# Vertical Height and Horizontal Length and Slope

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## Vertical Height and Horizontal Length and Slope

 Subject: Algebra I
 Sub-Subject: An investigation between steepness and the slope of a line
 Length/Duration: 2 - 4 days
 Technologies Used: YouTube World's Largest Skateboard Ramp Ski Jumping Video Mathwarehouse (algebra) Interactive software to explore slopes of lines

## Learning Goals

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.

Specifically students:

1. Investigate and make sense of the relationship between the vertical heights to the horizontal length.
2. Find the slope of a line given the coordinates of two points on the line.
3. Connect the learning's of slope to real life.

Mathematical Proficiency Skills

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.

• Conceptual Understanding: Comprehension of mathematical concepts, operations, and relations.
• Procedural Fluency: Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
• Strategic Competence: Ability to formulate, represent, and solve mathematical problems.
• Adaptive Reasoning: Capacity for logical thought, reflection, explanation, and justification.
• Productive Disposition: Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy.

21st Century Skills

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:

• Critical Thinking and Problem Solving Skills
• Communication and Collaboration Skills

## 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."

## Lesson Flow

Prerequisite

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.

Introduction

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.

A key connection between either of these videos and the topic of slope is steepness and how it relates to speed and distance. Allow students to consider different aspects of the ramp with minimal guidance from the teacher. Have the teacher inject comments and direction as needed. The objective is to promote the student's observations and thoughts to transition into the remainder of the lesson plan.

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.

Instructional

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:

1. If speed is important, what is the relationship between the vertical height to the horizontal length?
2. If the distance you travel is important, what is the relationship between the vertical height to the horizontal length?
3. As the vertical height increases, what happens to the horizontal length?
4. How might the coefficient of friction play a factor on speed and distance the car travels?
5. What other outside factors might influence speed and distance?
6. Where might we place the origin of a coordinate plane in relation to the line of the ramp in order to make calculations to find a slope?
7. What real-world situations incorporate the concept of slope(s)? Examples might be, construction and architecture.

The "second part" of the instructional activity is to scaffold the students into the calculation of slope using coordinates and the coordinate plane. Students interactively learn about slopes using the computer. The program from math warehouse allows students to learn about slopes in relation to the coordinate plane, how to calculate slope, and manipulate points to visually watch the slope of a line change along with the slope value. At the completion of the discussion of slopes, the students will be able to practice calculating on the computer.

NOTE: If the teacher does not have access to a computer for every student, they can use one computer and lead the activity with the class, otherwise have the students learn about slopes with their partner(s) and the teacher assist students as needed.

The "third part" of the instructional activity connects part one and part two. Discuss the following prompts with the students:

1. If speed is important, what is the relationship between the vertical height and the horizontal length?
2. If the distance you travel is important, what is the relationship between the vertical hight to the horizontal length?
3. How does the slope change when considering speed vs distance?
4. As the vertical height increases, what happens to the horizontal length?
5. Where might we place the origin of a coordinate plane in relation to the line of the ramp in order to make calculations to find a slope?

Close

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.

Materials

The following materials will be used during the lesson:

1. YouTube Video: World's Largest Skateboard Ramp, Ski Jumping
2. TBD ramps, cars and rulers for group manipulative activity
3. Means to measure time (watch, calculator, phone, wall clock...)
4. Slope discussion using interactive software from math warehouse
5. computer (at least one), projector
6. Handouts for students

Assessment Strategy

This lesson can be assessed with class participation and exit slip.

Class Participation

There will be ample opportunity to assess the students during the following activities:

1. Conceptual discussion of slope and steepness during the group manipulative activity and as a class (using suggested prompts or others of your choice)
2. Extension of the manipulative activity to the group interactive computer activity and exercise.

Exit Slip

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.

Optional Assessment Strategy

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.

## Related Articles

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:

1. The student environment promotes a genuine experience, where they can develop an interest for "its own sake";
2. Students engage in a genuine-problem that stimulates thought;
3. Students are able to posses or are provided the information to work the problem;
4. Students are able to develop and suggest solutions; and
5. Students have the opportunity to test and validate their ideas, thus allowing meaning to become clear.

In this lesson plan students learning (thinking) progression of slopes evolves in the following manner:

1. Wow students with skateboard video and the concept of steepness and speed (develop interest);
2. Engage students by allowing them to explore steepness using cars and a ramp, hands on (problem to stimulate thought);
3. Allow students to interactively learn about slopes on the computer (information to work problems);
4. Students connect interactive with hands on activities (develop suggestions to solutions); and
5. Students reflect on big ideas of the lesson (test & validate ideas).

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, M.T.H., De Leeuw, N., Chiu, M.-H., & Lavancher, C. Eliziting self-explanation improves understanding. Cognitive Science, 18(3), 439-477.

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.

Barab, S., Thomas, M., Dodge, T., Carteaux, R., & Tuzun, H. (2005). Making learning fun: Quest Atlantis, a game without guns. "Educational Technology Research and Development", 53(1), 86-107.

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.

Answers to the three main questions of the course:

How does learning occur?

Leaning occurs thru the use of manipulatives, computer interactive tools, and the social engagement of inquiring prompts.

In what environment can technology promote learning?

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.

What is the process by which technology enhances learning?

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.

1. (1,2), (4,3)
2. (7,2), (3,5)
3. (-3,-2), (4,-2)

Prompts

1. What happens to the slope of a line if your increase the vertical height and keep the horizontal length fixed?
2. What happens to the slope of a line if you fix the vertical height and increase the horizontal length?
3. Comparing the slope from question 1 versus the slope from question 2, which would be larger
4. You want to design two skateboard ramps. The first is for you and would allow you to go as fast as possible. The second is for your friend who does not want to go fast. Describe using the vocabulary that discussed in this lesson, what the two ramp designs might look like. Provide and example of what each of the ramp's slopes might look like.