Vertical Height and Horizontal Length and Slope
Vertical Height and Horizontal Length and Slope
Lesson / Content Skills
Students will be able to recognize applications of slope and see the relationship between the dependent and independent variables. Specifically students will focus on slope within the context of a rate of change, where students will learn how to find the slope of a line, given the coordinates of two points on a line.
There are five strands of Mathematical Proficiency that are essential for students to develop math skills that can be extended across many facets. Students are exposed to the skills of all five strands which are listed below.
The buzzword in education today is 21st Century Skills. At a time when increasingly advanced skills are required for success in life and work, people of all ages are seeking a diverse range of learning experiences to inspire, guide, and enhance their personal and professional lives. This lesson provides students the opportunity to develop a subset of the 21st Century Skills.
Specifically students develop:
CA Content Standards
The lesson addresses California Algebra I standard 7.0 (for 8th - 12th grade) - "Students verify that a point lies on a line, given an equation of a line. Students are able to derive linear equations by using the point-slope formula."
It is assumed that students have prior knowledge in coordinate planes, including graphing and locating points and addition and subtraction of integers. Exposure to ratios and the relationship between the numerator and denominator and how it affects the magnitude is helpful, but not essential.
The subject of slope is introduced using videos from youtube or a video of your choice. Two videos from youtube are provided. The first, lets the students think about slopes in the context of the world's largest skateboard ramp. The second, lets the students consider slopes viewing ski jumping. You can show one of these videos, both of them, or one of your choice, based on your class interests.
skate board ramp: http://www.youtube.com/watch?v=3_1Y8UoLlu4
ski jumping: http://www.youtube.com/watch?v=6ZeLbTYeNk4
This introduction is intended to "wow" the students and stimulate their interest in learning about slopes, its relationship to steepness and the real world. It assumes the students have an interest in skate boarding or skiing or TBD which could be indirectly applied.
The instruction of slope is divided into three parts.
The "first part" the teacher and the students engage in the idea of slope in a hands on manner. Students are provided the opportunity to play with cars and a ramp to assess the effect of the steepness on speed and distance.
This activity requires the students to divide into groups of 2-3. Each group is provided a piece of cardboard (or other fairly rigid flat material) of the same length between 12 - 20 inches, a matchbox car or other, a ruler, means to measure time and a sheet to record data. Students will have the opportunity to vary the height and assess its affect on the speed of the car and the distance it travels. Students record the speed and distance of the car at a measured vertical height and horizontal length. As the students explore the concept of slope, the teacher should wander the classroom and observe and probe the groups on observations they are seeing.
Points the teacher might want to incorporate into the discussion are:
At the end of the class, have the students complete three practice problems about slope and write a reflection based on provided prompts. After the students finish, have students share responses with the class for further discussion and clarification. The students work becomes an exit slip for the teacher to use for student assessment.
The following materials will be used during the lesson:
This lesson can be assessed with class participation and exit slip.
There will be ample opportunity to assess the students during the following activities:
Here the teacher can assess each students' individual understanding. Students are asked questions pertaining to slope along with a written reflection based on prompts provided. This exit slip is a direct reflection of the main lesson goals.
Studies have shown that one of the most effective ways for students to learn and for teachers to assess if students understood the material covered is to have students explain the concepts to other students. "The best way to learn something is to have to teach it." In this assessment the students, working with their same groups during the lesson, generate a video explaining the concept of slope. It is up to the students to determine how they want to explain this concept. The students will show their videos to the class for feedback and then, with teacher assistance, place on the school website, you tube, teacher tube, or TBD location. The intent is to place the videos in a location such that students from the school and outside of the school can view when they need help in understanding slopes.
Students can use the technology of their choice to make their videos. The website for JingProject is listed below. This website can be downloaded and allows students to easily produce videos, capturing both sound and video.
Dewey, J. (1916). "Democracy and education: An introduction to the philosophy of education (pp. 179 - 192). New York, NY: The Macmillan Company.
Dewey emphasizes the theoretical idea that "good habits of thinking" should be fostered in school. However, actual practice does not support this theory of developing students ability to think. Typical classroom environments support knowledge, which "develops the poison of conceit" rather than "further growth in the grace of intelligence." Intelligent learning promotes thinking, which uses and rewards the mind. Dewey defines the essential methods of thinking as:
In this lesson plan students learning (thinking) progression of slopes evolves in the following manner:
Sigler, J.W., & Hiebert, J. (2009). The teaching gap: Best ideas from the world's teachers for improving education in the classroom (pp. 85 - 101). New York, NY: Free Press.
