math_education

Case Number: 1
Subject: Mathematics Course: Integrated Math I IAS Standard Number: IM1.5.4 • IM1.5.4: Use the Law of Large Numbers to understand situations involving chance.

//Learners//: This case takes place in a freshmen-level Integrated Mathematics I course. This course is designed for students who may tend to not be college bound, and is seen as a general-level means of fulfilling one’s mathematics requirements for graduation. Most students are freshmen (9th grade), while there are 10th grade students – both as first time and as “repeat” students. Additionally, there are a small number of special needs students in the teacher’s Integrated Math I classes, each with different learning, emotional and behavioral needs.

//Context//: There are multiple computer labs available for teachers to use if one reserves a room with enough advanced notice. There is a basic complement of office, productivity and other software tools installed on these computers, and there are departmental sets of graphing calculators available for teachers to reserve and use in their classes.

Objective(s): to compare/contrast the numerical results of different sets of probability trials using a coin and draw conclusions as to the relationship betwen these results as the number of trials per set increases (to teach the Law of Large Numbers).

Description: Mr. Pearson tries to involve his freshmen-level Integrated Math I students in hands-on activities for the many different areas of mathematics which they explore in his class. He once tried to have students flip coins by hand in trials of 2, 10, 50, 100 and 250, but this activity caused great classroom management issues (noisy, flying quarters). It is his hope that these students can see how the results of these trials, by the Law of Large Numbers, will eventually tend towards the theoretical probability in a more tangible manner - without flying quarters!

Case Number: 2
Subject: Mathematics Course: Discrete Mathematics IAS Standard Number: • DM.4.3: Use graph coloring techniques to solve problems.

//Learners//: These are juniors and seniors in an elective math course entitles Discrete Mathematics (see Indiana's Academic Standards for Math/Discrete Mathematics if not familiar with "Discrete Math" by name)

//Context//: This course is to be taught online (see description below for more details), and students will be expected to provide their own access to a computer and the Internet (see description below for more details).

//Objective//: to use trial-and-error techniques to solve a problem; to cooperatively evaluate each group member's techniques and create an algorithm for solving this problem

Description: Ms. Haaken and Mr. Appel team-teach a junior/senior level mathematics course entitled Discrete Mathematics. As an introduction-to-the-class activity, these two teachers usually introduce their students to discrete mathematics through a hands-on application of the Four-Color Theorem, as part of learning about pictorial representations of problems in discrete mathematics. In the past, groups of three students each were given an 11x17 laminated outline map of the (contiguous) United States and four differing-color markers. Each team was to collaborate on finding a means of coloring the map which would only require four colors.

This course only has four senior-level students enrolled, and the administration does not want to use a classroom for this course but still want to offer the class. The students, due to their busy schedules, cannot all meet at the same time before or after school, so the teachers have proposed conducting this class online. Students would have to provide their own access to a computer for this course and would conduct class meetings both in real-time and asynchronously as possible. The teachers plan on using the school district’s course management system for this course.

Therefore, two teachers want to devise an internet-based version of this activity, where students are able to both color the map and do so in a more collaborative manner – forcing students to talk with each other about how to best color their group’s map. The problem is that they are unaware of a way that their students can both color the map digitally AND communicate with each other.

Case Number: 3
Subject: Mathematics Course: Probability and Statistics IAS Standard Numbers: PS.1.1, PS.1.2 • PS.1.1: Create, compare, and evaluate different graphic displays of the same data, using histograms, frequency polygons, cumulative frequency distribution functions, pie charts, scatterplots, stem-and-leaf plots, and box-and-whisker plots. Draw these by hand or use a computer spreadsheet program.

PS.1.2: Compute and use mean, median, mode, weighted mean, geometric mean, harmonic mean, range, quartiles, variance, and standard deviation.

Description:

Mr. Bayes normally has his AP Statistics classes do two different surveys, each dealing in-depth with descriptive (not inferential) statistics, including charts/graphs, measures of central tendency and measures of dispersion. The first one is very formative: He helps his students learn both basic survey methodology and how to both graph/plot and how to calculate/derive the other measures. The second one is more focused on calculations. Now, he has just been given access to several radar-based velocity-detecting tools for his calculus classes, and he would like to find a way to use this tool in his other classes if possible. Mr. Bayes wants to create an activity using this tool, but he has absolutely no idea how to both use this tool, and have a project where the students can collect numerical (technically, interval/ratio data, for the stats-savvy people) data that is meaningful and students can perform all of the normal descriptive stats tools to and create charts, graphs and plots on. What can he do?

