upstream 2. Which way is the water going in relation to the boat in each picture? 3. How are the rate of the water and the rate of the boat shown in your picture? How do you show direction? 4. Look up the definition of vector. Do you have vectors in your picture? Where? 5. Compare your picture to a neighbor’s. What is the same? What is different? 6. When are the rates of the water and boat working together? When are they working against each other? CREATING AN HONORS CURRICULUM 64 7. How will you find the total rate when they are working together? How will you find the total rate when they are working against each other? 8. Use Distance = (rate)time to write an equation for: a. Boat going downstream b. Boat going upstream 9. Use the system of equations you created to find the speed of the boat in still water and the speed of the current. 10. Which method did you use? (graphing, substitution, or elimination) 11. Which method did your neighbor use? 12. Did you both get the same answer? CREATING AN HONORS CURRICULUM 65 How Fast? Key A boat traveled 300 miles downstream in 10 hours. The trip back took 15 hours. What is the speed of the boat in still water? What is the speed of the current? 1. Draw a picture with labels of: a. Boat going downstream b. Boat going upstream 2. Which way is the water going in relation to the boat in each picture? Same direction downstream opposite directions upstream 3. How are the rate of the water and the rate of the boat shown in your picture? How do you show direction? Arrows show direction 4. Look up the definition of vector. Do you have vectors in your picture? Where? Vector has magnitude and direction. Arrows are the vectors 5. Compare your picture to a neighbor’s. What is the same? What is different? 6. When are the rates of the water and boat working together? When are they working against each other? Same direction=working together opposite directions = working against each other CREATING AN HONORS CURRICULUM 66 7. How will you find the total rate when they are working together? How will you find the total rate when they are working against each other? Add when they are working together. Subtract when they work against each other. 8. Use Distance = (rate)time to write an equation for: a. Boat going downstream b. Boat going upstream 300 = (B + C)10 300 = (B-C)15 9. Use the system of equations you created to find the speed of the boat in still water and the speed of the current. Boat in still water is 25 mph Current is 5 mph 10. Which method did you use? (graphing, substitution, or elimination) 11. Which method did your neighbor use? 12. Did you both get the same answer? CREATING AN HONORS CURRICULUM 67 Guided Notes Airplanes: Distance = (_____________________________________) Time Distance = (_____________________________________) Time CREATING AN HONORS CURRICULUM 68 Boat: Distance = (_____________________________________) Time Distance = (_____________________________________) Time Rate of Boat Rate of Current Rate of Boat Rate of Current CREATING AN HONORS CURRICULUM 69 Notes Key Airplanes: Distance = (Rate of Plane – Rate of Wind) Time Distance = (Rate of Plane + Rate of Wind) Time CREATING AN HONORS CURRICULUM 70 Boat: Distance = (Rate of Boat + Rate of Current) Time Distance = (Rate of Boat – Rate of Current) Time Rate of Boat Rate of Current Rate of Boat Rate of Current CREATING AN HONORS CURRICULUM 71 Labels For each problem draw, if needed, and label symbols that represent all rates, distance, and time. Find missing values. 1. An airplane took 2 hours to fly 1040 miles with a tailwind of 20 mph. What is the rate of the plane be in still air? 2. An airplane took 5 hours to fly 2575 miles into a wind. If the plane in still air is 525 mph, what is the rate of the wind? 3. It took Sam 2 hours to row 14 miles downstream. If she can row 6 mph in still water. What is the rate of the current? Distance=(rate of plane/boat + rate of wind/current)Time OR Distance=(rate of plane/boat – rate of wind/current)Time Distance = Time= CREATING AN HONORS CURRICULUM 72 4. Derek can row 5 mph in still water. It took him 3 hours to row 9 miles upstream. What is the rate of the current? 5. It took an airplane 7 hours to fly 3150 miles into the wind. If the plane wind was blowing at 25 mph, what is the rate of the plane in still air? 6. An airplane flew 2000 miles with a tailwind in 4 hours. If the plane can fly 485 mph in still air, what is the rate of the wind? 7. It took Ian 3 hours to row 18 miles upstream in a current of 2 mph. What is the rate he can row in still water? CREATING AN HONORS CURRICULUM 73 Labels Key For each problem draw, if needed, and label symbols that represent all rates, distance, and time. Find missing values. 1. An airplane took 2 hours to fly 1040 miles with a tailwind of 20 mph. What is the rate of the plane be in still air? 1040= (P+20)2 Rate of plane is 500 mph 2. An airplane took 5 hours to fly 2575 miles into a wind. If the plane in still air is 525 mph, what is the rate of the wind? 3. It took Sam 2 hours to row 14 miles downstream. If she can row 6 mph in still water. What is the rate of the current? Distance=(rate of plane/boat + rate of wind/current)Time OR Distance=(rate of plane/boat – rate of wind/current)Time Rate of plane Rate of wind = 20 mph Distance = 1040 miles Time = 2 hours Rate of plane = 525 mph Rate of wind Distance = 2575 miles Time = 5 hours 2575 = (525-W)5 Rate of wind = 10 mph Rate of boat = 6 mph Rate of current 14 = (6+C)2 Rate of current = 1 mph CREATING AN HONORS CURRICULUM 74 4. Derek can row 5 mph in still water. It took him 3 hours to row 9 miles upstream. What is the rate of the current? 