School Administrative District #4

School Administrative District #4

Unity of purpose

CORE CURRICULUM: SUPPLEMENTAL PAGES

Department: Mathematics Department

APPLIED MATHEMATICS: A PROGRAM MODIFICATION

The applied mathematics program at Piscataquis Community High School is designed for students, 9-12, who need a more practical approach to day-to-day instruction than the traditional abstract theoretical approach used in most other mathematics courses. The focus is on helping students develop and refine job-related mathematics skills. The course includes material from each of the ten core curricular strands in mathematics, but the emphasis is placed on the ability of students to understand and apply functional mathematics to solve problems in the world of work. The teacher, therefore, acts as a facilitator as students discover concepts in a laboratory environment.

There are three courses in the applied mathematics program, and each course is made up of units comprised of six elements: (1) student interaction with text, (2) videos showing the application of mathematics to the world of work, (3) skill drill, (4) lab activities, (5) problem solving, and (6) unit tests. The problem-solving component consists of forty problems for each unit, and each set of problems addresses six occupational fields: (1) general, (2) agricultural/agribusiness, (3) business and marketing, (4) health occupations, (5) home economics, and (6) industrial technology.

Since the applied mathematics program is designed to meet the same content standards as the core curriculum in mathematics, no new content standard nor performance indicators are offered here. Instead, a list of the course content across the three courses is provided below:

Modular Units of Study	Strand Connections
¬ Getting to know a calculator	Computation
¬ Naming numbers in different ways	Numbers and number sense Computation
¬ Finding answers with a calculator	Computation
¬ Learning problem-solving techniques	Across all strands
¬ Estimating answers	Computation
¬ Measuring in U.S. customary and metric units	Measurement
¬ Using graphs, charts, and tables	Data analysis and statistics
¬ Dealing with data	Data analysis and statistics
¬ Working with lines and angles	Geometry
¬ Working with shapes in two dimensions	Geometry
¬ Working with shapes in three dimensions	Geometry
¬ Using ratios and proportions	Computation
¬ Working with scale drawings	Measurement
¬ Using signed numbers and vectors	Computation Measurement
¬ Using scientific notation	Computation Measurement
¬ Precision, accuracy, and tolerance	Measurement
¬ Solving problems with powers and roots	Computation
¬ Using formulas to solve problems	Algebra Geometry
¬ Solving problems that involve linear equations	Algebra
¬ Graphing data	Data analysis and statistics
¬ Solving problems that involve nonlinear equations	Algebra
¬ Working with statistics	Data analysis and statistics
¬ Working with probabilities	Probability
¬ Using right-triangle relationships	Geometry Patterns, relations, and functions
¬ Using trigonometric functions	Patterns, relations, and functions
¬ Factoring	Algebra
¬ Patterns and functions	Patterns, relations, and functions
¬ Quadratics	Algebra
¬ Systems of equations	Algebra
¬ Inequalities	Algebra
¬ Geometry in the workplace	Geometry
¬ Solving problems with computer spreadsheets	Data analysis and statistics Discrete mathematics
¬ Solving problems with computer graphics	Data analysis and statistics
¬ Quality assurance and process control	Data analysis and statistics Discrete mathematics
¬ Spatial visualization	Geometry
¬ Coordinate geometry	Geometry
¬ Logic	Mathematical reasoning
¬ Transformations	Geometry

School Administrative District #4

Unity of purpose

CORE CURRICULUM: SUPPLEMENTAL PAGES

Department: Mathematics Department

PRE-CALCULUS: A PROGRAM MODIFICATION

Calculus is an abstract system consisting of certain assumptions, called axioms or postulates, and some of the logical consequences of those assumptions, called theorems. The assumptions at the foundation of calculus are a set of statements about real numbers; however, calculus uses properties of real numbers largely ignored in other mathematics courses, to develop the notions of limit and of limiting processes, derivatives, and integrals. For students who have mastered the core in twelve or fewer years and who want to pursue post-secondary training requiring a beginning knowledge of calculus, a course in pre-calculus is the ideal culmination of a high-school mathematics program. Such a course utilizes the knowledge and skills acquired in previous years and applies them to diverse problems and situations encountered in the physics, engineering, economics, and biology. In S.A.D. #4, pre-calculus is one of several modified program offerings developed to meet the needs of students who need remediation or acceleration.

