The First Four Standards - Grades 9-12

Overview

New Jersey's Mathematics Standards calls for a shift in emphasis from a high school curriculum often dominated by memorization of isolated facts and procedures and by proficiency with paper-and-pencil skills to one that stresses understanding of concepts, multiple representations and connections, mathematical modeling, and mathematical problem solving.

The distinction between mathematical problem solving and doing mathematics should begin to blur in the high school grades. The problem-solving strategies learned in the earlier grades should have become increasingly internalized and integrated to form a broad basis for doing mathematics, regardless of the specific topic being addressed. From this perspective, problem solving is much more than solving word problems; it is the process by which mathematical ideas are constructed and reinforced. There is more emphasis in high school on introducing new mathematical concepts and tools as responses to problem situations in mathematics, and on developing students' ability to pose problems themselves.

Through extensive experiences with mathematical communication, students improve their understanding of mathematics. Students must be able to describe how they obtain an answer or the difficulties they encounter in trying to solve a problem. Facility with mathematical language enables students to form multiple representations of ideas, express relationships within and among these representations, and form generalizations.

High school students should continue to experience two types of mathematical connections - those within mathematics, and those to other areas. First, students should make connections between different mathematical representations of the same concept or process. Students who are able to apply and translate among different representations of the same problem situation or of the same mathematical concept have not only a powerful, flexible set of tools for solving problems but also a deeper appreciation of the consistency and beauty of mathematics. Unifying ideas within mathematics to be emphasized in the high school grades include mathematical modeling, variation (how a change in one thing is associated with a change in another), algorithmic thinking (developing, interpreting, and analyzing mathematicalprocedures), mathematical argumentation, and a continued focus on multiple representations.

Second, students in high school should regularly discuss the connections between mathematics and other subjects and the real world. Connections between mathematics and science are particularly plentiful. Examples of such activities are:

• Students use computer-aided design (CAD) to produce scale drawings or models of three-dimensional objects such as houses.

• Students use statistical techniques to predict and analyze election results.

• Cooling a cup of coffee provides an opportunity for students to collect and analyze data while learning about the process of cooling: Why does cooling occur? Does the coffee cool at a constant rate? Does the temperature follow an asymptotically decreasing pattern? By using temperature probes connected to a computer or graphing calculator (CBL), students can collect data on the temperature of a cup of hot coffee as it cools, plot this data on a coordinate graph, and describe the pattern verbally and with an equation.

• Students might explore models using logarithmic scales by studying the Richter scale used to measure the strength of earthquakes, the decibel scale for sound loudness, or the scale used to describe the brightness of stars.

• Students might develop experiments in which they collect data in order to investigate polynomials of higher degree. They can examine the collected data by using "finite differences" to predict the degree of the polynomial. In addition, this graphic approach helps students develop a better understanding of the concept of "zeros of a function."

• Students at the precalculus level might use science experiments to investigate trigonometry. Students enrolled in physics classes often use the Law of Sines and Law of Cosines prior to their development and discussion in precalculus classes. Thus, there is a need to reexamine the order of presentation of topics both in science and in mathematics.

• Trigonometric functions can be applied not only to modeling terrestrial and astronomical problems requiring indirect measurement but also to describing the motion of water waves, waves in a rope, sound waves, and light waves.

• Students can collect data to model rational functions, including not only y = 1/x but also more complex equations such as y = (x-1)(x+2)/(x+2)(x-3). Students should see why the data excludes the domain value of -2 as well as why there is an asymptote at x=3.

• Students might integrate the study of geometric sequences or logarithmic and exponential functions with science topics involving growth and decay. (See Breaking the Mold at the end of the Introduction of this Framework in which students look at the growth patterns of living things and the vignette involving carbon dating at the beginning of the Introduction.)

• Students might conduct experiments involving velocity with constant acceleration, such as dropping a ball, to study parabolas and quadratic equations.

• Students might study vectors in conjunction with complex numbers.

• Students can relate three-dimensional figures to the geometry of molecules, crystals, and symmetry.

