Rate of Change, Linear Relationships, and the Real World
Jared Schuetter
Unit Overview
Summary
This unit is designed to ultimately teach students about linear equations. Rather than bombarding students with formulas to memorize and countless math problems with no context, this unit will help students to understand the actual concepts behind linear relationships. Critical to this goal is a study of the slope of a line and the part it plays in linear relationships. Rather than studying slope apart from its real world applications, this unit targets its real world counterpart: rate of change. Thus students will now have a context in which to talk about slope out of the realm of the mathematics classroom. By providing useful examples that are easily accessible to students' schemata, mathematics may become more interesting and fun!
Goals
1. Students will fully understand what "rate of change" means.
2. Students will be able to recognize real world examples in which rate of change plays a role.
3. Students will be able to plot data points on a graph.
4. Students will be able to draw a best fit line through their data and understand that it approximates the relation between the two variables.
5. Students will connect rate of change to its mathematical counterpart: slope.
6. Students will recognize all of the parts of a linear equation.
7. Students will be able to describe in their own words what an equation is and what it says about the two variables involved.
8. Students will be able to use a graphing calculator to plot data points, find a best fit line, and calculate the slope of that line.
9. Students will be able to draw connections between graphing calculator data and hand-written data based on mathematical calculations.
10. Students will be able to shift back and forth between describing relationships with tables, equations, or graphs.
11. Students will be able to observe a linear relationship in their own world and use mathematics to derive an equation exhibiting this relationship. This will be done using experimentation and graphing skills as well.
Concepts
1. "Rate" means amount of something as something else changes.
"Change" (for our purposes) is an increase or decrease in some measurable quantity.
2. Rate of change, therefore, is the amount of increase or decrease in some measurable quantity as another quantity varies.
3. Examples of rate of change: Car Speed (miles / hour), Fuel Consumption (gallons / mile), Salary (dollars / hour), Commercials (times seen / hour), Burning Candle (height / minute), etc.
4. Experiments can be set up to describe the behavior of this rate of change between variables.
5. Data from these experiments can be plotted on a graph.
6. Essential parts of a graph are: Graph Title, Axis Titles (with units of measure), and plotted points.
7. To plot points, follow a line up from the scale for values of the independent variable and find where it meets the horizontal line on the scale for the dependent variable values (values are on their respective axes).
8. After plotting points, a best fit line can be drawn (estimated) through the data.
9. "Slope" is (Rise / Run) = (Change in D.Var. / Change in I. Var.)
But rate of change is the same thing!
10. Slope can be negative, in which case the rise is actually the "sink."
11. Rate of change between two variables can be best approximated by measuring the Rise / Run of the best fit line of the data after experimentation.
12. To easily calculate slope: find two points on the grid that the best fit line passes through. Count the squares for the rise and run, then - making sure to count each square for the amount of measure it is worth - calculate the slope of the line.
13. To graph data on a graphing calculator, use the STAT, PLOT, Y =, and WINDOW keys.
14. The equation of the best fit line on the graphing calculator should be y = mx or y = mx + b, depending on what the starting value of the data is. This m value should be about the same number as the rate of change of the two variables. The b value is actually the starting value of the data, and in the y = mx case, the data started at zero, so the equation is still of the form y = mx + b.
15. Parts of a linear equation: y => dependent variable, x => independent variable, m => slope / rate of change, b => starting point of the data
*Students will understand the goals set forth for this unit.
*Students will have a respect for one another and will work with team members effectively.
2. Content Objectives/Content Analysis/Concepts for this lesson:
*This unit will encompass discussions about rate of change and linear relationships. Most importantly, it will explore how mathematics can relate to real life.
*Respect is a key piece to fostering a good learning environment.
*Students need to help each other out and not marginalize any classmates.
*Mathematics can be fun, especially when we work together!
3. How the content of this lesson builds on what was learned previously:
Students have some background in mathematics. They should be able to do basic algebraic manipulations such as proficiency with the common operations (+, - , * , / , exponents), familiarity with fractions and roots, and an understanding of ratios. Other than prerequisite knowledge, this lesson is designed mainly to set up the unit and prepare the students for the tasks that lay ahead.
4. How the content of this lesson relates to what students will be learning in the future:
This lesson has everything to do with what the students will learn in the future. By setting up a positive learning environment and summarizing what will be done in the next few weeks, students will be ready to dive into this exploration of mathematics concepts.
5. Literacy approach to be used:
N / A
6. How student characteristics will be used and/or accommodated:
Gender - It is a well known fact that girls often feel silenced in the classroom, or are reluctant to
participate in class proceedings because of aggressive male classmates. In order to
assist these students, I plan to make sure to treat the concerns of all classmates
equally, regardless of gender. If I am asking questions to the class, I will alternate the
gender of the person I ask. If people are asking me questions, I will alternate girl / boy
if possible.
Race/ethnicity - As with the gender issues, I will do my best to treat minority students' concerns
as the having the same - if not more - importance. Their perspective is crucial
to maintaining the learning environment of the classroom.
English language proficiency - Any difficulties as far as proficiency can be remedied
through extra time spent with students or repeating some of
what has been said. Also, a handout could be distributed
that outlines the goals of the unit so that non-native
speakers could have time to decipher the English and make
sense out of the writing.
Economic status - Economic status does not figure into this lesson very much, but will be
included in the discussion of respect for fellow students. In the future, I will
refrain from using examples of real life mathematics that involve objects or
events that are specific to strictly upper-class individuals. Some examples
would be using sports cars to demonstrate rate of change or talking about
rate of change as it applies to horseback riding.
Skill level - The students are not all at the same level of skill in mathematics, but being teamed
up will allow these setbacks to have less of an impact on the learning process.
Exceptionalities - Some students have slight learning disabilities. To help out, Ms. Warner, the
trained special-education teacher assistant, will be present. She knows the
students and their needs, and will assist them in the learning process.
7. Teaching method(s) to be employed:
Lecture with questions and discussion.
8. Sequence of activities and approximate time schedule:
Pass out optional handouts
Outline the goals of the course on the board or off of the handout
Lead a group discussion on the importance of respecting classmates
9. Instructional materials (may be attached):
Optional Handout
10. Technology to be used (optional) and an explanation of why it will be used:
Portable Erasable Nib Cryptic Intercommunication Language Stylus
Plastic Encoding iNvention
Pliable And Portable Encryption Receiver
Dry-Erase Board
These will be used to convey information in a clear and concise way.
11. If grouping is to be used:
Group Name Number of Students Basis for Group Membership
or Number
Students will be formed into teams of an undetermined size (2-4 per group would be nice)
12. Evaluation Plan (how you will know whether the students have learned what you
intended them to learn; this could be in the form of a list of oral questions, written
quiz, student demonstration of a skill, or any other evaluation strategy):
Verbal communication by the students indicating understanding (i.e. head nod, show of hands, etc.)