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.
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.
2.1 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.
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.
2.1 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.
3.1 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.
4.1 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 multi step problems involving rate, average speed, distance,
and time or a direct variation.
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.
2.1 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]).
3.1 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).
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 scatter plot 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.
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.
2.1 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.
3.1 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.