Learning Links
Spreadsheets / Advertisements / Shopping / Computer Programming / Timeshare / Travel Plans / Networks / Flowcharts / Ecology / Pay Schedules / Income Tax / Universal Law / Scientific Principles / Law
Related Concepts
Mathematical Expression / Mental Mathematics / Technology / Logic / Estimation / Proportion / Computational Algorighms / Development of Algebraic Processes / Proof / Order of Operations
Demonstrators should be read from top to bottom, but need not be demonstrated sequentially.
Elementary Demonstrators
• Manipulate objects tl model and communicate operations in a rich variety of problems.
• Investigate relationships through the use of patterns to construct operations and algorithms.
• Translate concrete and pictorial problem situations into mathematical language and symbols, and vice versa.
• Investigate and discover a variety of mental computation and estimation techniques.
• Explore the use of open sentences to express real-life situations.
• Explore logical thinking strategies.
Middle School Demonstrators
• Use models and investigations to construct algorithms using rational numbers.
• Interpret and organize information for logical deductions.
• Translate from concrete, pictorial, and verbal expressions to mathematical expressions and vice versa.
• Translate real-world proportional relationships into mathematical expressions and vice versa. • Solve equations with concrete, pictorial, and abstract methods.
• Apply and justify computational methods (calculator, paper/pencil, mental math, estimation, computer).
• Formulate examples and counterexamples.
High School Demonstrators
• Use concrete, pictorial, and abstract models to develop and/or generalize a procedure.
• Use concrete, pictorial, and abstract models to develop and/or solve algebraic problems.
• Select and apply appropriate strategies (e.g., equations, inequalities, matrices, and networks) to solve problems.
• Make and test conjectures through investigations.
• Validate mathematical assertions and judge the validity of real-world statements using deductive arguments.
Sample Teaching/Assessment Strategies
Collaborative Process: Cooperative Learning, Peer Tutoring / Graphic Organizers: Matrix / Problem Solving: Heuristics, Future Problem Solving, Research, Formulating Models / Technology/Tools: Manipulatives, Computers, Games, Calculators
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
• Work with local social services representatives to compile data and determine average number of homeless who are fed in the community and plan low-cost nutritious meals for them.
• Prepare income tax returns for senior citizens.
• Serve as interns in local projects with civil engineer.
• Survey a variety of white and blue collar workers to determine if and how mathematics is used in their vocations.
Core Concept - Mathematical Procedures
Sample Elementary Activities
• Use a storyboard to model the action in a number sentence. PE, OE, P
• Compare the strategies in computer software programs which address logical thinking (e.g., Gertrude's Puzzles, Moptown Parade, Iggy's Gnees, The Pond, or Gnees or Not Gnees.) PE
• Use a ten-frame chart to show various strategies for adding 9 + 8 (e.g., shift to make 10, doubles + 1, neighbors). PE, OE
• Use a calculator with a constant function to explore counting and operational patterns. OE
• Show how to divide different groups of items such as 5 candy bars, 10 donuts, or 11 comic books among 4 students. PE
• Create algorithms for adding, subtracting, multiplying, and dividing whole numbers. OE, P
Applications Across the Curriculum
Language Arts
· Sort out the jumbled pieces in an action sequence and list them in order. PE
· Locate a poem or story that shows a mathematical sequence. OE
Science
· Determine, calculate, and demonstrate the mechanical advantage of a pulley system or inclined plane. PE, OE, P
Social Studies
· Investigate time zones throughout the world. Calculate current time in countries/cities in the daily news (e.g., London, Tokyo, New York). PE
Practical Living
· Keep a log documenting regular participation in a physical fitness activity and graph time spent on your fitness program (e.g., jogging, bicycling, etc.). PE, OE, P
· Estimate and verify the total amount of trash in the school for a day. Graph results. PE, OE, P
Vocational Education
· Compose real-world problems using information from the newspaper or almanac (e.g., "What are the factors that affect milk prices?"). OE
