Authentic Math Applications

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Audubon Public Schools

Engaging Students ~ Fostering Achievement ~ Cultivating 21st Century Global Skills Written By: Patricia Martel

Course Title: Authentic Math Applications Unit Name: Number & Numerical Operations Grade Level: 12

Content Statements

This unit consolidates all of the rules for combining real, imaginary, rational, and irrational numbers including the use of conjugates to rationalize denominators.

Cumulative Progress Indicators (CPI) N-RN.1-3, N-Q.1-3, N-Q.1-3, 5, 6,

Overarching Essential Questions

How can numbers of any kind be combined?

How do we deal with division and imaginary or irrational numbers?

Overarching Enduring Understandings

Numbers, whether real, imaginary, rational or irrational, follow the same set of rules when being combined arithmetically. We are not allowed to divide by zero, irrational numbers and imaginary numbers so there are special procedures in place to deal with these situations.

Unit Essential Questions How are all numbers related?

How can irrational numbers be combined? How can radicals be simplified?

How can radicals be approximated with squares to two decimal places?

What is an imaginary number and how can they be combined?

How are radicals, fractions and exponents related? How can exponents be simplified?

How can I evaluate my answers before and after solving word problems?

How is estimation worthwhile in the real world? How do percents decimals and fractions relate to real world problems?

How do fractional exponents relate to radicals and solving problems?

How can tables be use to organize information from word problems?

Unit Enduring Understandings

The number system is the basis of mathematics. Numbers are used for counting, ordering, comparing and labeling objects in the physical world: Real numbers, Imaginary and complex numbers. Fluency in computation is essential. Computational algorithms use place value and equivalence to simplify

calculations in authentic situations: Fractional and negative exponents and operations with complex numbers.

Unit Rationale

This unit “wraps up” all concepts of computation of numbers from simple arithmetic to exponent rules and links these computations to real, imaginary, complex, rational and irrational numbers. These concepts are frequently found on college math placement tests and can be found in more challenging SAT questions.

Unit Overview

All students will understand the meaning of numbers, how they may be represented and the relationships among them. They will perform computations and acquire knowledge of the physical world from the point of view of quantitative relationships.

Key Terms

Complex Number- a number composed of a real term and an imaginary term

Dimensional Analysis- the use of cancellation and dimensions to determine how to structure a problem Unit Price- cost per oz, pound, 100 count etc. standardized based on item, used to compare like items Resources

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Internet research

www.kuta.com for practice worksheets on exponents and complex numbers Materials gleaned from Chemistry teachers on dimensional analysis

Suggested Activities for Inclusion in Lesson Planning Identify and use the properties of real numbers.

Identify imaginary and complex numbers.

Describe how real and complex numbers are related both arithmetically and graphically (Plot complex numbers as points in a plane.).

Compare real number expressions.

Use the laws of exponents for rational expressions.

Use dimensional analysis to organize conversions and computations.

Simplify numerical expressions with powers and roots, including fractional and negative exponents. Add, subtract, multiply and divide complex numbers.

Find the approximate value for solutions to problems involving cube roots.

Use estimation to judge the reasonableness of results of computations and solutions to problems involving real numbers.

Using dimensional analysis convert rates from metric to standard and hours to minutes or seconds. Review college placement tests application of numeration and computation problems without calculators. Apply these concepts in small groups or paired work on word problems.

Small group project: track frequently purchased food, cleaning, personal care and household items for two weeks. Gather price point data from two to three stores for these items comparing unit, sale and full price. Calculate total cost to purchase comparing sale to full prices and store to store. Use this data and analysis to determine where it is best or better to shop. Use internet research and in person visits. Share results in presentation to class using technology.

Course Title: Authentic Math Applications Unit Name: Data Analysis and Discrete Mathematics Grade Level: 12

This unit includes: methods of collecting and analyzing data, make predictions and decisions based on data and probabilities, and using special constructions like networks to make decisions.

Cumulative Progress Indicators (CPI) S-ID.1-9, S-IC.1-6, S-CP.1-9, S-MD.1-7

How can I use data to make better decisions?

How is probability applied outside of casinos and game strategies?

How can probabilities be shown visually?

Data is constantly being collected, organized and analyzed and the results being used to better market materials, reduce waste, predict future needs etc. There are whole divisions of mathematics devoted to the study of probability whether it is game theory or combinatorics and its applications in politics, business, or marketing.

Unit Essential Questions

How can the method used to collect data affect the data itself?

How can data be analyzed? How can it be displayed visually?

How can I take raw numbers and construct an equation? How can I evaluate the validity of this equation?

