| Curriculum Map 2006-2007 | |||
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The Dwight School |
| Period | Content | Purpose/ Objectives | Activities & Resources | Areas of Interaction | Assessments | |
| Linear Equations and Functions : | Functions and their graphs, slope and rate of change, Writing equations of lines, correlation and best fitting lines. Linear Inequalities in two variables, piecewise functions, absolute value functions. |
In this section the goals are to graph ordered pairs, relations, functions, linear equations and inequalities in two variables, piecewise functions and absolute value functions, how to write equations of lines and to solve real life problems using graphs and equations. |
Graphing equations that show a linear function in situations involving collecting state taxes, predicting future land covered by rainforests, and writing equations showing the number of minutes a sailboat can use its sails and its motor to get to an island. |
A study of transatlantic voyages. This involves History and Math. The Titanic, the length of its voyage, the equation relating its distance to NYC and identifying its domain and range. What would the graph of this equation look like? (HF) How can you predict membership enrollments for an organization? Why create linear models? What is the scope of understanding and interpreting rate of change in real-life situations? How are linear equations related to other fields of study? |
Formative: Daily hwk problems for the selected section. Writing, explain what lines have a slope and what lines dont and why. Summative: End of section test where students graph a relation and tell whether it is a function, evaluate a function at a given value of x, write and equation of a line given characteristics , and answer situations like finding the acceleration of a rollercoaster during a time, interval, and drawing a scatterplot to show the correlationa nd best fitting equation of a line between patents issued to residents since 1985. |
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| Systems of Linear Equations and Inequalities : | Solving Linear systems by graphing. Solving linear systems algebraically. Graphing and solving systems of Linear inequalities, Linear programming, Solving linear equations in three variables. |
This section includes learning how to solve linear systems in two or three variables by graphing and by using algebraic methods, and how to write and use linear systems to solve real world problems. |
Science Connection: Weights of atoms and molecules are measured in atomic mass units. We will look at a molecule of ethane and a molecule of propane and their individule components and then use linear systems to algebraically find the weights of the carbon and the hydrogen atom. |
How is the new time saving linear programming method of Narendra Karmarkar used by industries such as telephone companies, airlines and manufacturers to allocate their resources? A look at the changes of linear programming from the 40's until now.(HF,ATL) How is a linear equation graphed? How do you make sense of a linear relationship? How can you combine swinmming and inline skating to burn 300 calories? Where do we experience inequalities in life? Why study systems? |
Formative:Daily Hwk examples from this section as well as graphing exercises on the calculator. Summative: End of section test where students will graph a linear system and tell how many solutions it will have. Solve a system using any algebraic method, for two and three variables, and solve problems involving business applications like finding the least number of freezers a business beeds in order to minimize costs given their parameters. |
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| Quadratic Functions : | Graphing quadratic functions. Solving quadratic equations by factoring, solving quadratic equations by square roots, completing the square, the quadratic formula and the discriminant. Graphing and solving quadratic inequalities. |
Many real life situations can be modeled by quadratic equations. This section will show four ways to solve quadratic equaitons and how to graph quadratic functions and inequalities. |
Graphing calculator activity: Given a quadratic equation studetns will use the minimum and maximum functions on their calculator to detect these points. Then apply this to determine on a single lane highway what traffic concentration is traffic flow minimized and what is the maximum flow. |
What is the best fitting quadratic model for given data? Using the concepts of creating a quadratic we will write equations of quadratics based on given data for situations that involve fuel economy, botany, transportation, running, agriculture,and baseball. (HF, H&S, E, ATL) |
Formative:Daily hwk examples plus graphing exercises. Summative: End of section test, students will be asked to graph a quadratic function, factor a quadratic expression, solve a quadratic using the appropriate method. Use the discriminant to determine if the graph of a quadratic will pass the x- axis. They will have to write a quadratic model to determine a premium insurance function based on certain characteristics. |
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| Polynomials and Polynomial Functions : | Using properties of exponents, Evaluating and graphing polynomial functions, Operations with polynomials, Finding rational zeroes, analyzing graphs of polynomial functions. |
In this section the students will learn to perform operations with polynomials and how to evaluate, graph, and find zeroes of polynomial functions. |
Exploring finite differences: How are the finite differences for a polynomial function related to the function's degree? for given functions students will follow 4 steps and then draw conclusions for each function giving the number of paths through a gris and for each function state the degree and the number of times differences were calculated, and decide what they notice. Then answer which order differences will be constant ofr a given function. |
What type of function models the speed of the space shuttle? (HF,ATL) How will my knowledge of this material help me with more complex mathematics? |
Formative: Daily hwk assignment problems Summative: End of chapter test where students will simplify an expression and tell which properties of exponnets were used. Tehy will perform an indicated operation, and solve an equation, identify local maxima and minima, and show that a fourth degree polynomial has non zero constant fourth order differences. Thy will aslo be given a function based on a situation and they will have to write a polynomial function that patterns the one given. |
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| Powers roots and Radicals : | Nth roots and rational exponents, properties of rational exponents, inverse functions, graphing cube root functions, solving radical equations, statistical graphs. |
