Diocese of
MATHEMATICS
ACADEMIC CONTENT
STANDARDS
for grades K - 12
MATHEMATICS
STANDARDS
Kindergarten
Number Sense
1.0 Number Relationships
1.1 Compare two or
more sets of objects (up to 10 objects per group) and identify which set is
equal to, more than, or less than the other.
1.2 Count, recognize,
represent, name, and order number of objects up to 30.
1.3 Know that the
larger numbers describe sets with more objects in them than the smaller numbers
have.
2.0 Addition and Subtraction
2.1 Use concrete objects to determine the
answers to addition and subtraction problems with two numbers, each less than
10.
3.0 Estimation
3.1 Recognize when
an estimate is reasonable.
Algebra and Functions
1.0 Sorting and Classifying Objects
1.1 Identify, sort, and classify objects by
attribute and identify which objects do not belong to a particular group.
Measurement and Geometry
1.0 Measurement
1.1 Compare the
length, weight, and capacity of objects (e.g., shorter, longer, taller,
lighter, heavier, holds more).
1.2 Understand
concepts of time (e.g., morning, afternoon, evening, today, yesterday,
tomorrow, week, month, year) and the tools used to measure time (e.g., clock,
calendar).
1.3 Name the days
of the week.
1.4 Identify the
time (to the nearest hour) of everyday events (e.g., lunch time is 12
o’clock).
2.0 Geometry
2.1 Identify and
describe common geometric objects such as the circle, triangle, square,
rectangle, cube, sphere, and cone.
2.2 Compare
familiar plane (e.g., square, triangle) and solid objects (e.g., cube, sphere) by
common attributes such as position, shape, size, roundness, and number of
corners.
Statistics, Data Analysis,
and Probability
1.0 Collecting Information
1.1 Pose
informational questions, collecting data, then record
the results using objects, pictures, and/or picture graphs.
1.2 Identify, describe, and extend simple
patterns (e.g., circle, square, circle) by referring to their shapes, sizes, or
colors.
Mathematical Reasoning
1.0 Making Decisions about a Problem
1.1 Determine the
approach, materials, and strategies to be used.
1.2 Use tools and
strategies, such as manipulatives or sketches, to
model problems.
2.0 Solve Problems & Justify Reasoning
2.1 Explain their reasoning when using concrete
objects and/or pictorial representations to solve a problem.
2.2 Make precise
calculations when solving a problem, and check the validity of the results in
the context of a problem.
MATHEMATICS STANDARDS
Grade One
Number Sense
1.0 Number Relationships
1.1 Count, read, and write whole numbers to 100.
1.2 Compare and order whole numbers to 100 by
using the symbols for “less than”, “equal to”, or
“greater than” (<, =, >).
1.3 Represent equivalent forms of the same number
to 20, using physical models, diagrams, and number expressions (e.g., 8 may be
represented as 4 + 4, 5 + 3, 2 + 2 + 2 + 2, 10 - 2, 11 - 3).
1.4 Count and group objects into ones and tens
(e.g., three groups of 10 and 4 equals 34, or 30 + 4).
1.5 Identify and know the value of coins then
show different combinations of coins equaling the same value.
2.0 Addition and Subtraction
2.1 Know and memorize the addition facts (sums to
20) and the corresponding subtraction facts.
2.2 Use the inverse relationship (e.g., checking
a subtraction problem using addition).
2.3 Identify one more than, one less than, 10
more than, and 10 less than a given number.
2.4 Count by 2s, 5s, and 10s to 100.
2.5 Show the meaning of addition (putting
together) and subtraction (taking away, compare, find the difference).
2.6 Solve addition and subtraction problems with
one- and two-digit numbers (e.g., 5 + 58 = __).
2.7 Find the sum of three one-digit numbers.
3.0 Estimation
3.1 Make reasonable estimates when comparing
larger or smaller numbers.
1.0 Number Sentences
1.1 Write and solve number sentences from problem
situations that express relationships involving addition and subtraction.
1.2 Understand the meaning of the symbols for
addition, subtraction, and equal to (+, -, =).
1.3 Create problem situations that might lead to
a given number sentence involving addition and subtraction.
1.0 Measurement
1.1 Compare the length, width, and volume of two
or more objects by using
standard or nonstandard units.
1.2 Tell time to the nearest half hour and relate
time to events (e.g., before/after, shorter/longer).
2.0 Geometry
2.1 Identify, describe, and compare triangles,
rectangles, squares, and circles, including the faces of three-dimensional
objects.