• Students might link the study of solving linear equations in mathematics classes to thebalancing of equations in chemistry.

• The study of direct and inverse variation and different types of functions might be linked to the study of volcanic action and earthquakes.

• The study of calculus and physics might be integrated. In fact, a team-taught course might be more appropriate than the present approach, especially for the applications of calculus. This might prove more appropriate for those students who do not need a theoretical approach to calculus at this time.

• Students might design and construct a container that will sound an alarm when opened. In completing this task, they must use measurement, geometry, numerical operations, algebra, and mathematical reasoning as well as knowledge of electrical circuits, wiring, switches, electricity, insulators, conductors, Ohm's law and its use, and cost estimation.

A student who is doing mathematics often makes a conjecture by generalizing from a pattern of observations made in specific cases (inductive reasoning) and then tests the conjecture by constructing either a logical verification or a counterexample (deductive reasoning). High school students need to appreciate the role of both forms of reasoning. They should also learn that deductive reasoning is the method by which the validity of a mathematical assertion is finally established. Much inductive reasoning may take place in algebra, with students looking for patterns that arise in number sequences, making conjectures about general algebraic properties based on their observations, and verifying their conjectures with numerical substitutions. Students can be introduced to deductive reasoning by examining everyday situations, such as advertising, in which logic arises. Logical arguments in mathematical situations need not follow any specific format and may be presented orally or in writing in the student's own words.

High school students focus on mathematical problem solving, using multiple representations of mathematical concepts, mathematical connections, and reasoning throughout all of their mathematics learning. As they learn and do mathematics, they should regularly encounter situations where they are expected to discuss and solve problems, develop mathematical models, explain their results, and justify their reasoning. The content of these four standards is inextricably interwoven with the fabric of mathematics.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.

Standards: In addition to the First Four Standards, this vignette highlights Standards 7 (Spatial Sense), 11 (Patterns), and 13 (Algebra).

The problem: Yesterday, Mrs. Ellis' class finished a unit in which they used algebra tiles to help them develop procedures for multiplying binomials. Today, Mrs. Ellis began by asking the students to consider the following problem: Suppose we have a collection of red x² tiles, orange x tiles, and yellow unit tiles. Can we put them together to form a rectangle? For example, if we have one red tile, five orange tiles, and four yellow tiles, we can make the rectangle at the right. What combinations of tiles can form a rectangle, and what combination cannot?

The discussion: Mrs. Ellis encouraged the students to share their ideas for how they might go about solving this problem. Some of their questions were: What materials might be helpful in working on this problem? How could we use the algebra tiles? What might be some combinations of tiles that we might try? How could we keep track of what we have tried? Can we use more than one of the red x² tiles? Will we be able to try out all of the possible combinations? Do you suppose that we will find some sort of pattern that can help us predict which combinations will work and which will not?

Solving the problem: The students worked in groups of three or four over a period of three days. They tried different combinations of tiles, recording how many of each tile they used, whether or not that combination could be used to form a rectangle, and, if so, the dimensions of the rectangle. As they worked, they began to notice patterns. Some of their comments were: This seems like multiplying binomials but in reverse. When it works, the product of the number of 1s on the top times the number of 1s on the sides equals the number of yellow unit tiles. If there's just one of the red tiles, then the sum of the number of 1s on the top plus the number of 1s on the side equals the number of orange tiles. Each group summarized its conclusions and the patterns they found in a report.

Summary: Mrs. Ellis asked the groups to exchange reports with another group, then read, review, and comment on the other group's report. Each group then had an opportunity to review the comments on their report. Mrs. Ellis asked the students about the patterns they had found. Are there some patterns that both of your groups found? Are there others that only one of the groups found? She recorded the findings on the board. How can we be sure all of these statements are correct? The students suggested that, if everyone agreed with a statement and could justify their reasoning, then it should be accepted as correct. They discussed each of the statements, explaining their reasoning and arguing about some of the statements. For homework, Mrs. Ellis asked the students to use their findings to make some predictions about other combinations of tiles and to relate their results to the idea of factoring binomials. Mrs. Ellis expects that in the next classes she will connect the problem of making a rectangle from a red x² tiles, b orange x tiles, and c yellow unit tiles to the problem of factoring ax² + bx + c.