The purpose of this section is to show how to use rational exponents and the nth roots of numbers, and how to perform operations with and find inverses of functions, and also how to solve radical equations. |
Model amusement park rides with circular features using a radical function and then figure out the radius, also model the shoulder height of elephants using a cube root function and use that to determine the age of the elephant. Geometry connection: Study the equation for the volume of aright cylindrical cone and then find its surface area with the found information. |
Biology Connection: Studying the structure of animals.How can the structure of an animals form be represented mathematically? They will study the forms of antelopes and find that the measurement of an antelopes bone is a raation power and they determine the length of the bone understanding the discovered relationship. (E, H&S) How can you estimate the weight of a dinosaur? How do you apply your knowledge of radicals in problem solving involving distance and geometric situations? |
Formative:Daily hwk problems assigned from the current section Summative: End of section test where students will evaluate and simplify expressions, and perform indicated operations and state the domain of composing a function, find the inverse of a function , and be given a spread of data that they will use to determine measures of central tendency and dispersion. |
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| Exponential and logarithmic Functions : | Exponential growth, exponential decay, the number e, logarithmic functions, properties of logarithms, Solving exponential and logarithmic equations. Logistic growth functions. |
The goal of this section is to graph exponential, logarithmic functions and use the number e and the definition and properties of logarithms, and finally how to solve exponential and logarithmic equations. |
Writing models, where students write an exponential decay model that describes a situation from depreciation to population growth to medicine, and radioactive decay of plutonium. Writing questions: Is the product of two exponential decay functions another decay function. Justify your answer verbally in a written statement that includes mathematical proof. |
What relationships exist between exponential growth and exponential decay when a piece of paper is folded repeatedly?(ATL) How does altitude affect the air? (E) How do you apply the laws of exponents in the cocntext of factoring polynomials? |
Formative; Daily hwk problems. Summative: End of section test where students will write an exponential decay model for the value of a car's depreciation and figure out how many years it will be before it reaches a certain value. Also, take two comparing values and draw a scatterplot and see if an exponential model is a good fit for the original data, and then do some predicting after a period of time of specific data. |
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| Rational Equations and Functions : | Graphing simple rational functions, graphing general rational functions, multiplying and dividing rational expressions, adding and subtracting complex fractions and solving rational equations. |
Students will learn how to simplify and perform operations with rational expressions, graph rational functions and solve rational equations that model real life situations. |
Science connection: Look at fulcrums and how for a balanced lever the distance an object is from the fulcrum varies inversely with the objects weight. We will calculate how far the fulcrum has to be from the object to keep the lever balanced. |
A look at Deep water diving. The discovery of the diving bell, and the rational function used to determine the percent of oxygen used by a diver. Graph the relationship. (HF, H&S, E). How have the technological advances for diving changed the function used to determinea diver's reccommended percent oxygen? How does volume and surface area affect a skydiver's speed? |
Formative: Daily set assignments. Summative: End of section exam where students will graph a rational function perform an indicated operation and simplify the result, simplify a complex fraction, and solve an equation using any method. |
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| Probability and Statistics : | The fundamental counting principle and permutations. combinations and probability of compound events, probability of independent and dependent events, binomial distributions and normal distributions. |
This section will show how to count the number of ways an event can happen, and how to calculate and use probablilities, and finally how to use binomial and normal distributions. |
Graphics calculator activity: Generating Random numbers: Students will learn how to use the random number generator to record numbers, and then we will use this to simulate the drawing probablility of a deck of cards with replacement. |
In how many ways can you attend part of a summer concert series? (H&S) Investigate binomial distributions: What are the characteristics of a histogram that displays the results of a binomial distribution? (ATL) Are these events really independent? What constitutes the best fit line? |
Formative: Daily practice sets. Summative:End of section test where studetns must find the number of permutations or combinations, expand the power of a binomial and find the indicated probability depending on the given characteristics. Also, solve problems related to situations with the supreme court astronomy and health involving probablilities. |
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| Trigonometric ratios, functions and their graphs : | Right triangle trigonometry, general angles and radian measure, trigonometric functions of any angle, the law of Sines, the Law of Cosines, graphing sine cosine and translations and reflections of these graphs. |
How to evaluate trigonometric functions and inverse trigonometric functions, and how to find side length and angle measure, and areas of triangles, and also draw the periodic curves of these functions. |
Cartography, with specific focus on Columbus' voyage. Using old beliefs we will calculate the supposed radius of the earth and true distances from the Canary Islands to japan and discuss how aerial photography is what is used now to make accurate maps. |
How can you find the width of an opening between two halves of a bridge? (HF,E) Study of the Duquense Incline, determining height of its rise and vertical speed. (HF) How can you apply the rules of trigonometry in other areas of mathematics as an aide to solve problems? |
Formative:Daily practice sets with assigned problems. Summative: End of section test that will include having the students evaluate the six trigonometric functions fo theta, rewrite degrees in radians, solve for a missing side length or angle and find areas of non right triangles and draw curves based on given functions. |
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