2.2 Classify familiar plane and solid objects by
common attributes, such as color, position, shape, size, roundness, or number
of corners, and explain which attributes are being used for classification.
2.3 Give and follow directions about location.
2.4 Arrange and describe objects in space by
proximity, position, and direction (e.g., near, far, below, above, up, down,
behind, in front of, next to, left or right of).
1.0 Data
1.1 Sort objects and data by common attributes
and describe the categories.
1.2 Represent
and compare data (e.g., largest, smallest, most often, least often) by using pictures,
bar graphs, tally charts, and picture graphs.
2.0 Patterning
2.1 Describe, extend, and explain ways to get to
the next element in simple repeating patterns (e.g., rhythmic, numeric, color,
shape).
Mathematical Reasoning
1.0 Making Decisions about a Problem
1.1 Determine the approach, materials, and
strategies to be used.
1.2 Use tools, such as manipulatives
or sketches, to model problems.
2.0 Solve Problems & Justify Reasoning
2.1 Explain reasoning used and justify the
procedures selected.
2.2 Make precise calculations and check the
validity of the results from the context of a problem.
3.0 Make Connections
3.1 Note the
connection between one problem and another
MATHEMATICS STANDARDS
Grade Two
1.0 Number Relationships
1.1 Count, read, and
write whole numbers to 1,000 and identify the place value for each digit.
1.2 Use words,
models, and expanded forms (e.g., 45 = 4 tens + 5) to represent numbers to
1,000.
1.3 Order and
compare whole numbers to 1,000 by using the symbols
<, =, >.
2.0 Addition and
Subtraction
2.1 Understand and
use the inverse relationship between addition and subtraction to solve problems
and check solutions (e.g., an opposite number sentence for 8 + 6 = 14 is 14 - 6 = 8).
2.2 Find the sum or
difference of two whole numbers up to three digits.
2.3 Use mental math
to find the sum or difference to two-digit numbers.
3.0 Multiplication and Division
3.1 Use repeated
addition, arrays, and count by multiples to do multiplication.
3.2 Use repeated
subtraction, equal sharing, and form equal groups with remainders to do
division.
3.3 Know/memorize
multiplication tables of 2s, 5s, and 10s to “10 X 10.”
4.0 Fractions and Decimals
4.1 Recognize, name, and compare unit fractions
from 1/12 to 1/2.
4.2 Recognize fractions of a whole and parts of a
group.
4.3 Know that all fractional parts together
(e.g., four fourths) equal one whole.
5.0 Computation with Money
5.1 Solve problems using combinations of coins
and bills.
5.2 Know and use decimal notation and the dollar
and cent symbols for money.
6.0 Estimation
6.1 Recognize when an estimate is reasonable in
measurements.
Algebra and Functions
1.0 Number Relationships
1.1 Use commutative
and associative rules to simplify mental calculations and to check results.
1.2 Relate problem
situations to number sentences involving addition and subtraction.
1.3 Solve addition
and subtraction problems using data from simple charts, picture graphs, and
number sentences.
Measurement and Geometry
1.0 Measurement
1.1 Measure the
length of objects by repeating a nonstandard or standard unit.
1.2 Use different units
to measure the same object and predict whether the measure will be greater or
smaller when a different unit is used.
1.3 Measure the
length of an object to the nearest inch and/or centimeter.
1.4 Tell time to the nearest quarter hour and know relationships of time (e.g., minutes in an hour, days in a month).
1.5 Determine the
duration of intervals of time in hours (e.g.,11:00
a.m.-4:00 p.m.).
2.0 Geometry
2.1 Describe and classify plane and solid geometric shapes (e.g., circle, triangle) according to the number and shape of faces, edges, and vertices.
2.2 Put shapes
together and take them apart to form other shapes.
Statistics, Data Analysis,
and Probability
1.0 Data
1.1 Record numerical
data in systematic ways, keeping track of what has been counted.
1.2 Represent the
same data in more than one way.
1.3 Identify range
and mode.
1.4 Ask and answer
simple questions related to data representations.
2.0 Patterning
2.1 Recognize,
describe, and extend patterns and determine a text term in linear patterns.
2.2 Solve problems
in simple number patterns.
Mathematical Reasoning
1.0 Making Decisions
about a Problem
1.1 Determine the
approach, materials, and strategies to be used.
1.2 Use tools, such as manipulatives or sketches, to model problems.
2.0 Solving Problems
and Justify Reasoning
2.1 Defend the
approach, materials, and strategies to be used.
