September 22, 2008 Page 1 of 60
Minnesota K-12 Academic Standards in Mathematics
2007 Version
This official standards document contains the mathematics standards revised in 2007 and put into rule effective September 22, 2008.
The Minnesota Academic Standards in Mathematics set the expectations for achievement in mathematics for K-12 students in Minnesota. This
document is grounded in the belief that all students can and should be mathematically proficient. All students should learn important
mathematical concepts, skills, and relationships with understanding. The standards and benchmarks presented here describe a connected body
of mathematical knowledge that is acquired through the processes of problem solving, reasoning and proof, communication, connections, and
representation. The standards are placed at the grade level where mastery is expected with the recognition that intentional experiences at
earlier grades are required to facilitate learning and mastery for other grade levels.
The Minnesota Academic Standards in Mathematics are organized by grade level into four content strands: 1) Number and Operation, 2)
Algebra, 3) Geometry and Measurement, and 4) Data Analysis and Probability. Each strand has one or more standards, and the benchmarks for
each standard are designated by a code. In reading the coding, please note that for 3.1.3.2, the first 3 refers to the third grade, the 1 refers to
the Number and Operation strand, the next 3 refers to the third standard for that strand, and the 2 refers to the second benchmark for that
standard.
Grade Strand Standard Benchmark Code Benchmark
3 Number
and
Operation
Understand
meanings and
uses of
fractions in
the real-world
and
3.1.3.1 Read and write fractions with words and
symbols. Recognize that fractions can be
used to represent parts of a whole, parts
of a set, points on a number line, or
3.1.3.1 distances on a number line.
September 22, 2008 Page 2 of 60
Grade Strand Standard Benchmark Code Benchmark
mathematical
situations
For example: Parts of a shape (3/4 of a
pie), parts of a set (3 out of 4 people),
and measurements (3/4 of an inch).
3 Number
and
Operation
Understand
meanings and
uses of
fractions in
the real-world
and
mathematical
situations
3.1.3.2 Understand that the size of a fractional
part is relative to the size of the whole.
For example: One-half of a small pizza is
smaller than one-half of a large pizza, but
both represent one-half.
3 Number
and
Operation
Understand
meanings and
uses of
fractions in
the real-world
and
mathematical
situations
3.1.3.3 Order and compare unit fractions and
fractions with like denominators by using
models and an understanding of the
concept of numerator and denominator.
Please refer to the Frequently Asked Questions (FAQ) document for the Academic Standards for Mathematics for further information. This FAQ
document can be found under Academic Standards on the website
for the Minnesota Department of Education at
https://education.state.mn.gov.
September 22, 2008 Page 3 of 60
Minnesota K-12 Academic Standards in Mathematics
Grade
Strand
Standard
Code
K
Number &
Operation
Understand the relationship
between quantities and whole
numbers up to 31.
K.1.1.1
in a set or to represent the position of an object in a sequence.
For example: Count students standing in a circle and count the same
students after they take their seats. Recognize that this rearrangement
does not change the total number, but may change the order in which
students are counted.
K
Number &
Operation
Understand the relationship
between quantities and whole
numbers up to 31.
K.1.1.2
Representations may include numerals, pictures, real objects and picture
graphs, spoken words, and manipulatives such as connecting cubes.
For example: Represent the number of students taking hot lunch with
tally marks.
K
Number &
Operation
Understand the relationship
between quantities and whole
numbers up to 31.
K.1.1.3
K
Number &
Operation
Understand the relationship
between quantities and whole
numbers up to 31.
K.1.1.4
K
Number &
Operation
Understand the relationship
between quantities and whole
numbers up to 31.
K.1.1.5
20.
For example: Put the number cards 7, 3, 19 and 12 in numerical order.
K
Number &
Operation
Use objects and pictures to
represent situations involving
combining and separating.
K.1.2.1
numbers between 0 and 10.
K
Number &
Operation
Use objects and pictures to
represent situations involving
combining and separating.
K.1.2.2
For example: A group of 7 objects can be decomposed as 5 and 2 objects,
or 2 and 3 and 2, or 6 and 1.
September 22, 2008 Page 4 of 60
Grade
Strand
Standard
Code
K Algebra
Recognize, create, complete, and
extend patterns.
K.2.1.1
size, number, sounds and movements. Patterns may be repeating,
growing or shrinking such as ABB, ABB, ABB or ●, ●●, ●●●.
K
Geometry &
Measurement
Recognize and sort basic two- and
three-dimensional shapes; use them
to model real-world objects.
K.3.1.1
circles, triangles, rectangles, trapezoids, hexagons, cubes, cones, cylinders
and spheres.
K
Geometry &
Measurement
Recognize and sort basic two- and
three-dimensional shapes; use them
to model real-world objects.
K.3.1.2
K
Geometry &
Measurement
Recognize and sort basic two- and
three-dimensional shapes; use them
to model real-world objects.
K.3.1.3
For example: A cylinder can be used to model a can of soup.
Another example: Find as many rectangles as you can in your classroom.
Record the rectangles you found by making drawings.
K
Geometry &
Measurement
Compare and order objects
according to location and
measurable attributes.
K.3.2.1
position.
For example: Use same, lighter, longer, above, between and next to.
Another example: Identify objects that are near your desk and objects
that are in front of it. Explain why there may be some objects in both
groups.
K
Geometry &
Measurement
Compare and order objects
according to location and
measurable attributes.
K.3.2.2
and weight.
1
Number &
Operation
Count, compare and represent
whole numbers up to 120, with an
emphasis on groups of tens and
ones.
1.1.1.1
of tens and ones.
For example: Recognize the numbers 21 to 29 as 2 tens and a particular
number of ones.
September 22, 2008 Page 5 of 60
Grade
Strand
Standard
Code
1
Number &
Operation
Count, compare and represent
whole numbers up to 120, with an
emphasis on groups of tens and
ones.
1.1.1.2
may include numerals, addition and subtraction, pictures, tally marks,
number lines and manipulatives, such as bundles of sticks and base 10
blocks.
1
Number &
Operation
Count, compare and represent
whole numbers up to 120, with an
emphasis on groups of tens and
ones.
1.1.1.3
number up to 120.
1
Number &
Operation
Count, compare and represent
whole numbers up to 120, with an
emphasis on groups of tens and
ones.
1.1.1.4
For example: Using a hundred grid, find the number that is 10 more than
27.
1
Number &
Operation
Count, compare and represent
whole numbers up to 120, with an
emphasis on groups of tens and
ones.
1.1.1.5
1
Number &
Operation
Count, compare and represent
whole numbers up to 120, with an
emphasis on groups of tens and
ones.
1.1.1.6
For example: Use the words equal to, not equal to, more than, less than,
fewer than, is about, and is nearly to describe numbers.
1
Number &
Operation
Count, compare and represent
whole numbers up to 120, with an
emphasis on groups of tens and
ones.
1.1.1.7
tally charts.
For example: Make a bar graph of students' birthday months and count
to compare the number in each month.
1
Number &
Operation
Use a variety of models and
strategies to solve addition and
subtraction problems in real-world
and mathematical contexts.
1.1.2.1
numerals and number lines to model and solve addition and subtraction
problems in part-part-total, adding to, taking away from and comparing
situations.
September 22, 2008 Page 6 of 60
Grade
Strand
Standard
Code
1
Number &
Operation
Use a variety of models and
strategies to solve addition and
subtraction problems in real-world
and mathematical contexts.
1.1.2.2
ten.
For example: Given 3 blocks, 7 more blocks are needed to make 10.
1
Number &
Operation
Use a variety of models and
strategies to solve addition and
subtraction problems in real-world
and mathematical contexts.
1.1.2.3
subtraction. Skip count by 2s, 5s, and 10s.
1 Algebra
Recognize and create patterns; use
rules to describe patterns.
1.2.1.1
Identify possible rules to complete or extend patterns. Patterns may be
repeating, growing or shrinking. Calculators can be used to create and
explore patterns.
For example: Describe rules that can be used to extend the pattern 2, 4,
6, 8, __, __, __ and complete the pattern 33, 43, __, 63, __, 83 or 20,
__,__, 17.
1 Algebra
Use number sentences involving
addition and subtraction basic facts
to represent and solve real-world
and mathematical problems; create
real-world situations corresponding
to number sentences.
1.2.2.1
facts, using objects and number sentences.
For example: One way to represent the number of toys that a child has
left after giving away 4 of 6 toys is to begin with a stack of 6 connecting
cubes and then break off 4 cubes.
1 Algebra
Use number sentences involving
addition and subtraction basic facts
to represent and solve real-world
and mathematical problems; create
real-world situations corresponding
to number sentences.
1.2.2.2
For example: Determine if the following number sentences are true or
false
7 = 7
7 = 8 – 1
5 + 2 = 2 + 5
September 22, 2008 Page 7 of 60
Grade
Strand
Standard
Code
1 Algebra
Use number sentences involving
addition and subtraction basic facts
to represent and solve real-world
and mathematical problems; create
real-world situations corresponding
to number sentences.
1.2.2.3
objects and number lines, to identify the missing number in an equation
such as:
2 + 4 = __
3 +__ = 7
1 Algebra
Use number sentences involving
addition and subtraction basic facts
to represent and solve real-world
and mathematical problems; create
real-world situations corresponding
to number sentences.
1.2.2.4
situation using a number sentence.
For example: 5 + 3 = 8 could be used to represent a situation in which 5
red balloons are combined with 3 blue balloons to make 8 total balloons.
1
Geometry &
Measurement
Describe characteristics of basic
shapes. Use basic shapes to
compose and decompose other
objects in various contexts.
1.3.1.1
triangles, squares, rectangles, circles, rectangular prisms, cylinders, cones
and spheres.
For example: Triangles have three sides and cubes have eight vertices
(corners).
1
Geometry &
Measurement
Describe characteristics of basic
shapes. Use basic shapes to
compose and decompose other
objects in various contexts.
1.3.1.2
dimensional figures such as triangles, squares, rectangles, circles,
rectangular prisms and cylinders.
For example: Decompose a regular hexagon into 6 equilateral triangles;
build prisms by stacking layers of cubes; compose an ice cream cone by
combining a cone and half of a sphere.
Another example: Use a drawing program to find shapes that can be
made with a rectangle and a triangle.
1
Geometry &
Measurement
Use basic concepts of measurement
in real-world and mathematical
situations involving length, time and
money.
1.3.2.1
object.
For example: Measure a table by placing paper clips end-to-end and
counting.
September 22, 2008 Page 8 of 60
Grade
Strand
Standard
Code
1
Geometry &
Measurement
Use basic concepts of measurement
in real-world and mathematical
situations involving length, time and
money.
1.3.2.2
1
Geometry &
Measurement
Use basic concepts of measurement
in real-world and mathematical
situations involving length, time and
money.
1.3.2.3
coins, up to one dollar.
2
Number &
Operation
Compare and represent whole
numbers up to 1000 with an
emphasis on place value and
equality.
2.1.1.1
may include numerals, addition, subtraction, multiplication, words,
pictures, tally marks, number lines and manipulatives, such as bundles of
sticks and base 10 blocks.
2
Number &
Operation
Compare and represent whole
numbers up to 1000 with an
emphasis on place value and
equality.
2.1.1.2
terms of hundreds, tens and ones. Know that 100 is 10 tens, and 1000 is
10 hundreds.
For example: Writing 853 is a shorter way of writing
2
Number &
Operation
Compare and represent whole
numbers up to 1000 with an
emphasis on place value and
equality.
2.1.1.3
100 less than a given three-digit number.
For example: Find the number that is 10 less than 382 and the number
that is 100 more than 382.
2
Number &
Operation
Compare and represent whole
numbers up to 1000 with an
emphasis on place value and
equality.
2.1.1.4
to the nearest 10 and 100.
For example: If there are 17 students in the class and granola bars come
10 to a box, you need to buy 20 bars (2 boxes) in order to have enough
bars for everyone.
2
Number &
Operation
Compare and represent whole
numbers up to 1000 with an
emphasis on place value and
equality.
2.1.1.5
