TITLE: Systems of Equations - Practical Applications and Problem Solving
TASK DEVELOPER: James Ring
CONTENT AREA AND GRADE:10th and 11th Grade Foundations of Algebra II
SCOPE AND SEQUENCE: Section 8.8,8.9,10.10 Algebra Text
TARGET TEACHING DATE: March 22,23,26,27,28
SCHOOL: John F. Kennedy High School
PATTERNS AND ALGEBRA - GRADE 9-12
STANDARD 4.3 PATTERNS AND ALGEBRA: All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.
Strand A. Patterns: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
3. Use inductive reasoning to form generalizations.
Strand B. Functions and Relationships: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.
2. Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.
- Slope of a line or curve
- Domain and range
- Estimating roots of equations
- Intersecting points as solutions of systems of equations
- Rates of change
Strand C. Modeling: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
1. Use functions to model real-world phenomena and solve problems that involve varying quantities.
- Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)
- Direct and inverse variation
- Absolute value
- Expressions, equations and inequalities
- Same function can model variety of phenomena
- Growth/decay and change in the natural world
- Applications in mathematics, biology, and economics (including compound interest)
Strand D. Procedures: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
2. Select and use appropriate methods to solve equations and inequalities.
- Linear equations - algebraically
- Quadratic equations - factoring (when the coefficient of X2 is 1) and using the quadratic formula
- All types of equations using graphing, computer, and graphing calculator techniques
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- The students will write a job description for each practical application problem.
- The students will write the key words from this section's word wall.
- The students will write sample test problems for this unit.
- The students will use systems of Equations to perform a practical application Real Life problem.
- The students will solve various type of practical application Real Life problems.
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Real World Setting: Chemist
You are a chemist. You are faced with a problem. You must prepare a solution mixing two given solutions. You need to find how much of each given solution should be used to make your new solution. Once you have completed your problem, you will present your solution and the process used to the class.
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Level I: Acquiring Data - Data students will acquire in this standards-based task:
- Vocabulary: Students will learn the definition of acid solution, mixtures, problem solving, quantity, quality, and solution box.
- Skills: Students will use "thought process" in solving problems:
- What do you know
- What are you looking for
- What does it mean
- Processes: Students will follow a multi - step procedure.
Level II: Visualizing Information - Data from Level I that are visualized as information in this standards-based task:
- Organizing: Students will organize their thoughts and work and place in their notebooks.
- Creating patterns: students will use inductive and deductive reasoning skills from patterns found in each of the problems.
- Creating meaning: Students will analyze problem to determine the system of equations to use.
Level III: Applying Knowledge - Visualized information from Level II that is applied knowledge in this standards-based task:
- Making decisions: Students will determine type of solution to use in solving systems of equations.
- Solving problems: Students will generalize this specific example and others like it to a hypothesis about mixture problems.
- Creating solutions: Students will create a solution box for solving practical application solution and mixing problems.
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Student Involvement - The students will complete the task individually and as a cooperative group in a whole class group setting
Instruction - Activities will be organized and delivered by differentiating the activities or strategies to offer appropriate ways for students to learn, as a teacher-facilitated set of hands-on activitiesin a student booklet during class time
Special Education Accommodations - Students with special needs will require the following electronic devices: Calculators
Special Education Accommodations - Students with special needs will team up with a student partner.
Special Education Accommodations - Students with special needs will require extra processing and response time, written copies of orally presented instructional or assessment materials, and written or photocopied notes of orally presented instruction or assessment materials
Use of Resources - The school will provide classroom materials such as pencil, paper, notebooks.
Use of Resources - The students will provide classroom materials such as pencils, paper, notebooks and calculators.
Customer for Student Work - The student will present their work as evidence of task completion to peers,
teachers, and administrators.
Assessment of Student Work - The following people will be involved in assessing student work generated to complete the task: The student's teacher and peers.
Reporting Results - The assessment results will be reported as a score point on a rubric and
as a letter grade.
Timeline - The estimated time needed to plan, teach, and score this task is from 8 to 10 class periods.
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Activity 1: Review yesterday's lesson (Est. time: 10 min)
A) Ask students to define "Thought Process" in solving practical application problems.
- What do you know
- What are you looking for
- What does it mean
B) Everyone should know what the problem is about and what you are looking for.
C) Review IMAX MOVIE PRICES homework problem.
Technology for this activity: Calculators
Materials for this activity: Notebooks, pencils, and Practical Applications Problem Packet.
Student product or performance for this activity: Students will produce yesterday's HW, provide procedures used, and write new procedures in their notebooks.
