# Numbers and Relationships and

Algebra & Functions,

## Concepts and Outcomes from

SFAA - Science for All Americans

NAEP - from the National Assessment for Educational Progress

for Numbers and relationships

### SFAA - Science for All Americans

- SFAA - Numbers and relationships among them can be represented in symbolic statements, which provide a way to model, investigate, and display real-world relationships.
- SFAA - Seldom are we interested in only one quantity or category; rather, we are usually interested in the relationship between them—the relationship between age and height, temperature and time of day, political party and annual income, sex and occupation.
- SFAA - Rrelationships can be expressed by using pictures (typically charts and graphs), tables, algebraic equations, or words.
- SFAA - Graphs are especially useful in examining the relationships between quantities.
- SFAA - Algebra is a field of mathematics that explores the relationships among different quantities by representing them as symbols and manipulating statements that relate the symbols.
- SFAA -Sometimes a symbolic statement implies that only one value or set of values will make the statement true. For example, the statement 2A+4 = 10 is true if (and only if) A = 3. More generally, however, an algebraic statement allows a quantity to take on any of a range of values and implies for each what the corresponding value of another quantity is. For example, the statement A = s2 specifies a value for the variable A that corresponds to any choice of a value for the variables.
- SFAA - There are many possible kinds of relationships between one variable and another. A basic set of simple examples includes:
- directly proportional (one quantity always keeps the same proportion to another),
- inversely proportional (as one quantity increases, the other decreases proportionally),
- accelerated (as one quantity increases uniformly, the other increases faster and faster),
- converging (as one quantity increases without limit, the other approaches closer and closer to some limiting value),
- cyclical (as one quantity increases, the other increases and decreases in repeating cycles), and
- stepped (as one quantity changes smoothly, the other changes in jumps).

- SFAA - Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. For example, we could state symbolically the relationship between the width of a page, P, the length of a line of type, L, and the width of each vertical margin, m: P = L+2m. This equation is a useful model for determining page makeup. It can be rearranged logically to give other true statements of the same basic relationship: for example, the equations L = P-2m or m = (P-L)/2, which may be more convenient for computing actual values for L or m.
- SFAA - In some cases, we may want to find values that will satisfy two or more different relationships at the same time. For example, we could add to the page-makeup model another condition: that the length of the line of type must be 2/3 of the page width: L = 2/3P. Combining this equation with m = (P-L)/2, we arrive logically at the result that m = 1/6P. This new equation, derived from the other two together, specifies the only values for m that will fit both relationships. In this simple example, the specification for the margin width could be worked out readily without using the symbolic relationships. In other situations, however, the symbolic representation and manipulation are necessary to arrive at a solution—or to see whether a solution is even possible.
- SFAA - Often, the quantity that interests us most is how fast something is changing rather than the change itself. In some cases, the rate of change of one quantity depends on some other quantity (for example, change in the velocity of a moving object is proportional to the force applied to it). In some other cases, the rate of change is proportional to the quantity itself (for example, the number of new mice born into a population of mice depends on the number and gender of mice already there).

## Indicators or Outcomes

for Numbers and relationships and algebraic concepts

### NAEP - from the National Assessment for Educational Progress,

This strand extends from work with simple patterns at grade 4 to basic algebra concepts at grade 8 and sophisticated analysis at grade 12; it involves not only algebra but also precalculus and some topics from discrete mathematics. Algebraic concepts are developed throughout the grades, emphasizing informal modeling at the elementary level and functions at the secondary level.

Students are expected to use algebraic notation and thinking in meaningful contexts to solve mathematical and real-world problems, specifically addressing an increasing understanding of the use of functions (including algebraic and geometric) as a representational tool.

The assessment at all levels includes the use of open sentences and equations as representational tools. Students are expected to use equivalent representations to transform and solve number sentences and equations of increasing levels of complexity.

The *grade 4 assessment* involves the informal demonstration of students’ abilities to generalize from patterns and justify their generalizations. Students are expected to translate mathematical representations, use simple equations, and demonstrate basic graphing.

The *grade 8 assessment* includes more algebraic notation, stressing the meaning of variables and an informal understanding of the use of symbolic representations in problem-solving contexts. Students at this level are asked to use variables to represent a rule underlying a pattern. They should have a beginning understanding of equations as a modeling tool, and they should solve simple equations and inequalities through various methods, including both graphical and basic algebraic methods. Students should begin to use basic concepts of functions to describe relationships.