High-school students come to pre-calculus at many levels of intellectual maturity and with varying amounts of mathematical motivation. For this reason, the course provides a review of many concepts taught in algebra, geometry, functions, statistics, and trigonometry and leads to a discussion of concepts and methods used in proofs. The emphasis in the course, however, is on techniques of problem-solving, not on rigorous theory.

CONTENT STANDARD: PRE-CALCULUS

Students will understand and apply concepts from pre-calculus. Calculus makes use of plane geometry, algebra, and the notion of limit and limiting processes. From the idea of limit arises the two principal concepts that form the nucleus of calculus: the derivative and the integral. The derivative can be thought of as a rate of change (e.g., velocity of objects, maximum and minimum values of a function) and is important in such disciplines as physics, engineering, economics, and biology. The integral enables people to calculate areas of plane regions whose boundaries are curves other than circles and to calculate volume, centers of gravity, lengths of curves, work, and hydrostatic force.

Performance Indicators: The learner will·

Grade 12

1. Review and extend prior learning

2. Develop sketches of functions

á Circle

á Parabola

á Hyperbola

á Ellipse

á Logarithms

á Exponential

3. Understand the conceptual foundation of limit

á Convergent

á Divergent

á Infinite

4. Find and use limits of various functions

á Find limits by direct substitution

á Find limits of a function as the independent variable approaches infinity

á Find limits of a function by examining its graph

á Recognize the limits of special functions

á Recognize the limits that have non-existent limits

5. Differentiate

á Monomials

á Polynomials

á Logarithms

á e^x

á Inverse functions

á Trigonometric functions

á Derivatives of higher order

6. Use techniques of differentiation

á Product Rule

á Quotient Rule

á Chain Rule

á Implicit Rule

á L'Hopital's Rule

7. Apply concepts of derivative

á Find slope of a curve

á Find the tangent line to a curve

á Use the differential to approximate values

á Use Newton's method to approximate zeros of a function

á Determine where a function is increasing and where it is decreasing

á Find critical points, relative and absolute maximums, and minimum points

á Determine the concavity and points of inflection of a function

á Use the graph of the derivative of a function to identify information about the function

á Solve extreme value problems

á Find velocity and acceleration of a particle moving along a line

á Find average rates of change

á Find instantaneous rates of change

á Determine related rates of change

8. Integrate

á Monomials

á Polynomials

á Logarithms

á e^x

á Inverse functions

á Trigonometric functions

9. Use techniques of integration

á Integration by parts

á Trigonometric substitutions

á Partial fractions

á Integration by tables

á Symbolic integration

á Trapezoidal Rule

á Simpson's Rule

10. Apply derivative and integrals to investigate

á Mean value

á Exponential growth and decay

á Extreme values

á Concavity and inflection points

á Limits of infinity

á Graphing

á Volume of a solid of revolution about the axes or lines parallel to the axes

á Length of curves

á Area under a curve using rectangles or trapezoids

á Area between curves

á Work

á Gravity

á Hydrostatic force

School Administrative District #4

Unity of purpose

CORE CURRICULUM: SUPPLEMENTAL PAGES

Department: Mathematics Department

Students of all ages solve problems every day. Often these problems involve the use of mathematics and demand that students follow a process to find solutions. The following four-step process is recommended:

Problem-solving Steps

æ Understand the problem:

á What do you know?

á What do you need to find out?

á What do the words used mean?

á What information is important?

unimportant?

á What information, if any, is missing?

á How can missing information be found?

á Can the problem be simplified?

á Develop a plan:

á Have you organized your information?

á What process(es) and/or operation(s) are needed?

á Have you ever solved a similar problem?

á What strategies can you use?

á What is an estimate for the answer?

ü Solve the problem:

á Do you need to try another strategy?

á What is the solution?

Õ Look Back (check the answer):

á Did you answer the right question?

á Is your answer reasonable?

á Did you test your answer?

á Is there more than one solution?

á Does your answer contain the proper unit labels, if needed?

á Could you have solved the problem another way?