September 22, 2008 Page 9 of 60
Grade
Strand
Standard
Code
2
Number &
Operation
Compare and represent whole
numbers up to 1000 with an
emphasis on place value and
equality.
2.1.2.1
making tens, fact families, doubles plus or minus one, counting on,
counting back, and the commutative and associative properties. Use the
relationship between addition and subtraction to generate basic facts.
For example: Use the associative property to make tens when adding
2
Number &
Operation
Demonstrate mastery of addition
and subtraction basic facts; add and
subtract one- and two-digit numbers
in real-world and mathematical
problems.
2.1.2.2
facts.
2
Number &
Operation
Demonstrate mastery of addition
and subtraction basic facts; add and
subtract one- and two-digit numbers
in real-world and mathematical
problems.
2.1.2.3
For example: Know that 23 + 48 is about 70.
2
Number &
Operation
Demonstrate mastery of addition
and subtraction basic facts; add and
subtract one- and two-digit numbers
in real-world and mathematical
problems.
2.1.2.4
and equality to add and subtract two-digit numbers. Strategies may
include decomposition, expanded notation, and partial sums and
differences.
For example: Using decomposition, 78 + 42, can be thought of as:
78 + 2 + 20 + 20 = 80 + 20 + 20 = 100 + 20 = 120
and using expanded notation, 34 - 21 can be thought of as:
2
Number &
Operation
Demonstrate mastery of addition
and subtraction basic facts; add and
subtract one- and two-digit numbers
in real-world and mathematical
problems.
2.1.2.5
involving whole numbers with up to 2 digits.
September 22, 2008 Page 10 of 60
Grade
Strand
Standard
Code
2
Number &
Operation
Demonstrate mastery of addition
and subtraction basic facts; add and
subtract one- and two-digit numbers
in real-world and mathematical
problems.
2.1.2.6
tables, bar graphs and tally charts.
2
Algebra
Recognize, create, describe, and use
patterns and rules to solve real-
world and mathematical problems.
2.2.1.1
addition or subtraction, skip counting and arrays of objects such as
counters or tiles. Use patterns to solve problems in various contexts.
For example: Skip count by 5s beginning at 3 to create the pattern 3, 8,
13, 18, …
Another example: Collecting 7 empty milk cartons each day for 5 days will
generate the pattern 7, 14, 21, 28, 35, resulting in a total of 35 milk
cartons.
2 Algebra
Use number sentences involving
addition, subtraction and unknowns
to represent and solve real-world
and mathematical problems; create
real-world situations corresponding
to number sentences.
2.2.2.1
subtraction and unknowns represented by letters. Use objects and
number lines and create real-world situations to represent number
sentences.
For example: One way to represent n + 16 = 19 is by comparing a stack of
16 connecting cubes to a stack of 19 connecting cubes; 24 = a + b can be
represented by a situation involving a birthday party attended by a total
of 24 boys and girls.
2
Algebra
Use number sentences involving
addition, subtraction and unknowns
to represent and solve real-world
and mathematical problems; create
real-world situations corresponding
to number sentences.
2.2.2.2
represent given problem situations. Use number sense and properties of
addition and subtraction to find values for the unknowns that make the
number sentences true.
For example: How many more players are needed if a soccer team
requires 11 players and so far only 6 players have arrived? This situation
can be represented by the number sentence 11 – 6 = p or by the number
sentence 6 + p = 11.
September 22, 2008 Page 11 of 60
Grade
Strand
Standard
Code
2
Geometry &
Measurement
Identify, describe and compare basic
shapes according to their geometric
attributes.
2.3.1.1
according to number and shape of faces, and the number of sides, edges
and vertices (corners).
2
Geometry &
Measurement
Identify, describe and compare basic
shapes according to their geometric
attributes.
2.3.1.2
squares, circles, triangles, rectangles, trapezoids, hexagons, cubes,
rectangular prisms, cones, cylinders and spheres.
For example: Use a drawing program to show several ways that a
rectangle can be decomposed into exactly three triangles.
2
Geometry &
Measurement
Understand length as a measurable
attribute; use tools to measure
length
2.3.2.1
and the number of units needed to measure the length of an object.
For example: It will take more paper clips than whiteboard markers to
measure the length of a table.
2
Geometry &
Measurement
Understand length as a measurable
attribute; use tools to measure
length
2.3.2.2
the numbers on a ruler by using a ruler to measure lengths to the nearest
centimeter or inch.
For example: Draw a line segment that is 3 inches long.
2
Geometry &
Measurement
Use time and money in real-world
and mathematical situations.
2.3.3.1
2
Geometry &
Measurement
Use time and money in real-world
and mathematical situations.
2.3.3.2
coins and determine combinations of coins that equal a given amount.
For example: 50 cents can be made up of 2 quarters, or 4 dimes and 2
nickels, or many other combinations.
3
Number &
Operation
Compare and represent whole
numbers up to 100,000 with an
emphasis on place value and
equality.
3.1.1.1
Representations may include numerals, expressions with operations,
words, pictures, number lines, and manipulatives such as bundles of
sticks and base 10 blocks.
September 22, 2008 Page 12 of 60
Grade
Strand
Standard
Code
3
Number &
Operation
Compare and represent whole
numbers up to 100,000 with an
emphasis on place value and
equality.
3.1.1.2
terms of ten thousands, thousands, hundreds, tens and ones.
For example: Writing 54,873 is a shorter way of writing the following
sums:
5 ten thousands + 4 thousands + 8 hundreds + 7 tens + 3 ones
3
Number &
Operation
Compare and represent whole
numbers up to 100,000 with an
emphasis on place value and
equality.
3.1.1.3
more or 1000 less than a given four- or five-digit. Find 100 more or 100
less than a given four- or five-digit number.
3
Number &
Operation
Compare and represent whole
numbers up to 100,000 with an
emphasis on place value and
equality.
3.1.1.4
round down to estimate sums and differences.
For example: 8726 rounded to the nearest 1000 is 9000, rounded to the
nearest 100 is 8700, and rounded to the nearest 10 is 8730.
Another example: 473 – 291 is between 400 – 300 and 500 – 200, or
between 100 and 300.
3
Number &
Operation
Compare and represent whole
numbers up to 100,000 with an
emphasis on place value and
equality.
3.1.1.5
3
Number &
Operation
Add and subtract multi-digit whole
numbers; represent multiplication
and division in various ways; solve
real-world and mathematical
problems using arithmetic.
3.1.2.1
procedures based on knowledge of place value, including standard
algorithms.
3
Number &
Operation
Add and subtract multi-digit whole
numbers; represent multiplication
and division in various ways; solve
real-world and mathematical
problems using arithmetic.
3.1.2.2
problems involving whole numbers. Use various strategies, including the
relationship between addition and subtraction, the use of technology,
and the context of the problem to assess the reasonableness of results.
For example: The calculation 117 – 83 = 34 can be checked by adding 83
and 34.
September 22, 2008 Page 13 of 60
Grade
Strand
Standard
Code
3
Number &
Operation
Add and subtract multi-digit whole
numbers; represent multiplication
and division in various ways; solve
real-world and mathematical
problems using arithmetic.
3.1.2.3
repeated addition, equal-sized groups, arrays, area models, equal jumps
on a number line and skip counting. Represent division facts by using a
variety of approaches, such as repeated subtraction, equal sharing and
forming equal groups. Recognize the relationship between multiplication
and division.
3
Number &
Operation
Add and subtract multi-digit whole
numbers; represent multiplication
and division in various ways; solve
real-world and mathematical
problems using arithmetic.
3.1.2.4
division, including both "how many in each group" and "how many
groups" division problems.
For example: You have 27 people and 9 tables. If each table seats the
same number of people, how many people will you put at each table?
Another example: If you have 27 people and tables that will hold 9
people, how many tables will you need?
3
Number &
Operation
Add and subtract multi-digit whole
numbers; represent multiplication
and division in various ways; solve
real-world and mathematical
problems using arithmetic.
3.1.2.5
equality and properties of addition and multiplication to multiply a two-
or three-digit number by a one-digit number. Strategies may include
mental strategies, partial products, the standard algorithm, and the
commutative, associative, and distributive properties.
For example: 9 × 26 = 9 × (20 + 6) = 9 × 20 + 9 × 6 = 180 + 54 = 234.
3
Number &
Operations
Understand meanings and uses of
fractions in real-world and
mathematical situations.
3.1.3.1
fractions can be used to represent parts of a whole, parts of a set, points
on a number line, or distances on a number line.
For example: Parts of a shape (3/4 of a pie), parts of a set (3 out of 4
people), and measurements (3/4 of an inch).
3
Number &
Operations
Understand meanings and uses of
fractions in real-world and
mathematical situations.
3.1.3.2
whole.
For example: One-half of a small pizza is smaller than one-half of a large
pizza, but both represent one-half.
September 22, 2008 Page 14 of 60
Grade
Strand
Standard
Code
3
Number &
Operations
Understand meanings and uses of
fractions in real-world and
mathematical situations.
3.1.3.3
using models and an understanding of the concept of numerator and
denominator.
3
Algebra
Use single-operation input-output
rules to represent patterns and
relationships and to solve real-world
and mathematical problems.
3.2.1.1
addition, subtraction and multiplication to solve problems in various
contexts.
For example: Describe the relationship between number of chairs and
number of legs by the rule that the number of legs is four times the
number of chairs.
3
Algebra
Use number sentences involving
multiplication and division basic
facts and unknowns to represent
and solve real-world and
mathematical problems; create real-
world situations corresponding to
number sentences
3.2.2.1
and division basic facts and unknowns. Create real-world situations to
represent number sentences.
For example: The number sentence 8 × m = 24 could be represented by
the question "How much did each ticket to a play cost if 8 tickets totaled
$24?"
3
Algebra
Use number sentences involving
multiplication and division basic
facts and unknowns to represent
and solve real-world and
mathematical problems; create real-
world situations corresponding to
number sentences
3.2.2.2
situation using a number sentence. Use number sense and multiplication
and division basic facts to find values for the unknowns that make the
number sentences true.
For example: Find values of the unknowns that make each number
sentence true
6 = p ÷ 9
24 = a × b
5 × 8 = 4 × t.
Another example: How many math teams are competing if there is a total
of 45 students with 5 students on each team? This situation can be
represented by 5 × n = 45 or
45
5
= n or
45
n
= 5.
3
Geometry &
Measurement
Use geometric attributes to describe
and create shapes in various
contexts.
3.3.1.1
to describe and create geometric shapes, such as right triangles,
rectangles, parallelograms and trapezoids.
September 22, 2008 Page 15 of 60
Grade
Strand
Standard
Code
3
Geometry &
Measurement
Use geometric attributes to describe
and create shapes in various
contexts.
3.3.1.2
as pentagons, hexagons and octagons.
3
Geometry &
Measurement
Understand perimeter as a
measurable attribute of real-world
and mathematical objects. Use
various tools to measure distances.
3.3.2.1
For example: Measure a person's height to the nearest half inch.
3
Geometry &
Measurement
Understand perimeter as a
measurable attribute of real-world
and mathematical objects. Use
various tools to measure distances.
3.3.2.2
3
Geometry &
Measurement
Understand perimeter as a
measurable attribute of real-world
and mathematical objects. Use
various tools to measure distances.
3.3.2.3
For example: Measure the distance around a classroom, or measure a
person's wrist size.
3
Geometry &
Measurement
Use time, money and temperature
to solve real-world and
mathematical problems.
3.3.3.1
elapsed time to the minute.
For example: Your trip began at 9:50 a.m. and ended at 3:10 p.m. How
long were you traveling?
3
Geometry &
Measurement
Use time, money and temperature
to solve real-world and
mathematical problems.
3.3.3.2
For example: Know the number of minutes in an hour, days in a week and
months in a year.
3
Geometry &
Measurement
Use time, money and temperature
to solve real-world and
mathematical problems.
3.3.3.3
few coins as possible.
For example: A chocolate bar costs $1.84. You pay for it with $2. Give two