Standards: In addition to the First Four Standards, this vignette highlights Standards 6 (Number Sense), 7 (Geometry), 8 (Numerical Operations), 9 (Measurement), 11 (Patterns), 13 (Algebra), and 15 (Building Blocks of Calculus).

The problem: Ms. Longhart began class by posing the following problem for her students: Suppose that you are setting up a water ice stand for the summer and are trying to decide how to make the cones in which you will serve the water ice. You've found some circles of radius 10 cm that are the right type of paper and have figured out that by cutting on a radius, you can make cones. You decide that you would like to make cones that hold as much water ice as possible, so you can charge a higher price. What will be the radius of the base of your cones? What will be the height?

The discussion: Ms. Longhart asked the students what materials might be useful in helping them solve the problem. Some students suggested making models out of paper circles, while others thought that writing equations and using graphing calculators to find the maximum volume would be best. After some discussion of the relative merits of each approach, Ms. Longhart suggested that they do both and compare their answers.

Solving the problem: The students separated into groups to work on the problem. Most students remembered that the volume of a cone is 1/3 pi r²h. They made a variety of paper cones out of circles with radius 10cm, and measured the radius and height of those cones. Using the formula, and a calculator, they generated a table of values, trying to find the maximum volume. They wanted to graph the formula using their calculator, but realized that they needed to solve for h in terms of r. After some initial difficulties, they decided that, since the original circles had a radius of 10 cm, the height of the resulting cone must be SQRT (100 - r²). Then they graphed the equation they had generated on the graphing calculator and used the graph to find the maximum volume. Finally, each group summarized its findings in writing.

Summary: Each group listed its actual measurements and its results generated on the graphing calculator on the board and then explained any discrepancies that might have occurred. The class as a whole discussed the accuracy of the solutions. One of the students noticed that, in the cone of maximum volume, the radius was much larger than the height of the cone, and asked why that happened. Ms. Longhart asked the class to think about some possible reasons. To summarize the lesson, Ms. Longhart asked the students to list in their journals all of the mathematical concepts that they used to solve the problem in class that day. For homework, she asked the students to (1) describe how the function they generated would change if the radius of the circle was 8 cm, 9 cm, 11 cm, or 12 cm; (2) find the maximum volumes and corresponding radius for each of the new functions; and (3) determine whether there is a relationship between the radius and the maximum volume for each of the five functions.

Standards: In addition to the First Four Standards, this vignette highlights Standards 7 (Geometry), 11 (Patterns), and 13 (Algebra).

The problem: Before this session, students in Mr. Evans' class investigated different situations that can be modeled using quadratic functions. They looked at how to maximize the area of a yard given a fixed amount of fencing, and how to predict the path of a rocket. They graphed many quadratic functions, some by plotting points and some on the graphing calculator, and discovered that all of the quadratic functions have graphs that are parabolas. For this session, they went to the computer lab to investigate the relationship of each of the constants in the general form of the parabola to the graph of that equation.

The discussion: Before beginning work on the computers, the students reviewed the general shape of a parabola and discussed the differences between one quadratic function and another: the width of the parabola, how high up or down the vertex is, and whether the parabola opens up or down. Mr. Evans introduced the general form of the equation of a parabola, y = a (x - h)² + k , and asked the students to predict how each of a, h, and k will affect the graph of the parabola. He asked them to explain their reasoning and record their predictions individually in their notebooks, and then he led a discussion of their predictions.

Solving the problem: Each pair of students used a program on Green Globs software to predict the equation for a series of parabolas, keeping notes on how the different constants seemed to affect the graphs. There was considerable excitement in the room, as well as some disagreements and some frustration at times. Some pairs of students found that several of the graphs required many attempts before the correct equation was found. At the conclusion of the computer activity, the students compared their results to their predictions and discussed their findings with each other in pairs. They then individually wrote a description of how the values of a, h, and k in the general equation y = a (x - h)² + k affect the graph of y = x².

