Course Overview

Semester B of Mathematical Models is designed for high school math students after the completion of Mathematical Models Semester A. The semester looks at applying mathematical modeling concepts to architecture, engineering, fine art, photography, and music. Each of the five units includes between seven and fourteen lessons, and one project. Each lesson has a minimum of five formative assessment questions to enable students and their teacher to gauge student understanding. Each project uses concepts covered in the unit. Summative assessments include three quizzes in each unit, a test for each unit, and a semester exam covering all five units.

  • Unit 1: Identify and apply appropriate algebraic processes and models to solve problems and analyze data in science contexts.
  • Unit 2: Identify and apply appropriate algebraic and geometric processes and models to solve problems and analyze data in architecture and engineering contexts.
  • Unit 3: Identify and apply appropriate algebraic and geometric processes and models to examine patterns and techniques in fine arts contexts.
  • Unit 4: Identify and apply appropriate models and techniques to solve problems and analyze data in social sciences.
  • Unit 5: Identify and apply appropriate probability models to solve problems and analyze data in various contexts.
 
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Semester B: Curriculum Content and Skills Focus

Unit 1: Applications in Science

  • Recognize nonlinear functions.
  • Construct and compare linear, quadratic, and exponential models.
  • Solve problems with different function models and interpret function expressions in terms of the situation they model.
  • Recognize and solve direct and indirect variation equations.
  • Graph direct and indirect variations.
  • Describe the properties of direct and inverse variation graphs.
  • Determine the constant of variation/constant of proportionality.
  • Calculate and graph direct and indirect variation equations using universal laws including Hooke’s law, Newton’s second, and Boyle’s law.
  • Define growth and decay factors and determine factors from percents of increase or decrease.
  • Apply growth and decay factors to solve problems involving percents of increase or decrease.
  • Determine annual growth or decay rate of an exponential function represented by a table of values or an equation.
  • Graph an exponential function for population growth.
  • Apply exponential decay to real world situations involving radioactive decay.
  • Analyze and graph quadratic functions in any form.
  • Explain the meaning of the vertex and the intercepts of a quadratic function in the context of a problem.
  • Fit a quadratic function to an image of motion and use the model to make predictions.
  • Collect, organize, analyze, and present data using knowledge of motion and the path of an object.

Unit 2: Applications in Architecture and Engineering

  • Solve for missing sides, perimeter, area, and surface area of similar figures using proportions in architectural contexts for 2D and 3D objects.
  • Solve for volumes of similar figures using proportions in architectural contexts for 3D objects.
  • Use translations, reflections, and rotations to describe mathematical patterns in architecture.
  • Identify symmetric elements in architectural contexts.
  • Use dilations and tessellations to describe mathematical patterns in architecture.
  • Identify and apply one-point and two-point perspective to describe mathematical patterns in architecture.
  • Identify and apply perspective to describe mathematical patterns in architecture.
  • Compare one and two-point perspective.
  • Create a one-point perspective drawing and a two-point perspective drawing of the same room.
  • Apply the Pythagorean theorem to calculate distances in an architectural or engineering context.
  • Use special right triangle theorems to calculate distances in an architectural or engineering context.
  • Use sine, cosine, and tangent to calculate distances and side lengths in an architectural or engineering context.
  • Use the angle of elevation in an architectural or engineering context.
  • Use sine, cosine, and tangent to calculate angle measures in an architectural or engineering context.
  • Use the angle of elevation in an architectural or engineering context.
  • Use sine, cosine, and tangent to calculate angle measures in an architectural or engineering context.

Unit 3: Applications in Fine Art

  • Use similarity to describe fractals in art and photography .
  • Using transformations, describe patterns and tessellations in art and photography.
  • Identify symmetric elements in art and photography.
  • Use natural perspective to portray three dimensional objects in two dimensions
  • Use one and two-point perspective to demonstrate mathematical patterns and structure in art and photography.
  • Use transformations, proportions, and periodic motion to describe mathematical patterns in music.
  • Use scale factors to demonstrate proportional and non-proportional changes in surface areas of 2-D and 3-D objects in fine arts.
  • Use scale factors to demonstrate proportional and non-proportional changes in volume of 3 D objects in fine arts.
  • Use trigonometric ratios and functions to model periodic behavior in art.
  • Use trigonometric functions to describe periodic patterns and behavior in music.

Unit 4: Applications in Statistics

  • Organize and put data into a line graph, or a stem-and-leaf plot.
  • Interpret data presented in a bar graph, circle graph, or table.
  • Organize data into a frequency table and then graph the data using a histogram.
  • Plot data in a scatter plot. Estimate and draw lines of best fit. Estimate errors of lines of best fit.
  • Recognize characteristics of the data using a dot plot. Interpret and draw conclusions from dot plots.
  • Determine mean, median, and mode using formulas, frequency tables, dot plots, and histograms.
  • Recognize symmetric and skewed frequency distributions.
  • Measure the variability of frequency distributions using range, interquartile range, and standard deviation.
  • Use standard deviation to understand the variability of a set of data.
  • Create and interpret box-and-whisker plots.
  • Explain the difference between surveys, experiments, and observational studies and identify the appropriate research method for a study. Use data from a sample to estimate population mean or population proportion.
  • Identify increasing, decreasing, extremes (maxima and minima) and constant parts of graphs.
  • Recognize how scaling the axis on a graph could make the data misleading.
  • Use regression methods to create various models for data such as linear, quadratic, and exponential. Select the most appropriate model for a set of data and use the model to interpret the information.
  • Formulate a meaningful question. Determine the data needed to answer the question. Gather appropriate data. Analyze the data. Draw reasonable conclusions. Create a written report, a visual display, an oral report, or a multimedia presentation to present findings.

Unit 5: Mathematical Applications in Probability

  • Develop and use a probability model to find probabilities of events.
  • Develop a uniform probability model by assigning equal probabilities to all outcomes.
  • Find probabilities of compound events using organized lists, tables, tree diagrams, and simulations.
  • Represent sample spaces for compound events using organized lists, tables, and tree diagrams.
  • Explain the properties of probability. Determine relative frequency for a set of data.
  • Determine theoretical and experimental probability. Simulate an experiment.
  • Recognize components of binomial experiments and calculate binomial probabilities.
  • Define the fundamental counting principal.
  • Recognize situations where the fundamental counting principle can be used to solve problems.
  • Determine the number of ways an event can occur using combinations or permutations.
  • Apply the fundamental counting principle to solve for possible combinations and permutations.
  • Create and a game of chance, determining the theoretical and experimental probability.
 
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Additional Resources

In addition to the default course content, some projects may require paper and pencil or drawing supplies to complete the assignment. Writing assignments may require a graphic organizer to be printed out and used in the writing process. Projects (such as book reports or informational essays) may require students to acquire outside resources for research or reading.

The following lessons require specific materials that are not included in this course and must be acquired separately:

Unit Assignment Resource
All General
requirement
  • Access to the Internet
  • A graphing calculator or web-based graphic calculator
1 Project
  • Materials necessary to create a projectile such as rubber bands, spoon, rocks, paperclips, etc.
  • Ruler or measuring tape
  • Stopwatch
2 Project
  • Colored pencil
  • Drawing paper
  • Ruler
3 Project
  • Computer program that allows students to hear and see sound waves
  • Printout of sound wave graph file
4 Project
  • Poster or slideshow software (optional)
5 Project
  • Materials necessary to play a game such as spinner, six-sided number cube, cards, game pieces, etc.
 
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