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Mathematics is a language we use every day,
often without knowing it. It builds and draws on conceptual understanding
and skills, and helps us make decisions and solve problems. In this draft
document we have tried to connect the notion of problem solving to conceptual
understanding and skill development by embedding it within the content strands
at every grade. Given here are only some of the ways you will see this
exemplified. Students are asked to:
- ask relevant questions about problem situations
- decide between relevant and extraneous information
- choose appropriate operations tools and approaches to
problem situations
- decide whether an exact or approximate answer is called
for
- apply specific techniques in new situations
- explain, check, justify, prove, and judge the reasonableness
of results
- create new approaches and connect knowledge and understanding
in new ways
Number Sense
1. Students understand the real number system as a coherent set
of elements, operations, and properties.
1.1 identify and apply field properties (including closure), axioms
of equality and inequality, and properties of order that are valid for
the set of real numbers and its subsets.
1.2 describe relationships among the subsets of the set of real numbers
1.3 use properties of numbers to construct simple valid arguments (direct
and indirect) of, or formulate counter-examples to, claimed assertions
2. Students understand and use operations such as opposite, reciprocal,
raising to a power, and taking a root.
2.l. use identities, opposites, reciprocals, integral and rational powers
to simplify expressions and justify steps in an algebraic process.
Task/Assignment
If an integer, x, is a perfect square of n (x = n^2) find a formula
for the least integer after x that is a perfect square. Explain and justify
your reasoning.
Symbols and Algebra
1. Students write and solve linear equations and justify the process.
1.1 simplify and solve linear equations and inequalities in one variable
(e.g., solve (3(2x - 5) + 4(x - 2) = 12x + 17),
using inverse operations, integers, and fractions and graph the solutions
on a number line
1.2 use properties of real numbers to justify steps used in simplifying
expressions, solving equations and solving inequalities
1.3 describe the logic of algebraic procedures (e.g., why the sense
of an inequality changes when both sides are multiplied or divided by a
negative number)
1.4 solve word problems including those involving direct and inverse
variation
2. Students apply the operations and properties of the real number
system to simplify polynomial expressions and justify the process used.
2.1 expand or combine polynomial expressions and justify the steps by
referring to a particular property
2.2 apply basic factoring techniques to second- and simple third-degree
polynomials (common factor, difference of squares, perfect square trinomials,
sum and difference of cubes, and general trinomials) including expressions
involving variables expressed as quantities rather than simply as a letter
(e.g. (3x - 2)^2 - (y + 4)^2, x^6 5x^3 - 6)
2.3 divide a polynomial by a polynomial by expressing as a rational
fraction, factoring the numerator and denominator, and reducing to lowest
terms
2.4 apply factoring techniques and properties of fractions to add, subtract,
multiply, and divide rational expressions
2.5 simplify square roots of monomials that are not perfect squares
(e.g., sqrt32x^8, sqrt18x^5)
2.6 solve equations containing rational expressions and equations containing
radical expressions algebraically and graphically
3. Students solve systems of linear equations and simple factorable
polynomial equations of higher degree.
3.1 solve simple higher degree polynomial equations (e.g., a^2 - 64
= 0, x^4 7x^3 + 3x^2 21 = 0)
in one variable by setting equal to zero and factoring
3.2 formulate, interpret, and solve systems of two linear equations
in two variables, both algebraically and graphically
Measurement and Geometry
1. Students use coordinate geometry to determine attributes of
lines and line segments.
1.1 determine the length, midpoint and slope of a line segment on a
coordinate graph
1.2 find the distance between two points, between a point and a vertical
or horizontal line and between two vertical or horizontal lines
1.3 use slope to determine whether two lines are perpendicular or parallel
1.4 explain and justify the Pythagorean Theorem using coordinates
2. Students model situations geometrically to answer questions
about length, area, or angle measure.
2.1 draw diagrams to interpret practical situations geometrically (e.g.,
the amount of wood needed to frame a 3' x 2.5' portrait, the number of
plants that would fit in a garden plot if the plants were to be placed
6' apart)
2.2 write and solve equations involving perimeter, area (e.g., If the
perimeter of rectangle  is less than 20, what might the
area be?), or angle measure (e.g., Find the measures of each angle in the
triangle  )
Functions
1. Students discover, describe, and generalize patterns, including
linear, exponential, and simple quadratic relationships, and represent them
with expressions, equations, and graphs.
1.1 interpret and use linear functions as a mathematical representation
of proportional relationships and of non-proportional relationships having
a constant rate of change
1.2 write the equation of a line, given two points on the line, one
point and the slope, or the slope and y-intercept
1.3 discover, describe and generalize linear, exponential, and simple
(factorable) quadratic functions and graph them on the coordinate plane
2. Students understand the concepts of a relation and a function,
determine whether a given relation defines a function, and give pertinent
information about given relations and functions.
2.1 determine the domain and range of a relation defined by a graph,
a set of ordered pairs, or a symbolic expression
2.2 determine whether a relation defined by a graph, a set of ordered
pairs, or a symbolic expression is a function and justify the conclusion
2.3 represent linear, exponential, and simple quadratic relationships
using formulas, tables, and graphs and translate between these forms
3. Students determine and interpret the slope and intercept(s)
of a line.
3.1 find the slope of a line given by a table of values, a graph or
symbolically
3.2 interpret the meaning of the slope as a constant rate of change
or in the context of a verbal problem (e.g., the cost of mailing a letter
increases 24¢ for each additional half ounce)
3.3 understand the relationship between the solution of a linear equation
in one variable and the x-intercept of a related linear equation in two
variables (e.g., If 3x + 12 = 0, then x = -4 and (-4,0) is the x-intercept
of y = 3x + 12)
Statistics, Data Analysis, and Probability
1. Students make inferences and predictions based on the analysis
of a set of data.
1.1 collect, organize and display single-variable data in appropriately
designed and labeled tables, charts, and graphs (e.g., use a bar graph,
histogram, or frequency table for grouped data)
1.2 use information displayed in graphs (line, bar circle, and picture
graphs and histograms) to make comparisons, predictions, inferences, and
critique the conclusions (e.g., assess the accuracy of graphs taken from
the media; analyze possible bias in survey questions)
1.3 determine a line of best fit ("eyeball," median-median,
or linear regression by calculator) for a set of bi-variate data, write
its equation, and extrapolate or interpolate to make predictions
1.4 explain what measure of central tendency is most representative
for a given set of data
1.5 describe the effect on the mean, mode, median, quartiles, and range
when the same number is added to each data point or when each data point
is multiplied by the same number
2. Students estimate and find the possible outcomes and probability
of simple and compound events.
2.1 construct an appropriate sample space and apply addition and multiplication
principles of probability for a simple or compound chance event (e.g.,
games of chance, board games, spinners, dice, coins, cards)
2.2 estimate the probability of simple and compound events using a series
of trials
2.3 explain why two results are equally likely or why one is more likely
than another
Credits
The Draft Standards were prepared by:
The State of California Academic Standards Commission
The Commission for the Establishment of Academic Content and Performance Standards
Comments may be addressed to The Commission
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