Summary: For homework, Mr. Evans asked the students to use their findings to sketch the graphs of several parabolas without using graphing calculators and then to check their sketches by using graphing calculators. He suggested that they revise their journal entries if they find that some of their hypotheses don't work. Mr. Evans began class the next day by having pairs of students play the computer game Green Globs, allowing the students to use only parabolas to hit the globs on the coordinate grid. After about 15 minutes, he led a discussion of the students' findings about parabolas, asking them how they arrived at their hypotheses, what steps they took to verify them, and whether they modified their hypotheses based on their experiences with the homework and the computer game. He then asked the students to think of other areas of mathematics that seem to be related, making connections between their findings about parabolas and what they learned last year about geometric transformations.

The cumulative progress indicators for grade 12 for each of the First Four Standards appear in boldface type below the standard. Each indicator is followed by a brief discussion of how the preceding grade level vignettes might address the indicator in the classroom in grades 9-12. The Introduction to this Framework contains three vignettes describing lessons for grades 9-12 which also illustrate the indicators for the First Four Standards; these are entitled On the Boardwalk, A Sure Thing!?, and Breaking the Mold.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 9-12 will be such that all students:

4^*. Pose, explore, and solve a variety of problems, including non-routine problems and open-ended problems with several solutions and/or solution strategies.

• In Making Rectangles, the students explore and solve an open-ended question. In Ice Cones, the students explore and solve a problem using several different solution strategies (concrete materials or writing equations). In Building Parabolas, the students explore an open-ended question.

5^*. Construct, explain, justify, and apply a variety of problem-solving strategies in both cooperative and independent learning environments.

• In all three vignettes, the students work in small groups to construct, explain, and justify their strategies.

6^*. Verify the correctness and reasonableness of results and interpret them in the context of the problems being solved.

• In all three situations, the students verify their results by sharing them and interpreting them in a whole-class discussion.

7^*. Know when to select and how to use grade-appropriate mathematical tools and methods (including manipulatives, calculators and computers, as well as mental math and paper-and-pencil techniques) as a natural and routine part of the problem-solving process.

• Students in Making Rectangles use manipulatives to help them solve their problem. Some students in Investigating Cones choose to use manipulatives, while others use graphing calculators to help them solve the problem. Students in Building Parabolas use computers to help develop their ideas.

8^*. Determine, collect, organize, and analyze data needed to solve problems.

• The students in Making Rectangles organize and analyze their results. Some of the students in Ice Cones decide to approach the problem by looking at some specific cones and finding their volume. (Note that this was not a particularly efficient approach to solving this problem.) The students in Building Parabolas decide which parabolas to try out in playing Green Globs and then collect, organize, and analyze their data.

12^*. Construct and use concrete, pictorial, symbolic, and graphical models to represent problem situations and effectively apply processes of mathematical modeling in mathematics and other areas.

• Students in Making Rectangles use concrete, pictorial, and symbolic models to represent their problem situation. Students in Ice Cones use concrete, symbolic, and graphical models to represent the problem and develop a mathematical model to describe the height of the desired cone. Students in Building Parabolas use symbolic and graphical models to represent the problem situation.

14^*. Persevere in developing alternative problem-solving strategies if initially selected approaches do not work.

• The students who experience initial difficulties in Ice Cones are often the ones who try a "guess-and-check" strategy. They decide to try a different approach, using an equation, when they encounter problems. The students in Building Parabolas must use alternative approaches to predict the correct equations from the Green Globs program. There is considerable frustration for some students and perseverance is required as they make numerous attempts to find the correct function.

15. Use discovery-oriented, inquiry-based, and problem-centered approaches to investigate and understand the mathematical content appropriate to the high-school grades.

• The students in Making Rectangles are learning about factoring by solving a problem. The students in Ice Cones are exploring how to find a maximum in a problem involving both algebra and geometry. In Building Parabolas, the students are using a question about parabolas to learn about how the coefficients of the equation affect the graph.

