# Slope

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```					 Overview and Background: Unit: Slope
Stephanie Sneyd : Griffin RESA
Mathematics : Middle School Mathematics : 6-8th Patterns and Relationships/Algebra
Griffin RESA : Grades 6 - 8 : Aug. - May.
Title:             Slope
Topics:            linear equations, intercepts, slope, functions, patterns, coordinate system
Time Frame:
Start Date:             Mar. 1 - Mar. 28
Status:                 Draft
Date Revised:           May 8
Other Designers: Debbie Megrue, Jennifer Couch, Pam Shroyer
Summary:
Students will encounter concepts such as slope, intercepts, graphing lines, etc. as they solve problems involving
these ideas. They will be asked to create a roller coaster and explain the problems that go along with zero and
undefined slopes. Students will create a drawing on a coordinate graph for placement in a time capsule. They will
also be given the opportunity to explore the best way to earn money given choices.
Print Materials Needed:
Resources:
Concept map can be found at Link 1.
Resource Attachments:              Listed with Individual Performance Tasks.
Stage 1: Identify Desired Results
State:             GA      Pre Algebra QCC: 1, 2, 3, 4, 29, 30, 31, 37, 39
Title:             Algebra
Standard(s):       GA QCC: 1 Solves problem, reasons, and estimates throughout mathematics: Selects and uses
problem-solving strategies such as reading the problem, drawing a picture or diagram, using trial
and error, making a table or chart, looking for patterns, making a simpler problem and then
generalizing, and working backwards, etc.
Selects and uses appropriate tools in solving problems.
Uses estimating to check the reasonableness of results.
Solves non-routine problems for which the answer is not obvious.
Relates concepts and skills to practical applications.
2: Selects and uses appropriate estimation strategies, such as rounding, truncating, front-end,
adjusting, compensation, compatible numbers, clustering, and reference point, and recognizes
situations in which estimates are more appropriate than exact numbers.
3. Selects and uses appropriate mental computational strategies such as multiples of ten, multiples
of one tenth, and powers of ten.
4. Expresses, orders, and categorizes rational numbers in various forms, such as fractions,
decimals, percent, and scientific notation using tools such as calculators and number lines.
#29: Collects and organizes information or data by classifying or identifying patterns, and
organizes data into tables, charts, and graphs.
#30: Graphs points in the coordinate plane, identifies coordinates of points, graphs linear
equations, and solves problems using these concepts.
#31: Reads and interprets tables, charts, graphs, and diagrams.
#37: Solves equations and applied problems of the form ax=b, ax+b=c, ax+b=cx+d, x/a=b,
x/a+b=c.
#39: Models the concept of division (as rate, ratio comparison, and missing factors) using physical
models and pictorial and algebraic representations.
National Standard 8: Understands and applies basic and advanced properties of functions and
algebra.
McCrel Level III(grades 6-8) #7: Understands special values of patterns, relationships, and
functions.
Understandings:
user           The student will understand that patterns and relationships exist in slopes and in functions.
Essential Questions:
user           How do patterns allow us to make predictions in real world situations?
user           How are slopes, x-intercepts, and y-intercepts related?
Knowledge and Skills:
The student will know the following key vocabulary terms: coordinate system, ratios, equations, intercept, slope,
parabola, linear, non-linear, and function.

The student will be able to:
graph linear and nonlinear equations
determine the slope of a line from its graph
predict the x and y intercepts of a linear equation
determine slope using the slope formula
write an equation in slope-intercept form
perform the vertical line test to identify functions
Stage 2: Determine Acceptable Evidence
Assessment Summary:
Students have an opportunity for 3 performance assessments in this unit. "Screaming Slopes" requires students to design a roller
coaster. "What will you Choose?" is a problem solving activity using patterns. Students will receive one-on-one slope and graph
training with an on-line chameleon in "Force and Motion with Slope." Once trained, the students will use FOSS airplanes and the
rectangular coordinate graph to predict the fuel required for a successful airplane flight.

Topics: slope

Summary:
The student will design a roller coaster for an amusement park. They must show slope for all portions of the roller coaster, including
those sections with zero slope.