possible ways to make change.
September 22, 2008 Page 16 of 60
Grade
Strand
Standard
Code
3
Geometry &
Measurement
Use time, money and temperature
to solve real-world and
mathematical problems.
3.3.3.4
degree in Fahrenheit and Celsius.
For example: Read the temperature in a room with a thermometer that
has both Fahrenheit and Celsius scales. Use the thermometer to compare
Celsius and Fahrenheit readings.
3 Data Analysis
Collect, organize, display, and
interpret data. Use labels and a
variety of scales and units in
displays.
3.4.1.1
picture graphs and number line plots having a variety of scales. Use
appropriate titles, labels and units.
4
Number &
Operation
Demonstrate mastery of
multiplication and division basic
facts; multiply multi-digit numbers;
solve real-world and mathematical
problems using arithmetic.
4.1.1.1
4
Number &
Operation
Demonstrate mastery of
multiplication and division basic
facts; multiply multi-digit numbers;
solve real-world and mathematical
problems using arithmetic.
4.1.1.2
1000.
4
Number &
Operation
Demonstrate mastery of
multiplication and division basic
facts; multiply multi-digit numbers;
solve real-world and mathematical
problems using arithmetic.
4.1.1.3
procedures, based on knowledge of place value, including standard
algorithms.
4
Number &
Operation
Demonstrate mastery of
multiplication and division basic
facts; multiply multi-digit numbers;
solve real-world and mathematical
problems using arithmetic.
4.1.1.4
rounding, benchmarks and place value to assess the reasonableness of
results.
For example: 53 × 38 is between 50 × 30 and 60 × 40, or between 1500
and 2400, and 411/73 is between 5 and 6.
September 22, 2008 Page 17 of 60
Grade
Strand
Standard
Code
4
Number &
Operation
Demonstrate mastery of
multiplication and division basic
facts; multiply multi-digit numbers;
solve real-world and mathematical
problems using arithmetic.
4.1.1.5
of addition, subtraction and multiplication of multi-digit whole numbers.
Use various strategies, including the relationship between operations, the
use of technology, and the context of the problem to assess the
reasonableness of results.
4
Number &
Operation
Demonstrate mastery of
multiplication and division basic
facts; multiply multi-digit numbers;
solve real-world and mathematical
problems using arithmetic.
4.1.1.6
equality and properties of operations to divide multi-digit whole numbers
by one- or two-digit numbers. Strategies may include mental strategies,
partial quotients, the commutative, associative, and distributive
properties and repeated subtraction.
For example: A group of 324 students is going to a museum in 6 buses. If
each bus has the same number of students, how many students will be on
each bus?
4
Number &
Operation
Represent and compare fractions
and decimals in real-world and
mathematical situations; use place
value to understand how decimals
represent quantities.
4.1.2.1
set, fraction circles, fraction strips, number lines and other manipulatives.
Use the models to determine equivalent fractions.
4
Number &
Operation
Represent and compare fractions
and decimals in real-world and
mathematical situations; use place
value to understand how decimals
represent quantities.
4.1.2.2
whole numbers and fractions, including mixed numbers and improper
fractions.
For example: Locate
and 1
on a number line and give a comparison
statement about these two fractions, such as "
is less than 1
."
4
Number &
Operation
Represent and compare fractions
and decimals in real-world and
mathematical situations; use place
value to understand how decimals
represent quantities.
4.1.2.3
in real-world and mathematical situations. Develop a rule for addition and
subtraction of fractions with like denominators.
September 22, 2008 Page 18 of 60
Grade
Strand
Standard
Code
4
Number &
Operation
Represent and compare fractions
and decimals in real-world and
mathematical situations; use place
value to understand how decimals
represent quantities.
4.1.2.4
describe decimals in terms of thousands, hundreds, tens, ones, tenths,
hundredths and thousandths.
For example: Writing 362.45 is a shorter way of writing the sum:
3 hundreds + 6 tens + 2 ones + 4 tenths + 5 hundredths,
which can also be written as:
4
Number &
Operation
Represent and compare fractions
and decimals in real-world and
mathematical situations; use place
value to understand how decimals
represent quantities.
4.1.2.5
number line and models such as grids and base 10 blocks.
4
Number &
Operation
Represent and compare fractions
and decimals in real-world and
mathematical situations; use place
value to understand how decimals
represent quantities.
4.1.2.6
using words and symbols; know the fraction and decimal equivalents for
halves and fourths.
For example
= 0.5 = 0.50 and
= 1
= 1.75, which can also be written as
one and three-fourths or one and seventy-five hundredths.
4
Number &
Operation
Represent and compare fractions
and decimals in real-world and
mathematical situations; use place
value to understand how decimals
represent quantities.
4.1.2.7
For example: The number 0.36 rounded to the nearest tenth is 0.4.
September 22, 2008 Page 19 of 60
Grade
Strand
Standard
Code
4
Algebra
Use input-output rules, tables and
charts to represent patterns and
relationships and to solve real-world
and mathematical problems.
4.2.1.1
multiplication and division to solve problems in various contexts. Record
the inputs and outputs in a chart or table.
For example: If the rule is "multiply by 3 and add 4," record the outputs
for given inputs in a table.
Another example: A student is given these three arrangements of dots:
Identify a pattern that is consistent with these figures, create an input-
output rule that describes the pattern, and use the rule to find the
number of dots in the 10th figure.
4
Algebra
Use number sentences involving
multiplication, division and
unknowns to represent and solve
real-world and mathematical
problems; create real-world
situations corresponding to number
sentences.
4.2.2.1
division and unknowns. Use real-world situations involving multiplication
or division to represent number sentences.
For example: The number sentence a × b = 60 can be represented by the
situation in which chairs are being arranged in equal rows and the total
number of chairs is 60.
4
Algebra
Use number sentences involving
multiplication, division and
unknowns to represent and solve
real-world and mathematical
problems; create real-world
situations corresponding to number
sentences.
4.2.2.2
situation using a number sentence. Use number sense, properties of
multiplication, and the relationship between multiplication and division
to find values for the unknowns that make the number sentences true.
For example: If $84 is to be shared equally among a group of children, the
amount of money each child receives can be determined using the
number sentence 84 ÷ n = d.
Another example: Find values of the unknowns that make each number
sentence true:
12 × m = 36
4
Geometry &
Measurement
Name, describe, classify and sketch
polygons.
4.3.1.1
and acute triangles. Recognize triangles in various contexts.
September 22, 2008 Page 20 of 60
Grade
Strand
Standard
Code
4
Geometry &
Measurement
Name, describe, classify and sketch
polygons.
4.3.1.2
trapezoids, rhombuses, parallelograms and kites. Recognize quadrilaterals
in various contexts.
4
Geometry &
Measurement
Understand angle and area as
measurable attributes of real-world
and mathematical objects. Use
various tools to measure angles and
areas.
4.3.2.1
protractor or angle ruler.
4
Geometry &
Measurement
Understand angle and area as
measurable attributes of real-world
and mathematical objects. Use
various tools to measure angles and
areas.
4.3.2.2
obtuse.
For example: Compare different hockey sticks according to the angle
between the blade and the shaft.
4
Geometry &
Measurement
Understand angle and area as
measurable attributes of real-world
and mathematical objects. Use
various tools to measure angles and
areas.
4.3.2.3
counting the total number of same size square units that cover a shape
without gaps or overlaps. Justify why length and width are multiplied to
find the area of a rectangle by breaking the rectangle into one unit by one
unit squares and viewing these as grouped into rows and columns.
For example: How many copies of a square sheet of paper are needed to
cover the classroom door? Measure the length and width of the door to
the nearest inch and compute the area of the door.
4
Geometry &
Measurement
Understand angle and area as
measurable attributes of real-world
and mathematical objects. Use
various tools to measure angles and
areas.
4.3.2.4
divided into rectangular shapes. Use square units to label area
measurements.
4
Geometry &
Measurement
Use translations, reflections and
rotations to establish congruency
and understand symmetries.
4.3.3.1
4
Geometry &
Measurement
Use translations, reflections and
rotations to establish congruency
and understand symmetries.
4.3.3.2
lines and relate reflections to lines of symmetry.
September 22, 2008 Page 21 of 60
Grade
Strand
Standard
Code
4
Geometry &
Measurement
Use translations, reflections and
rotations to establish congruency
and understand symmetries.
4.3.3.3
4
Geometry &
Measurement
Use translations, reflections and
rotations to establish congruency
and understand symmetries.
4.3.3.4
congruency and use them to show that two figures are congruent.
4
Data Analysis
Collect, organize, display and
interpret data, including data
collected over a period of time and
data represented by fractions and
decimals.
4.4.1.1
The data may include fractions or decimals. Understand that spreadsheet
tables and graphs can be used to display data.
5
Number &
Operation
Divide multi-digit numbers; solve
real-world and mathematical
problems using arithmetic.
5.1.1.1
based on knowledge of place value, including standard algorithms.
Recognize that quotients can be represented in a variety of ways,
including a whole number with a remainder, a fraction or mixed number,
or a decimal.
For example: Dividing 153 by 7 can be used to convert the improper
fraction
153
7
to the mixed number 21
.
5
Number &
Operation
Divide multi-digit numbers; solve
real-world and mathematical
problems using arithmetic.
5.1.1.2
useful form of the quotient for the solution and use the context to
interpret the quotient appropriately.
For example: If 77 amusement ride tickets are to be distributed equally
among 4 children, each child will receive 19 tickets, and there will be one
left over. If $77 is to be distributed equally among 4 children, each will
receive $19.25, with nothing left over.
5
Number &
Operation
Divide multi-digit numbers; solve
real-world and mathematical
problems using arithmetic.
5.1.1.3
reasonableness of results.
September 22, 2008 Page 22 of 60
Grade
Strand
Standard
Code
5
Number &
Operation
Divide multi-digit numbers; solve
real-world and mathematical
problems using arithmetic.
5.1.1.4
subtraction, multiplication and division of multi-digit whole numbers. Use
various strategies, including the inverse relationships between
operations, the use of technology, and the context of the problem to
assess the reasonableness of results.
For example: The calculation 117 ÷ 9 = 13 can be checked by multiplying
9 and 13.
5
Number &
Operation
Read, write, represent and compare
fractions and decimals; recognize
and write equivalent fractions;
convert between fractions and
decimals; use fractions and decimals
in real-world and mathematical
situations.
5.1.2.1
of groups from millionths to millions.
For example: Possible names for the number 0.0037 are:
37 ten thousandths
3 thousandths + 7 ten thousandths;
a possible name for the number 1.5 is 15 tenths.
5
Number &
Operation
Read, write, represent and compare
fractions and decimals; recognize
and write equivalent fractions;
convert between fractions and
decimals; use fractions and decimals
in real-world and mathematical
situations.
5.1.2.2
than a number and 0.01 less than a number. Find 0.001 more than a
number and 0.001 less than a number.
5
Number &
Operation
Read, write, represent and compare
fractions and decimals; recognize
and write equivalent fractions;
convert between fractions and
decimals; use fractions and decimals
in real-world and mathematical
situations.
5.1.2.3
fractions, and locate on a number line.
For example: Which is larger 1.25 or
?
Another example: In order to work properly, a part must fit through a
0.24 inch wide space. If a part is
inch wide, will it fit?
September 22, 2008 Page 23 of 60
Grade
Strand
Standard
Code
5
Number &
Operation
Read, write, represent and compare
fractions and decimals; recognize
and write equivalent fractions;
convert between fractions and
decimals; use fractions and decimals
in real-world and mathematical
situations.
5.1.2.4
and improper fractions in various contexts.
For example: When comparing 1.5 and