16. Recognize, formulate, and solve problems arising from mathematical situations, everyday experiences, applications to other disciplines, and career applications.

• The problem in Ice Cones arises from career applications. The problems in Building Parabolas and Making Rectangles arise from mathematics itself.

17. Monitor their own progress toward problem solutions.

• The students in Making Rectangles keep a record of the different combinations of tiles they try, whether or not the combination could be used to form a rectangle, and the dimensions of the rectangle; they begin to notice patterns. The students in Ice Cones try a different strategy when they find that they are not making progress. In Building Parabolas the very use of the computer game Green Globs, helps students monitor their own progress towards a solution.

18. Explore the validity and efficiency of various problem-posing and problem-solving strategies, and develop alternative strategies and generalizations as needed.

• The students in all three vignettes share the strategies they use to solve their problems. In Tiling a Floor, the students select a variety of materials, including a computer program, to solve the problem. In Sharing Cookies, students use different methods to solve the same problem.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 9-12 will be such that all students:

1^*. Discuss, listen, represent, read, and write as vital activities in their learning and use of mathematics.

• Students in all three vignettes use a variety of communication activities: listening, discussing in small and large groups, representing in algebraic and graphical or concrete contexts, and reading and writing about their solutions.

2^*. Identify and explain key mathematical concepts, and model situations using oral, written, concrete, pictorial, and graphical methods.

• In Ice Cones, the teacher asks the students to list all of the mathematical concepts they use in solving the problem in their journals. In all three vignettes, students model situations using different methods: oral, written, concrete, and graphical.

3^*. Represent and communicate mathematical ideas through use of learning tools such as calculators, computers, and manipulatives.

• Some students in Ice Cones use graphing calculators; others use manipulatives (paper circles). The students in Building Parabolas use a computer program. Students in Making Rectangles use manipulatives.

4^*. Engage in mathematical brainstorming and discussions by asking questions, making conjectures, and suggesting strategies for solving problems.

• In all three vignettes, the students engage in brainstorming before solving the problem.

5^*. Explain their own mathematical work to others, and justify their reasoning and conclusions.

• In all three vignettes, the students explain their work to the others in the class.

6^*. Identify and explain key mathematical concepts and model situations using geometric and algebraic methods.

• In Making Rectangles, most of the students draw pictures of the various tiles they can make and write about the tiles using algebraic notation. For example, they write that they can use one red x² tile, two orange x tiles, and one yellow unit tile to make a larger square tile and record this as x² + 2x + 1 = (x +1) (x +1). The students in Ice Cones model the situation using both geometric (paper circles) and algebraic (graphing calculator) methods. In Building Parabolas, the students model the situation both algebraically with equations and geometrically with graphs.

7^*. Use mathematical language and symbols to represent problem situations, and recognize the economy and power of mathematical symbolism and its role in the development of mathematics.

• The students in all three vignettes use mathematical language and symbols to represent the problem situation.

8^*. Analyze, evaluate, and explain mathematical arguments and conclusions presented by others.

• The students in all three vignettes present their results to the class, evaluating and explaining their conclusions.

9. Formulate questions, conjectures, and generalizations about data, information, and problem situations.

• In Making Rectangles, the students use their findings to make predictions about combinations of tiles. In Ice Cones, the students make conjectures and generalizations, including determining whether there is a general relationship between the radius and the maximum volume for each of five different cones. The students in Building Parabolas make conjectures and generalizations about the effects of the coefficients in the general form of the equation on the graph.

10. Reflect on and clarify their thinking so as to present convincing arguments for their conclusions.

• Students in all three vignettes are asked to reflect on and clarify their thinking by sharing with others in small groups and by summarizing their findings in writing.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 9-12 will be such that all students:

1^*. View mathematics as an integrated whole rather than as a series of disconnected topics and rules.

• In Making Rectangles, factoring binomials is related to area and multiplication. In Ice Cones, the students are drawing from concepts in algebra, geometry, and calculus. In Building Parabolas, the students have already investigated different situations that can be modeled by parabolas. At the end, they also relate their work to previous work with geometric transformations.

