|
By the end of grade seven, students are adept at manipulating numbers
and equations and understand the general principles at work. Students
understand and use factoring of numerators and denominators and
properties of exponents. They know the Pythagorean theorem and solve
problems in which they compute the length of an unknown side. Students
know how to compute the surface area and volume of basic
three-dimensional objects and understand how area and volume change with
a change in scale. Students make conversions between different units of
measurement. They know and use different representations of fractional
numbers (fractions, decimals, and percents) and are proficient at
changing from one to another. They increase their facility with ratio
and proportion, compute percents of increase and decrease, and compute
simple and compound interest. They graph linear functions and understand
the idea of slope and its relation to ratio.
Number Sense
1.0 Students know the properties of, and compute
with, rational numbers expressed in a variety of forms:
1.1 Read, write, and compare rational numbers in
scientific notation (positive and negative powers of 10) with
approximate numbers using scientific notation.
1.2 Add, subtract, multiply, and divide rational numbers (integers,
fractions, and terminating decimals) and take positive rational numbers
to whole-number powers.
1.3 Convert fractions to decimals and percents and use these
representations in estimations, computations, and applications.
1.4 Differentiate between rational and irrational numbers.
1.5 Know that every rational number is either a terminating or repeating
decimal and be able to convert terminating decimals into reduced
fractions.
1.6 Calculate the percentage of increases and decreases of a quantity.
1.7 Solve problems that involve discounts, markups, commissions, and
profit and compute simple and compound interest.
2.0 Students use exponents, powers, and roots and use
exponents in working with fractions:
2.1 Understand negative whole-number exponents.
Multiply and divide expressions involving exponents with a common base.
2.2 Add and subtract fractions by using factoring to find common
denominators.
2.3 Multiply, divide, and simplify rational numbers by using exponent
rules.
2.4 Use the inverse relationship between raising to a power and
extracting the root of a perfect square integer; for an integer that is
not square, determine without a calculator the two integers between
which its square root lies and explain why.
2.5 Understand the meaning of the absolute value of a number; interpret
the absolute value as the distance of the number from zero on a number
line; and determine the absolute value of real numbers.
Algebra and Functions
1.0 Students express quantitative relationships by
using algebraic terminology, expressions, equations, inequalities, and
graphs:
1.1 Use variables and appropriate operations to write
an expression, an equation, an inequality, or a system of equations or
inequalities that represents a verbal description (e.g., three less than
a number, half as large as area A).
1.2 Use the correct order of operations to evaluate algebraic
expressions such as 3(2x + 5)2.
1.3 Simplify numerical expressions by applying properties of rational
numbers (e.g., identity, inverse, distributive, associative,
commutative) and justify the process used.
1.4 Use algebraic terminology (e.g., variable, equation, term,
coefficient, inequality, expression, constant) correctly.
1.5 Represent quantitative relationships graphically and interpret the
meaning of a specific part of a graph in the situation represented by
the graph.
2.0 Students interpret and evaluate expressions
involving integer powers and simple roots:
2.1 Interpret positive whole-number powers as repeated
multiplication and negative whole-number powers as repeated division or
multiplication by the multiplicative inverse. Simplify and evaluate
expressions that include exponents.
2.2 Multiply and divide monomials; extend the process of taking powers
and extracting roots to monomials when the latter results in a monomial
with an integer exponent.
3.0 Students graph and interpret linear and some
nonlinear functions:
3.1 Graph functions of the form y = nx2
and y = nx3 and use in solving problems.
3.2 Plot the values from the volumes of three-dimensional shapes for
various values of the edge lengths (e.g., cubes with varying edge
lengths or a triangle prism with a fixed height and an equilateral
triangle base of varying lengths).
3.3 Graph linear functions, noting that the vertical change (change in
y- value) per unit of horizontal change (change in x-
value) is always the same and know that the ratio ("rise over run") is
called the slope of a graph.
3.4 Plot the values of quantities whose ratios are always the same
(e.g., cost to the number of an item, feet to inches, circumference to
diameter of a circle). Fit a line to the plot and understand that the
slope of the line equals the quantities.
4.0 Students solve simple linear equations and
inequalities over the rational numbers:
4.1 Solve two-step linear equations and inequalities
in one variable over the rational numbers, interpret the solution or
solutions in the context from which they arose, and verify the
reasonableness of the results.
4.2 Solve multistep problems involving rate, average speed, distance,
and time or a direct variation.
Measurement and Geometry
1.0 Students choose appropriate units of measure and
use ratios to convert within and between measurement systems to solve
problems:
1.1 Compare weights, capacities, geometric measures,
times, and temperatures within and between measurement systems (e.g.,
miles per hour and feet per second, cubic inches to cubic centimeters).