This article emphasizes the concept that teaching is a cultural activity, that evolves over long periods of time in ways that are consistent with the beliefs and assumptions that are part of the culture. A comparison is made between the culture in the United States versus Japan. The typical U.S. lesson follows the belief that the mathematics is a set of procedures that emphasizes students developing skills. In Japan, lessons follow the belief that mathematics is a set of relationships between concepts, facts, and procedures, that emphasize students seeing relationships between mathematical ideas. This lesson was developed with the Japanese culture in mind. The students are provided the opportunity to explore, make mistakes and reflect on the mathematical concept of slope. The students are not shown procedures and equations to calculate slopes, rather they explore the underlying meaning of slope and how two variables can deliver different outcomes.
This video highlights how video games can organize deep conceptual understanding, as well as a variety of 21st century skills, that are rarely offered in schools today. An important concept Gee talks about is "situated meaning". Typical classrooms use verbal (words thru words), rather than the desired situated (image, action, dialogue, words) which is more suited for problem solving. The most effective way for a student to learn is thru action and experience and then read the text (instructions). Providing students with equations without mental images of what they represent is an ineffective way of learning. It is preferable to provide the student with a situated meaning and then provide the equations that describe that situation. In this lesson, even though the initial work is hands on and not thru a video game, the students are provided the opportunity to play with the idea of slopes, speed and distance prior to learning the equation for slope. When students have the opportunity to interactively learn the equation for slope, they now have the mental model of how vertical / horizontal changes effect the slope and ultimately speed and distance.
Chi talks about the importance of self-explanation in promoting student learning. The article discusses prompts that the students considered while reading about a subject matter. This philosophy was used in this lesson by having the teacher provide prompts for the students to consider at the start of the instructional activities, along with the middle and the end. It is the hopes that these prompts promote student thinking on a deeper level and not just memorizing a formula. In addition, these prompts should provide connection a theme throughout the lesson.
Even though this lesson does not employ computer games, it does apply some of the key concepts that the authors note are important for student learning. First, this lesson uses more than one form of media that connects and provides meaning to students as they participate in the lesson. Students work in groups of 2-3, thus they are participating in social engagement that taps on their experiences and ideas as they progress thru the lesson. The authors derived their educational games from Vygotsky where children's play creates a zone of proximal development for the child. This lesson is designed for understanding using a participatory framework that stresses hands-on action and reflection as components central to the learning process. The authors emphasize that learners should be involved in doing domain-related activities, not simply receiving the results of someone else's activities as summarized in texts or heard in lectures. This lesson does not use a textbook or lecture. Students learn best when the learning process involves inquiry as opposed to the memorization of the facts and principals that were generated from someone else's inquiry. In this activity the students explore the relationship of vertical height and horizontal length in regards to speed, distance, and steepness, before they even learn the formula to calculate the slope.
Robinson notes that education is supposed to take us into the future that we cannot grasp at this time. The best way to approach the unknown is thru creativity. He notes that students have extraordinary capacity for creativity and talent that we squash with the current education environment. The classroom should allow students to explore, take chances, and not be frightened of being wrong. In fact, that is when students can be the most creative. Schools are educating students out of their creativity, as emphasis is placed more on the right answer. Students are now afraid to be wrong. The only hope for the future is to adopt a new ecology on education, recognize the gift of human imagination. This lesson does not ask students to show their creativity by inventing new things in many different facets, however, it does allow students to inquire about the concept of slope and how it might apply to everyday life. The topic is math, but aspects of art, social science, and English can be incorporated by each student, depending on their interests and desires.
How does learning occur?
Leaning occurs thru the use of manipulatives, computer interactive tools, and the social engagement of inquiring prompts.
Technology is being used in the classroom as the students engage in an interactive math program on slopes. The students can work alone or in groups and change two points ultimately changing the slope of the line. During this activity the students can visually see the slope of a line change and the slope formula calculate the slope at the same time.
In this lesson students can develop a better understanding of what the slope equation represents. Before the students learn the equation, they develop a mental model of what slope represents and how vertical height and horizontal length change this value. This visual representation of slope is interactive by the students. Now as they calculate the slope of a line, they have a deeper understanding of the relationship between the x and y coordinates.
Educational Technologies: http://en.wikiversity.org/wiki/Educational_Technology
Distance Education Lesson Plans: http://en.wikiversity.org/wiki/Distance_Education_Lesson_Plans
Appendix - Possible Exit Slip
Possible exit slip with prompts for teacher to use:
Find the slope of the line that passes through each pair of points.