5. It took an airplane 7 hours to fly 3150 miles into the wind. If the plane wind was blowing at 25 mph, what is the rate of the plane in still air? 6. An airplane flew 2000 miles with a tailwind in 4 hours. If the plane can fly 485 mph in still air, what is the rate of the wind? 7. It took Ian 3 hours to row 18 miles upstream in a current of 2 mph. What is the rate he can row in still water? Distance = 14 miles Time =2 hours Rate of boat = 5 mph Rate of current Distance = 9 miles Time = 3 hours 9 = (5 - C)3 Rate of current = 2 mph Rate of wind = 25 mph Rate of plane Distance = 3150 miles Time = 7 hours 3150 = (P - 25)7 Rate of plane = 475 mph Rate of plane = 485 mph Rate of wind Distance = 2000 mph Time = 4 hours 2000 = (485 + w)4 Rate of wind = 15 mph Rate of boat Rate of current = 2 mph Distance = 18 mph Time = 3 hours 18 = (B - 2)3 Rate of boat = 8 mph CREATING AN HONORS CURRICULUM 75 Word Problem Practice Remember: Use the formula Distance = (Total Rate) Time to solve the following word problems. 1. A boat traveled 450 miles downstream in 25 hours. The trip back took 30 hours. What is the speed of the boat in still water? What is the speed of the current? 2. Flying to Las Vegas with a tailwind a plane took 1.75 hours to fly the 658 miles. On the return trip the plane took 2 hours while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air. When the rate of the wind and plane or rate of the boat and current are represented by vectors going opposite directions the Total rate = Rate of Plane – Rate of wind or Total rate = Rate of Boat – Rate of Current When the rate of the wind and plane or rate of the boat and current are represented by vectors going the same direction the Total rate = Rate of Plane + Rate of wind or Total rate = Rate of Boat + Rate of Current CREATING AN HONORS CURRICULUM 76 3. Flying the 2205 miles to New York City into the wind took 5.25 hours. On the return trip the plane took 4.5 hours while flying with same wind acting as a tailwind. Find the speed of the wind and the speed of the plane in still air. 4. A boat traveled 56 miles upstream in 8 hours. The trip back took 8 hours. What is the speed of the boat in still water? What is the speed of the current? 5. What could you use vectors to represent other than a boat in a current or a plane in a wind? 6. Write a word problem. 7. Act as a coach as your neighbor solves your story problem. CREATING AN HONORS CURRICULUM 77 Word Problem Practice Key Remember: Use the formula Distance = (Total Rate) Time to solve the following word problems. 1. A boat traveled 450 miles downstream in 25 hours. The trip back took 30 hours. What is the speed of the boat in still water? What is the speed of the current? Speed of boat in still water is 16.5 mph. Speed of current is 1.5 mph. 2. Flying to Las Vegas with a tailwind a plane took 1.75 hours to fly the 658 miles. On the return trip the plane took 2 hours while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air. Speed of plane in still air is 352.5 mph. Speed of wind is 23.5 mph. When the rate of the wind and plane or rate of the boat and current are represented by vectors going opposite directions the Total rate = Rate of Plane – Rate of wind or Total rate = Rate of Boat – Rate of Current When the rate of the wind and plane or rate of the boat and current are represented by vectors going the same direction the Total rate = Rate of Plane + Rate of wind or Total rate = Rate of Boat + Rate of Current CREATING AN HONORS CURRICULUM 78 3. Flying the 2205 miles to New York City into the wind took 5.25 hours. On the return trip the plane took 4.5 hours while flying with same wind acting as a tailwind. Find the speed of the wind and the speed of the plane in still air. Speed of plane in still air is 455 mph. Speed of wind is 35 mph. 4. A boat traveled 56 miles upstream in 8 hours. The trip back took 8 hours. What is the speed of the boat in still water? What is the speed of the current? Speed of boat in sill water is 7 mph. Speed of current is 0 mph. 5. What could you use vectors to represent other than a boat in a current or a plane in a wind? Any force or rate 6. Write a word problem. 7. Act as a coach as your neighbor solves your story problem. CREATING AN HONORS CURRICULUM 79 Unit 2 Assessment Create a presentation that explains the use of vectors to the class. Your presentation can be a slide show, video, art project, or anything that gets across the required information. Turn in this rubric with the presentation. The presentation must answer and explain the following questions: 1. What is important about vectors? 2. What kind of problems are vectors useful in solving? 3. How are vectors used? Project Rubric 3 points 2 points 1 points 0 point Content Answered the questions correctly and completely. Answered the questions, but showed gaps in understanding for one question. Answered the questions, but showed gaps in understanding for two questions. Did not answer the questions. Creativity Project was presented very creatively. It made the audience want to learn more. Project was presented with some creativity. It kept the audience listening. Project was presented with little creativity. The audience listened for parts. Project did not show the student’s creativity. Design Design on the project helped portray the meaning. Design was sufficient. It neither hurt nor helped understanding. Design made the information difficult to read/understand. Design made the information misleading. Total /9 CREATING AN HONORS CURRICULUM 80 Unit 3 Teacher Notes The determinant of a matrix is non-zero if and only if the matrix has an inverse. However, the inverse of a matrix is not a concept taught this year. Therefore, this unit interprets a non-zero determinant as coming from a system of equations that has exactly one solution, which is a consequence of having an inverse. Solving