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Mathematics Curriculum Framework
Achieving Mathematical Power  January 1996
 "Mathematics is an exploratory science that seeks to understand every kind of patternpatterns that occur in nature, patterns invented by the human mind, and even patterns created by other patterns. To grow mathematically, children must be exposed to a rich variety of patterns appropriate to their own lives through which they can see variety, regularity, and interconnections."
 Lynn Arthur Steen, On the Shoulders of Giants
Patterns, Relations, and Functions form the base of mathematics. Recognizing patterns and describing their relations mathematically by using geometry, number sequences, and functions, helps us interact with and make sense of our world. To appreciate fully the intrinsic value of such pleasures as poetry, art, music, plants, and animals, lifelong learners should know the mathematics of patterns and use mathematical representations to describe them. The terminology of patterns, relations, and functions has become a part of our culture. Headlines and news reports speak of exponential growth of the national debt, a variable rate mortgage, or a balanced budget.
Students discover patterns on leaves and stems as they observe plants growing at the science table. They marvel at the patterns on fish they see in aquariums. Quilt patterns made with fabric or on dot paper are other examples. These realworld examples of patterns serve as segues to numerical and geometrical investigations of mathematical patterns and functions, such as skipcounting, odd and even numbers, square numbers, triangular numbers, Pascal's triangle, and the Fibonacci sequence.
Concepts for algebra are first formulated in PreK. By the end of eighth grade, algebraic concepts should be welldeveloped. Algebra learning continues through adulthood. Furthermore, it not only serves as a connector of the topics in mathematics, it is also recommended as a venue for interdisciplinary curriculum.
Younger students identify, translate, and extend repeating rhythmic, verbal, and visual patterns. They recognize patterns and relationships among objects when they sort and classify them, identifying similarities and differences. Groups of students may make a list of insects and the corresponding sequential total number of legs, or fellow students and the number of shoes, as they begin developing their own understanding of functions and multiples.
As students develop, their work with patterns and functions moves concurrently toward mathematical sophistication. They identify and represent numerical relationships in tables, charts, and graphs. Perhaps students work in groups to plan and map a trip from their home to either coast, showing the relationship between constant speed and distance with a linear graph. They may also collect time and distance data of a local trip that show how average speed represents only one aspect of the trip, displaying these real data on a nonlinear graph. Software programs featuring spreadsheets and graphing capabilities are ideal resources for teachers and students.
Patterns and functions are the building blocks for transformational geometry, algebra, discrete mathematics, trigonometry, and calculus. Underpinnings of calculus can be seen in algebra and geometry during earlier grades. Older students will explore ideas about continuity, discontinuity, maximum and minimum. Real world optimization problems, such as those related to maximum profit earned or minimum height achieved, provide an opportunity to see the relevance of these concepts. Technology such as mathematical software and graphing calculators can be used to build understanding of function, explaining their domain and range, and comparing and translating among various representations.
Life experience offers adult basic education learners a broad base of realworld ties that are readily linked to the concepts of equation, function, variable, and graph. Each of these are representations of patterns, relations, and functions. From baby formulas to chemical formulas, algebra offers a succinct way to define realworld situations that can aid adults in the home and in the workplace.
PreK4 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 identify, describe, extend, and create a wide variety of patterns.
 represent and describe mathematical relationships.
 explore the use of variables and open sentences to express relationships.
Examples of Student Learning
 Make a group chart, counting the number of eyes in the class. Give students some type of manipulative for them to use when representing eyes. Fill in the pattern on a 099 chart. Discuss the numerical pattern. Use the constant key on the calculator to extend the pattern.
 Introduce Fibonacci number patterns. Beforehand, the teacher uses different colors of poster paint on pine cones to identify spirals. Small working groups count the numbers of bracts of the same colors that form the spirals, and record their data on charts. Some spirals are steep, others are gradual. What they discover is an example of the Fibonacci number sequence. The activity is extended when students cut spirals from colored squares and hang them from the ceiling.
 Students look at and discuss patterns and symbolism of the United States flag and those of other countries. They design a mathematics flag for the classroom. They may make it any size they wish, choosing from an array of art materials. The only requirement is that it shows a pattern that is repeated. Transformations (flips, slides, and turns) may be encouraged. Pairs of students identify and describe each other's patterns.
PreK4 Learning Standards
 Students engage in problem solving, communicating, reasoning, and connecting to:
 discover how to form, then write, number sentences for real problems.
 investigate and describe ways to find missing components in number sentences.
 demonstrate through handson activities, an understanding of maintaining balances in number sentences.
Examples of Student Learning
 If you were to make cheese sandwiches for yourself and a friend, how many more pieces of bread would you need than cheese? Is there a mathematical way to write this? If so, what is it?
 Students use shoe boxes to make an input/output mathematics machines, which they decorate. Using counters as tokens, they can explore fact families, properties, functions, and computational skills. When students are ready to move to a more abstract model, they can use a function machine that is represented on paper to play Guess the Rule.