Print Materials Needed:
cm Graph paper

Resources:
www.funderstandingrollercoasters.com
Concept map can be found at http://griffinresa.net/UBDforms/dwarfhouse/slope.doc

Resource Attachments:

Notes:

Student Directions:
You are a roller coaster designer for Six Flags Over Georgia. You have been asked to design a new and exciting roller coaster for the
park. Along with a drawing of your design(on grid paper), you must include measurements for each incline/decline. Drawing a line on
each incline/decline, include the slope of each. In a summary, point out the zero slope areas and explain why they are necessary. In a
report, discuss some of the problems a roller coaster designer might encounter. Is it possible to build a roller coaster containing a hill
with an undefined slope? Why or Why not? Your final product to be presented to Six Flags will be your drawing (labeled approriately)
and a summary report detailing your roller coaster and describing the problems you dealt with.

Rubric(s)

Rubric: Screaming Slope

Summary:
Students will design a roller coaster and identify possible design problems.

Trait: Presentation
Performance Type: Oral.
Level 1: poor
Level 2: good
Level 3: excellent
Level 4: superior

Presentation is not audible or student is not knowledgeable of the material being presented.
Presentation is minimal with much information needed to clearly explain project.
Presentation is clearly audible and adequate information is provided to explain the project.
Presentation is well rehearsed and every detail is explained clearly.

Trait: product/drawing
Performance Type: Display.

Level 1: poor
Level 2: good
Level 3: excellent
Level 4: superior

Drawing leaves out important critera and is poorly done. Slopes may be calculated inaccurately.
Drawing contains all criteria but with no concern to details. Slopes are accurately calculated.
All criteria are met. Drawing is completed with reasonable care. Slopes are computed accurately.
Drawing is completed with all criteria met. Attention to detail is obvious as well as accurate calculations. Student went beyond what is
expected.

Trait: Summary Report
Performance Type: Written.

Level 1: poor
Level 2: good
Level 3: excellent
Level 4: superior

Report is poorly written with inaccurate explanations. It is apparent that the student doesn't understand the task.
Paper is written with a few errors (spelling, grammar,etc) but shows understanding by giving accurate explanations.
Paper clearly shows understanding of slope by answering criteria in a well-written summary. Grammar and spelling have been edited
and mistakes are minimal.
Summary is written in a way that shows understanding beyond expectations. Student has produced a paper that shows great pride and
attention to detail.

Trait: Participation
Performance Type: Process.

Level 1: poor
Level 2: good
Level 3: excellent
Level 4: superior
Student showed no interest in learning the needed understandings to produce a quality result.
Student did what was required but some were not complete or incorrect. Student occasionally participated in class.
Student asked questions and participated in class lessons and discussions leading up to the final product.
Student was an active learner, particpating in discussions, editing and correcting work to make it a quality product.

Task/Prompt: Force and Motion with Slope

Topics: slope, equations, x-intercept, y-intercept, function,

Summary:
Students will receive one-on-one slope and graph training from an on-line chameleon. Once trained, students will use FOSS airplanes
and the rectangular coordinate graph to predict required fuel for a successful transcontinental flight.

Print Materials Needed:
FOSS Science Materials:
Grades 5-6 Investigation 3: Plane Sense
and FOSS airplane kits
MAY BE ORDERED THROUGH:
Delta Education
P.O. Box 3000
80 NW Blvd
Nashua, NH 03063-4067
1-800-258-1302

Resources:
Check with your school's science department for any variation of a student-made rubberband fueled model airplane. The planes are
made with straws, craft sticks, rubber bands, and a plastic propeller.

Resource Attachments:

Notes:
This lesson is great when integrated with science! If time is limited, ask the science teacher to perform the tests during
a motion and force science lesson, and then you can have the students create the charts and make predictions based
on the science lab.

Student Directions:
You are the captain of a 767 airliner. You are clearing for take-off as your co-pilot asks you, "Are you sure we have enough fuel to make
this trip?" Not a question you want to ignore!

Chameleon. Pay close attention to Carl's explanation of slope. When your training is complete, build a replica of your 767 using the
FOSS airplane kit provided. Test the airplane's fuel requirements. Fuel in this case is the number of times you wind the rubber band.
Your FOSS airplane must travel 5 meters. How many winds of the rubber band does this take?