, note that 1.5 = 1
= 1

, =


,
so 1.5 <


.
5
Number &
Operation
Read, write, represent and compare
fractions and decimals; recognize
and write equivalent fractions;
convert between fractions and
decimals; use fractions and decimals
in real-world and mathematical
situations.
5.1.2.5
For example: Fifth grade students used a calculator to find the mean of
the monthly allowance in their class. The calculator display shows
25.80645161. Round this number to the nearest cent.
5
Number &
Operation
Add and subtract fractions, mixed
numbers and decimals to solve real-
world and mathematical problems.
5.1.3.1
procedures, including standard algorithms.
5
Number &
Operation
Add and subtract fractions, mixed
numbers and decimals to solve real-
world and mathematical problems.
5.1.3.2
of representations.
For example: Represent
+
and
-
by drawing a rectangle divided into
4 columns and 3 rows and shading the appropriate parts or by using
fraction circles or bars.
5
Number &
Operation
Add and subtract fractions, mixed
numbers and decimals to solve real-
world and mathematical problems.
5.1.3.3
reasonableness of results.
For example: Recognize that 12
- 3
is between 8 and 9 (since
<
).
5
Number &
Operation
Add and subtract fractions, mixed
numbers and decimals to solve real-
world and mathematical problems.
5.1.3.4
subtraction of decimals, fractions and mixed numbers, including those
involving measurement, geometry and data.
For example: Calculate the perimeter of the soccer field when the length
is 109.7 meters and the width is 73.1 meters.
September 22, 2008 Page 24 of 60
Grade
Strand
Standard
Code
5
Algebra
Recognize and represent patterns of
change; use patterns, tables, graphs
and rules to solve real-world and
mathematical problems.
5.2.1.1
patterns of change and solve problems.
For example: An end-of-the-year party for 5th grade costs $100 to rent
the room and $4.50 for each student. Know how to use a spreadsheet to
create an input-output table that records the total cost of the party for
any number of students between 90 and 150.
5
Algebra
Recognize and represent patterns of
change; use patterns, tables, graphs
and rules to solve real-world and
mathematical problems.
5.2.1.2
graph these ordered pairs on a coordinate system.
5
Algebra
Use properties of arithmetic to
generate equivalent numerical
expressions and evaluate
expressions involving whole
numbers.
5.2.2.1
of operations to generate equivalent numerical expressions and to solve
problems involving whole numbers.
For example: Purchase 5 pencils at 19 cents and 7 erasers at 19 cents.
The numerical expression is 5 × 19 + 7 × 19 which is the same as (5 + 7) ×
19.
5
Algebra
Understand and interpret equations
and inequalities involving variables
and whole numbers, and use them
to represent and solve real-world
and mathematical problems.
5.2.3.1
or false for a given value of the variable.
For example: Determine whether the inequality 1.5 + x < 10 is true for
x = 2.8, x = 8.1, or x = 9.2.
5
Algebra
Understand and interpret equations
and inequalities involving variables
and whole numbers, and use them
to represent and solve real-world
and mathematical problems.
5.2.3.2
variables. Create real-world situations corresponding to equations and
inequalities.
For example: 250 – 27 x a = b can be used to represent the number of
sheets of paper remaining from a packet of 250 sheets when each
student in a class of 27 is given a certain number of sheets.
September 22, 2008 Page 25 of 60
Grade
Strand
Standard
Code
5
Algebra
Understand and interpret equations
and inequalities involving variables
and whole numbers, and use them
to represent and solve real-world
and mathematical problems.
5.2.3.3
for the variables are given.
For example: Using the formula, A= ℓw, determine the area when the
length is 5, and the width 6, and find the length when the area is 24 and
the width is 4.
5
Geometry &
Measurement
Describe, classify, and draw
representations of three-
dimensional figures.
5.3.1.1
and pyramids by the number of edges, faces or vertices as well as the
types of faces.
5
Geometry &
Measurement
Describe, classify, and draw
representations of three-
dimensional figures.
5.3.1.2
5
Geometry &
Measurement
Determine the area of triangles and
quadrilaterals; determine the
surface area and volume of
rectangular prisms in various
contexts.
5.3.2.1
parallelograms and figures that can be decomposed into triangles.
5
Geometry &
Measurement
Determine the area of triangles and
quadrilaterals; determine the
surface area and volume of
rectangular prisms in various
contexts.
5.3.2.2
of objects that are shaped like rectangular prisms.
For example: Use a net or decompose the surface into rectangles.
Another example: Measure the volume of a cereal box by using a ruler to
measure its height, width and length, or by filling it with cereal and then
emptying the cereal into containers of known volume.
5
Geometry &
Measurement
Determine the area of triangles and
quadrilaterals; determine the
surface area and volume of
rectangular prisms in various
contexts.
5.3.2.3
by counting the total number of same-sized cubic units that fill a shape
without gaps or overlaps. Use cubic units to label volume measurements.
For example: Use cubes to find the volume of a small box.
September 22, 2008 Page 26 of 60
Grade
Strand
Standard
Code
5
Geometry &
Measurement
Determine the area of triangles and
quadrilaterals; determine the
surface area and volume of
rectangular prisms in various
contexts.
5.3.2.4
volume of rectangular prisms. Justify why base area B and height h are
multiplied to find the volume of a rectangular prism by breaking the prism
into layers of unit cubes.
5
Data Analysis
Display and interpret data;
determine mean, median and range.
5.4.1.1
data. Know how to use a spreadsheet to find the mean, median and range
of a data set. Understand that the mean is a "leveling out" of data.
For example: The set of numbers 1, 1, 4, 6 has mean 3. It can be leveled
by taking one unit from the 4 and three units from the 6 and adding them
to the 1s, making four 3s.
5
Data Analysis
Display and interpret data;
determine mean, median and range.
5.4.1.2
understanding of whole numbers, fractions and decimals. Know how to
create spreadsheet tables and graphs to display data.
6
Number &
Operation
Read, write, represent and compare
positive rational numbers expressed
as fractions, decimals, percents and
ratios; write positive integers as
products of factors; use these
representations in real-world and
mathematical situations.
6.1.1.1
positive rational numbers on a coordinate grid.
6
Number &
Operation
Read, write, represent and compare
positive rational numbers expressed
as fractions, decimals, percents and
ratios; write positive integers as
products of factors; use these
representations in real-world and
mathematical situations.
6.1.1.2
symbols <, = and >.
For example:
> 0.36.
September 22, 2008 Page 27 of 60
Grade
Strand
Standard
Code
6
Number &
Operation
Read, write, represent and compare
positive rational numbers expressed
as fractions, decimals, percents and
ratios; write positive integers as
products of factors; use these
representations in real-world and
mathematical situations.
6.1.1.3
For example: 75% corresponds to the ratio 75 to 100, which is equivalent
to the ratio 3 to 4.
6
Number &
Operation
Read, write, represent and compare
positive rational numbers expressed
as fractions, decimals, percents and
ratios; write positive integers as
products of factors; use these
representations in real-world and
mathematical situations.
6.1.1.4
among these representations to solve problems.
For example: If a woman making $25 an hour gets a 10% raise, she will
make an additional $2.50 an hour, because $2.50 is

or 10% of $25.
6
Number &
Operation
Read, write, represent and compare
positive rational numbers expressed
as fractions, decimals, percents and
ratios; write positive integers as
products of factors; use these
representations in real-world and
mathematical situations.
6.1.1.5
factors with exponents.
For example: 24 = 2
x 3.
6
Number &
Operation
Read, write, represent and compare
positive rational numbers expressed
as fractions, decimals, percents and
ratios; write positive integers as
products of factors; use these
representations in real-world and
mathematical situations.
6.1.1.6
common factors and common multiples to calculate with fractions and
find equivalent fractions.
For example: Factor the numerator and denominator of a fraction to
determine an equivalent fraction.
September 22, 2008 Page 28 of 60
Grade
Strand
Standard
Code
6
Number &
Operation
Read, write, represent and compare
positive rational numbers expressed
as fractions, decimals, percents and
ratios; write positive integers as
products of factors; use these
representations in real-world and
mathematical situations.
6.1.1.7
numbers.
For example: Express as.



=
+
= 1
6
Number &
Operation
Read, write, represent and compare
positive rational numbers expressed
as fractions, decimals, percents and
ratios; write positive integers as
products of factors; use these
representations in real-world and
mathematical situations.
6.1.2.1
quantities using ratios is not the same as comparing quantities using
subtraction.
For example: In a classroom with 15 boys and 10 girls, compare the
numbers by subtracting (there are 5 more boys than girls) or by dividing
(there are 1.5 times as many boys as girls). The comparison using division
may be expressed as a ratio of boys to girls (3 to 2 or 3:2 or 1.5 to 1).
6
Number &
Operation
Understand the concept of ratio and
its relationship to fractions and to
the multiplication and division of
whole numbers. Use ratios to solve
real-world and mathematical
problems.
6.1.2.2
to solve problems in various contexts, including those involving mixtures
and concentrations.
For example: If 5 cups of trail mix contains 2 cups of raisins, the ratio of
raisins to trail mix is 2 to 5. This ratio corresponds to the fact that the
raisins are
of the total, or 40% of the total. And if one trail mix consists
of 2 parts peanuts to 3 parts raisins, and another consists of 4 parts
peanuts to 8 parts raisins, then the first mixture has a higher
concentration of peanuts.
6
Number &
Operation
Understand the concept of ratio and
its relationship to fractions and to
the multiplication and division of
whole numbers. Use ratios to solve
real-world and mathematical
problems.
6.1.2.3
For example: 60 miles for every 3 hours is equivalent to 20 miles for every
one hour (20 mph).
September 22, 2008 Page 29 of 60
Grade
Strand
Standard
Code
6
Number &
Operation
Understand the concept of ratio and
its relationship to fractions and to
the multiplication and division of
whole numbers. Use ratios to solve
real-world and mathematical
problems.
6.1.2.4
problems.
For example: If 5 items cost $3.75, and all items are the same price, then
1 item costs 75 cents, so 12 items cost $9.00.
6
Number &
Operation
Multiply and divide decimals,
fractions and mixed numbers; solve
real-world and mathematical
problems using arithmetic with
positive rational numbers.
6.1.3.1
generalizable procedures, including standard algorithms.
6
Number &
Operation
Multiply and divide decimals,
fractions and mixed numbers; solve
real-world and mathematical
problems using arithmetic with
positive rational numbers.
6.1.3.2
relationship between multiplication and division to make sense of
procedures for multiplying and dividing fractions.
For example: Just as
12
4
= 3  12 = 3 4, =
2
3
÷
4
5
=
5
6