systems of equations using row reduction with augmented matrices can be hard for students because it involves multiple steps and abstract thinking. However, students employ similar strategies when playing number puzzles like Sudoku. Therefore, it is introduced as a puzzle to solve. The students will be tempted to create their own rules to achieving the goal. The only operations that are permissible are: 1. multiply or divide either row by anything other than zero 2. add the rows of numbers together 3. switch the rows 4. use multiple steps to achieve the goals These rules will maintain balance in the equations. The rules are the same as the way equations can be manipulated while solving systems using elimination. The difference between this and elimination is creating 1’s and 0’s as coefficients instead of only looking to create 0’s. CREATING AN HONORS CURRICULUM 81 Unit 3 Vocabulary Additive Identity Definition: Example for Real Numbers Example for Matrices Multiplicative Identity Definition: Example for Real Numbers Example for Matrices CREATING AN HONORS CURRICULUM 82 Determinant Definition: Example Non-Example Augmented Matrix Definition: Example Non-Example CREATING AN HONORS CURRICULUM 83 Unit 3 Vocabulary Key Additive Identity Definition: An element when used with addition makes no change. Example for Real Numbers 0 𝑥+0=𝑥 Example for Matrices [0000] [𝑎𝑏𝑐𝑑]+[0000]=[𝑎𝑏𝑐𝑑] Multiplicative Identity Definition: An element when used with multiplication makes no change. Example for Real Numbers 1 𝑥×1=𝑥 Example for Matrices [1001] [𝑎𝑏𝑐𝑑]×[1001]=[𝑎𝑏𝑐𝑑] CREATING AN HONORS CURRICULUM 84 Determinant Definition: A number that can be found by following the pattern|𝐴|=|𝑎𝑏𝑐𝑑|=𝑎𝑑−𝑏𝑐. Example |1234|=1(4)−2(3)=4−6=−2 Non-Example |1234|=2(3)−1(4)=6−4=2 Augmented Matrix Definition: A matrix that has been created by combining two separate matrices. Example [𝑎𝑏𝑐𝑑],[𝑒𝑓]→[𝑎𝑏𝑒𝑐𝑑𝑓] Non-Example [1234]+[5678]=[681012] CREATING AN HONORS CURRICULUM 85 Is there a pattern? Step 1: Find the determinant. Determinant of [𝑎𝑏𝑐𝑑] = ad – bc For example determinant of [645−4] = 6(-4) – 5(4) = -24 – 20 = -44 Step 2: Solve the system of equations. Step 3: Record Determinant and solution side. 1. [−4−248] 2. [1−2−48] 3. [51102] 4. [−491−3] 5. What patterns do you see? 1. Determinant_______ Solution__________ 2. Determinant_______ Solution__________ 3. Determinant_______ Solution__________ 4. Determinant_______ Solution__________ CREATING AN HONORS CURRICULUM 86 6. Compare with a neighbor. How are their patterns different? How are they the same? Make predictions. 7. 8. 9. Test your predictions. 10. Where they right? 11. What rule can you make about determinants? CREATING AN HONORS CURRICULUM 87 Is there a pattern? Key Step 1: Find the determinant. Determinant of [𝑎𝑏𝑐𝑑] = ad – bc For example determinant of [645−4] = 6(-4) – 5(4) = -24 – 20 = -44 Step 2: Solve the system of equations. Step 3: Record Determinant and solution side. 1. [−4−248] 2. [1−2−48] 3. [51102] 4. [−491−3] 5. What patterns do you see? The numbers in the matrix are the coefficients in the equations. When the determinant is not zero, there is one solution. When the determinant is zero, there is IMS or no solution. (Answers may vary) 1. Determinant -24 Solution (6, -6) 2. Determinant 0 Solution IMS 3. Determinant 0 Solution No Solution 4. Determinant 3 Solution (9,5) CREATING AN HONORS CURRICULUM 88 6. Compare with a neighbor. How are their patterns different? How are they the same? Make predictions. 7. a. 48 b. yes 8. a. 0 b. no 9. Test your predictions. 7. (-4, -4) one solution 8. No Solution 10. Where they right? 11. What rule can you make about determinants? Determinant is zero if and only if there is not exactly one solution. (Answers will vary) CREATING AN HONORS CURRICULUM 89 Puzzle it out You will be given a Matrix. The matrix puzzle is solved when: 1. The first row starts with a 1. 2. The second row starts with a 0 1 Rules: 5. You may multiply or divide either row by anything other than zero. 6. You may add the rows of numbers together. 7. You may switch the rows. 8. You may use multiple steps to achieve the goals. Example: Given Matrix [26103−42] 1. I can divide the first row by 2. My matrix is now [1353−42] 2. I can multiply the first row by -3 and add it to the second row (I can do these two steps mentally to avoid changing my first row). My matrix is now [1350−13−13] 3. I can divide the second row by -13. My matrix is now [133011] and I have solved the puzzle. You can probably find a different ways to solve this puzzle. Any way you choose to solve it is great as long as you follow the rules. It’s your turn to solve some puzzles. 1. [1−1112119] 2. [51910−7−18] CREATING AN HONORS CURRICULUM 90 3. [3−2−2122] 4. [−23943−9] 5. [2162−12] 6. [2171517] 7. [327−345] 8. Compare your answers to your neighbor’s. The bottom rows should be the same, but the top rows may be different. Take turns explaining the steps that were taken. Make sure you both followed the rules. What do you notice? CREATING AN HONORS CURRICULUM 91 Puzzle it out Key You will be given a Matrix. The matrix puzzle is solved when: 3. The first row starts with a 1. 4. The second row starts with a 0 1 Rules: 9. You may multiply or divide either row by anything other than zero. 10. You may add the rows of numbers together. 