Test your FOSS airplane at 1m, 2m, 3m, 4m, and 5m. Carefully record your fuel needs. Use a rectangular coordinate graph to plot the
points. Study the slope created by your tests. Predict 6m, 7m, 8m fuel requirements. Compare your results with the other pilots in your
class.

Rubric(s)

Rubric: Predicting Fuel

Summary:
Students will train with Carl the Chameleon, build an airplane, perform actual fuel tests, and then predict fuel requirements.

Trait: Train with Carl the Chameleon
Performance Type: Oral.

Level 1: Proficient
Level 2: Developing
Level 3: Beginning

All glossary terms have been correctly defined. All of Carl's questions have been answered correctly.
At least 90% of glossary terms have been correctly defined. At least 90% of Carl's questions have been answered correctly.
Less than 90% of glossary terms have been correctly defined. Less than 90% of Carl's questions have been answered correctly.

Trait: Build an Airplane
Performance Type: Display.

Level 1: Proficient
Level 2: Developing
Level 3: Beginning

The airplane flies along the flightline without any malfunctions and requires the average number of rubber band winds for a 5m flight.
The airplane flies along the flightline without any malfunctions, but requires more than the average number of rubber band winds for a
5m flight.
The airplane displays malfunctions and/or requires more than the average number of rubber band winds for a 5m flight.

Trait: Actual Fuel Test
Performance Type: Display.

Level 1: Proficient
Level 2: Developing
Level 3: Beginning
The graph reveals a distinct and accurate pattern for all actual fuel tests. The graph's x and y-axis display flight in meters and number of
winds in the rubber band (fuel).
The graph reveals an inconsistent pattern for fuel tests and/or the graph's x and y-axis lack clear deliniation of flight in meters and
number of winds in the rubber band (fuel).
The graph lacks accurate coordinates with little to no pattern for fuel tests and/or the x and y axis are not accurately labeled.

Trait: Predicting Fuel
Performance Type: Display.

Level 1: Proficient
Level 2: Developing
Level 3: Beginning

Predictions clearly follow the pattern created from actual fuel requirements and the predicted coordinates are accurately indicated.
Predictions follow a pattern similar to actual fuel requirements, but lack consistency. And/or the predicted coordinates are inaccurately
indicated.
No pattern exists and the predicted coordinates are inaccurately indicated.

Topics: exponents, linear equations, functions

Summary:
Students will choose the best rate of pay given 3 choices. The answer is not obvious and will require a little investigation to find a good
choice.

Print Materials Needed:
One Grain of Rice: A Mathematical Folktale by Demi

Resources:

Resource Attachments:

Notes:

When using internet links above, use the key words "exponential growth."

Student Directions:
You have taken a job building a fence around a swimming pool. You have 30 days to complete the task. The owner who hired you has
offered to pay you in one of three ways. 1) He will give you a flat rate of \$1000. 2) Pay you \$10 each day that you work. 3) He will give
you a penny the first day and double your salary each day. The pattern will continue throughout the 30 days. (.01, .02, .04, .08)
Which method of payment will you choose? Create a table and a graph to prove that you have chosen the best way to make the most
money. Then write an equation for each payment option and decide if it is a funtion.

Rubric(s)

Rubric: What will you choose?

Summary:
Students are given an option of three pay plans. They are to choose the one that they think will pay them the most and provide evidence
to back their reasoning.

Trait: Create a Table
Performance Type: Display.

Level 1: Proficient
Level 2: Developing
Level 3: Beginner
Level 4: Needs Improvement

Students create an accurate table for each payment plan. The table shows an accurate amount for each of the 30 days and total
payment he/she would receive with each plan. Students are able to easily decide which payment plan they would choose.
The table shows each payment plan and the patterns evident. Very few mathematical mistakes were made and students were able to
determine which plan would give them the most money.
Students were able to develop a table for all three payment plans but mathematical mistakes were made and the choice of which plan to
take was not obvious.
Students create a table but it is inaccurate, does not include all 30 days, and an payment choice is not obvious.

Trait: Graph it!
Performance Type: Display.

