Learning Links
Nature / Evolution / Chaos / Topology / Human Development / Mutations / Chemical Reactions / Geopolitical / Fractals / Rock Formations / Migration
Related Concepts
Transformations / Trigonometric Functions / Continuous vs. Discrete / Algebraic Representations / Limit / Sequences / Functions / Matrix Representations / Series / Patterns
Demonstrators should be read from bottom to top, but need not be demonstrated sequentially.
Elementary Demonstrators
• Observe patterns of change (e.g., seasons, height, weather) in everyday life and discuss causes and effects.
• Extend and create patterns and generalize structures from patterns (e.g., square numbers, geometric patterns, patterns in nature) in number sequences.
• Explore the concepts of unknown quantities and effects of change (functions).
• Use variables, represented by manipulatives, to express relationships involving change.
Middle School Demonstrators
• Represent patterns in several ways (e.g., graphs, ordered pairs, verbal statements, algebraic rules).
• Investigate patterns in number sequences and relate to real world experiences.
• Explore functions that can be derived from physical models, data, and other mathematical representations.
High School Demonstrators
• Analyze the effects of parametric changes on graphs.
• Model a variety of problem situations with similar functions.
• Explain how a change in one quantity affects a change in another.
• Use curve fitting to predict change.
• Determine appropriate model to represent change in data (i.e., discrete or continuous).
• Investigate the properties of various types of functions, (e.g., linear, quadratic, logarithmic, trigonometric, etc.).
Sample Teaching/Assessment Strategies
Collaborative Process: Cooperative Learning / Community-Based Instruction: Mentoring / Continuous Progress Assessment: Observation, Performance Events, Portfolio Development / Graphic Organizers: Graphic Representations, Flowchart / Problem Solving: Inquiry, Questioning, Case Studies, Creative Problem Solving, Future Problem Solving, Formulating Models, Simulations / Technology/Tools: Manipulatives, Computers, Multimedia
These sample strategies offer ideas and are not meant to limit teacher resourcefulness. More strategies are found in the resource section.
Ideas for Incorporating Community Resources
• Contact public agencies for data specific to the community (e.g., water company for information on water pressure and flow rate, U.S. Census Bureau for demographic data).
• Take a field trip to the local automobile dealership and have a salesperson explain how the NADA (Blue Book) standards are used to determine the price of used cars.
• Survey businesses to discover how future sales/services are projected from current sales figures and how that affects future staffing.
Core Concept - Change
Sample Elementary Activities
• Use a working set of interlocking gears with 24, 12, 8, and 4 teeth, and predict the number of revolutions for each. Use the computer program "Gears" for a follow-up program. PE, P
• Read "Caps for Sale." Use a simple spreadsheet program to calculate money received if a variable number of caps are sold at $.50 a cap. OE
• Use a calculator with a constant to play "What's My Rule." Enter "+ 4 = " to store. Press "4 = "(the number 8 will show). Give to partner, who presses a number and " = " to determine the rule (the number that is stored). OE
• Place one bean plant in sunlight, and another bean plant of the same size in a closet. After one week, measure both, record the changes and discuss the reasons for any differences. PE, OE
• Build 2, 3, and 5 span bridges using red and yellow Cessionaire rods (1 yellow and 2 end red rods = 1 span). Explain how many red and yellow rods you would use to make a 20 span bridge. Develop a rule for building bridges. PE, OE, P
• Describe patterns found on addition, subtraction, multiplication, and division tables. OE
• Describe patterns on a hundreds chart. OE
Applications Across the Curriculum
Language Arts
• Write a recipe book, including measurements, for enlarging or reducing the recipes. P
Science
• Create a collage time line of science-related toys over a period of fifty years. P
• Construct flip-books of the life cycle of a plant (bean) or animal (butterfly). OE
Social Studies
• Select a popular vacation spot. Predict seasonal or event related population changes. Gather data that supports or rejects the prediction. Present findings using graphic representatives. OE, P
Arts and Humanities
• Make a value scale using black and white tempera paint. Record the value changes by amount of white paint added. OE, P
Practical Living