PROBLEM-SOLVING STRATEGIES
Strategies	Examples
¬ Use manipulatives or act it out				"Here's one for you and you and you. Here's another one for you and·"				Mrs. Brown has six apples and three students. How many apples would she give each student if she passed them all out?
¬ Draw a picture or a diagram								The Hawaiian volcano Mauna Kea is 33,480 ft. tall. The Hawaiian volcano Kilauea rises 4,077 ft. above sea level. Mauna Kea has 19,684 ft. of its height below sea level. How much farther above sea level is Mauna Kea than Kilauea?
¬ Look for a pattern								A display of boxes has 1 box in the top row, 2 boxes in the second row, 3 boxes in the third row, and so on. If there are ten rows in all, how many boxes are in the display?
¬ Classify	s A triangle has three sides and is, therefore, not a quadrilateral.							A quadrilateral is a four-sided figure. Is a triangle a quadrilateral?
¬ Make an organized list	Dale Rock Rock Rock Paper Paper Paper Scissors Scissors Scissors			Ali Rock Paper Scissors Rock Paper Scissors Rock Paper Scissors			Winner None Ali Dale Dale None Ali Ali Dale None	Dale and Ali play Rock, Paper, Scissors. Each player plays by putting out either a fist (rock), a flat hand (paper), or two fingers (scissors). Paper defeats (covers) rock, rock defeats (dulls) scissors, and scissors defeats (cuts) paper. No one wins if each person plays the same item. How many different ways can Dale and Ali play one game.
¬ Make a table	Year 1990 1991 1992 1993 1994		Redwood (height in feet) 6 8 10 12 14				Willow (height in feet) 2 5 8 11 14	In 1990, Beth bought a 6 ft. tall redwood seedling. The tree grew 2 ft. each year. In 1990, she bought a 2 ft. tall willow seedling. The tree grew 3 ft. each year. In what year were the trees the same height?
¬ Guess and check	Guess #1: .25 x 3 = .75 .10 x 3 = .30 .05 x 3 = .15 1.20 Guess #2: .25 x 4 = 1.00 .10 x 4 = .40 .05 x 4 = .20 1.60							At a library book sale, Mary used an equal number of quarters, dimes, and nickels to buy a book for $1.60. How many of each coin did she use?
¬ Work backward	ü Karen took 8, leaving none á Seth took 2/3, leaving 8 ¦ Laura took half, leaving 24							Laura, Seth, and Karen volunteered to wash a box of test tubes in the chemistry lab. Laura took half the total, Seth took two-thirds of what was left, and Karen took the remaining 8. How many test tubes were in the box?
¬ Use logical reasoning								Polly, Ike, and Fred have a dog, cat, and a bird for pets, though not necessarily in that order. Ike is allergic to feathers. The dog owner is Polly's friend and Ike's classmate. Match the owners with their pets.
¬ Check for reasonable-ness	100,000 km is more than 4,000 km. Since the Bells have only completed part of their trip, they cannot have traveled 100,000 km.							The Bell family is on vacation visiting national parks. They plan to travel about 4000 km. Mr. Bell said that they had completed about .25 of their trip. One of the children said, "That's 100,000 km!" Mrs. Bell said, "100,000 km is not reasonable." Was Mrs. Bell correct?
¬ Solve a simpler problem	x x x	You could sketch all 25 squares and count the 1 m sections of fence, but all you need to do is look at a three squares. The first square would require 4 m of fence; the second square and all squares thereafter would require 3 m of fence. 4 + (24 x3) = 76						Farmer Jones plants Christmas tree seedlings in square plots that are side by side. Each square side is a 1 m section of fence. What is the total length of fence needed for a row of 25 tree plots?
¬ Use a formula	P = 2 x (l + w) = 2 x (54 + 36) = 2 x 90 = 180 cm							Tom needs to put molding around the outside of a picture 54 cm long and 36 cm wide. Use a formula to find the perimeter of the picture.
¬ Write an equation	c = number of cats 2c + c = 12 3c = 12 c = 4 2c = number of dogs 2 (4) = 8							There are twice as many dogs as cats in Mr. Black's pet shop. There are a total of 12 dogs and cats. How many dogs are there?
¬ Choose the operation	17 + 9 + 12 + 23 = 61 17 - 9 = 8 15 x 9 = 135 3,082 h 23 = 134							When Camp Granola posted the sign-ups for activities, 17 campers had signed up for volleyball, 9 for hiking, 12 for dramatics, and 23 for crafts. How many campers signed up for an activity? How many more chose volleyball than hiking? If each hiker picked 15 wild flowers, how many flowers were picked in all? If there are 3,082 beads for crafts, how many beads can each camper have?
¬ Use estimation	Estimate: $ .90 .70 .50 6.00 $8.10							You have $10. Can you buy a loaf of bread for $.92, a pint of milk for $.65, a bottle of juice for $.54, and a package of chicken for $6.17?
¬ Be aware of too much information	$7.25 1.20 6.05 $14.35 million dollars for the town budget÷unnecessary information.							The annual budget for the town of Lowton is 14.35 million dollars. $1.2 million is spent on the town's recreation program. Another $7.25 million is spent for education. About how much more is spent for education than for recreation?
¬ Be aware of too little information	You cannot solve this problem without information about last year's budget.							The annual budget for the town of Lowton is 14.35 million dollars. $1.2 million is spent on the town's recreation program. Another $7.25 million is spent for education. How much more did Lowton spend on recreation last year?
¬ Be prepared to apply more than one step (two- or multi-step problems)	Step #1: 15 x 12.50 = 187.50 Step #2: 5 x 26.99 = 134.95 Step #3: 187.50 + 134.95 = 322.45 Step #4 322.45 + 10 = $332.45							The Hampton Athletic Club is buying new soccer equipment from Gold Medal Sporting Company. They are ordering 15 soccer uniforms for $12.50 each and 5 soccer balls at $26.99 each. They must pay a $10 shipping and handling fee. What is the total cost of the order?
¬ Interpret the answer	150 h 60 = 2 ¸ Answer: 3 boats (You can't use half a boat!)							There is room for 60 people on a boat. How many boats are needed for 150 people?
¬ Use rates	D = R x T D = 50 x 5 = 250 mi.							What is the distance in miles covered by a car that traveled at 50 miles per hour for 5 hours?
¬ Use percents	20% = .20 $325 x .20 = $65.00 $325 - $65 = $260.							To attract customers, the Thrifty Travel Agency gives a 20% discount on the cost of a trip. What is the final cost of a trip that costs $325?
¬ Interpret a chart	Wind speed (10 km/h) 18 14 9 5 0 -4 -8 -13					Temperature (^oC) 20 16 12 8 4 0 -4 -8		How cold will it feel if the temperature is 12 ^oC and the wind speed in 10 km/h?
¬ Be aware that some problems are open-ended	CARS (4 wheels) 10 9 8 7 6 5 4 3 2 1 0				MOTORCYCLES (2 wheels) 1 3 5 7 9 11 13 15 17 19 21			Tim counted 42 wheels in a parking lot for cars and motorcycles. What possible combinations of vehicles could he have seen?