5
6
4
5
=
2
3
6
Number &
Operation
Multiply and divide decimals,
fractions and mixed numbers; solve
real-world and mathematical
problems using arithmetic with
positive rational numbers.
6.1.3.3
number is of another number to solve problems in various contexts.
For example: If John has $45 and spends $15, what percent of his money
did he keep?
6
Number &
Operation
Multiply and divide decimals,
fractions and mixed numbers; solve
real-world and mathematical
problems using arithmetic with
positive rational numbers.
6.1.3.4
decimals, fractions and mixed numbers.
September 22, 2008 Page 30 of 60
Grade
Strand
Standard
Code
6
Number &
Operation
Multiply and divide decimals,
fractions and mixed numbers; solve
real-world and mathematical
problems using arithmetic with
positive rational numbers.
6.1.3.5
decimals and use the estimates to assess the reasonableness of results in
the context of the problem.
For example: The sum
+ 0.25 can be estimated to be between ½ and 1,
and this estimate can be used to check the result of a more detailed
calculation.
6
Algebra
Recognize and represent
relationships between varying
quantities; translate from one
representation to another; use
patterns, tables, graphs and rules to
solve real-world and mathematical
problems.
6.2.1.1
change, often in relationship to another changing quantity. Use variables
in various contexts.
For example: If a student earns $7 an hour in a job, the amount of money
earned can be represented by a variable and is related to the number of
hours worked, which also can be represented by a variable.
6
Algebra
Recognize and represent
relationships between varying
quantities; translate from one
representation to another; use
patterns, tables, graphs and rules to
solve real-world and mathematical
problems.
6.2.1.2
rules, graphs and tables; translate between any two of these
representations.
For example: Describe the terms in the sequence of perfect squares
t = 1, 4, 9, 16, ... by using the rule
2
tn=
for n = 1, 2, 3, 4, ....
6
Algebra
Use properties of arithmetic to
generate equivalent numerical
expressions and evaluate
expressions involving positive
rational numbers.
6.2.2.1
of operations to generate equivalent expressions and to solve problems
involving positive rational numbers.
For example:
32 5 2 16 5
32 5 16 5
16
2
15 6 15 6 3 5 3 2 9 2 5 9
× ××
×= = = ××=
× ×××
.
Another example: Use the distributive law to write:
( )
9 15 9 15 3 5 5 3
11 11 1 1
21
232 8 23238 228 8 8
+ − =+×−× =+−=−=
.
September 22, 2008 Page 31 of 60
Grade
Strand
Standard
Code
6
Algebra
Understand and interpret equations
and inequalities involving variables
and positive rational numbers. Use
equations and inequalities to
represent real-world and
mathematical problems; use the
idea of maintaining equality to solve
equations. Interpret solutions in the
original context.
6.2.3.1
inequalities involving variables and positive rational numbers.
For example: The number of miles m in a k kilometer race is represented
by the equation m = 0.62.
6
Algebra
Understand and interpret equations
and inequalities involving variables
and positive rational numbers. Use
equations and inequalities to
represent real-world and
mathematical problems; use the
idea of maintaining equality to solve
equations. Interpret solutions in the
original context.
6.2.3.2
properties of arithmetic and the idea of maintaining equality on both
sides of the equation. Interpret a solution in the original context and
assess the reasonableness of results.
For example: A cellular phone company charges $0.12 per minute. If the
bill was $11.40 in April, how many minutes were used?
6
Geometry &
Measurement
Calculate perimeter, area, surface
area and volume of two- and three-
dimensional figures to solve real-
world and mathematical problems.
6.3.1.1
units, such as cm2 and cm3. Justify the formulas used. Justification may
involve decomposition, nets or other models.
For example: The surface area of a triangular prism can be found by
decomposing the surface into two triangles and three rectangles.
6
Geometry &
Measurement
Calculate perimeter, area, surface
area and volume of two- and three-
dimensional figures to solve real-
world and mathematical problems.
6.3.1.2
rectangles, rhombuses, parallelograms, trapezoids and kites. When
formulas are used, be able to explain why they are valid.
For example: The area of a kite is one-half the product of the lengths of
the diagonals, and this can be justified by decomposing the kite into two
triangles.
September 22, 2008 Page 32 of 60
Grade
Strand
Standard
Code
6
Geometry &
Measurement
Calculate perimeter, area, surface
area and volume of two- and three-
dimensional figures to solve real-
world and mathematical problems.
6.3.1.3
cannot be decomposed into common figures and use correct units, such
as cm and
.
6
Geometry &
Measurement
Understand and use relationships
between angles in geometric
figures.
6.3.2.1
intersecting lines.
For example: If two streets cross, forming four corners such that one of
the corners forms an angle of 120˚, determine the measures of the
remaining three angles.
Another example: Recognize that pairs of interior and exterior angles in
polygons have measures that sum to 180˚.
6
Geometry &
Measurement
Understand and use relationships
between angles in geometric
figures.
6.3.2.2
sum of the interior angles of a triangle is 180˚. Use models of triangles to
illustrate this fact.
For example: Cut a triangle out of paper, tear off the corners and
rearrange these corners to form a straight line.
Another example: Recognize that the measures of the two acute angles
in a right triangle sum to 90˚.
6
Geometry &
Measurement
Understand and use relationships
between angles in geometric
figures.
6.3.2.3
by decomposing them into triangles.
6
Geometry &
Measurement
Choose appropriate units of
measurement and use ratios to
convert within measurement
systems to solve real-world and
mathematical problems.
6.3.3.1
capacities, geometric measurements and times within measurement
systems using appropriate units.
6
Geometry &
Measurement
Choose appropriate units of
measurement and use ratios to
convert within measurement
systems to solve real-world and
mathematical problems.
6.3.3.2
benchmarks in measurement systems with appropriate units.
For example: Estimate the height of a house by comparing to a 6-foot
man standing nearby.
September 22, 2008 Page 33 of 60
Grade
Strand
Standard
Code
6
Data Analysis &
Probability
Use probabilities to solve real-world
and mathematical problems;
represent probabilities using
fractions, decimals and percents.
6.4.1.1
experiment and determine which members of the sample space are
related to certain events. Sample space may be determined by the use of
tree diagrams, tables or pictorial representations.
For example: A 6 6 table with entries such as (1,1), (1,2), (1,3), …, (6,6)
can be used to represent the sample space for the experiment of
simultaneously rolling two number cubes.
6
Data Analysis &
Probability
Use probabilities to solve real-world
and mathematical problems;
represent probabilities using
fractions, decimals and percents.
6.4.1.2
the event and the size of the sample space; represent probabilities as
percents, fractions and decimals between 0 and 1 inclusive. Understand
that probabilities measure likelihood.
For example: Each outcome for a balanced number cube has probability
, and the probability of rolling an even number is
.
6
Data Analysis &
Probability
Use probabilities to solve real-world
and mathematical problems;
represent probabilities using
fractions, decimals and percents.
6.4.1.3
compare the resulting relative frequencies with the known probabilities;
know that there may be differences.
For example: Heads and tails are equally likely when flipping a fair coin,
but if several different students flipped fair coins 10 times, it is likely that
they will find a variety of relative frequencies of heads and tails.
6
Data Analysis &
Probability
Use probabilities to solve real-world
and mathematical problems;
represent probabilities using
fractions, decimals and percents.
6.4.1.4
percents, fractions and decimals between 0 and 1 inclusive. Use
experimental probabilities to make predictions when actual probabilities
are unknown.
For example: Repeatedly draw colored chips with replacement from a
bag with an unknown mixture of chips, record relative frequencies, and
use the results to make predictions about the contents of the bag.
7
Number &
Operation
Read, write, represent and compare
positive and negative rational
numbers, expressed as integers,
fractions and decimals.
7.1.1.1
integers or as a terminating or repeating decimal. Recognize that π is not
rational, but that it can be approximated by rational numbers such as

and 3.14.
September 22, 2008 Page 34 of 60
Grade
Strand
Standard
Code
7
Number &
Operation
Read, write, represent and compare
positive and negative rational
numbers, expressed as integers,
fractions and decimals.
7.1.1.2
number. Use this information to interpret the decimal result of a division
problem when using a calculator.
For example:


gives 4.16666667 on a calculator. This answer is not
exact. The exact answer can be expressed as 4
, which is the same as
4.16
. The calculator expression does not guarantee that the 6 is repeated,
but that possibility should be anticipated.
7
Number &
Operation
Read, write, represent and compare
positive and negative rational
numbers, expressed as integers,
fractions and decimals.
7.1.1.3
understand the concept of opposites, and plot pairs of positive and
negative rational numbers on a coordinate grid.
7
Number &
Operation
Read, write, represent and compare
positive and negative rational
numbers, expressed as integers,
fractions and decimals.
7.1.1.4
forms using the symbols < , > , = , ≤ , ≥ .
For example:
< 0.36.
7
Number &
Operation
Read, write, represent and compare
positive and negative rational
numbers, expressed as integers,
fractions and decimals.
7.1.1.5
negative rational numbers, including equivalent fractions.
For example:


=


=

= 33.
7
Number &
Operation
Calculate with positive and negative
rational numbers, and rational
numbers with whole number
exponents, to solve real-world and
mathematical problems.
7.1.2.1
that are integers, fractions and terminating decimals; use efficient and
generalizable procedures, including standard algorithms; raise positive
rational numbers to whole-number exponents.
For example: 3
(
)
=

.
September 22, 2008 Page 35 of 60
Grade
Strand
Standard
Code
7
Number &
Operation
Calculate with positive and negative
rational numbers, and rational
numbers with whole number
exponents, to solve real-world and
mathematical problems.
7.1.2.2
and subtraction to explain why the procedures of arithmetic with
negative rational numbers make sense.
For example: Multiplying a distance by -1 can be thought of as
representing that same distance in the opposite direction. Multiplying by
-1 a second time reverses directions again, giving the distance in the
original direction.
7
Number &
Operation
Calculate with positive and negative
rational numbers, and rational
numbers with whole number
exponents, to solve real-world and
mathematical problems.
7.1.2.3
truncate or round numbers.
For example: A decimal that repeats or terminates after a large number
of digits is truncated or rounded.
7
Number &
Operation
Calculate with positive and negative
rational numbers, and rational
numbers with whole number
exponents, to solve real-world and
mathematical problems.
7.1.2.4
and negative rational numbers and positive integer exponents, including
computing simple and compound interest.
7
Number &
Operation
Calculate with positive and negative
rational numbers, and rational
numbers with whole number
exponents, to solve real-world and
mathematical problems.
7.1.2.5
contexts.
For example: A recipe calls for milk, flour and sugar in a ratio of 4:6:3 (this
is how recipes are often given in large institutions, such as hospitals). How
much flour and milk would be needed with 1 cup of sugar?
7
Number &
Operation
Calculate with positive and negative
rational numbers, and rational
numbers with whole number
exponents, to solve real-world and
mathematical problems.
7.1.2.6
value of a rational number and distance on a number line. Use the symbol
for absolute value.
For example: |
−
3| represents the distance from
−
3 to 0 on a number
line or 3 units; the distance between 3 and
9
2
on the number line is | 3
−
9
2
| or
3
2
.
September 22, 2008 Page 36 of 60
Grade
Strand
Standard
Code
7
Algebra
Understand the concept of
proportionality in real-world and
mathematical situations, and
distinguish between proportional
and other relationships.
7.2.1.1
proportional if it can be expressed in the form
=k or y=kx. Distinguish
proportional relationships from other relationships, including inversely
proportional relationships (xy=k or y=
).
For example: The radius and circumference of a circle are proportional,
whereas the length x and the width y of a rectangle with area 12 are
inversely proportional, since xy= 12 or equivalently, y=