· Compare the price of a cafeteria lunch versus a brown-bag lunch. Consider additional factors. PE, OE, P
Sample Middle School Activities
• Write a set of directions for a younger student, explaining how to add two fractions (e.g., 2/3 + 1/4). Use computer graphics to draw pictures or diagrams in the explanation. PE, OE, P
• Tell a story based on a selected picture that generates a multi-step mathematical problem. OE
• Investiage the Fibonacci sequence as applied to nature (e.g., seeds in a sunflower, whorls on a pineapple, birth of rabbits). Present findings using multimedia. PE, OE, P
• Verify the theorem: The sum of the anbles of a triangle is 180 degrees. Cut a triangle out of a piece of paper. Tear off the corners of the triangle and assemble them in a straight angle. Or, fold a paper triangle so that the vertices meet at a point. P
• Compare the answers of a multi-operational exercise using paper/pencil and various brands of calculators. Determine the necessity for order of operations (e.g., 2 + 3x5 on some calculators equals 25, rather than the correct answer of 17). PE, OE, P
Applications Across the Curriculum
Variations on a theme: Commerce
Language Arts
· Use ads from newspapers to figure out the savings on sale items. Compare similar products and justify the selection of the best buy. OE, P
Science
· Estimate the cost of gasoline for the average driver over a one-year period. Reevaluate estimation if there were an additional 10 cents increase per gallon. Discuss impact of increases on a fixed salary. OE, P
Social Studies
· Track the change in the cost of a single item over an extended period. Calculate the cost of the item in a foreign currency using the current rate of exchange. OE
Arts and Humanities
· Trace the value of a work of art from the time it was created through each sale. OE
Practical Living
· Calculate the cost of smoking a pack of cigarettes a day for one week, one month, and one year. OE
Vocational Education
· Establish and maintain a classroom bank. PE
Sample High School Activities
• Explain the appropriate strategy to determine how many ways there are to travel among three points in a city using a series of one-way streets. Also consider the number of ways to travel from one point to another using one stopover. Use graph theory and matrices. P
• Obtain data of beginning and ending inventory from a local department store for a month including profit and sales commission. Use matrix operations to determine the number of items sold, profits, and commissions earned. Display data using spreadsheets. PE, P
• Solve an algebraic equation using algebra titles, pictures of tiles, progressing to algebraic symbolism. PE
• Analyze advertisements for "if-then" statements embedded within them. Present findings using multimedia. PE, P
• Investigate perimeters of various rectangles with area of 24 square cm using models and drawings. Generalize to answer such problems as "Is there a rectangle of minimum perimeter with specified area?" Give dimensions. OE, P
• Design an algorithm to solve a difficult equation by using approximations. OE, P
• Design a method to approximate the area between two curves. OE, P
Applications Across the Curriculum
Language Arts
· Solve word problems by ordering the sequence of steps needed to complete the problem. Create a flowchart to match the steps. OE
· Create a database of words that have double meanings, one specific to mathematics and the other in another area (e.g., root, square). PE, P
Science
· Using a spreadsheed which incorporates birth, death, immigration, and emigration rates, predict the population of the United States at five year intervals over the next 25 years. PE, P
· Design an experiment to investigate a scientific principle such as Boyle's Law. PE, OE, P
Social Studies
· Investigate insurance actuarial charts to determine how insurance rates in the United States are established. Research the same information in another country. PE, P
Arts and Humanities
· Cut a matte of given dimensions for a picture. PE
Practical Living
· Research the incidence of HIV infection in the United States population. Choose a factor (e.g., age, place of residence, number of partners) and calculate the risk of infection. OE, P
Vocational Education
· Prepare a spreadsheet comparing grocery brands for price, nutritional value, and convenience. PE
· Calculate cost of setting up a medical, dental, or other health care practice, including malpractice insurance and other costs. OE