2.2 Make precise
calculations and check the validity of the results from the context of the
problem.
3.0 Make Connections
3.1 Note connections between one problem and another.
MATHEMATICS
STANDARDS
Grade Three
1.0 Place Value
1.1 Count, read, and
write whole numbers to 10,000.
1.2 Compare and
order whole numbers to10,000.
1.3 Identify the
place value for each digit in numbers to 10,000.
1.4 Round
off numbers to 10,000 to the nearest ten, hundred, and thousand.
1.5 Use expanded
notation to represent numbers
(e.g., 3,206 = 3,000 + 200 + 6).
2.0 Computation
2.1 Find the sum or
difference of two whole numbers between 0 and 10,000.
2.2 Memorize
multiplication tables from 1 to 10.
2.3 Use the inverse
relationship of multiplication and division to compute and check results.
2.4 Solve
multiplication problems when multiplying by one-digit numbers.
2.5 Solve division
problems when dividing by a one-digit number with no remainder.
2.6 Understand the
special properties of 0 and 1 in multiplication and division.
2.7 Determine the
unit cost when given the total cost and number of units.
2.8 Solve problems
that require two or more of the skills mentioned above.
3.0 Fractions and
Decimals
3.1 Compare
equivalent fractions using drawings or concrete materials.
3.2 Add and subtract
simple fractions.
3.3 Solve problems
involving addition, subtraction, multiplication, and division of money amounts..
3.4 Understand that
fractions and decimals are two different representations of the same concept
(e.g., 50 cents is 1/2 of a dollar).
1.0 Number Sentences
1.1 Represent relationships of
quantities in the form of mathematical expressions, equations, or inequalities.
1.2 Solve problems
involving numeric equations or inequalities.
1.3 Select the
appropriate operation to make an expression true (e.g., 4 x 3 = 12).
1.4 Express simple
unit conversions in symbolic form (e.g., in. = ___ feet x 12).
1.5 Recognize and
use the commutative and associative properties of multiplication (e.g., if 5 x
7 x 3 = 105, then what is 7 x 3x 5?).
2.0 Functional Relationships
2.1 Solve simple
problems involving a functional relationship between two quantities (e.g., find
the total cost of multiple items given the cost per unit).
2.2 Extend and
recognize a linear pattern.
1.0 Measurement
1.1 Choose the
appropriate tools and units (metric and
1.2 Estimate or
determine the area and volume of solid figures by covering them with squares or
by counting the number of cubes that would fill them.
1.3 Find the
perimeter of a polygon with integer sides.
1.4 Carry out simple
unit conversions within a system of measurement (e.g., centimeters and meters,
hours and minutes).
2.0 Geometry
2.1 Identify, describe, and classify polygons.
2.2 Identify
attributes of triangles (e.g., two equal sides for the isosceles triangle).
2.3 Identify
attributes of quadrilaterals (e.g., parallel sides for the parallelogram, right
angles for the rectangle).
2.4 Identify right
angles in geometric figures or in appropriate objects and determine whether
other angles are greater or less than a right angle.
2.5 Identify,
describe, and classify common three-dimensional geometric objects (e.g., cube,
rectangular solid, sphere, prism, pyramid, cone, cylinder).
2.6 Identify common
solid objects that are the components needed to make a more complex solid
object.
Statistics, Data
Analysis, and Probability
1.0 Data
1.1 Identify whether common events are certain,
likely, unlikely, or improbable.
1.2 Record the possible outcomes for a simple event
(e.g., tossing a coin) and systematically keeping track of the outcomes when
the event is repeated many times.
1.3 Summarize and display the results of
probability experiments in a clear and organized way (e.g., using a bar graph).
1.4 Use the results
of probability experiments to predict future events.
Mathematical
Reasoning
1.0 Make Decisions about a Problem
1.1 Analyze problems
by identifying relationships, distinguishing relevant from irrelevant
information, sequencing and prioritizing information, and observing patterns.
1.2 Determine when
and how to break a problem into simpler parts.
2.0 Solve Problems, Justify Reasoning
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 Use a variety of
methods (e.g., words, numbers, symbols) to explain mathematical reasoning.
2.4 Express the
solution clearly and logically.
2.5 Indicate the
relative advantages of exact and approximate solutions to problems.
2.6 Make precise
calculations and check the validity of the results.
3.0 Make Connections
3.1 Evaluate the
reasonableness of the solution.
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 apply them in other circumstances.
MATHEMATICS STANDARDS
Grade Four
1.0 Place Value
1.1 Read and write
whole numbers to millions.
