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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.
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