7
Algebra
Understand the concept of
proportionality in real-world and
mathematical situations, and
distinguish between proportional
and other relationships.
7.2.1.2
the origin whose slope is the unit rate (constant of proportionality). Know
how to use graphing technology to examine what happens to a line when
the unit rate is changed.
7
Algebra
Recognize proportional relationships
in real-world and mathematical
situations; represent these and
other relationships with tables,
verbal descriptions, symbols and
graphs; solve problems involving
proportional relationships and
explain results in the original
context.
7.2.2.1
symbols, equations and graphs; translate from one representation to
another. Determine the unit rate (constant of proportionality or slope)
given any of these representations.
For example: Larry drives 114 miles and uses 5 gallons of gasoline. Sue
drives 300 miles and uses 11.5 gallons of gasoline. Use equations and
graphs to compare fuel efficiency and to determine the costs of various
trips.
7
Algebra
Recognize proportional relationships
in real-world and mathematical
situations; represent these and
other relationships with tables,
verbal descriptions, symbols and
graphs; solve problems involving
proportional relationships and
explain results in the original
context.
7.2.2.2
numerous contexts.
For example: Distance-time, percent increase or decrease, discounts, tips,
unit pricing, lengths in similar geometric figures, and unit conversion
when a conversion factor is given, including conversion between different
measurement systems.
Another example: How many kilometers are there in 26.2 miles?
September 22, 2008 Page 37 of 60
Grade
Strand
Standard
Code
7
Algebra
Recognize proportional relationships
in real-world and mathematical
situations; represent these and
other relationships with tables,
verbal descriptions, symbols and
graphs; solve problems involving
proportional relationships and
explain results in the original
context.
7.2.2.3
For example: Recognize that it would be unreasonable for a cashier to
request $200 if you purchase a $225 item at 25% off.
7
Algebra
Recognize proportional relationships
in real-world and mathematical
situations; represent these and
other relationships with tables,
verbal descriptions, symbols and
graphs; solve problems involving
proportional relationships and
explain results in the original
context.
7.2.2.4
inequalities involving variables and positive and negative rational
numbers.
For example: "Four-fifths is three greater than the opposite of a number"
can be represented as
= -n+3, and "height no bigger than half the
radius" can be represented as h ≤
.
Another example: "x is at least -3 and less than 5" can be represented as
-3≤x<5, and also on a number line.
7
Algebra
Apply understanding of order of
operations and algebraic properties
to generate equivalent numerical
and algebraic expressions containing
positive and negative rational
numbers and grouping symbols;
evaluate such expressions.
7.2.3.1
expressions containing rational numbers, grouping symbols and whole
number exponents. Properties of algebra include associative,
commutative and distributive laws.
For example: Combine like terms (use the distributive law) to write
3 7 1 (3 7) 1 4 1xx x x− += − +=− +
.
7
Algebra
Apply understanding of order of
operations and algebraic properties
to generate equivalent numerical
and algebraic expressions containing
positive and negative rational
numbers and grouping symbols;
evaluate such expressions.
7.2.3.2
number exponents at specified values of their variables.
For example: Evaluate the expression
2
1
(2 5)
3
x −
at x = 5.
September 22, 2008 Page 38 of 60
Grade
Strand
Standard
Code
7
Algebra
Apply understanding of order of
operations and algebraic properties
to generate equivalent numerical
and algebraic expressions containing
positive and negative rational
numbers and grouping symbols;
evaluate such expressions.
7.2.3.3
using calculators and other technologies.
For example: Recognize the conventions of using a caret (^ raise to a
power) and asterisk (* multiply); pay careful attention to the use of
nested parentheses.
7
Algebra
Represent real-world and
mathematical situations using
equations with variables. Solve
equations symbolically, using the
properties of equality. Also solve
equations graphically and
numerically. Interpret solutions in
the original context.
7.2.4.1
variables and positive and negative rational numbers. Use the properties
of equality to solve for the value of a variable. Interpret the solution in
the original context.
For example: Solve for w in the equation P = 2w + 2ℓ when P = 3.5 and ℓ
= 0.4
Another example: To post an Internet website, Mary must pay $300 for
initial set up and a monthly fee of $12. She has $842 in savings, how long
can she sustain her website?
7
Algebra
Represent real-world and
mathematical situations using
equations with variables. Solve
equations symbolically, using the
properties of equality. Also solve
equations graphically and
numerically. Interpret solutions in
the original context.
7.2.4.2
contexts.
For example: Given the side lengths of one triangle and one side length of
a second triangle that is similar to the first, find the remaining side
lengths of the second triangle.
Another example: Determine the price of 12 yards of ribbon if 5 yards of
ribbon cost $1.85.
7
Geometry &
Measurement
Use reasoning with proportions and
ratios to determine measurements,
justify formulas and solve real-world
and mathematical problems
involving circles and related
geometric figures.
7.3.1.1
the diameter and circumference of a circle and that the unit rate
(constant of proportionality) is
π. Calculate the circumference and area of
circles and sectors of circles to solve problems in various contexts.
September 22, 2008 Page 39 of 60
Grade
Strand
Standard
Code
7
Geometry &
Measurement
Use reasoning with proportions and
ratios to determine measurements,
justify formulas and solve real-world
and mathematical problems
involving circles and related
geometric figures.
7.3.1.2
formulas used.
For example: Justify the formula for the surface area of a cylinder by
decomposing the surface into two circles and a rectangle.
7
Geometry &
Measurement
Analyze the effect of change of
scale, translations and reflections on
the attributes of two-dimensional
figures.
7.3.2.1
similarity, and determine scale factors.
For example: Corresponding angles in similar geometric figures have the
same measure.
7
Geometry &
Measurement
Analyze the effect of change of
scale, translations and reflections on
the attributes of two-dimensional
figures.
7.3.2.2
lengths and areas of similar geometric figures.
For example: If two similar rectangles have heights of 3 and 5, and the
first rectangle has a base of length 7, the base of the second rectangle has
length

.
7
Geometry &
Measurement
Analyze the effect of change of
scale, translations and reflections on
the attributes of two-dimensional
figures.
7.3.2.3
and conversions of measurement units.
For example: 1 square foot equals 144 square inches.
Another example: In a map where 1 inch represents 50 miles,
inch
represents 25 miles.
7
Geometry &
Measurement
Analyze the effect of change of
scale, translations and reflections on
the attributes of two-dimensional
figures.
7.3.2.4
grid and determine the coordinates of the vertices of the figure after the
transformation.
For example: The point (1, 2) moves to (-1, 2) after reflection about y-
axis.
September 22, 2008 Page 40 of 60
Grade
Strand
Standard
Code
7
Data Analysis &
Probability
Use mean, median and range to
draw conclusions about data and
make predictions.
7.4.1.1
and range for quantitative data and from data represented in a display.
Use these quantities to draw conclusions about the data, compare
different data sets, and make predictions.
For example: By looking at data from the past, Sandy calculated that the
mean gas mileage for her car was 28 miles per gallon. She expects to
travel 400 miles during the next week. Predict the approximate number
of gallons that she will use.
7
Data Analysis &
Probability
Use mean, median and range to
draw conclusions about data and
make predictions.
7.4.1.2
mean and the median of a data set. Know how to create data displays
using a spreadsheet to examine this impact.
For example: How does dropping the lowest test score affect a student's
mean test score?
7
Data Analysis &
Probability
Display and interpret data in a
variety of ways, including circle
graphs and histograms.
7.4.2.1
graphs (pie charts) and histograms. Choose the appropriate data display
and know how to create the display using a spreadsheet or other
graphing technology.
7
Data Analysis &
Probability
Calculate probabilities and reason
about probabilities using
proportions to solve real-world and
mathematical problems.
7.4.3.1
from a table to simulate situations involving randomness, make a
histogram to display the results, and compare the results to known
probabilities.
For example: Use a spreadsheet function such as RANDBETWEEN(1, 10)
to generate random whole numbers from 1 to 10, and display the results
in a histogram.
7
Data Analysis &
Probability
Calculate probabilities and reason
about probabilities using
proportions to solve real-world and
mathematical problems.
7.4.3.2
Express probabilities as percents, decimals and fractions.
For example: Determine probabilities for different outcomes in game
spinners by finding fractions of the area of the spinner.
September 22, 2008 Page 41 of 60
Grade
Strand
Standard
Code
7
Data Analysis &
Probability
Calculate probabilities and reason
about probabilities using
proportions to solve real-world and
mathematical problems.
7.4.3.3
relative frequencies of outcomes based on probabilities.
For example: When rolling a number cube 600 times, one would predict
that a 3 or 6 would be rolled roughly 200 times, but probably not exactly
200 times.
8
Number &
Operation
Read, write, compare, classify and
represent real numbers, and use
them to solve problems in various
contexts.
8.1.1.1
root of a positive integer is not an integer, then it is irrational. Know that
the sum of a rational number and an irrational number is irrational, and
the product of a non-zero rational number and an irrational number is
irrational.
For example: Classify the following numbers as whole numbers, integers,
rational numbers, irrational numbers, recognizing that some numbers
belong in more than one category:
6
3
,
3
6
,
3.6
,
2
π
,
4−
,
10
,
6.7−
.
8
Number &
Operation
Read, write, compare, classify and
represent real numbers, and use
them to solve problems in various
contexts.
8.1.1.2
the square root of a positive integer as an integer, or if it is not an integer,
locate it as a real number between two consecutive positive integers.
For example: Put the following numbers in order from smallest to largest:
2,
3
,
−
4,
−
6.8,
37−
.
Another example:
68
is an irrational number between 8 and 9.
8
Number &
Operation
Read, write, compare, classify and
represent real numbers, and use
them to solve problems in various
contexts.
8.1.1.3
real numbers.
For example: A calculator can be used to determine that
7
is
approximately 2.65.
Another example: To check that
5
1
12
is slightly bigger than
2
, do the
calculation
( ) ( )
22
5 17 289
1
12
12 12 144 144
= = =
Another example: Knowing that
10
is between 3 and 4, try squaring
numbers like 3.5, 3.3, 3.1 to determine that 3.1 is a reasonable rational
approximation of
10
.
September 22, 2008 Page 42 of 60
Grade
Strand
Standard
Code
8
Number &
Operation
Read, write, compare, classify and
represent real numbers, and use
them to solve problems in various
contexts.
8.1.1.4
to generate equivalent numerical expressions.
For example:
( ) ( )
( )
3
53
1
2
1
3
27
33 3
−−
×==
=
.
8
Number &
Operation
Read, write, compare, classify and
represent real numbers, and use
them to solve problems in various
contexts.
8.1.1.5
scientific notation; understand how calculators display numbers in
scientific notation. Multiply and divide numbers expressed in scientific
notation, express the answer in scientific notation, using the correct
number of significant digits when physical measurements are involved.
For example:
43 8
(4.2 10 ) (8.25 10 ) 3.465 10×× ×= ×
, but if these numbers
represent physical measurements, the answer should be expressed as
8
3.5 10×
because the first factor,
4
4.2 10×
, only has two significant digits.
8
Algebra
Understand the concept of function
in real-world and mathematical
situations, and distinguish between
linear and non-linear functions.
8.2.1.1
variable and a dependent variable in which the value of the independent
variable determines the value of the dependent variable. Use functional
notation, such as f(x), to represent such relationships.
For example: The relationship between the area of a square and the side
length can be expressed as
2
()fx x=
. In this case,
(5) 25f =
, which
represents the fact that a square of side length 5 units has area 25 units
squared.
8
Algebra
Understand the concept of function
in real-world and mathematical
situations, and distinguish between
linear and non-linear functions.
8.2.1.2
input variable by some amount leads to a change in the output variable
that is a constant times that amount.
For example: Uncle Jim gave Emily $50 on the day she was born and $25
on each birthday after that. The function
( ) 50 25fx x= +
represents the
amount of money Jim has given after x years. The rate of change is $25
per year.
September 22, 2008 Page 43 of 60
Grade
Strand
Standard
Code
8
Algebra
Understand the concept of function
in real-world and mathematical
situations, and distinguish between
linear and non-linear functions.
8.2.1.3
its graph is a straight line.
For example: The function
2
()fx x=
is not a linear function because its
graph contains the points (1,1), (-1,1) and (0,0), which are not on a
straight line.
8
Algebra
Understand the concept of function
in real-world and mathematical
situations, and distinguish between
linear and non-linear functions.
8.2.1.4
expressed in the form , where () =  + , where = 0, 1, 2, 3, …
For example: The arithmetic sequence 3, 7, 11, 15,…, can be expressed as
() = 4 + 3.
8
Algebra
Understand the concept of function
in real-world and mathematical
situations, and distinguish between
linear and non-linear functions.
8.2.1.5
be expressed in the form
(
)
= 
, where x = 0, 1, 2, 3,….
For example: The geometric sequence 6, 12, 24, 48, … , can be expressed
in the form () = 6(2
).
8
Algebra
Recognize linear functions in real-
world and mathematical situations;
represent linear functions and other
functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions and explain results in the
original context.
8.2.2.1
equations and graphs; translate from one representation to another.
8
Algebra
Recognize linear functions in real-
world and mathematical situations;
represent linear functions and other
functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions and explain results in the
original context.
8.2.2.2
intercepts. Know that the slope equals the rate of change, and that the y-
intercept is zero when the function represents a proportional
relationship.
September 22, 2008 Page 44 of 60
Grade
Strand
Standard
Code
8
Algebra
Recognize linear functions in real-
world and mathematical situations;
represent linear functions and other
functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions and explain results in the
original context.
8.2.2.3
the graphs of linear functions. Know how to use graphing technology to
examine these effects.
8
Algebra
Recognize linear functions in real-
world and mathematical situations;
represent linear functions and other
functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions and explain results in the
original context.
8.2.2.4
verbal descriptions, and use them to solve problems.
For example: If a girl starts with $100 in savings and adds $10 at the end
of each month, she will have 100 + 10 dollars after months.
8
Algebra
Recognize linear functions in real-
world and mathematical situations;
represent linear functions and other
functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions and explain results in the
original context.
8.2.2.5
verbal descriptions, and use them to solve problems.
For example: If a girl invests $100 at 10% annual interest, she will have
100(1. 1
) dollars after years.
8
Algebra
Generate equivalent numerical and
algebraic expressions and use
algebraic properties to evaluate
expressions.
8.2.3.1
and absolute values, at specified values of their variables.
For example: Evaluate 2 when = 3  = 0.5, and then use an
approximation of π to obtain an approximate answer.
8
Algebra
Generate equivalent numerical and
algebraic expressions and use
algebraic properties to evaluate
expressions.
8.2.3.2
properties used, including the properties of algebra. Properties include
the associative, commutative and distributive laws, and the order of
operations, including grouping symbols.
September 22, 2008 Page 45 of 60
Grade
Strand
Standard
Code
8
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear expressions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
8.2.4.1
change, including proportional and non-proportional relationships.
For example: For a cylinder with fixed radius of length 5, the surface area
= 2(5) + 2(5)
= 10 + 50, is a linear function of the
height , but it is not proportional to the height.
8
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear expressions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
8.2.4.2
multi-variable equation in terms of the other variables. Justify the steps
by identifying the properties of equalities used.
For example: The equation 10 + 17 = 3 can be changed  7 +
17 = 0, and then to 7 = 17 by adding/subtracting the same
quantities to both sides. These changes do not change the solution of the
equation.
Another example: Using the formula for the perimeter of a rectangle,
solve for the base in terms of the height and perimeter.
8
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear expressions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
8.2.4.3
forms, and convert between these forms. Given sufficient information,
find an equation of a line.
For example: Determine an equation of the line through the points
(1,6) and (2/3, 3/4).
8
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear expressions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
8.2.4.4
For example: A gas station charges $0.10 less per gallon of gasoline if a
customer also gets a car wash. Without the car wash, gas costs $2.79 per
gallon. The car wash is $8.95. What are the possible amounts (in gallons)
of gasoline that you can buy if you also get a car wash and can spend at
most $35?
September 22, 2008 Page 46 of 60
Grade
Strand
Standard
Code
8
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear expressions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
8.2.4.5
solutions on a number line.
For example: The inequality 3 < 6 is equivalent to > 2, which
can be represented on the number line by shading in the interval to the
right of 2.
8
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear expressions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
8.2.4.6
inequalities involving the absolute value of a linear expression. Solve such
equations and inequalities and graph the solutions on a number line.
For example: A cylindrical machine part is manufactured with a radius of
2.1 cm, with a tolerance of 1/100 cm. The radius r satisfies the inequality
|
– 2.1| .01.
8
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear expressions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
8.2.47
equations. Solve systems of linear equations in two variables symbolically,
graphically and numerically.
For example: Marty's cell phone company charges $15 per month plus
$0.04 per minute for each call. Jeannine's company charges $0.25 per
minute. Use a system of equations to determine the advantages of each
plan based on the number of minutes used.
8
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear expressions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
8.2.4.8
solution, or an infinite number of solutions. Relate the number of
solutions to pairs of lines that are intersecting, parallel or identical. Check
whether a pair of numbers satisfies a system of two linear equations in
two unknowns by substituting the numbers into both equations.
September 22, 2008 Page 47 of 60
Grade
Strand
Standard
Code
8
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear expressions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
8.2.4.9
solve problems.
For example: If 
2
= 5, then
5
x
π
=
, or equivalently,
5
x
π
=
or
5
x
π
= −
. If
is understood as the radius of a circle in this example, then the negative
solution should be discarded and
5
x
π
=
.
8
Geometry &
Measurement
Solve problems involving right
triangles using the Pythagorean
Theorem and its converse.
8.3.1.1
For example: Determine the perimeter of a right triangle, given the
lengths of two of its sides.
Another example: Show that a triangle with side lengths 4, 5 and 6 is not
a right triangle.
8
Geometry &
Measurement
Solve problems involving right
triangles using the Pythagorean
Theorem and its converse.
8.3.1.2
line in a coordinate system. Use the Pythagorean Theorem to find the
distance between any two points in a coordinate system.
8
Geometry &
Measurement
Solve problems involving right
triangles using the Pythagorean
Theorem and its converse.
8.3.1.3
diagrams and computer software.
8
Geometry &
Measurement
Solve problems involving parallel
and perpendicular lines on a
coordinate system.
8.3.2.1
lines and between the slopes of perpendicular lines. Dynamic graphing
software may be used to examine these relationships.
8
Geometry &
Measurement
Solve problems involving parallel
and perpendicular lines on a
coordinate system.
8.3.2.2
their sides.
For example: Given the coordinates of four points, determine whether
the corresponding quadrilateral is a parallelogram.
8
Geometry &
Measurement
Solve problems involving parallel
and perpendicular lines on a
coordinate system.
8.3.2.3
the line, find lines through that point that are parallel and perpendicular
to the given line, symbolically and graphically.
September 22, 2008 Page 48 of 60
Grade
Strand
Standard
Code
8
Data Analysis &
Probability
Interpret data using scatterplots and
approximate lines of best fit. Use
lines of best fit to draw conclusions
about data.
8.4.1.1
scatterplot to informally estimate a line of best fit and determine an
equation for the line. Use appropriate titles, labels and units. Know how
to use graphing technology to display scatterplots and corresponding
lines of best fit.
8
Data Analysis &
Probability
Interpret data using scatterplots and
approximate lines of best fit. Use
lines of best fit to draw conclusions
about data.
8.4.1.2
change and to make predictions about values not in the original data set.
For example: Given a scatterplot relating student heights to shoe sizes,
predict the shoe size of a 5'4" student, even if the data does not contain
information for a student of that height.
8
Data Analysis &
Probability
Interpret data using scatterplots and
approximate lines of best fit. Use
lines of best fit to draw conclusions
about data.
8.4.1.3
interpreting them in the original context.
For example: A set of data may show that the number of women in the
U.S. Senate is growing at a certain rate each election cycle. Is it
reasonable to use this trend to predict the year in which the Senate will
eventually include 1000 female Senators?
9-11 Algebra
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.2.1.1
evaluate a function at a given point in its domain.
For example: If
(
)
=