11. You may switch the rows. 12. You may use multiple steps to achieve the goals. Example: Given augmented Matrix [26103−42] 4. I can divide the first row by 2. My matrix is now [1353−42] 5. I can multiply the first row by -3 and add it to the second row (I can do these two steps mentally to avoid changing my first row). My matrix is now [1350−13−13] 6. I can divide the second row by -13. My matrix is now [133011] and I have solved the puzzle. You can probably find a different ways to solve this puzzle. Any way you choose to solve it is great as long as you follow the rules. It’s your turn to solve some puzzles. (Bottom row will be the same. Top row may vary.) 9. [1−1112119] = [1−11101−1] 10. [51910−7−18]= [11/59/5014] CREATING AN HONORS CURRICULUM 92 11. [3−2−2122] = [1−6−6011] 12. [−23943−9] = [1−3/2−9/2011] 13. [2162−12] = [11/23012] 14. [2171517] = [1−4−10013] 15. [327−345] = [12/37/3012] 16. Compare your answers to your neighbor’s. The bottom rows should be the same, but the top rows may be different. Take turns explaining the steps that were taken. Make sure you both followed the rules. What do you notice? CREATING AN HONORS CURRICULUM 93 Puzzles solve systems The matrices that were just puzzles yesterday represent systems of equations. The example matrix [26103−42] is an augmented matrix that represents the system When it is simplified to [133011] it represents 1. Why is easier to solve than ? What solution method does this remind you of? 2. How would you find the value of x? Write the equations represented by the following augmented matrices. 3. [51910−7−18] 4. [1−1112119] Write the augmented matrices for the following systems. 5. 3x - 2y = 2 x + 2y = 2 6. -2x + 3y = 9 4x + 3y = -9 CREATING AN HONORS CURRICULUM 94 7. Compare to your neighbor. What do you notice? Solve the following system of equations using augmented matrices. Using the following rules to simplify: 13. You may multiply or divide either row by anything other than zero. 14. You may add the rows of numbers together. 15. You may switch rows. 16. You may use multiple steps to achieve the goals. (first row start with 1, second row start with 0 then 1) 8. x + 2y = 13 -x + y = 5 9. x + y = 2 2x + 7y = 9 10. 12x – 6y = -6 4x + 2y = 10 11. x + 2y = 6 2x – y = 0 CREATING AN HONORS CURRICULUM 95 12. 2x + y = 6 2x – y = 2 13. 5x + 2y = -4 x + 2y = 4 14. What would you have to do differently to use augmented matrices to solve the following? 3x + 6y = −4 x + y − z = −2 9x − 12y + 15z = 28 15. Compare your answer to number 14 to your neighbor’s. What do you notice? Use augmented matrices to solve. 16. −2x − 5y + 4z = 21 −5x − 5y + z = 21 −4y − 4z = 8 17. 6x + 4y − z = −31 x + y + 6z = 14 –y − 6z = −20 CREATING AN HONORS CURRICULUM 96 Puzzles solve systems Key The matrices that were just puzzles yesterday represent systems of equations. The example matrix [26103−42] is an augmented matrix that represents the system When it is simplified to [133011] it represents 18. Why is easier to solve than ? What solution method does this remind you of? Y is solved for with the first system. It is like elimination. 19. How would you find the value of x? Use substitution. Write the equations represented by the following augmented matrices. 20. [51910−7−18] = 5x + y = 9 10x – 7y = -18 21. [1−1112119] x – y = 11 2x + y = 19 Write the augmented matrices for the following systems. 22. 3x - 2y = 2 [3−22122] x + 2y = 2 23. -2x + 3y = 9 [−23943−9] 4x + 3y = -9 CREATING AN HONORS CURRICULUM 97 24. Compare to your neighbor. What do you notice? Solve the following system of equations using augmented matrices. Using the following rules to simplify: 17. You may multiply or divide either row by anything other than zero. 18. You may add the rows of numbers together. 19. You may switch rows. 20. You may use multiple steps to achieve the goals. (first row start with 1, second row start with 0 then 1) 25. x + 2y = 13 -x + y = 5 (1, 6) 26. x + y = 2 2x + 7y = 9 (1, 1) 27. 12x – 6y = -6 4x + 2y = 10 (1, 3) 28. x + 2y = 6 2x – y = 0 (1.2, 2.4) CREATING AN HONORS CURRICULUM 98 29. 2x + y = 6 2x – y = 2 (2, 2) 30. 5x + 2y = -4 x + 2y = 4 (-2, 3) 31. What would you have to do differently to use augmented matrices to solve the following? Larger augmented matrix, rows would start with 1, 0 1, 0 0 1, and use substitution twice. 3x + 6y = −4 x + y − z = −2 9x − 12y + 15z = 28 32. Compare your answer to number 14 to your neighbor’s. What do you notice? Use augmented matrices to solve. 33. −2x − 5y + 4z = 21 −5x − 5y + z = 21 −4y − 4z = 8 (-1, -3, 1) 34. 6x + 4y − z = −31 x + y + 6z = 14 –y − 6z = −20 (-6, 2, 3) CREATING AN HONORS CURRICULUM 99 Unit 3 Quiz 1. What is the significance of [0000]? 2. What is the significance of [1001]? Find the determinant of the matrix. 3. [−481−2] 4. [09−1−2] 5. What is the significance of the determinant? Solve the system using augmented matrices and row reduction. 6. x – 3y = -25 x + 5y = 47 7. 2x + y = -6 4x – y = -36 8. –x – y + 2z = −11 3x – 2y + z = −18 −4x + 5y = 34 CREATING AN HONORS CURRICULUM 100 Unit 3 Quiz Key 1. What is the significance of [0000]? It is the additive identity of matrices. 2. What is the significance of [1001]? It is the multiplicative identity of matrices. Find the determinant of the matrix. 3. [−481−2] = 0 4. [09−1−2] = 9 5. What is the significance of the determinant? If the determinant is zero, the system of equations that it came from does not have exactly one solution. Solve the system using augmented matrices and row reduction. 6. x – 3y = -25 x + 5y = 47 (2, 9) 7. 