1.2 Order and
compare whole numbers and decimals to two decimal places.
1.3 Round whole
numbers through the millions.
1.4 Decide/explain
when a rounded solution is appropriate.
1.5 Explain
different interpretations of fractions (e.g., parts of a whole, parts of a set,
and division of whole numbers).
1.6 Write tenths and
hundredths in decimal and fraction notations and know the fraction and decimal
equivalents for halves and fourths (e.g., 1/2 = 0.5 or .50; 7/4 = 1 3/4 =
1.75).
1.7 Write the
fraction represented by a drawing of parts of a figure; represent a given
fraction by using drawings; and relate a fraction to a simple decimal on a
number line.
1.8 Use concepts of
negative numbers.
1.9 Identify, on a
number line, the relative position of positive fractions, positive mixed
numbers, and positive decimals to two decimal places.
2.0 Computation -
Decimals
2.1 Estimate and
compute the sum or difference of whole numbers and positive decimals to two
places.
2.2 Round two-place
decimals to one decimal or the nearest whole number and judge the
reasonableness of the rounded answer.
3.0 Computation -
Whole Numbers
3.1 Solve addition
and subtraction problems with multi-digit numbers.
3.2 Demonstrate an
understanding of, and the ability to use, standard algorithms for multiplying a
multi-digit number by a two-digit number and for dividing a multi-digit number
by a one-digit number; use relationships between them to simplify computations
and to check results.
3.3 Solve problems
involving multiplication of multi-digit numbers by two-digit numbers.
3.4 Solve problems
involving division of multi-digit numbers by one-digit numbers.
4.0 Factoring
4.1 Understand that
many whole numbers break down in different ways (e.g., 12 = 4 x 3 = 2 x 6 = 2 x
2 x 3).
4.2 Know that
numbers such as 2, 3, 5, 7, and 11 do not have any factors except 1 and themselves and that such numbers are called prime numbers.
1.0 Number Sentences
1.1 Use letters, boxes,
or other symbols to stand for any number in simple expressions or equations
(e.g., demonstrating an understanding and the use of the concept of a
variable).
1.2 Interpret and
evaluate mathematical expressions that now use parentheses.
1.3 Use parentheses
to indicate which operation to perform first when writing expressions
containing more than two terms and different operations.
1.4 Use and
interpret formulas (e.g., area = length x width or A = lw) to answer questions about quantities
and their relationships.
1.5 Understand that
an equation such as y = 3x + 5 is a
prescription for determining a second number when a first number is given.
2.0 Manipulate Equations
2.1 Know equals
added to equals are equal.
2.2 Know equals
multiplied by equals are equal.
Measurement
and Geometry
1.0 Area and Perimeter
1.1 Measure the area of rectangular shapes by
using appropriate units, such as square centimeter (cm2), square meter (m2),
square inch (in2), square yard (yd2), or square mile (mi2).
1.2 Recognize that rectangles
that have the same area can have different perimeters.
1.3 Understand that rectangles that have the
same perimeter can have different areas.
1.4 Understand and
use formulas to solve problems involving perimeters and areas of rectangles and
squares. Use those formulas to find the
areas of more complex figures by dividing the figures into basic shapes.
2.0 Coordinate Grids
2.1 Draw the points
corresponding to linear relationships on graph paper (e.g., draw 10 points on
the graph of the equation y = 3x and connect them by using a straight
line).
2.2 Understand that
the length of a horizontal line segment equals the difference of the x-coordinates.
2.3 Understand that
the length of a vertical line segment equals the difference of the y-coordinates.
3.0 Geometry
3.1 Identify lines
that are parallel and perpendicular.
3.2 Identify the
radius and diameter of a circle.
3.3 Identify
congruent figures.
3.4 Identify figures
that have bilateral and rotational symmetry.
3.5 Know the definitions
of a right angle, an acute angle, and an obtuse angle. Understand that 90˚, 180˚, 270˚,
and 360˚ are associated, respectively, with 1/4, 1/2, 3/4, and full turns.
3.6 Visualize,
describe, and make models of geometric solids (e.g., prisms, pyramids) in terms
of the number and shape of faces, edges, and vertices; interpret
two-dimensional representations of three-dimensional objects; and draw patterns
(of faces) for a solid that, when cut and folded, will make a model of the
solid.
3.7 Know the definitions
of different triangles (e.g., equilateral, isosceles, scalene) and identify
their attributes.