Scoring tool for this activity: Teacher checklist
Activity 2: Mixture of Solution Problem (Est. time 10 min)
The teacher directs students to read the problem(#6 in packet) aloud. Ask a student to start the process.
DO NOT START IT FOR THEM.
WAIT for them; they will get it started.
What you know: Label unknowns "x" and "y", a skill learned in past lessons.
"Thought Process" - from English sentences (the problem) make Mathematical sentences. The teacher will prompt them by asking and reinforcing "What are Math sentences?"Their response will be Math. sentences are EQUATIONS.
So let's make equations.
Activity 3: Synthesis and Analysis of Problem ( Est. time 20 min)
A) Have them reread the problem aloud and ask for the equations.
B) WAIT for them to determine what sentences will be used to determine the equations to be used in solving the problem.
C) Write 1st equation. x + y = 200. Ask what that represents? The response you are looking for is "quantity" - amount of stuff.
D) Ask "what is the 2nd equation?" and "what does it represent?"
E) WAIT! They will have trouble with this part which leads us to Activity 4.
Activity 4: Side Bar Example (Est. time 10 min)
A) Give this problem: you have 6 quarters and 3 dimes. What do you have? Let them answer.
B) WAIT!! Let them answer. A majority will respond $1.80.
C) Ask them to now think like a 1st grader and the response will be 9 coins.
D) Let them see that BOTH answers are correct.
E) Have them write equations:
- 6 + 3 = 9 quantity of stuff
- 6(.$25) + 3($.10) = $1.80 quality or value of stuff
F) Tell them to use this side bar exercise in figuring out what the 2nd equation in their problem will be
G) WAIT!!!!! They will come up with it.
Activity 5: Using Applications to Solve (Est. time 10 min)
A) Ask for both equations:
- An equation about "stuff" will be:
x + y = 200
- An equation about "value" will be:
80%(x) + 30%(y) = 62%(200)
B) Once you have both equations, change % values to decimal values
C) Ask how to remove decimals? Their response will be multiply by 100. A skill previously learned.
D) With both equations in "workable" (easier terms) form allow them to complete the "thought process" by solving these two equations. a skill they possess from previous lessons.
Activity 6: Recap the Problem (Est. time 10 min)
A) Once solutions are found ask various students to present their work and give the "thought process" they used in solving this problem
B) This will ensure they know the process needed to solve these problems
Activity 7: Discussion (Est. time 10 min)
A0 Initiate a group discussion of the process in solving these types of Real Life practical application problems
B) These discussions should lead to the presentation of a solution box for mixture problems
Materials for Activities 2-7: Notebooks, pencils, calculators, and Practical Application Packet (included below).
Scoring Tool for Activities 2-7: Assign problem 7 or 8 for homework and check notebooks
PRACTICAL APPLICATION PACKET
Applications and Problem Solving
- Pizza and Soda Prices. A campus vendor charges $3.50 for one slice of pizza and one medium soda and $9.15 for three slices of pizza and two medium sodas. Determine the price of one medium soda and the price of one slice of pizza.
- Film Processing. Cord Camera charges $9.00 for processing a 24-exposure roll of film and $12.60 for processing a 36-exposure roll of film. After Jack's field trip he took 17 rolls of film to be developed at Cord Camera. He paid $171 for processing the film. How many rolls of each type were processed?
- IMAX Movie Prices. There were 322 people at a recent showing of the IMAX 3D movie "300". Admission was $8.75 each for adults and $6.00 each for children. Receipts totaled $2531.50. How many adults and how many children attended?
- The Butterfly Exhibit. On one day during a weekend, 1630 people visited The Butterfly Exhibit at White River Gardens in Indianapolis, Indiana. Admission was $7 each for adults and $6 each for children. The receipts totaled $11,080. How many adults and how many children visited that day?
- Printing. A printer knows that a page of print contains 830 words if large type is used and 1050 words if small type is used. A document containing 11,720 words fills exactly 12 pages. How many pages are in large type? In small type?
- Mixture of Solutions. A chemist has one solution that is 80% acid (that is 8 parts acid and 2 parts water) and one solution that is 30% acid. What is needed is 200L of a solution that is 62% acid. The chemist will prepare it by mixing the two solutions. How much of each should be used?
- Paint Mixtures. At a local "paint swap," Kari found large supplies of Skylite Pink (12.5% red pigment) and MacIntosh Red (20% red pigment). How many gallons of each color should Kari pick up in order to mix a gallon of Summer Rose (17% red pigment).
- Candy Mixtures. A bulk wholesaler wishes to mix some candy worth $.45 per pound and some worth $.80 per pound to make 350 lb. of a mixture worth $.65 per pound. How much of each type of candy should be used?