2x + y = -6 4x – y = -36 (-7, 8) 8. –x – y + 2z = −11 3x – 2y + z = −18 −4x + 5y = 34 (-1, 6, -3) CREATING AN HONORS CURRICULUM 101 Unit 4 Teacher Notes Component form of a vector is written as using angle brackets with the x-component first and the y-component second. Example: Vector v is written as ⟨3,−2⟩ in component form. Magnitude is the length of a vector found by using the Pythagorean Theorem. Vector addition can be done graphically by graphing the vectors starting subsequent vectors to the terminal point of the previous vector. The solution is a vector that runs from the first initial point the final terminal point. Example: Vector addition can also be done by finding the sum the x-components and y-components. Example: Multiplying by a scalar is the same for vectors as it is for matrices. Multiply each component by the scalar. Note that, multiplying by a negative scalar reverses the direction of the vector. Use this when being asked to subtract vectors. Rotations and Reflections can be done with matrix multiplication. When using this technique, the transformation matrix must be multiplicand and the point(s) must be the multiplier. Note that, as many points can be transformed simultaneously with the coordinate values of each point represented in a column in the multiplier. Example: for a reflection over the x-axis. For a list of all transformation matrices that are used in Secondary 1 see the “Put it Together” key. CREATING AN HONORS CURRICULUM 102 Investigate For each of these questions, use your chrome book to look up information, record your findings, share with a neighbor, and compare your answers. Last year, you learned about transformations with pictures and diagrams. This year, you have spent time and will spend more time learning about transformations with algebraic equations. This unit will give you the opportunity to learn about transformations using matrices and vectors. 1. Why is it important to learn so many strategies to solve the same problems? a. Your ideas: b. Neighbor’s ideas: c. Compare: 2. Where could you use transformations outside of math class? a. Your ideas: b. Neighbor’s ideas: c. Compare: 3. When would vectors and matrices be beneficial over the other strategies to solve transformations? a. Your ideas: b. Neighbor’s ideas: c. Compare: CREATING AN HONORS CURRICULUM 103 4. What can you do during this unit to help yourself learn? a. Your ideas: b. Neighbor’s ideas: c. Compare: 5. Who can you turn to when you have trouble? a. Your ideas: b. Neighbor’s ideas: c. Compare: CREATING AN HONORS CURRICULUM 104 Investigate Key For each of these questions, use your chrome book to look up information, record your findings, share with a neighbor, and compare your answers. Last year, you learned about transformations with pictures and diagrams. This year, you have spent time and will spend more time learning about transformations with algebraic equations. This unit will give you the opportunity to learn about transformations using matrices and vectors. 1. Why is it important to learn so many strategies to solve the same problems? a. Your ideas: Develop thinking skills and some strategies work better in some situations b. Neighbor’s ideas: c. Compare: 2. Where could you use transformations outside of math class? a. Your ideas: graphic art, construction, decorating, city planning, etc. b. Neighbor’s ideas: c. Compare: 3. When would vectors and matrices be beneficial over the other strategies to solve transformations? a. Your ideas: Computers and calculators can manipulate matrices so problems can be solved that involve many different parameters and variables efficiently. b. Neighbor’s ideas: c. Compare: CREATING AN HONORS CURRICULUM 105 4. What can you do during this unit to help yourself learn? a. Your ideas: b. Neighbor’s ideas: c. Compare: 5. Who can you turn to when you have trouble? a. Your ideas: b. Neighbor’s ideas: c. Compare: CREATING AN HONORS CURRICULUM 106 Magnitude and Direction Earlier this year, you used Vectors to show direction when drawing pictures and writing equations for word problems. Now, you are going to learn how to use vectors more precisely in preparation to use them to represent translations. A vector is defined by its length (magnitude) and direction. A vector’s placement on a graph is not important. 1. Circle the vectors that appears multiple times. 2. How can you tell? 3. Compare to a neighbor. Component form of a vector uses angle brackets to show the change in x and change in y from the initial point of a vector to the terminal point. Examples: Vector v is written as ⟨3, −2⟩ and Vector w is written as⟨−4, 1⟩. 4. Write the vectors in component form. a. b. c. d. 5. How did you find the component form? a b a c d CREATING AN HONORS CURRICULUM 107 6. Compare to a neighbor. Did you both use the same process? Is there more than one way to get the answer? 7. If you were given the initial point and terminal point, how would you find the component form of the vector? 8. Write the component form for the vector with an initial point at (3, -5) and terminal point at (7, 2). 9. Compare to a neighbor. Did you both use the same process? Is there more than one way to get the answer? CREATING AN HONORS CURRICULUM 108 Key Magnitude and Direction Earlier this year, you used Vectors to show direction when drawing pictures and writing equations for word problems. Now, you are going to learn how to use vectors more precisely in preparation to use them to represent translations. A vector is defined by its length (magnitude) and direction. A vector’s placement on a graph is not important. 1. Circle the vectors that appears multiple times. 2. How can you tell? They are the same length and direction. 3. Compare to a neighbor. Component form of a vector uses angle brackets to show the change in x and change in y from the initial point of a vector to the terminal point. Examples: Vector v is written as ⟨3, −2⟩ and Vector w is written as⟨−4, 1⟩. 4. Write the vectors in component form. a. ⟨3, 4⟩ b. ⟨5, −1⟩ c. ⟨3, 2⟩ d. ⟨2, −1⟩ 5. How did you find the component form? a b a c d CREATING AN HONORS CURRICULUM 109 6. Compare to a neighbor. Did you both use the same process? Is there more than one way to get the answer? 