PreK4 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 explore and demonstrate an understanding of commutative properties for addition and
 multiplication.
Examples of Student Learning
 Students think of commutative and noncommutative tasks. For example, does it matter whether I brush my hair first or my teeth first? Does it matter whether I brush my teeth first or eat my chocolate cremefilled cookie first?
 Students explore commutative in the sense of commuting to work or school. Does it matter which way you go when you determine the distance of the commute?
Grades 58 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 describe, extend, analyze, and create a wide variety of patterns.
 describe and represent relationships with models, tables, graphs, and rules, using sentences and algebraic expressions.
 analyze functional relationships to explain how a change in one quantity results in a change in another.
 use patterns and functions to represent and solve problems.
Examples of Student Learning
 Students use color interlocking cubes to build a sequence of towers, pyramids, or other patterns. As each sequence is built, students describe the pattern they see and predict, then build, the next two shapes. They organize their information by making a table, describing the pattern in writing, writing a rule, and plotting points on a coordinate graph.
 Reduce, Reuse, Recycle. What does this mean in your classroom and at your school? Could more be done to encourage recycling? Small groups of students gather data that explain the present recycling practices in their classroom and support their suggestions for improvement. They work together to write a proposal for garbage and trash reduction. It includes graphs, tables and charts as well as cost analysis. Students make posters for display and present their findings to the community and administration.
Grades 58 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 understand and apply the concepts of variable, expression, and equation.
 represent situations and number patterns with tables, graphs, verbal rules, and equations and explore the interrelationships of these representations.
 analyze tables and graphs to identify properties and relationships.
 demonstrate an ability to solve linear equations, using concrete, informal, and methods.
 describe the strategies used to explore inequalities and nonlinear equations.
 apply algebraic methods to solve a variety of realworld and theoretical problems.
 construct expressions or equations that model problems.
 explore and describe a variety of ways to solve equations, including handson activities, trial and error, and numerical analysis.
 know and apply algebraic procedures for solving equations and inequalities.
 The snowy tree cricket sings a distinctive song on summer nights. Scientists have discovered that when 37 is added to the number of chirps heard in 15 seconds, the current Fahrenheit temperature is known. Students make tables and graphs, then write an equation and a rule for this phenomenon. They conduct library research or observe nature to discover another such phenomenon, then write about it.
 Students begin by working in groups and dropping a ball from a specified height. They observe how high it rebounds on the first bounce. They find the average first rebound from the given height and predict the height of the second rebound. They test their predictions. Students develop a model for the situation and predict the height after 4, 5, 25, 100, 500 falls. They compare the elasticity of similar and different balls. How much variation is there among tennis balls from the same can? How much variation is there among different brands of tennis balls? How do tennis balls compare with basketballs and baseballs?
Grades 910 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 formulate problems that involve variable quantities with expressions, equations, and inequalities.
 use tables and graphs as tools to interpret expressions, equations, and inequalities.
 simplify algebraic expressions to solve equations and inequalities.
Examples of Student Learning
 One "Ask Marilyn" feature in Parade magazine read: Half the employees in a firm went to lunch at noon. Since then, 25 have returned and 7 others have gone out. At this point, there are twice as many people working as there are people out to lunch. How many people are employed by the firm? Students explore this problem in groups, comparing solutions and various methods for determining them. They write equations and explanations describing their solutions. Groups of students make up a similar problem, then solve each other's problems.
 The 807 airliner holds a maximum of 200 passengers. A carrier has a price structure that states if the airline is full, each passenger pays a $200 fare. For each drop of 5 passengers, the price per passenger increases by $10. Find the price that will maximize the income of the airline.
Grades 910 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 model realworld phenomena with a variety of functions.
 represent and analyze relationships, using tables, verbal rules, equations, and graphs.
 translate among tabular, symbolic, and graphical representations of functions.
Examples of Student Learning
 Algebra for the Twentyfirst Century: Proceedings of the August 1992 Conference presents this problem: A patient is taking approximately 16 mL of medicine every four hours for a long period of time. How much of the medicine will eventually be in her blood? Assume the body eliminates 25% of the medicine in every fourhour interval.
 Students can investigate this problem in several different ways. They may carry out a simulation, set up a table by using arithmetic calculations, use a spreadsheet, apply the formula for the sum of a geometric series, or use the recursive mode on their graphing calculator.
 Students explore and discuss the patterns and exponential representations in Pascal's triangle. They devise a rule for finding the sum of the nth row in the triangle. Using what they have learned, they formulate and solve their own probability questions that can be answered by using Pascal's triangle.
Grades 910 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 apply trigonometry to problem situations involving right triangles.