1.2 Construct and read drawings and models made to scale.
1.3 Use measures expressed as rates (e.g., speed, density) and measures
expressed as products (e.g., person-days) to solve problems; check the
units of the solutions; and use dimensional analysis to check the
reasonableness of the answer.
2.0 Students compute the perimeter, area, and volume
of common geometric objects and use the results to find measures of less
common objects. They know how perimeter, area, and volume are affected
by changes of scale:
2.1 Use formulas routinely for finding the perimeter
and area of basic two-dimensional figures and the surface area and
volume of basic three-dimensional figures, including rectangles,
parallelograms, trapezoids, squares, triangles, circles, prisms, and
cylinders.
2.2 Estimate and compute the area of more complex or irregular two-and
three-dimensional figures by breaking the figures down into more basic
geometric objects.
2.3 Compute the length of the perimeter, the surface area of the faces,
and the volume of a three-dimensional object built from rectangular
solids. Understand that when the lengths of all dimensions are
multiplied by a scale factor, the surface area is multiplied by the
square of the scale factor and the volume is multiplied by the cube of
the scale factor.
2.4 Relate the changes in measurement with a change of scale to the
units used (e.g., square inches, cubic feet) and to conversions between
units (1 square foot = 144 square inches or [1 ft2]
= [144 in2], 1 cubic inch is approximately
16.38 cubic centimeters or [1 in3] = [16.38 cm3]).
3.0 Students know the Pythagorean theorem and deepen
their understanding of plane and solid geometric shapes by constructing
figures that meet given conditions and by identifying attributes of
figures:
3.1 Identify and construct basic elements of geometric
figures (e.g., altitudes, mid-points, diagonals, angle bisectors, and
perpendicular bisectors; central angles, radii, diameters, and chords of
circles) by using a compass and straightedge.
3.2 Understand and use coordinate graphs to plot simple figures,
determine lengths and areas related to them, and determine their image
under translations and reflections.
3.3 Know and understand the Pythagorean theorem and its converse and use
it to find the length of the missing side of a right triangle and the
lengths of other line segments and, in some situations, empirically
verify the Pythagorean theorem by direct measurement.
3.4 Demonstrate an understanding of conditions that indicate two
geometrical figures are congruent and what congruence means about the
relationships between the sides and angles of the two figures.
3.5 Construct two-dimensional patterns for three-dimensional models,
such as cylinders, prisms, and cones.
3.6 Identify elements of three-dimensional geometric objects (e.g.,
diagonals of rectangular solids) and describe how two or more objects
are related in space (e.g., skew lines, the possible ways three planes
might intersect).
Statistics, Data Analysis, and Probability
1.0 Students collect, organize, and represent data
sets that have one or more variables and identify relationships among
variables within a data set by hand and through the use of an electronic
spreadsheet software program:
1.1 Know various forms of display for data sets,
including a stem-and-leaf plot or box-and-whisker plot; use the forms to
display a single set of data or to compare two sets of data.
1.2 Represent two numerical variables on a scatterplot and informally
describe how the data points are distributed and any apparent
relationship that exists between the two variables (e.g., between time
spent on homework and grade level).
1.3 Understand the meaning of, and be able to compute, the minimum, the
lower quartile, the median, the upper quartile, and the maximum of a
data set.
Mathematical Reasoning
1.0 Students make decisions about how to approach
problems:
1.1 Analyze problems by identifying relationships,
distinguishing relevant from irrelevant information, identifying missing
information, sequencing and prioritizing information, and observing
patterns.
1.2 Formulate and justify mathematical conjectures based on a general
description of the mathematical question or problem posed.
1.3 Determine when and how to break a problem into simpler parts.
2.0 Students use strategies, skills, and concepts in
finding solutions:
2.1 Use estimation to verify the reasonableness of
calculated results.
2.2 Apply strategies and results from simpler problems to more complex
problems.
2.3 Estimate unknown quantities graphically and solve for them by using
logical reasoning and arithmetic and algebraic techniques.
2.4 Make and test conjectures by using both inductive and deductive
reasoning.
2.5 Use a variety of methods, such as words, numbers, symbols, charts,
graphs, tables, diagrams, and models, to explain mathematical reasoning.
2.6 Express the solution clearly and logically by using the appropriate
mathematical notation and terms and clear language; support solutions
with evidence in both verbal and symbolic work.
2.7 Indicate the relative advantages of exact and approximate solutions
to problems and give answers to a specified degree of accuracy.
2.8 Make precise calculations and check the validity of the results from
the context of the problem.
3.0 Students determine a solution is complete and
move beyond a particular problem by generalizing to other situations:
3.1 Evaluate the reasonableness of the solution in the
context of the original situation.
3.2 Note the method of deriving the solution and demonstrate a
conceptual understanding of the derivation by solving similar problems.
3.3 Develop generalizations of the results obtained and the strategies
used and apply them to new problem situations.
|