7. If you were given the initial point and terminal point, how would you find the component form of the vector? ⟨𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥,𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦⟩ 8. Write the component form for the vector with an initial point at (3, -5) and terminal point at (7, 2). ⟨4,7⟩ 9. Compare to a neighbor. Did you both use the same process? Is there more than one way to get the answer? CREATING AN HONORS CURRICULUM 110 Translations You have seen translations represented by a picture, a description, and an equation. All of these methods feature a movement of all points of a figure moving the same distance and same direction. This can be represented with movement along a vector because a vector is defined by its magnitude (distance) and its direction. Example: The transformation along ⟨2,5⟩ is shown by this picture can be described as 2 to the right and 5 up, and can be described with the equation(𝑥,𝑦)→(𝑥+2,𝑦+5). What vector represents the following translations? 1. 2. 3. 3 left and 10 up 4. 2 right and 7 down 5. (𝑥,𝑦)→(𝑥+6,𝑦−4) 6. (𝑥,𝑦)→(𝑥−5,𝑦−1) CREATING AN HONORS CURRICULUM 111 Finding the magnitude of the vector that represents the translation will tell you how far the object moved. To find the magnitude, use the Pythagorean Theorem. Example V = ⟨2,5⟩ Find the magnitude of the following vectors. 7. W = ⟨−5,−4⟩ 8. P= ⟨3,5⟩ 9. R = ⟨2,1⟩ 10. Q = ⟨3,−2⟩ 11. S = ⟨6,3⟩ 2 5 Vector V can be represented by this right triangle with a=2 and b=5. 22+52=𝑐2 4+25=𝑐2 29=𝑐2 √29=√𝑐2 √29=c Pythagorean Theorem 𝑎2+𝑏2=𝑐2 |𝑉|= √29 |𝑉|≈5.39 Magnitude of V CREATING AN HONORS CURRICULUM 112 Translations Key You have seen translations represented by a picture, a description, and an equation. All of these methods feature a movement of all points of a figure moving the same distance and same direction. This can be represented with movement along a vector because a vector is defined by its magnitude (distance) and its direction. Example: The transformation along ⟨2,5⟩ is shown by this picture can be described as 2 to the right and 5 up, and can be described with the equation(𝑥,𝑦)→(𝑥+2,𝑦+5). What vector represents the following translations? 2. 2. ⟨3,−4⟩ ⟨−5,1⟩ 4. 3 left and 10 up 4. 2 right and 7 down ⟨−3,10⟩ ⟨2,−7⟩ 5. (𝑥,𝑦)→(𝑥+6,𝑦−4) 6. (𝑥,𝑦)→(𝑥−5,𝑦−1) ⟨6,−4⟩ ⟨−5,−1⟩ CREATING AN HONORS CURRICULUM 113 Finding the magnitude of the vector that represents the translation will tell you how far the object moved. To find the magnitude, use the Pythagorean Theorem. Example V = ⟨2,5⟩ Find the magnitude of the following vectors. 7. W = ⟨−5,−4⟩ |𝑊| = √41 ≈ 6.40 𝑢𝑛𝑖𝑡𝑠 8. P= ⟨3,5⟩ |𝑃| = √34 ≈5.83 𝑢𝑛𝑖𝑡𝑠 9. R = ⟨2,1⟩ |𝑅| = √5 ≈ 2.24 𝑢𝑛𝑖𝑡𝑠 10. Q = ⟨3,−2⟩ |𝑄| = √13 ≈ 3.61 𝑢𝑛𝑖𝑡𝑠 11. S = ⟨6,3⟩ |𝑆| = √45 ≈ 6.71 𝑢𝑛𝑖𝑡𝑠 2 5 Vector V can be represented by this right triangle with a=2 and b=5. 22+52=𝑐2 4+25=𝑐2 29=𝑐2 √29=√𝑐2 √29=c Pythagorean Theorem 𝑎2+𝑏2=𝑐2 |𝑉|= √29 Magnitude of V |𝑉|≈5.39 units CREATING AN HONORS CURRICULUM 114 Move it! Move it! Before, you used vectors to represent a translation. Today, you are going to add vectors together by drawing multiple transformations. Example: Starting with the point A (-2, 4) and translating along vector ⟨1,−2⟩ and vector ⟨3,1⟩ looks like this Draw a picture of the following pairs of translations. Draw a dotted line from the initial point to the final point that represents vector addition. 1. Point A (1, -1) 2. Point B (-2, 4) Translate along ⟨3,−2⟩ Translate along ⟨−3,−4⟩ Then along ⟨−6,−2 ⟩ Then along ⟨7,1⟩ 3. Compare your answer to a neighbor. Work with your neighbor to find a short cut. Explain your short cut. 4. How would you use your short cut to move Point A in question 1 to its final destination? 5. How would you use your short cut to move Point B in question 2 to its final destination? A A’ A’’ The dotted line from A to A’’ represents the vector created when adding together ⟨1,−2⟩ and ⟨3,1⟩. CREATING AN HONORS CURRICULUM 115 Add the following vectors by drawing diagrams starting at (0, 0). Write solution in component form. Check using component addition. 6. ⟨−4,2⟩+⟨2,−5 ⟩ 7. ⟨−1,2⟩+⟨−2,−1 ⟩ 8. ⟨5,1⟩+⟨−2,4 ⟩ 9. 2⟨2,1⟩ 10. 3⟨1,3⟩ Use a diagram to simplify the following expressions. Write the solution in component form. CREATING AN HONORS CURRICULUM 116 11. Find the magnitude of ⟨1,−2⟩ plus the magnitude of ⟨5,3 ⟩. 12. Find the magnitude of ⟨6,1⟩ (which equals ⟨1,−2⟩+⟨5,3 ⟩). 13. Did you get a larger value for question 11 or 12? Why do you think that happened? CREATING AN HONORS CURRICULUM 117 Move it! Move it! Key Before, you used vectors to represent a translation. Today, you are going to add vectors together by drawing multiple transformations. Example: Starting with the point A (-2, 4) and translating along vector ⟨1,−2⟩ and vector ⟨3,1⟩ looks like this Draw a picture of the following pairs of translations. Draw a dotted line from the initial point to the final point that represents vector addition. 1. Point A (1, -1) 2. Point B (-2, 4) Translate along ⟨3,−2⟩ Translate along ⟨−3,−4⟩ Then along ⟨−6,−2 ⟩ Then along ⟨7,1⟩ 3. Compare your answer to a neighbor. Work with your neighbor to find a short cut. Explain your short cut. 4. How would you use your short cut to move Point A in question 1 to its final destination? 5. How would you use your short cut to move Point B in question 2 to its final destination? A A’ A’’ The dotted line from A to A’’ represents the vector created when adding together ⟨1,−2⟩ and ⟨3,1⟩. A A’ A’’ B’ B B’’ CREATING AN HONORS CURRICULUM 118 Add the following vectors by drawing diagrams starting at (0, 0). Write solution in component form. Check using component addition. 6. ⟨−4,2⟩+⟨2,−5 ⟩ 7. ⟨−1,2⟩+⟨−2,−1 ⟩ 8. ⟨5,1⟩+⟨−2,4 ⟩ ⟨−2,−3⟩ ⟨−3,1⟩ ⟨3,5⟩ 9. 2⟨2,1⟩ 10. 