Mathematical data can be collected, organized, described and displayed in frequency tables, circle graphs and histograms. Distinct data sets can be compared using statistical measures, such as mean (including weighted averages), median, mode, quartiles and box-and-whisker plots. Probabilities can be estimated by lines of best fit. Expected values and

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How can I use math to decide whether to purchase a warrantee for a product?

How do insurance companies use probabilities to decide how much I pay for a policy?

How can I illustrate probabilities?

probabilities of conditional events can be computed. Algorithms and visual tools, such as networks (vertex-edge graphs) and Venn diagrams can be used to solve problems and answer real-world questions. Certain paths within vertex-edged graphs can be classified as ones which include every edge.

Unit Rationale

This unit builds upon prior knowledge of data collection, analysis and probability and applies it to more intense and engaging applications.

Unit Overview

All students will be able to collect, organize, and display relevant data to answer questions that can be addressed with data; use appropriate statistical methods to develop and evaluate inferences and predictions that are based on data; and apply basic concepts of

probability in order to make informed choices and reasonable decisions about information presented. Key Terms

Expected value- using the probability of an outcome and its price, determine what its value to the user is and based on this cost determine if it is worth purchasing or playing

Normal Distribution- the spread to a set of data that is centrally located and follows a specific pattern making it highly reliable for making predictions and drawing conclusions

Standard Deviation- a measure of the spread or range of the data telling how far apart or close together the set of data is

Resources

Algebra with Pizzazz www.kuta.com worksheets

TI website for explorations with Data and the graphing calculator Media center for use of excel spreadsheets to manipulate data CBL temperature sensor

CBL distance sensor

Projects from Mathalicious.com web site relate probability and the counting principle, expected value to warrantees and health insurance policies

Black line master maps of NJ, the US, Africa, and the Middle East Suggested Activities for Inclusion in Lesson Planning

Design surveys and apply random sampling techniques to avoid bias in the data collection.

Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data.

Apply regression results and curve fitting to make predictions from data.

Describe and use a set of frequency distribution data spread (i.e. variance and standard deviation), skew, symmetry, number of modes, or other characteristics.

Use combinatorics to compute probabilities of compound events. Use conditional probability and Bayes’ Theorem to solve problems.

Use the fundamental counting principle to find the number of outcomes in a problem situation. Use combinatorial reasoning to solve probability problems.

Use simulations to solve counting and probability problems. Use critical path analysis to solve scheduling problems. Use graph coloring techniques to solve problems.

Use fair division techniques to solve apportionment problems. Use geometric techniques to solve optimization problems.

Construct and administer a survey student’s meal choices in the cafeteria. Explore options for distributing and collecting this data including online forms and advertising the survey through school media. Explore graphic methods of organizing this data to find one or ones that give the clearest picture of the results. Share the results with appropriate administration.

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Use CBL temperature sensor and distance measurement tools to collect data on heating and cooling of liquids or pizza. Use this data to generate a line of best fit and regression model to create an equation. Determine which equation is more accurate in this situation.

Collect data from a variety of sources and analyze this data using graphing calculators and excel to see if the data can be used to draw conclusions. Explore a variety of graphic organizers to determine the best fit for this data. Also explore the results of the data when standard deviation and the rules of a normal distribution are applied. Conduct probability experiments to reinforce concepts of combination, permutation and the counting principle. Apply these discoveries to solving problems.

Explore the interconnectedness of probabilities and health insurance, warrantees, and expected value using mathalicious.com activities.

Invite a guest speaker to explain how the master course schedule is arranged so that all students can be scheduled into their necessary courses of study (critical path analysis) to graduate within four years.

Explore how map coloring requires a combinatory approach to determine the number of colors necessary to color the map; different maps require different numbers of colors. Determine the number colors necessary to color the counties of NJ, the states of the US, the countries of Africa, and the countries of the Middle East.

Course Title: Authentic Math Applications Unit Name: Geometry & Measurement Grade Level: 12

This unit explores the applications of geometry from area, perimeter and volume through transformations, similar triangles and congruence.

Cumulative Progress Indicators (CPI)

G.CO.1-7, SRT.6-11, GPE.4-7, GMD.3-4, G-MG.1-3

How does theoretical geometry apply to the real world? How is geometry and measurement used in areas other than mathematics?

The rules of geometry are used to define our physical space. Whether it is creating a custom tire insert for a “chopper” or machining a part in industry or wood working, geometry and its algebraic representations are used across all industries.

How do circles and their parts: arcs, diameters, chords relate to the real world?

How are area, surface area, and volume used and compared in the real world?

How are the algebraic expressions of area, surface area, perimeter, volume etc used in the real world?

What are nets and how do they relate two dimensions to three?

How can I create a two dimensional image to relate to a three dimensional figure and vice versa?

How can I find irregular areas and how do they apply in practice?