Level 1: Proficient
Level 2: Developing
Level 3: Beginner
Level 4: Needs Improvement

Students are able to use the table to create an accurate graph. They can determine a slope (if there is one) and determine if the
equation is linear.
Students can create a graph but some of the points are not accurate. They can tell whether or not an equation has a slope and if it is
linear.
Students points are not accurate. Students can tell which plan will produce the most amount of money.
Students graphs are unacceptable and no answer choice is obvious.

Trait: Write an Equation
Performance Type: Process.

Level 1: Proficient
Level 2: Developing
Level 3: Beginner
Level 4: Needs Improvement

Students wrote accurate equations for each of the three payment options and were able to determine whether or not it was a function.
Students wrote an accurate equation for each payment plan.
Students wrote an equation but it was not accurate. They were able to explain their reasoning behind the equation.
Students wrote an inaccurate equation and were not able to explain their work.

Other assessment evidence to be collected:
The students will take a quiz on slope and intercepts.

Process check
The student will demonstrate, either by written explanation or oral explanation/example, the relation between the slope of a line and
the x,y-intercepts.

Stage 3: Plan Learning Experiences and Instruction
Learning Activities:
Read "0ne Grain of Rice" by Demi to students after they have completed the task. Ask them to predict the outcome
of the story.

Introduce unit with essential question. Brainstorm examples of mathematical patterns and how those patterns help
make predictions. Include a discussion of stock market rates and how the stock market works.

Students follow a particular stock for one month, charting and graphing the stock's highs and lows. Based on the
month's results, predictions are made, recorded, then actual results are compared. (This is ongoing throughout the
unit).

Use current legislation and news to parallel specific stocks. Determine how legislation and daily news drive particular
stocks.

Review graphing points on a coordinate graph. Students are to graph points and name points already on a graph.

Performance Task: What will you Choose?

Introduce the concept of positive and negative slope. Give students individual coordinate graphs to form lines with
the slope called out by teacher.(This is a hands on activity to get them to understand positive and negative slope.
Directions for making these graphs are in the notes and a very easy!)

Algebra Aerobics
Teach students to show simple linear equations with their arms. For example: the equation X=3 could be shown with
arms going vertical and students move 3 steps to the right (they would move left for negatives). The equation Y=-5
could be shown with arms extended horizontally and students squat to show that the line is below the X axis.
Students will then be able to show what other equations look like and understand what a slope does to a line.
This is a great "warm-up" for class each day!

Teach a lesson on how to "count" slope from a graph. Have students work in pairs with their individual graphs and
create slope problems for each other by placing the lines in various positions on the graph.

Teach a traditional lesson on finding the slope of a line using the slope formula.

Performance Task: Screaming Slopes (A traditional quiz may be given beforehand on calculating/counting slope)

Review graphing lines using xy charts. Move into a lesson on how to write an equation in slope-intercept form. Model
the process using individual students graphs. Call out problems such as: "Show me a line with a slope of 2 and a y-
intercept of-4". Then model the equation of the line or have students write to equation of the line.

Teach Lesson on the topic of linear and nonlinear equations. The main points of instruction will be parabolas,
identifying a linear/nonlinear equation based on degree of the variable, x-y chart patterns, and the vertical line test.
Be sure to include many types of equations including one-step, two-step, and equations with variable on both sides of
the equal sign.

Culminating Performance Task: Force and Motion with Slope

Notes:    Directions for making individual student graphs: Make small copies of rectangular coordinate graphs and
glue on constructions paper. Graphs should be about 4" by 5". Laminate these graphs. Cut overhead
transparency film or excess laminating film into small strips. Using a permanent marker and ruler, draw a
line that is long enough to go the length of each student graph. These fit nicely into index card containers
for storage. After teaching positive and negative slope, you can then ask students to "show" a positive
slope, negative slope, zero slope, or undefined slope and can quickly check for understanding.
Key v ocabulary :
coordinate sy stem
ratios
equations
intercept
Slope
slope
parabola
linear
nonlinear
f unction

How are slopes,                           How do patterns
x-intercepts, and                         allow us to make
y -intercepts                          predictions in real
related?                            world situations?

graph linear and
nonlinear
equations

determine the
slope of a line
f rom its graph

predict the x and
y intercepts of a
linear equation

write an equation
in slope-intercept
f orm

perf orm the
v ertical line test
to identif y
f unctions

```
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