School Administrative District #4

Unity of purpose

CORE CURRICULUM: SUPPLEMENTAL PAGES

Department: Mathematics Department

STRATEGIES FOR TEACHING STUDENTS TO CREATE THEIR OWN WORD PROBLEMS

Strategy	Example
Provide a topic and let the students supply the data	If the class has been working on percent, the teacher would suggest the topic and have students supply the data based on the material that has been presented in class.
Provide the numbers and let the students supply the operation.	Ask students to make up a problem, using the numbers 40, 20, and 60.
Provide a situation and let students provide a problem which the situation suggests.	What does a shed that measures five meters by six meters by three meters suggest?
Provide a condition.	The answer must be between 1 ² and 2 ¸ .
Provide experience with manipulatives.
Provide a picture as a basis for a problem.
Encourage students to create a problem around an operation.	Create a problem such that 35 x 5 is the answer.
Provide students with data from a newspaper, menu, etc.		MIKE'S SPECIAL Frankfurter. . . .$1.65 Coke. . . . . . . . $ .85 Frankfurter and Coke. . . . . . . . $2.25
Encourage students to rewrite an existing problem.	Select a problem from the text and have students rewrite it to include too much or too little information or information that would change the answer.

School Administrative District #4

Unity of purpose

CORE CURRICULUM: SUPPLEMENTAL PAGES

Department: Mathematics Department

USING JOURNALS IN THE MATH PROGRAM

In addition to being a place for noting questions, reflections, and insights, a math journal can be a record of a student's progress. It can provide the math teacher important feedback about what a student does and does not understand. Additionally, the math journal is the ideal place to ask students to respond to open-ended questions like those encountered on the Maine Educational Assessment Test.

Suggestions for keeping a journal:

˜ Use a standard hard-covered composition book of 100+ pages.

˜ Record name, course, and teacher's name on the front cover

˜ Date and title all entries.

˜ Write in complete thoughts.

˜ Use the math journal only for math.

Suggested journal topics:

˜ Notes on skills

˜ Notes on how to solve certain problems

˜ Reflections or impressions about math class

˜ Expression of frustration experienced trying to solve a problem

˜ Expression of joy/relief experienced after solving a problem