Standard units of measure (including both U.S. Customary System units and Metric System units) can be used to compare and order objects and to solve problems including arc measures and related angles, and areas of sectors. Shapes can be described, classified, and analyzed by geometric properties and attributes: Transformations, Congruence, Similarity, and Right Triangles.

Unit Rationale

This unit relates prior knowledge of geometry to tangible physical shapes and attributes of the world in which we live. Students will then apply this theory to

Unit Overview

All students will develop spatial sense and the ability to use geometric properties, relationships, and

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solve and discover in greater detail the world around them.

phenomena.

Key Terms

Arc- a portion of the circumference of a circle

Cosine- the ratio of the side adjacent to a specific angle to the hypotenuse of a right triangle Secant- the line perpendicular to a tangent line on a circle

Sine- the ratio of the side opposite a specific angle to the hypotenuse of a right triangle

Tangent- the ratio of the side opposite a specific angle to the side adjacent to the angle in a right triangle, also a line perpendicular to a circle at one point

Resources

Algebra with Pizzazz www.kuta.com worksheets

Information and visit to the teacher of CAD or mechanical drawing Punch out blocks for use with the cutter

Square block set Pattern Block set Rulers

Protractors Compass Plumb bob

Suggested Activities for Inclusion in Lesson Planning

Apply transformations (slides, flips, turns, dilations and contractions) to polygons to determine congruence, similarity, symmetry and tessellations.

Solve problems involving angles formed by transversals and coplanar lines.

Solve simple triangle problems using the triangle sum property and /or the Pythagorean Theorem.

Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. Use the triangle inequality theorem to solve problems involving triangles.

Use the properties of special right triangles to solve problems.

Define and use the trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) in terms of the angles of right triangles.

Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. Define and identify relationships among: radius, diameter, arc, the measure of an arc, chord, secant and tangent. Find the equation of a circle in the coordinate plane in terms of its center and radius.

Use coordinate geometry to find slopes, parallel lines, perpendicular lines and equations of lines. Use coordinate geometry to prove properties of polygons, such as congruence and similarity.

Describe solids that can be made from a given net and describe the net that can be made from a given solid. Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems.

Find measures of interior and exterior angles of polygons.

Find and use measures of sides, perimeters, and areas of polygons and relate these measures to each other using formulas.

Use properties of congruent and similar polygons to solve problems involving lengths and areas.

Find and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents) Find and use measures of circumference, arc length, and areas of circles and sectors and use these measures to solve problems.

Find and use measures of sides, volumes of solids, and surface areas of solids and relate these measures to each other using formulas.

Build structures with blocks and create orthogonal drawings based on front side and top view of construction, trade drawings with another team and challenge them to build your structure. Explore how these drawing relate to nets and how nets relate to the physical creation of the blocks.

Test the “squareness” of walls in the building by measuring length width and diagonal and checking them with the Pythagorean theorem.

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surface area of the boxes to the original area of the card board material. Can we make a comparison between the volume of the box and the surface are of the bag containing the cereal? Can we use this comparison to calculate the volume of a bag of chips or pretzels?

Investigate how CAD and computerized cutters use algebra to create and modify blocks of material to great precision. Explore how cutting wood and trim with a miter saw relates to the angles cut by a transversal.

Construct Polyhedron using punch outs, students may add a layer of difficulty by incorporating Escher like tessellations on the punch outs before assembly.

Solve problems relating circles, arcs, secants, chords, diameters, radius, and sectors to bridges, pizzas, tunnels, rocket launches, satellites and airplane travel paths.

Use pattern blocks to construct patterns modeling translations, rotations and reflections of shapes and polygons. Record these images on dot paper labeling each resulting image.

Use exploration of trig ratios to find unknown heights or distances when only an angle and one length are known. Use ratios to find an unknown length when similar figures are present. Measure students in comparison to object with unknown height. Explore the school and its grounds for objects for comparison and calculation.

Unit Project: Research artists who make use of geometry in their work and classify there use in terms of transformations, proportionality. Create a digital “gallery” of these artworks incorporating information about the artist and the type of geometry that they are using or illustrating.

Course Title: Authentic Math Applications Unit Name: Patterns and Algebra Grade Level: 12

This unit covers the manipulation, solution and graphing of functions from linear through exponential and their direct applications in the physical world.

Cumulative Progress Indicators (CPI)

A-SSE.1-4, A-APR.1-7, A-CED.1-4, A-REI.1-12

How can I determine the best investment for me? How can I determine a budget based on my income? How can I compare plans to purchase items or services?

Determine a best investment or best purchase, more than one factor needs to be considered. Algebraic expression, equations and graph can make the analysis and comparison process easier to visualize.

What strategies can be used to solve linear, quadratic and polynomial equations?

What strategies can be used to solve exponential and logarithmic equations?