, find f (-4).
9-11
Algebra
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.2.1.2
graphically or in tabular form.
9-11
Algebra
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.2.1.3
real-world context.
For example: The formula () = 2 can represent a function whose
domain is all real numbers, but in the context of the area of a circle, the
domain would be restricted to positive .
September 22, 2008 Page 49 of 60
Grade
Strand
Standard
Code
9-11
Algebra
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.2.1.4
other relations.
For example: If a graph shows the relationship between the elapsed flight
time of a golf ball at a given moment and its height at that same moment,
identify the time interval during which the ball is at least 100 feet above
the ground.
9-11
Algebra
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.2.1.5
corresponding to a quadratic function, using symbolic and graphical
methods, when the function is expressed in the form () =
2
+
 + , in the form () = ( – ℎ)
2
+ , or in factored form.
9-11
Algebra
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.2.1.6
decrease from the graph of a function.
9-11
Algebra
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.2.1.7
exponential functions and reciprocals of linear functions, using symbolic
and graphical methods.
9-11
Algebra
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.2.1.8
based on its graph or table of values.
For example: The function () = 3
increases for all , but it increases
faster when > 2 than it does when < 2.
9-11
Algebra
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.2.1.9
function. Know how to use graphing technology to examine translations.
For example: Determine how the graph of () = | – ℎ| + changes
as h and change.
September 22, 2008 Page 50 of 60
Grade
Strand
Standard
Code
9-12
Algebra
Recognize linear, quadratic,
exponential and other common
functions in real-world and
mathematical situations; represent
these functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions, and explain results in the
original context.
9.2.2.1
quadratic functions.
For example: Write a function that represents the area of a rectangular
garden that can be surrounded with 32 feet of fencing, and use the
function to determine the possible dimensions of such a garden if the
area must be at least 50 square feet.
9-11
Algebra
Recognize linear, quadratic,
exponential and other common
functions in real-world and
mathematical situations; represent
these functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions, and explain results in the
original context.
9.2.2.2
functions, such as investment growth, depreciation and population
growth.
9-11
Algebra
Recognize linear, quadratic,
exponential and other common
functions in real-world and
mathematical situations; represent
these functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions, and explain results in the
original context.
9.2.2.3
translate between graphs, tables and symbolic representations. Know
how to use graphing technology to graph these functions.
September 22, 2008 Page 51 of 60
Grade
Strand
Standard
Code
9-11
Algebra
Recognize linear, quadratic,
exponential and other common
functions in real-world and
mathematical situations; represent
these functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions, and explain results in the
original context.
9.2.2.4
Express the terms in a geometric sequence recursively and by giving an
explicit (closed form) formula, and express the partial sums of a
geometric series recursively.
For example: A closed form formula for the terms t
in the geometric
sequence 3, 6, 12, 24, ... is t
n
= 3(2)
n-1
, where n = 1, 2, 3, ... , and this
sequence can be expressed recursively by writing t
1
= 3 and t
n
= 2t
n-1
,
for
2
Another example: The partial sums s
n
of the series 3 + 6 + 12 + 24 + ... can
be expressed recursively by writing s
1
= 3 and s
n
= 3 + 2s
n-1
,
for 2.
9-11
Algebra
Recognize linear, quadratic,
exponential and other common
functions in real-world and
mathematical situations; represent
these functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions, and explain results in the
original context.
9.2.2.5
sequences and series, such as home mortgage and other compound
interest examples. Know how to use spreadsheets and calculators to
explore geometric sequences and series in various contexts.
9-11
Algebra
Recognize linear, quadratic,
exponential and other common
functions in real-world and
mathematical situations; represent
these functions with tables, verbal
descriptions, symbols and graphs;
solve problems involving these
functions, and explain results in the
original context.
9.2.2.6
( )
fx x=
( )
fx x=
,
( )
1
x
fx=
, f (x) = x
3
, and translations of these functions, such as
(
)
=
2
+ 4. Know how to use graphing technology to graph these
functions.
9-11
Algebra
Generate equivalent algebraic
expressions involving polynomials
and radicals; use algebraic
properties to evaluate expressions.
9.2.3.1
radicals and absolute values at specified points in their domains.
September 22, 2008 Page 52 of 60
Grade
Strand
Standard
Code
9-11
Algebra
Generate equivalent algebraic
expressions involving polynomials
and radicals; use algebraic
properties to evaluate expressions.
9.2.3.2
polynomial of equal or lower degree.
9-11
Algebra
Generate equivalent algebraic
expressions involving polynomials
and radicals; use algebraic
properties to evaluate expressions.
9.2.3.3
polynomials, and factor the difference of two squares.
For example: 9x
6
– x
4
= (3x
3
– x
2
)(3x
3
+ x
2
).
9-11
Algebra
Generate equivalent algebraic
expressions involving polynomials
and radicals; use algebraic
properties to evaluate expressions.
9.2.3.4
For example:
1
11
x
xx
+
−+
is equivalent to
2
2
12
1
xx
x
+−
−
.
9-11
Algebra
Generate equivalent algebraic
expressions involving polynomials
and radicals; use algebraic
properties to evaluate expressions.
9.2.3.5
equation by substituting it for the variable and evaluating the expression,
using arithmetic with complex numbers.
For example: The complex number
1
2
i+
is a solution of 2x
2
– 2x + 1 = 0,
since
( )
2
11
2 2 1 1 10
22
ii
ii