In *grade 12 *students are expected to be adept at appropriately choosing and applying a rich set of representational tools in various problem-solving situations. They should have an understanding of basic algebraic notation and terminology as they relate to representations of mathematical and real-world problem situations. Students should be able to use functions to represent and describe relationships.

## Indicators or Outcomes

for Numbers and relationships
and algebraic concepts

*Science for All Americans*

*Numbers and relationships* among them can be represented in symbolic statements, which provide a way to model, investigate, and display real-world relationships. Seldom are we interested in only one quantity or category; rather, we are usually interested in the relationship between them—the relationship between age and height, temperature and time of day, political party and annual income, sex and occupation. Such relationships can be expressed by using pictures (typically charts and graphs), tables, algebraic equations, or words. Graphs are especially useful in examining the relationships between quantities.

Algebra is a field of mathematics that explores the relationships among different quantities by representing them as symbols and manipulating statements that relate the symbols. Sometimes a symbolic statement implies that only one value or set of values will make the statement true. For example, the statement 2A+4 = 10 is true if (and only if) A = 3. More generally, however, an algebraic statement allows a quantity to take on any of a range of values and implies for each what the corresponding value of another quantity is. For example, the statement A = s2 specifies a value for the variable A that corresponds to any choice of a value for the variable s.

There are many possible kinds of relationships between one variable and another. A basic set of simple examples includes (1) directly proportional (one quantity always keeps the same proportion to another), (2) inversely proportional (as one quantity increases, the other decreases proportionally), (3) accelerated (as one quantity increases uniformly, the other increases faster and faster), (4) converging (as one quantity increases without limit, the other approaches closer and closer to some limiting value), (5) cyclical (as one quantity increases, the other increases and decreases in repeating cycles), and (6) stepped (as one quantity changes smoothly, the other changes in jumps).

Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. For example, we could state symbolically the relationship between the width of a page, P, the length of a line of type, L, and the width of each vertical margin, m: P = L+2m. This equation is a useful model for determining page makeup. It can be rearranged logically to give other true statements of the same basic relationship: for example, the equations L = P-2m or m = (P-L)/2, which may be more convenient for computing actual values for L or m.

In some cases, we may want to find values that will satisfy two or more different relationships at the same time. For example, we could add to the page-makeup model another condition: that the length of the line of type must be 2/3 of the page width: L = 2/3P. Combining this equation with m = (P-L)/2, we arrive logically at the result that m = 1/6P. This new equation, derived from the other two together, specifies the only values for m that will fit both relationships. In this simple example, the specification for the margin width could be worked out readily without using the symbolic relationships. In other situations, however, the symbolic representation and manipulation are necessary to arrive at a solution—or to see whether a solution is even possible.

Often, the quantity that interests us most is how fast something is changing rather than the change itself. In some cases, the rate of change of one quantity depends on some other quantity (for example, change in the velocity of a moving object is proportional to the force applied to it). In some other cases, the rate of change is proportional to the quantity itself (for example, the number of new mice born into a population of mice depends on the number and gender of mice already there).

## Indicators or Outcomes

for Algebraic and Functions

### National Assessment of Educational Progress (**NAEP) **

This strand extends from work with simple patterns at grade 4 to basic algebra concepts at grade 8 and sophisticated analysis at grade 12; it involves not only algebra but also precalculus and some topics from discrete mathematics. Algebraic concepts are developed throughout the grades, emphasizing informal modeling at the elementary level and functions at the secondary level.

Students are expected to use algebraic notation and thinking in meaningful contexts to solve mathematical and real-world problems, specifically addressing an increasing understanding of the use of functions (including algebraic and geometric) as a representational tool. The assessment at all levels includes the use of open sentences and equations as representational tools. Students are expected to use equivalent representations to transform and solve number sentences and equations of increasing levels of complexity.

The grade 4 assessment involves the informal demonstration of students’ abilities to generalize from patterns and justify their generalizations. Students are expected to translate mathematical representations, use simple equations, and demonstrate basic graphing.

The grade 8 assessment includes more algebraic notation, stressing the meaning of variables and an informal understanding of the use of symbolic representations in problem-solving contexts. Students at this level are asked to use variables to represent a rule underlying a pattern. They should have a beginning understanding of equations as a modeling tool, and they should solve simple equations and inequalities through various methods, including both graphical and basic algebraic methods. Students should begin to use basic concepts of functions to describe relationships.

In grade 12, students are expected to be adept at appropriately choosing and applying a rich set of representational tools in various problem-solving situations. They should have an understanding of basic algebraic notation and terminology as they relate to representations of mathematical and real-world problem situations. Students should be able to use functions to represent and describe relationships.

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