- Coffee Blends. Cafebucks coffee shop mixes Brazilian coffee worth $19 per pound with Turkish coffee worth $22 per pound. The mixture is to sell for $20 per pound. How much of each type of coffee should be used in order to make a 300 lb. mixture?
- Test Scores. You are taking a math test in which items in part A are worth 10 points and items in part B are worth 15 points. It takes 3 min. to complete each item of part A and 6 min. to complete each item in part B. The total time allowed is 60 min. and you do exactly 16 questions. How many questions of each part did you complete? Assuming that all your answers were correct, after all you had a GREAT Math Teacher, what was your score?
- Octane Ratings. In most areas of the United States, gas stations offer three grades of gasoline, indicated by octane ratings on the pumps, such as 87, 89, 93. When a tanker comes and delivers gas, it brings only two grades of gasoline, the highest and the lowest, filling two large underground tanks. If you purchase the middle grade, the pump's computer mixes the other two grades appropriately. How much of 87-octane gas and 93-octane gas should be blended in order to make 18 gal of 89-octane gas?
- Suntan Lotion. Lisa has a tube of Kinney's suntan lotion that is rated 15 spf and a second tube of Coppertone that is 30 spf. How many fluid ounces of each type of lotion should be mixed in order to create 50 fluid ounces that is rated 20 spf?
Phone Rates. Recently, AT&T offered an unlimited long-distance calling plan to anyone in the United States, 24 hours a day, 7 days a week, for $29.95 a month. Another plan charges $.07 a minute all day every day, but costs an additional $3.95 per month. For what number of minutes will the two plans cost the same? Remember "Thought Process."
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Student Performance One: Problem solving "thought process"
Assessment Benchmarking Example: The student's ability to make "math sentences" (equations), properly list unknowns, and solve equations.
Student Performance Two: Solving Practical Applications
Assessment Benchmarking Example: The student will produce a valid solution. Each step will follow logically from previous steps and each step will correctly identify the methods used.
Student performance Three: Group discussions.
Assessment Benchmarking Example: The student will participate actively in the group discussions. The student's comments will be mathematically valid, relevant, and courteous to classmates and teacher.
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High School Mathematics Rubric: Extended Constructed Response
|Level 4: A||The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.|
|Level 3: B|
The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.
|Level 2: C|
The response indicates an incomplete application of a reasonable strategy that may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.
|Level 1: D|
The response indicates little or no application of a reasonable strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.
|Level 0: F||The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know."|
|Notes: Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.|
Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.
Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.
Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.
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Cognitive Information: I will collect the following information in a survey at the end of the unit.
- Describe what skills you needed to complete this task.
- Explain how you solved the goal, problem, or issue in this task.
- Give "thought process" used that helped you solve this task.
- Explain why you completed the task your way.
Attitude Information: I will collect the following information in a survey at the end of the unit.
- Do you feel that you are good in performing practical applications of systems of algebraic equations?
- Did you find this task to be difficult?
- Do you still find it a difficult task?
- Did you see the usefulness of what you were asked to do in real life?
- Did you enjoy the task?
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Analyze: I will examine the data in my chart to look for trends, contributing factors, and implications of student performance over a series of assessments of the same learning standard.
Trends: I will look for improvement relative to previous lessons which included practical applications.
Reflect: I will consider two or more of the following stems to reflect on the results and instructional practices I used and others I might benchmark and apply in the future. Then, I'll write a brief summary about my findings, contributing factors, and implications for improvement.
As I relate my students' results with my lesson activities, I noticed that having the students understand the "thought process" and perform solving practical application problems has the most promise for becoming a best practice in my classroom because I find that the students have a better retention of algebraic concepts if they have a chance to see, understand, and perform real life practical applications rather than just reading, hearing, and copying them.
This connects to previous and subsequent lessons in the chapter on Systems of Equations. The students are becoming familiar with the techniques of problem solving and are using them correctly.
Action Plan: I will complete the following TaskBuilder Figure 8 Strategy Action Plan to prepare for my next standards-based task.
1. Plan - My next standards-based task will focus on:
- Title: Rational Expressions in Algebra
- Content Area:Algebra (10th and 11th grade)
- Learning Standard(s):NJCCS
- Intent:Define the concept of rational expressions and apply it to the simplification of algebraic expressions and practical real life applications.
5. Team or Grade Level Portfolio and School Web Site - I will insert the standards-based instruction or assessment task, results, samples of student work, and summary into my Team or Grade Level Portfolio and upload them to my School's Instructional Web Site on the following dates:
Target date for School Instructional Web Site:May 18, 2007
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