3⟨1,3⟩ ⟨4,2⟩ ⟨3,9⟩ Use a diagram to simplify the following expressions. Write the solution in component form. CREATING AN HONORS CURRICULUM 119 11. Find the magnitude of ⟨1,−2⟩ plus the magnitude of ⟨5,3 ⟩. √5+ √34 ≈ 8.07 units 12. Find the magnitude of ⟨6,1⟩ (which equals ⟨1,−2⟩+⟨5,3 ⟩). √37 ≈ 6.08 units 13. Did you get a larger value for question 11 or 12? Why do you think that happened? CREATING AN HONORS CURRICULUM 120 Add, Multiply, and Subtract Add the following vectors by adding the x and y components. 1. ⟨−4,8⟩+⟨−8,6 ⟩ 2. ⟨1,2⟩+⟨1,−6 ⟩ 3. ⟨−1,5⟩+⟨−9,−3 ⟩ 4. ⟨−1,2⟩+⟨−1,2 ⟩ 5. ⟨−1,2⟩+⟨−1,2 ⟩+⟨−1,2 ⟩ 6. ⟨−1,2⟩+⟨−1,2 ⟩+⟨−1,2 ⟩+⟨−1,2 ⟩ 7. Can you find a short cut to questions 4, 5, and 6? Describe your short cut. Use your pattern to simplify the following. 8. 4⟨−7,9⟩ 9. 8⟨5,−3⟩ 10. −2⟨4,−10⟩ CREATING AN HONORS CURRICULUM 121 11. What is 4 times the magnitude of ⟨−7,9⟩? 12. What is the magnitude of 4⟨−7,9⟩? 13. What do you notice about the answers for questions 11 and 12? Why do you think this happened? 14. How would you use the ideas of adding vectors and multiplying vectors by a scalar to subtract vectors? 15. Compare your answer to 13 with a neighbor. How were your answers different? How were they the same? Subtract the following vectors 16. ⟨−1,4⟩−⟨2,−4 ⟩ 17. ⟨8,6⟩−⟨9,5⟩ 18. ⟨1,8⟩−⟨−3,3⟩ 19. ⟨3,−6⟩−⟨10,4 ⟩ 20. ⟨7,9⟩−⟨3,−9⟩ 21. ⟨−3,8⟩−⟨10,5⟩ CREATING AN HONORS CURRICULUM 122 Add, Multiply, and Subtract Key Add the following vectors by adding the x and y components. 1. ⟨−4,8⟩+⟨−8,6 ⟩ 2. ⟨1,2⟩+⟨1,−6 ⟩ 3. ⟨−1,5⟩+⟨−9,−3 ⟩ ⟨−12,14⟩ ⟨2,−4⟩ ⟨−10,2⟩ 4. ⟨−1,2⟩+⟨−1,2 ⟩ 5. ⟨−1,2⟩+⟨−1,2 ⟩+⟨−1,2 ⟩ ⟨−2,4⟩ ⟨−3,6⟩ 6. ⟨−1,2⟩+⟨−1,2 ⟩+⟨−1,2 ⟩+⟨−1,2 ⟩ ⟨−4,8⟩ 7. Can you find a short cut to questions 4, 5, and 6? Describe your short cut. Use your pattern to simplify the following. 8. 4⟨−7,9⟩ 9. 8⟨5,−3⟩ 10. −2⟨4,−10⟩ ⟨−28,36⟩ ⟨40,−24⟩ ⟨−8,20⟩ CREATING AN HONORS CURRICULUM 123 11. What is 4 times the magnitude of ⟨−7,9⟩? 4√130 ≈45.61 12. What is the magnitude of 4⟨−7,9⟩? |⟨−28,36⟩|= √520 ≈45.61 13. What do you notice about the answers for questions 11 and 12? Why do you think this happened? 14. How would you use the ideas of adding vectors and multiplying vectors by a scalar to subtract vectors? 15. Compare your answer to 13 with a neighbor. How were your answers different? How were they the same? Subtract the following vectors 16. ⟨−1,4⟩−⟨2,−4 ⟩ 17. ⟨8,6⟩−⟨9,5⟩ 18. ⟨1,8⟩−⟨−3,3⟩ ⟨−3,8⟩ ⟨−1,11⟩ ⟨4,5⟩ 19. ⟨3,−6⟩−⟨10,4 ⟩ 20. ⟨7,9⟩−⟨3,−9⟩ 21. ⟨−3,8⟩−⟨10,5⟩ ⟨−7,−10⟩ ⟨4,18⟩ ⟨−13,3⟩ CREATING AN HONORS CURRICULUM 124 What Happened? So far, you have worked with vectors and matrices separately. Now, you are going to use vectors and matrices together. When using vectors and matrices together, you can do reflections, and rotations. 1. What transformation do you see? What is happening to the x and y-values of each point? 2. Use matrix multiplication on a graphing calculator to perform the same transformation on the image below. 3. What would happen if you used the matrix [0−110] instead of [01−10]? 4. A’ B’ C’ A B C CREATING AN HONORS CURRICULUM 125 5. What would you have to multiply by to rotate a figure 180°? Why? 6. What transformation do you see? What happens to the x and y-values of each point? 7. What matrix would you multiply by to reflect over the y-axis? Why? 8. What matrix would you multiply by to reflect over y=x? Why? 9. What matrix would you multiply by to reflect over y= - x? Why? A’ B’ C’ A’ B’ C’ CREATING AN HONORS CURRICULUM 126 10. Why is beneficial to use matrices for transformations? Bonus Round Matrices can be used to find the area of a parallelogram. Example [3015]= A Area = |3015|= 3×5−1×0=15 – 0 = 15 Find the Area of the parallelogram using matrices 11. 12. 13. ⟨3,0⟩ ⟨1,5⟩ Vector 1 Vector 1 Vector 2 Vector 2 Area of the parallelogram equals the absolute value of the determinant of A. Area is 15 square units. CREATING AN HONORS CURRICULUM 127 What Happened? Key So far, you have worked with vectors and matrices separately. Now, you are going to use vectors and matrices together. When using vectors and matrices together, you can do reflections, and rotations. 1. What transformation do you see? What is happening to the x and y-values of each point? Rotation 270° (x, y) -> (y, -x) or x and y switched places, x changed signs. 2. Use matrix multiplication on a graphing calculator to perform the same transformation on the image below. [01−10]×[−21−22−2−2]=[2−2−22−12] 3. What would happen if you used the matrix [0−110] instead of [01−10]? 4. 5. [0−110]×[−21−22−2−2]=[−222−21−2] 6. Rotate 90°, changed the sign on y instead of x. A’ B’ C’ A B C A’ B’ C’ A’ B’ C’ CREATING AN HONORS CURRICULUM 128 7. What would you have to multiply by to rotate a figure 180°? Why? 8. 9. [−100−1]×[−21−22−2−2]=[2−12−222] Do not switch x and y, just change signs. 10. What transformation do you see? What happens to the x and y-values of each point? Reflection over x-axis X-value stayed the same y-value changed sign 11. What matrix would you multiply by to reflect over the y-axis? Why? [−1001]×[135141]=[−1−3−5141] 12. What matrix would you multiply by to reflect over y=x? Why? [0110]×[135141]=[141135] Switch x and y-values without changing Any signs 13. What matrix would you multiply by to reflect over y= - x? Why? [0−1−10]×[135141]=[−1−4−1−1−3−5] Switch x and y-values and change both Signs A’ B’ C’ A’ B’ C’ A’ B’ C’ A’ B’ C’ A’ B’ C’ B’ C’ A’ CREATING AN HONORS CURRICULUM 129 14. Why is beneficial to use matrices for transformations? Bonus Round Matrices can be used to find the area of a parallelogram. Example [3015]= A Area = |3015|= 3×5−1×0=15 – 0 = 15 Find the Area of the parallelogram using matrices 15. |1340| = -12 Area is 12 square units. 12. |2−323| = 12 Area is 12 square units. 