What strategies can be used to solve systems of equations?

What strategies can be used to solve and display the solution to inequalities?

How does trigonometry apply to the real world?

How are exponential equations and logarithms related to banking, investing, and college loans?

How can I use logs to solve equations?

How do I convert to and from log and exponent notation?

Algebra is the language of patterns, rules and symbols. Functions may be used to describe real-world events and relationships. The same function may represent multiple situations.

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Unit Rationale

This unit bridges the gap between theoretical applications of algebraic processes and their actual usage. Students will use these processes to solve problems that they can relate to directly and immediately.

Unit Overview

All students understand how patterns, relations, and functions are interrelated; be able to represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to understand quantitative relationships; and analyze change in various contexts.

Key Terms

Annuity- a payment made or receive annually based upon an investment or loan

Credit- borrowing money with either a possession such as a car or home “securing” it or without security, such as a credit card

Sinusoidal- having a repetitive cyclical pattern, graphically bounded above and below, have a consistent period and amplitude

Resources

TI website – calculator explorations – solving systems of equations and inequalities, matrices, Algebra with Pizzazz

www.kuta.com for applicable worksheets

Internet research on phases of the moon, water levels at high and low tides etc.

Explore guest speaker opportunities from local banks on credit, credit history, bank etc. Suggested Activities for Inclusion in Lesson Planning

Describe, complete, extend, analyze, generalize and create a wide variety of patterns, including iterative, recursive (e.g. Fibonacci Numbers, Pascal’s Triangle), linear, quadratics and exponential functional relationships. Use a graph to estimate the solution of a system of linear equations in two variables.

Use a graph to find the solution of a system of linear inequalities in two variables.

Use elimination and matrices to solve systems of two or three linear equations in two and three variables. Use systems of linear equations and inequalities to solve problems.

Factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.

Find solutions to quadratic equations (with real coefficients and real and complex roots) and apply to the solutions of problems.

Graph quadratic functions and determine the maxima, minima, and zeros of the function. Explain the effect that changing a coefficient has on a graph of quadratic functions. Add, subtract and multiply polynomials.

Divide polynomials by others of lower degree.

Add, subtract, multiply, divide, reduce and evaluate rational expressions with monomial and polynomial denominators.

Simplify rational expressions, including those with negative exponents in the denominator.

Solve everyday problems that can be modeled using linear, quadratic, exponential, periodic (sine and cosine) and step functions.

Unit problem: Explore the costs involved with smoking cigarettes including price per pack, distance traveled as gas mileage and taxes. As more sophisticated function models are explored, develop the cost per week, per year, over a lifetime, and using exponents, adjust the cost to inflation over a lifetime. If this subject is too controversial for the class, change to a daily purchased cup of coffee from either Wawa or Dunkin Donuts. Use MapQuest to determine gas mileage.

Student exploration: of the costs of a cell phone. Create equations modeling the costs of different cell phone plans from different companies. Determine the cost per year etc. Have students investigate their own cell phone usage to develop inequality limits on these equations to determine which plan is most cost effective based on their won usage. Use a variety of methods including graphing, elimination, substitution and matrices to find the points of intersection between the plans. Use these critical points to determine the maximum and minimum value.

Use situational criteria to write equations for the trajectory of a ball based on the distance between two people, the height of the space that they are in, obstacles in the path etc. Use regression modeling, vertex form equations,

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graphing, foiling and factoring to solve and construct these problems. Have students find images with similar paths online, using the Smart board, construct equations that model these paths as well. Look for patterns in the constructed equations based on different parameters within the problem.

Use situational criteria to write equations for cyclical patterns including the phases of the moon, high and low tide, the position of a piston in a motor, and the position of a person on a Ferris wheel in terms of an appropriate sinusoidal function.

Small group project: investigate the types of bank and investment accounts available to them. Calculate the potential savings to them of investing several amounts of money over several periods of time. Investigate the costs of college or post high school education, paid for both outright and through loans. Use a combination of online loan calculators, annuity formulas, and rate tables to determine the lifetime cost of that education if paid immediately, over 10 years, 15 years, 20 years and 30 years. Create a digital presentation of their results.

Small Group Project: investigate the salary of positions and careers of interest. Based on these salaries, determine the expected income tax both federal and state, complete a 1040 EZ tax return, determine your potential rent or mortgage payment, locate a property within this budgeted amount, and calculate remaining income for living expenses including water, sewer, electricity, natural gas, fuel, car, insurance, cable, cell phone, food etc. Create a digital presentation of their results.

Focus Lesson: on the meaning of credit and development of credit history. Explore all of the ways in which a good credit history will help them with employment, reduce output of capitol etc. Including information on ways to monitor credit and get credit reports for free.