• Record height and weight of students over time. Do a chart comparison from the beginning, middle, and end of the year. P
Vocational Education
• Calculate the number of cups of ice required to meet the Recommended Daily Allowance (RDA) of water. P
Sample Middle School Activities
• Discuss the effect of coordinate changes on the shape of coordinate plot pictures (e.g., double both x and y coordinates and plot a new picture, double the x values and plot a new picture). PE, P
• Show graphically how overhead costs and expenditures for advertising affect profits of a pizza parlor. PE, OE
• Investigate relationships between height and length of radius bone of several students in the class. Develop a general rule. Suppose you are an anthropologist and have found a radius bone of 25 cm. long. Predict the height of the person. PE, P
• Use the school records from 2O and 10 years ago to determine the most popular first name of boys and girls. Survey the current school population to find the current top ten. Based on past trends, predict the ten most popular names for boys and for girls in the year 2000. P
Applications Across the Curriculum
Variations on a theme: Pollution and the Environment
Language Arts
• Research waste management control practices for public schools. Present your findings to the school council, school board or other decision-making body. PE, OE, P
Science
• Track the amount of paper or food-scrap waste in the school over specific periods of time. Graph the results. P
Social Studies
• As a group project, design a campaign to address waste management issues and convince classmates, teachers, administrators and staff to become involved with a waste reduction program at the school. PE, P
Arts and Humanities
• Inform people through a visual medium about the waste management campaign project and encourage continued involvement. PE
Practical Living
• Prepare a set of suggestions for reducing the amount of food-scraps and packaging waste. Graph results of the reduction in waste. OE, P
Vocational Education
• Investigate and present data showing the growth in waste management careers over the last decade. PE, P
Sample High School Activities
• Collect data on a particular car model from the NADA book (Blue Book). Plot the year (independent variable) with the price (dependent variable). Look for patterns. Compare different models. PE, P
• Investigate and compare graphs of families of functions using a graphing calculator. PE, P
• Discuss and act out a story based on interpretation of a graph of a traveling vehicle. PE, OE
• Use technology to graph the following situation: A ball is dropped from a height of 10 meters and loses 300/o of its height (y) on each successive bounce (x). The maximum height on each bounce is given by y = 10(O.7)x where x must be a positive integer. Compare this discrete graph to a continuous graph of the same equation where the domain is the set of reals. Discuss why a discrete graph (rather than a continuous graph) is the appropriate representation of this situation. OE, P
• Explain how changing the parameters of a function affects the graph of a function, e.g., how does f(x) differ from f(x+a) and from f(bx). OE
• Create a large right triangle on the floor using a large loop of string with student at each vertex. Investigate sine and cosine relationships by changing the angle at the center (maintain constant length of the hypotenuse). Follow up with use of scientific calculator. PE
Applications Across the Curriculum
Language Arts
• Write a valid argument to convince someone that adding the same number of points to each student's test score will increase the average score by the same amount. OE, P
Science
• Collect and graph pressure and volume data. Develop an equation relating pressure to volume. PE, OE, P
Social Studies
• Research the changes in the types and numbers of crime occurring since 1800. Draw conclusions as to why changes have occurred and make predictions for changes in the future. OE, P
Arts and Humanities
• Write an essay describing how color changes and relationships influence color clarity, setting, and mood. PE, OE, P
• Choose a well known artist's work (e.g., Starry Night, Whistler's Mother ). Reproduce the painting using opposite colors from the artist. P
Practical Living
• Calculate the average number of times people change jobs in the United States. Infer the consequences for your education as you prepare for a career. P
Vocational Education
• Contrast the root-growth rate of plants that have a 5.0 pH and those that have a 7.0 pH. PE, P
• Set up a graphical representation of an insurance actuarial table using spreadsheet software. PE, P