Alternate Cases If Needed
Case 4

Course: Geometry IAS Standard Number: G.1.2; G.1.3 • G.1.2: Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass, explaining and justifying the process used. • G.1.3: Understand and use the relationships between special pairs of angles formed by parallel lines and transversals.

//Learners//: This case takes place in the context of a (face-to-face) high school Geometry course. This is a required mathematics course, though this version of the course is college-preparatory in context. Students are drawn from a wide variety of educational, cultural and socioeconomic backgrounds.

//Context//:There are multiple computer labs available for teachers to use if one reserves a room with enough advanced notice. There is a variety of productivity software tools installed on these computers.

//Objectives//: to construct three different sets oftwo parallel lines that are intersected by a transversal; to measure with 100% accuracy the angles formed by these intersections; To compare these different values and write out conclusions about the relationships between the angles formed.

Description: Ms. Reimann is teaching a college-prep Geometry course to a mix of freshman and sophomore high-school students. As they begin to learn about the properties of angles formed by the intersection of different types of lines (parallel, perpendicular, oblique) and which angles at an intersection are equal, complementary and/or supplementary, she has students create (with a ruler and a pencil) two parallel lines, and then they are to draw an oblique line that intersects both lines. Next, they are to measure each of the (eight) angles formed and - using these measurements – conjecture which angles (adjacent, vertical, alternate interior, etc) are equal, complementary or supplementary. They are asked to do this five times, with a different slope of the intersecting line each time. The problem with this activity (in her eyes) is that it takes so much time for students to draw each set of parallel lines and measure each set of angles. She wishes that there were some computer-based tool in which students could do the same thing, but save time.

==Case Number: 5 Subject: Mathematics Course: Pre-calculus and Trigonometry IAS Standard Numbers: PC.4.6; PC.4.7 • PC.4.6: Find domain, range, intercepts, periods, amplitudes, and asymptotes of trigonometric functions. • PC.4.7: Draw and analyze graphs of translations of trigonometric functions, including period, amplitude, and phase shift. //Learners//: These are 11th/12th grade students in an elective math course //Context//: The school has access to computers with basic probability software and Internet access; There are also a set of graphing calculators available to use. //Objective//: to manipulate the graphs of functions that are trigonometric in nature through the modification of differing values in the corresponding equation; to evaluate the effect of changing values in a trigonometric expression, both in verbally and graphically Description: Dr. Gauss is teaching a one-semester elective course for juniors and seniors on Trigonometry in a local high school, and wants to give her students extended experience with how different coefficients and/or added values in a trigonometric equation can affect the graph of that equation. Traditionally, she has each student take a “base” equation, such as y = sin x, and graph it. Next, they are to modify the equation, such as y = 3sin x or y = sin (x-2), and graph these. Her hope is that students can learn that if a value is place in the equation at certain places, it affects the graph in different ways, and that they can learn what these ways are (changes in amplitude, frequency, phase shift and/or vertical shift). Her problem is that the process of creating the graphs most often takes away from the student’s ability to observe the changes made and a value’s effect on the graph. She wants to let her students see the effected changes, yet not be too distracted by the process of graphic these equations.== ==Case Number: 6 Subject: Mathematics Course: Geometry IAS Standard Number: G.2.5 • G.2.5: Find and use measures of sides, perimeters, and areas of polygons. Relate these measures to each other using formulas. //Learners//: These are 9-11 grade students in a general-level geometry class (one that emphasizes informal proof) with a WIDE range of previous experience and potential difficulty with "math" (as the students perceive it). //Context//: The teacher has access to a computer lab with 30 computers, general productivity software, and Internet access. You as a teacher cannot install or have anything installed on these computers, and students cannot save files to computers directly. Additionally, the teacher has access to rulers, pencils, paper, and scientific calculators as needed. Objectives: to measure an distance, and use these measurements (or their sums, differences, products and/or quotients) to calculate a variety of perimeters, areas, sums of perimeters and sums of areas. Description: Miss Euclid is teaching a unit on the area and perimeter of (convex) polygons, and always ends her unit with an individual project, a design-your-dream-house” project. As a means of assessing student learning in calculating the area and perimeter of assorted polygons, she has each student, using pencil-on-grid-paper, create a floor plan to a one-story house. Only straight lines are allowed. Once the floor plan has been created, students are required to calculate the square-footage (area) for each room and the perimeter of each room. While these drawings are quite nice, she would love for her students to produce digital versions of these floor plans, which can be easily modified and colored. The problem is that she does not know much about software that her students could use to do this!==