3.8 Know the
definition of different quadrilaterals (e.g., rhombus, square, rectangle,
parallelogram, trapezoid).
Statistics,
Data Analysis, and Probability
1.0 Data Analysis
1.1 Formulate survey questions; systematically
collecting and representing data on a number line; and coordinating graphs,
tables, and charts.
1.2 Identify the mode(s) for sets of categorical
data and the mode(s), median, and any apparent outliners for numerical data
sets.
1.3 Interpret one- and two-variable data graphs
to answer questions about a situation.
2.0 Making Predictions
2.1 Represent all possible outcomes for a simple probability
situation in an organized way (e.g., tables, grids, tree diagrams).
2.2 Express outcomes of experimental probability
situations verbally and numerically (e.g., 3 out of 4; 3/4).
1.0 Make Decisions about a Problem
1.1 Analyze problems by identifying
relationships, distinguishing relevant from irrelevant information, sequencing
and prioritizing information, and observing patterns.
1.2 Determine when and how to break a problem
into simpler parts.
2.0 Solve Problems and Justify Reasoning
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 Use a variety of methods, such as words,
numbers, symbols, charts, graphs, tables, diagrams, and models, to explain
mathematical reasoning.
2.4 Express the solution clearly and logically by
using the appropriate mathematical notation and terms and clear language;
supporting solutions with evidence in both verbal and symbolic work.
2.5 Indicate the relative advantages of exact and
approximate solutions to problems and give answers to a specified degree of
accuracy.
2.6 Make precise calculations and check the
validity of the results from the context of the problem.
3.0 Make Generalizations
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 applying them in other circumstances.
MATHEMATICS STANDARDS
Grade Five
Number Sense
1.0 Relative Magnitude of Numbers
1.1 Estimate, round, and manipulate very large
(e.g., millions) and very small (e.g., thousandths) numbers.
1.2 Interpret percents as a part of a hundred;
find decimal and percent equivalents for common fractions and explain why they
represent the same value; compute a given percent of a whole number.
1.3 Understand and compute positive integer
powers of nonnegative integers; compute examples as repeated multiplication.
1.4 Determine the prime factors of all numbers
through 50 and write the numbers as the product of their prime factors by using
exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3).
1.5 Identify and represent on a number line
decimals, fractions, mixed numbers, and positive and negative integers.
2.0 Computation
2.1 Add, subtract, multiply, and divide with
decimals; add with negative integers; subtract positive integers from negative
integers; and verify the reasonableness of the results.
2.2 Demonstrate proficiency with division,
including division with positive decimals and long division with multi-digit
divisors.
2.3 Solve simple problems, including ones arising
in concrete situations, involving the addition and subtraction of fractions and
mixed numbers (like and unlike denominators of 20 or less), and express answers
in the simplest form.
2.4 Understand the concept of multiplication and
division of fractions.
2.5 Compute and perform simple multiplication and
division of fractions, and apply these procedures to solving problems.
1.0 Simple Expressions
1.1 Use information
taken from a graph or equation to answer questions about a problem situation.
1.2 Use a letter to
represent an unknown number; write and evaluate simple algebraic expressions in
one variable by substitution.
1.3 Know and use the
distributive property in equations and expressions with variables.
1.4 Identify and
graph ordered pairs in the four quadrants of the coordinate plane.
1.5 Solve problems involving linear functions
with integer values; write the equation; and graph the resulting ordered pairs
of integers on a grid.
1.0 Area and Volume
1.1 Derive and use the
formula for the area of a triangle and of a parallelogram by comparing it with
the formula for the area of a rectangle (i.e., two of the same triangles make a
parallelogram with twice the area; a parallelogram is compared with a rectangle
of the same area by cutting and pasting a right triangle on the parallelogram).
1.2 Construct a cube
and rectangular box from two-dimensional patterns and use these patterns to
compute the surface area for these objects.
1.3 Understand the concept
of volume and use the appropriate units in common measuring systems (i.e.,
cubic centimeter [cm3], cubic meter [m3], cubic inch [in3], and cubic yard [yd3] ) to compute the volume of rectangular solids.
1.4 Differentiate
between, and use appropriate units of measures for, two- and three-dimensional
objects (i.e., find the perimeter, area, volume).
2.0 Geometry
2.1 Measure,
identify, and draw angles, perpendicular and parallel lines, rectangles, and
triangles by using appropriate tools (e.g., straightedge, ruler, compass,
protractor, drawing software).
2.2 Know that the sum of the angles of any
triangle is 180˚ and the sum of the angles of any quadrilateral is 360˚
and use this information to solve problems.