2^*. Relate mathematical procedures to their underlying concepts.

• Making Rectangles relates factoring to the underlying concepts of area and multiplication. Ice Cones relates finding a maximum to the underlying concepts of volume and solving equations. Building Parabolas relates the shape of a graph to its equation.

3^*. Use models, calculators, and other mathematical tools to demonstrate the connections among various equivalent graphical, concrete, and verbal representations of mathematical concepts.

• Students in Making Rectangles use their models to demonstrate the connection between the geometric topic of area and the algebraic topic of factoring. The students in Ice Cones use calculators to help demonstrate the connection between the equation and the maximum value for the volume. Students in Building Parabolas use computers to demonstrate the connection between the algebraic and graphical representations for parabolas.

8^*. Recognize and apply unifying concepts and processes which are woven throughout mathematics.

• Students in Making Rectangles and Building Parabolas are focusing on multiple representations. Students in Ice Cones are applying mathematical modeling to a real-life situation.

9^*. Use the process of mathematical modeling in mathematics and other disciplines, and demonstrate understanding of its methodology, strengths, and limitations.

• The Ice Cones vignette illustrates the process of mathematical modeling in real life. In Making Rectangles and Building Parabolas, students use mathematical modeling to discover patterns which ultimately help them to solve the problems.

10^*. Apply mathematics in their daily lives and in career-based contexts.

• Making Rectangles and Ice Cones involve applying mathematics in career-based contexts.

11^*. Recognize situations in other disciplines in which mathematical models may be applicable, and apply appropriate models, mathematical reasoning, and problem solving to those situations.

• All three vignettes focus on modeling within mathematics. The 9-12 Overview describes a number of situations in other disciplines in which mathematical models are applicable.

12. Recognize how mathematics responds to the changing needs of society, through the study of the history of mathematics.

• The problems discussed in the vignettes are not presented in a social or historical context. However, students can also investigate the role of the quadratic functions, discussed in Building Parabolas, in ballistics, and can extend their Ice Cones discussions to include other examples of packaging.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 9-12 will be such that all students:

2^*. Draw logical conclusions and make generalizations.

• Students in all three vignettes draw logical conclusions and make generalizations.

5^*. Analyze mathematical situations by recognizing and using patterns and relationships.

• Students in all three vignettes look for patterns and relationships in order to make their generalizations.

8^*. Follow and construct logical arguments, and judge their validity.

• In Making Rectangles, the students discuss how they will know that their work is correct. They judge the validity of each others' arguments in the subsequent discussion. In Ice Cones, the students worked in groups and explained their results and any discrepancies they encountered. The class as a whole discussed the accuracy of solutions. In Building Parabolas, also in a class discussion, students explained their hypotheses, how they verified them, and whether they had to modify them based on their experiences.

9^*. Recognize and use deductive and inductive reasoning in all areas of mathematics.

• All three vignettes deal primarily with inductive reasoning. However, building on theactivities in these vignettes, students can use deductive reasoning to show the effect of increasing k by 3 in Building Parabolas or to prove that certain collections of titles can or cannot form rectangles in Making Rectangles.

10^*. Utilize mathematical reasoning skills in other disciplines and in their lives.

• Making Rectangles and Ice Cones illustrate the use of mathematics in daily life.

11^*. Use reasoning rather than relying on an answer-key to check the correctness of solutions to problems.

• None of the students in any of the vignettes checks answers with an answer key; they all report their answers to the class and explain how they got them.

12. Make conjectures based on observation and information, and test mathematical conjectures, arguments, and proofs.

• The students in Making Rectangles make some initial conjectures about which tiles they can make and then test them. The students in Ice Cones make and test conjectures in their homework as they begin to generalize their results. The students in Building Parabolas make conjectures about the effects of each constant and then test these using the computer.

13. Formulate counter-examples to disprove an argument.

• This indicator is not explicitly addressed in these vignettes. However, in the discussion in any of the vignettes, it is possible that some of the analysis of solutions will lead students to present counterexamples to disprove another student's argument.