Examples of Student Learning
 Students replicate the experiment in which Eratosthenes calculated the circumference of the Earth and got a remarkably good answer. They locate some schools roughly due north or south and connect with them through electronic mail. Students in each school agree that on a given day, at high noon, they will measure the shadow cast by a vertical stick on level ground. After sharing the measurements of the stick and the shadow, students use trigonometric ratios to determine the angle of the sun's rays. Using this information along with approximate distances between schools, the students use proportions to find an approximation of the Earth's circumference. This example can be extended to sharing data with students from other states and countries.
Ms. Arroya has students bring in examples (from mail, newspaper, TV, or other advertisements) involving rates. Students work in small groups to analyze them. Here are some samples.
Recently, a bank in Boston advertised a special for customers who had a home equity account with them. The offer was stated, "Borrow at least $2500 by April 30, and receive a certificate which can be used for up to $25 in long distance telephone calls with a specified phone service." Is this a good deal?
Two banks in the Worcester area recently ran competing ads for their savings accounts. One offered a rate of 6% compounded monthly, while the other offered a rate of 5.75%, but compounded daily. If you had $100 to invest, what would you do? What are some of the factors that might affect your decision?
Mario did not bring in an ad, but he posed a problem from something his mother told him about interest. She said there is a "Rule of 72," which holds that the number of years it takes an investment to double in value when the interest is compounded can be approximated by dividing 72 by the rate (multiplied by 100). For example, if an amount is invested at 8%, it would take about 9 years to double in value. Does this rule work? The class explored the rule and decided its usefulness by trying to prove it.
Grades 1112 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 represent situations that involve variable quantities with matrices.
 use tables and graphs as tools to interpret higher order equations, inequalities, and matrices.
 use matrices to solve linear systems.
Example of Student Learning
 Students can use the matrix feature of graphing calculators or computer programs to solve systems of linear equations generated by real problems. They may use information gathered from medical and law enforcement authorities to solve problems connected with amounts of toxins in a person's system. They use graphing calculators or computer programs to create lists, graph functions, and to investigate how changes in one variable will affect others.
Grades 1112 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 model realworld phenomena with a variety of nonlinear functions.
 identify a variety of problem situations that can be modeled by the same type of function.
 analyze the effects of parameter changes on the graphs of functions.
Example of Student Learning
 Students use parametric equations to solve problems they have posed based on movement over time. One group of students may suggest the problem of ships passing (or colliding) in the night. Using graphing calculators and parametric equations, the students can simulate the actual passage of the ships, and can determine visually, as well as algebraically, how far apart they are at any given time and when they pass (or collide!).
Grades 1112 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 apply trigonometry to problem situations involving triangles.
 explore and describe periodic realworld phenomena, using the sine and cosine functions.
 apply general graphing techniques to trigonometric functions.
 express the relationship between trigonometric and circular functions.
Example of Student Learning
 Students use the local pier as their frame of reference to determine the time when a boat can safely moor. They determine from a newspaper when high and low tides occur, and study depth charts or talk with the harbor master to determine the depths at high and low tide. Students find a trigonometric equation that models the depth of the water and use graphing calculators or computers to help them find the best times for mooring.
Grades 1112 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 investigate and describe limiting processes by examining infinite sequences and series.
 determine maximum and minimum points of a graph and interpret the results in problem situations.
 investigate and describe limiting processes by examining areas under curves.
Example of Student Learning
 Students explore limiting sums for sequences of fractions and repeating decimals. They begin their explorations by shading grid paper and/or folding and cutting squares of paper to estimate the limits. After making conjectures about formal procedures for finding sums, they use graphing calculators to verify or refine these procedures.
ABE Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 explore, identify, analyze, and extend patterns in mathematical and realworld situations.
 articulate and represent number and data relationships, using words, tables, graphs, and rules.
 discover and use patterns and functions to represent and solve problems.
Example of Student Learning
 Using a spreadsheet, students explore the rules of operations with integers. They see what happens when they combine positive and negative numbers, using the four basic operations. After recording the results of their experimentation, students form groups to discuss why the procedures work as they do. They brainstorm to find reallife examples that might help them make sense of their results.
ABE Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 represent arithmetic patterns and realworld situations, using tables, graphs, verbal rules, equations, then explore and describe the interrelationships of these representations.
 recognize and use variables, expressions, and equations.
 solve equations in one or two variables, using concrete, informal, and formal methods.
 apply algebraic methods to solve and represent a variety of testrelated, workspecific, and realworld mathematical problems.
Example of Student Learning
 Students determine how far they live from class and record how long it takes each of them to get there from home. The class makes a chart that lists the distances and times, then uses the distance formula to determine each student's rate of travel. Students discuss reasons why some rates might be similar or different.
In a preGeneral Education Development (GED) classroom, students explore the relationship between the weight and the volume of water, using a set of calibrated measuring cups. They record their findings and write a rule that summarizes those findings. As a followup, students and teachers bring to class a variety of containers, measure the dimensions (to determine the volume) of each container, and then use their rule to predict the weight of water. Finally, each container is filled with water and weighed to check their predictions.
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Last Updated: January 1, 1996
