Student mathematics motivation and engagement in the first year of high school is critical because of its volatility (Mangu, Middleton, & Lee, 2016) and importance for future academic advancement and career opportunities (Trusty, & Niles, 2003). Following the quantitative longitudinal non-experimental design (O'Dwyer & Bernauer, 2013), the study uses the person-centered analytic approach (cluster analysis), which can complement the study of engagement and motivation in attending to the diversity of student experiences (Lawson & Lawson, 2013). The study has two goals: 1) examine the changes in students' perception of the mathematics classroom learning environment, and their mathematics motivation and engagement during one academic semester in the first year of high school, and 2) look into how students' perceptions of the mathematics classroom learning environment may influence their mathematics motivation, which may further influence their engagement in the longitudinal context of changes. This study is a secondary analysis of the survey data of the SMiLES (Secondary Mathematics, In-the-moment, Longitudinal Engagement Study) project. In Spring 2018, 199 students across fifteen ninth-grade mathematics classrooms in Delaware and Arizona participated in this study. These students filled out the validated survey of students' perceptions of the learning environment and mathematics motivation and engagement at the beginning and the end of the semester. The data analytic methods include cluster analysis, cluster transition analysis, t-test, multinomial logistic regression, and MANOVA. The study results revealed that students' mathematics motivation and engagement can change significantly within a semester at the critical stage of the first-year high school mathematics class. Regarding the time effects, both previous perception of the classroom learning environment and the changes in perception of the classroom learning environment predicted changes in motivation. In addition, both previous motivation, as well as the changes in motivation, predicted changes in engagement. One contribution of this study is a theoretical framework for how the classroom learning environment influences students' engagement and motivation built for mathematics subject in particular. The longitudinal design complements the inference constraints of traditional cross-sectional studies of engagement, and the person-centered analytic approach also helps unpack the variety of student experiences. Teachers may benefit from paying close attention to students' changes in mathematics motivation and adapting their feedback and other instructional support to differences in students' motivation. Finally, as the study has some limitations in sample size, type of data source, and length of study, future research could consider conducting a longer-term observation with a greater sample size, using additional qualitative data sources, and following up with other student learning outcome data. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]