13. |4−35−1| = 11 Area is 11 square units. ⟨3,0⟩ ⟨1,5⟩ Vector 1 Vector 1 Vector 2 Vector 2 Area of the parallelogram equals the absolute value of the determinant of A. Area is 15 square units. CREATING AN HONORS CURRICULUM 130 Put it together! Multiply What happens to x and y? Which transformation is it? Point (x, y) [100−1]×[𝑥𝑦]= Point (x, y) [−1001]×[𝑥𝑦]= Point (x, y) [0110]×[𝑥𝑦]= Point (x, y) [0−1−10]×[𝑥𝑦]= Point (x, y) [0−110]×[𝑥𝑦]= Point (x, y) [−100−1]×[𝑥𝑦]= Point (x, y) [01−10]×[𝑥𝑦]= CREATING AN HONORS CURRICULUM 131 Use Matrices to conduct the following transformations. 1. Rotate the point (-1, 4) 90°. 2. Reflect the point (-4, 6) over the x-axis. 3. Rotate the point (1, 7) 270°. 4. Rotate the point (6, -5) 270°. 5. Reflect the point (-2, 3) over the line y=x. 6. Reflect the point (4, -9) over the y-axis. 7. Rotate the point (2, -6) 180°. 8. Reflect the point (-1, -3) over the y-axis. 9. Reflect the point (6, -1) over the line y=-x. CREATING AN HONORS CURRICULUM 132 10. Rotate the point (-3, -2) 180°. 11. Rotate the point (3, 5) 90°. 12. Reflect the point (7, -3) over the x-axis. 13. Reflect the point (0, 5) over the y-axis. 14. Reflect the point (4, -6) over the line y=x. 15. Reflect the point (3, 7) over the line y=-x. CREATING AN HONORS CURRICULUM 133 Key Put it together! Multiply What happens to x and y? Which transformation is it? Point (x, y) [100−1]×[𝑥𝑦]=[𝑥−𝑦] X is the same y changes sign Reflect over x-axis Point (x, y) [−1001]×[𝑥𝑦]=[−𝑥𝑦] X changes sign y is the same Reflect over y-axis Point (x, y) [0110]×[𝑥𝑦]=[𝑦𝑥] X and Y switch places Reflect over y = x Point (x, y) [0−1−10]×[𝑥𝑦]=[−𝑦−𝑥] X and Y switch places and change signs Reflect over y = -x Point (x, y) [0−110]×[𝑥𝑦]=[−𝑦𝑥] X and Y switch places and Y changes sign Rotate 90 degrees Point (x, y) [−100−1]×[𝑥𝑦]=[−𝑥−𝑦] X and Y change signs Rotate 180 degrees Point (x, y) [01−10]×[𝑥𝑦]=[𝑦−𝑥] X and Y switch places and X changes sign Rotate 270 degrees CREATING AN HONORS CURRICULUM 134 Use Matrices to conduct the following transformations. 1. Rotate the point (-1, 4) 90°. [0−110]×[−14]=[−4−1] (-4,-1) 2. Reflect the point (-4, 6) over the x-axis. [100−1]×[−46]=[−4−6] 3. Rotate the point (1, 7) 270°. [01−10]×[17]=[7−1] 4. Rotate the point (6, -5) 270°. [01−10]×[6−5]=[−5−2] 5. Reflect the point (-2, 3) over the line y=x. [0110]×[−23]=[3−2] 6. Reflect the point (4, -9) over the y-axis. [100−1]×[4−9]=[4−9] 7. Rotate the point (2, -6) 180°. [−100−1]×[2−6]=[−2−6] 8. Reflect the point (-1, -3) over the y-axis. [−1001]×[−1−3]=[1−3] 9. Reflect the point (6, -1) over the line y=-x. [0−1−10]×[6−1]=[1−6] CREATING AN HONORS CURRICULUM 135 10. Rotate the point (-3, -2) 180°. [−100−1]×[−3−2]=[32] 11. Rotate the point (3, 5) 90°. [0−110]×[35]=[−53] 12. Reflect the point (7, -3) over the x-axis. [100−1]×[7−3]=[73] 13. Reflect the point (0, 5) over the y-axis. [−1001]×[05]=[05] 14. Reflect the point (4, -6) over the line y=x. [0110]×[4−6]=[−64] 15. Reflect the point (3, 7) over the line y=-x. [0−1−10]×[37]=[−7−3] CREATING AN HONORS CURRICULUM 136 Unit 4 Quiz Name the vector in component form that is shown and find its magnitude. 1. 2. Use a diagram to simplify the following expressions. Write the solution in component form. 3. ⟨3,5⟩+ ⟨1,−3⟩ 4. 4⟨1,2⟩ Use any method to simplify the following expressions. Write the solution in component form. 5. 2⟨3,−4⟩+ ⟨−5,7⟩ 6. 4⟨2,−1⟩−3⟨−1,3⟩ CREATING AN HONORS CURRICULUM 137 Use the vector to translate the image. 7. ⟨−3,2⟩ Use matrices to perform the following transformation. 8. Reflect the triangle with vertices A(2, 1), B(2, -3), C(4, -1) over the y-axis. 9. Rotate the triangle with vertices A(2, 1), B(2, -3), C(4, -1) 180°. Bonus point: 10. Use matrices to find the area of the parallelogram. CREATING AN HONORS CURRICULUM 138 Unit 4 Quiz Key Name the vector in component form that is shown and find its magnitude. 1. 2. ⟨5,2⟩,√29 ≈5.39 𝑢𝑛𝑖𝑡𝑠 ⟨5,−6⟩,√61≈7.81 𝑢𝑛𝑖𝑡𝑠 Use a diagram to simplify the following expressions. Write the solution in component form. 3. ⟨3,5⟩+ ⟨1,−3⟩ = ⟨4,2⟩ 4. 4⟨1,2⟩= ⟨4,8⟩ Use any method to simplify the following expressions. Write the solution in component form. 5. 2⟨3,−4⟩+ ⟨−5,7⟩ 6. 4⟨2,−1⟩−3⟨−1,3⟩ ⟨1,−1⟩ ⟨11,−13⟩ CREATING AN HONORS CURRICULUM 139 Use the vector to translate the image. 7. ⟨−3,2⟩ Use matrices to perform the following transformation. 8. Reflect the triangle with vertices A(2, 1), B(2, -3), C(4, -1) over the y-axis. [−1001] × [2241−3−1]= [−2−2−41−3−1] A’ (-2, 1), B’ (-2, -3), C’ (-4, -1) 9. Rotate the triangle with vertices A(2, 1), B(2, -3), C(4, -1) 180°. [−100−1] × [2241−3−1]= [−2−2−4−131] A’ (-2, -1), B’ (-2, 3), C’ (-4, 1) Bonus point: 10. Use matrices to find the area of the parallelogram. |5124|=18 𝐴𝑟𝑒𝑎 𝑖𝑠 18 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠. CREATING AN HONORS CURRICULUM 140 Appendix B: Curriculum Evaluation Tool CREATING AN HONORS CURRICULUM 141 CREATING AN HONORS CURRICULUM 142 CREATING AN HONORS CURRICULUM 143 CREATING AN HONORS CURRICULUM 144 Appendix C: Feedback from Evaluators Unit 1 CREATING AN HONORS CURRICULUM 145 CREATING AN HONORS CURRICULUM 146 CREATING AN HONORS CURRICULUM 147 CREATING AN HONORS CURRICULUM 148 CREATING AN HONORS CURRICULUM 149 CREATING AN HONORS CURRICULUM 150 CREATING AN HONORS CURRICULUM 151 Unit 2 CREATING AN HONORS CURRICULUM 152 CREATING AN HONORS CURRICULUM 153 CREATING AN HONORS CURRICULUM 154 CREATING AN HONORS CURRICULUM 155 CREATING AN HONORS CURRICULUM 156 CREATING AN HONORS CURRICULUM 157 CREATING AN HONORS CURRICULUM 158 Unit 3 CREATING AN HONORS CURRICULUM 159 CREATING AN HONORS CURRICULUM 160 CREATING AN HONORS CURRICULUM 161 CREATING AN HONORS CURRICULUM 162 CREATING AN HONORS CURRICULUM 163 CREATING AN HONORS CURRICULUM 164 CREATING AN HONORS CURRICULUM 165 Unit 4 CREATING AN HONORS CURRICULUM 166 CREATING AN HONORS CURRICULUM 167 CREATING AN HONORS CURRICULUM 168 CREATING AN HONORS CURRICULUM 169 CREATING AN HONORS CURRICULUM 170 CREATING AN HONORS CURRICULUM 171 CREATING AN HONORS CURRICULUM 172 Appendix D: IRB Permission Letter