2.3 Visualize and
draw two-dimensional views of three-dimensional objects made from rectangular
solids.
Statistics, Data
Analysis, and Probability
1.0 Data
1.1 Know the
concepts of mean, median, and mode; computing and comparing simple examples to
show that they may differ.
1.2 Organize and
display single-variable data in appropriate graphs and representations (e.g.,
histogram, circle graphs) and explain which types of graphs are appropriate for
various data sets.
1.3 Use fractions
and percentages to compare data sets of different sizes.
1.4 Identify ordered
pairs of data from a graph and interpret the meaning of the data in terms of
the situation depicted by the graph.
1.5 Know how to
write ordered pairs correctly; for example, (x,y).
Mathematical
Reasoning
1.0 Making Decisions about a Problem
1.1 Analyze problems
by identifying relationships, distinguishing relevant from irrelevant
information, sequencing and prioritizing information, and observing patterns.
1.2 Determine
when and how to break a problem into simpler parts.
2.0 Solve Problems and Justify Reasoning
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 Use a variety of
methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and
models to explain mathematical reasoning.
2.4 Express the
solution clearly and logically by using the appropriate mathematical notation
and terms, and clear language; supporting solutions with evidence in both
verbal and symbolic work.
2.5 Indicate the relative advantages of exact and
approximate solutions to problems and giving answers to a specified degree of
accuracy.
2.6 Make precise
calculations and check the validity of the results from the context of the
problem.
3.0 Make Connections
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 apply them in other circumstances.
MATHEMATICS STANDARDS
Grade Six
Number Sense
1.0 Comparing and Ordering Numbers
1.1 Compare and
order positive and negative fractions, decimals, and mixed numbers and place
them on a number line.
1.2 Interpret and
use ratios in different contexts (e.g., batting averages, miles per hour) to
show the relative sizes of two quantities, using appropriate notations (a/b, a to b,
a:b).
1.3 Use proportions
to solve problems (e.g., determining the value of N if 4/7 = N/21, finding the length of a side of a polygon similar
to a known polygon). Use
cross-multiplication as a method for solving such problems, understanding it as
the multiplication of both sides of an equation by a multiplicative inverse.
1.4 Calculate given percentages of quantities
and solve problems involving discounts at sales, interest earned, and tips.
2.0 Calculating
2.1 Solve problems
involving addition, subtraction, multiplication, and division of positive
fractions and explain why a particular operation was used for a given
situation.
2.2 Explain the
meaning of multiplication and division of positive fractions and perform the
calculations (e.g., 5/8 ÷ 15/16 = 5/8 x 16/15 = 2/3).
2.3 Solve addition,
subtraction, multiplication, and division problems, including those arising in
concrete situations, that use positive and negative
integers and combinations of these operations.
2.4 Determine the least
common multiple and the greatest common divisor of whole numbers; use them to
solve problems with fractions (e.g., to find a common denominator to add two
fractions or to find the reduced form for a fraction).
Algebra and
Functions
1.0 Writing Expressions
1.1 Write and solve
one-step linear equations in one variable.
1.2 Write and
evaluate an algebraic expression for a given situation, using up to three
variables.
1.3 Apply algebraic
order of operations and the commutative, associative, and distributive
properties to evaluate expressions; and justify each step in the process.
1.4 Solve problems manually by using the correct
order of operations or by using a scientific calculator.
2.0 Rates and Proportions
2.1 Convert one unit
of measurement to another (e.g., from feet to miles, from centimeters to
inches).
2.2 Demonstrate an
understanding that rate is a measure
of one quantity per unit value of another quantity.
2.3 Solve problems
involving rates, average speed, distance, and time.
3.0 Patterns
3.1 Use variables in
expressions describing geometric quantities (e.g., P = 2w + 21, A = 1/2 bh, C = π d - the formulas
for the perimeter of a rectangle, the area of a triangle, and the circumference
of a circle, respectively).
3.2 Express in
symbolic form simple relationships arising from geometry.
Measurement
and Geometry
1.0 Area and Volume
1.1 Understand the
concept of a constant such as π; knowing the formulas for the
circumference and the area of a circle.
1.2 Know common
estimates of π (3.14; 22/7) and
use these values to estimate and calculate the circumference and the area of
circles; compare with actual measurements.
1.3 Know and use the
formulas for the volume of triangular prisms and cylinders (area of base x
height); compare these formulas and explain the similarity between them and the
formula for the volume of a rectangular solid.