++
− +=− + +=
.
9-11
Algebra
Generate equivalent algebraic
expressions involving polynomials
and radicals; use algebraic
properties to evaluate expressions.
9.2.3.6
generate equivalent algebraic expressions, including those involving nth
roots.
For example:
11 1
22 2
2 7 2 7 14 14=×= =×
. Rules for computing directly
with radicals may also be used:
3
33
22xx
×=
.
9-11
Algebra
Generate equivalent algebraic
expressions involving polynomials
and radicals; use algebraic
properties to evaluate expressions.
9.2.3.7
properties used. Use substitution to check the equality of expressions for
some particular values of the variables; recognize that checking with
substitution does not guarantee equality of expressions for all values of
the variables.
September 22, 2008 Page 53 of 60
Grade
Strand
Standard
Code
9-11
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear, quadratic, exponential and
n
th
root functions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
9.2.4.1
inequalities. Solve quadratic equations and inequalities by appropriate
methods including factoring, completing the square, graphing and the
quadratic formula. Find non-real complex roots when they exist.
Recognize that a particular solution may not be applicable in the original
context. Know how to use calculators, graphing utilities or other
technology to solve quadratic equations and inequalities.
For example: A diver jumps from a 20 meter platform with an upward
velocity of 3 meters per second. In finding the time at which the diver hits
the surface of the water, the resulting quadratic equation has a positive
and a negative solution. The negative solution should be discarded
because of the context.
9-11
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear, quadratic, exponential and
n
th
root functions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
9.2.4.2
exponential functions; solve these equations graphically or numerically.
Know how to use calculators, graphing utilities or other technology to
solve these equations.
9-11
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear, quadratic, exponential and
n
th
root functions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
9.2.4.3
extended from whole numbers to integers, from integers to rational
numbers, from rational numbers to real numbers, and from real numbers
to complex numbers. In particular, non-real complex numbers are needed
to solve some quadratic equations with real coefficients.
September 22, 2008 Page 54 of 60
Grade
Strand
Standard
Code
9-11
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear, quadratic, exponential and
n
th
root functions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
9.2.4.4
inequalities; solve them graphically. Indicate which parts of the boundary
are included in and excluded from the solution set using solid and dotted
lines.
9-11
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear, quadratic, exponential and
n
th
root functions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
9.2.4.5
methods.
9-11
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear, quadratic, exponential and
n
th
root functions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
9.2.4.6
inequalities in two variables; solve them graphically.
For example: If a pipe is to be cut to a length of 5 meters accurate to
within a tenth of its diameter, the relationship between the length x of
the pipe and its diameter y satisfies the inequality | – 5| 0.1.
9-11
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear, quadratic, exponential and
n
th
root functions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
9.2.4.7
extraneous solutions may arise when using symbolic methods.
For example: The equation
9 = 9
may be solved by squaring both
sides to obtain – 9 = 81, which has the solution x = -

However, this
is not a solution of the original equation, so it is an extraneous solution
that should be discarded. The original equation has no solution in this
case.
Another example: Solve
 + 1
= -5.
September 22, 2008 Page 55 of 60
Grade
Strand
Standard
Code
9-11
Algebra
Represent real-world and
mathematical situations using
equations and inequalities involving
linear, quadratic, exponential and
n
th
root functions. Solve equations
and inequalities symbolically and
graphically. Interpret solutions in
the original context.
9.2.4.8
the solution to appropriate graphical or numerical estimates; interpret a
solution in the original context.
9-11
Geometry &
Measurement
Calculate measurements of plane
and solid geometric figures; know
that physical measurements depend
on the choice of a unit and that they
are approximations.
9.3.1.1
Use measuring devices or formulas as appropriate.
For example: Measure the height and radius of a cone and then use a
formula to find its volume.
9-11
Geometry &
Measurement
Calculate measurements of plane
and solid geometric figures; know
that physical measurements depend
on the choice of a unit and that they
are approximations.
9.3.1.2
decomposition to determine the perimeter, area, surface area and
volume of various figures.
For example: Find the volume of a regular hexagonal prism by
decomposing it into six equal triangular prisms.
9-11
Geometry &
Measurement
Calculate measurements of plane
and solid geometric figures; know
that physical measurements depend
on the choice of a unit and that they
are approximations.
9.3.1.3
be assigned units; apply such units correctly in expressions, equations and
problem solutions that involve measurements; and convert between
measurement systems.
For example: 60 miles/hour = 60 miles/hour × 5280 feet/mile × 1
hour/3600 seconds = 88 feet/second.
9-11
Geometry &
Measurement
Calculate measurements of plane
and solid geometric figures; know
that physical measurements depend
on the choice of a unit and that they
are approximations.
9.3.1.4
area and volume is to multiply each by ,
and
, respectively.
September 22, 2008 Page 56 of 60
Grade
Strand
Standard
Code
9-11
Geometry &
Measurement
Calculate measurements of plane
and solid geometric figures; know
that physical measurements depend
on the choice of a unit and that they
are approximations.
9.3.1.5
resulting from calculations involving measurements.
For example: Suppose the sides of a rectangle are measured to the
nearest tenth of a centimeter at 2.6 cm and 9.8 cm. Because of
measurement errors, the width could be as small as 2.55 cm or as large as
2.65 cm, with similar errors for the height. These errors affect
calculations. For instance, the actual area of the rectangle could be
smaller than 25
or larger than 26 
, even though 2.6 × 9.8 = 25.48.
9-11
Geometry &
Measurement
Construct logical arguments, based
on axioms, definitions and
theorems, to prove theorems and
other results in geometry.
9.3.2.1
theorems in logical arguments.
9-11
Geometry &
Measurement
Construct logical arguments, based
on axioms, definitions and
theorems, to prove theorems and
other results in geometry.
9.3.2.2
only if," "all," and "not." Recognize the logical relationships between an
"if…then" statement and its inverse, converse and contrapositive.
For example: The statement "If you don't do your homework, you can't
go to the dance" is not logically equivalent to its inverse "If you do your
homework, you can go to the dance."
9-11
Geometry &
Measurement
Construct logical arguments, based
on axioms, definitions and
theorems, to prove theorems and
other results in geometry.
9.3.2.3
disprove a statement.
9-11
Geometry &
Measurement
Construct logical arguments, based
on axioms, definitions and
theorems, to prove theorems and
other results in geometry.
9.3.2.4
results in geometry, including proofs by contradiction. Express proofs in a
form that clearly justifies the reasoning, such as two-column proofs,
paragraph proofs, flow charts or illustrations.
For example: Prove that the sum of the interior angles of a pentagon is
540˚ using the fact that the sum of the interior angles of a triangle is 180˚.
September 22, 2008 Page 57 of 60
Grade
Strand
Standard
Code
9-11
Geometry &
Measurement
Construct logical arguments, based
on axioms, definitions and
theorems, to prove theorems and
other results in geometry.
9.3.2.5
perform constructions and develop mathematical reasoning skills in
multi-step problems. The tools may include compass and straight edge,
dynamic geometry software, design software or Internet applets.
9-11
Geometry &
Measurement
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.3.1
properties of angles formed by a transversal, to solve problems and
logically justify results.
For example: Prove that the perpendicular bisector of a line segment is
the set of all points equidistant from the two endpoints, and use this fact
to solve problems and justify other results.
9-11
Geometry &
Measurement
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.3.2
interior, vertical, complementary and supplementary angles, to solve
problems and logically justify results.
For example: Prove that two triangles formed by a pair of intersecting
lines and a pair of parallel lines (an "X" trapped between two parallel
lines) are similar.
9-11
Geometry &
Measurement
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.3.3
to solve problems and logically justify results.
For example: Use the triangle inequality to prove that the perimeter of a
quadrilateral is larger than the sum of the lengths of its diagonals.
9-11
Geometry &
Measurement
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.3.4
logically justify results.
For example: When building a wooden frame that is supposed to have a
square corner, ensure that the corner is square by measuring lengths
near the corner and applying the Pythagorean Theorem.
September 22, 2008 Page 58 of 60
Grade
Strand
Standard
Code
9-11
Geometry &
Measurement
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.3.5
45-90 and 30-60-90 triangles, to solve problems and logically justify
results.
For example: Use 30-60-90 triangles to analyze geometric figures
involving equilateral triangles and hexagons.
Another example: Determine exact values of the trigonometric ratios in
these special triangles using relationships among the side lengths.
9-11
Geometry &
Measurement
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.3.6
problems and logically justify results.
For example: Analyze lengths and areas in a figure formed by drawing a
line segment from one side of a triangle to a second side, parallel to the
third side.
Another example: Determine the height of a pine tree by comparing the
length of its shadow to the length of the shadow of a person of known
height.
Another example: When attempting to build two identical 4-sided
frames, a person measured the lengths of corresponding sides and found
that they matched. Can the person conclude that the shapes of the
frames are congruent?
9-11
Geometry &
Measurement
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.3.7
polygons—to define them, classify them, solve problems and logically
justify results.
For example: Recognize that a rectangle is a special case of a trapezoid.
Another example: Give a concise and clear definition of a kite.
9-11
Geometry &
Measurement
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.3.8
justify results.
For example: Show that opposite angles of a quadrilateral inscribed in a
circle are supplementary.
September 22, 2008 Page 59 of 60
Grade
Strand
Standard
Code
9-11
Geometry &
Measurement
Solve real-world and mathematical
geometric problems using algebraic
methods.
9.3.4.1
trigonometric ratios to be defined, and determine the sine, cosine and
tangent of an acute angle in a right triangle.
9-11
Geometry &
Measurement
Solve real-world and mathematical
geometric problems using algebraic
methods.
9.3.4.2
such as determining lengths and areas in right triangles and in figures that
can be decomposed into right triangles. Know how to use calculators,
tables or other technology to evaluate trigonometric ratios.
For example: Find the area of a triangle, given the measure of one of its
acute angles and the lengths of the two sides that form that angle.
9-11
Geometry &
Measurement
Solve real-world and mathematical
geometric problems using algebraic
methods.
9.3.4.3
trigonometric ratios to find angle measures in right triangles in various
contexts.
9-11
Geometry &
Measurement
Solve real-world and mathematical
geometric problems using algebraic
methods.
9.3.4.4
polygons, including determining lengths, midpoints and slopes of line
segments.
9-11
Geometry &
Measurement
Solve real-world and mathematical
geometric problems using algebraic
methods.
9.3.4.5
(
,
)
, ( )
+ ( )
=
and justify this equation using the
Pythagorean Theorem and properties of translations.
9-11
Geometry &
Measurement
Solve real-world and mathematical
geometric problems using algebraic
methods.
9.3.4.6
two dimensions, such as reflections, translations, scale changes and
rotations about the origin by multiples of 90˚, to solve problems involving
figures on a coordinate grid.
For example: If the point (3,-2) is rotated 90˚ counterclockwise about the
origin, it becomes the point (2, 3).
9-11
Geometry &
Measurement
Solve real-world and mathematical
geometric problems using algebraic
methods.
9.3.4.7
geometry, such as solving for an unknown length in a figure involving
similar triangles, or using the Pythagorean Theorem to obtain a quadratic
equation for a length in a geometric figure.
September 22, 2008 Page 60 of 60
Grade
Strand
Standard
Code
9-11
Data Analysis &
Probability
Display and analyze data; use
various measures associated with
data to draw conclusions, identify
trends and describe relationships.
9.4.1.1
describe and compare data sets using summary statistics, including
measures of center, location and spread. Measures of center and location
include mean, median, quartile and percentile. Measures of spread
include standard deviation, range and inter-quartile range. Know how to
use calculators, spreadsheets or other technology to display data and
calculate summary statistics.
9-11
Data Analysis &
Probability
Display and analyze data; use
various measures associated with
data to draw conclusions, identify
trends and describe relationships.
9.4.1.2
For example: Understand how inserting or deleting a data point may
affect the mean and standard deviation.
Another example: Understand how the median and interquartile range
are affected when the entire data set is transformed by adding a constant
to each data value or multiplying each data value by a constant.
9-11
Data Analysis &
Probability
Display and analyze data; use
various measures associated with
data to draw conclusions, identify
trends and describe relationships.
9.4.1.3
two variables. Using technology, determine regression lines (line of best
fit) and correlation coefficients; use regression lines to make predictions
and correlation coefficients to assess the reliability of those predictions.
9-11
Data Analysis &
Probability
Display and analyze data; use
various measures associated with
data to draw conclusions, identify
trends and describe relationships.
9.4.1.4
distribution (bell-shaped curve) and to estimate population percentages.
Recognize that there are data sets for which such a procedure is not
appropriate. Use calculators, spreadsheets and tables to estimate areas
under the normal curve.
For example: After performing several measurements of some attribute
of an irregular physical object, it is appropriate to fit the data to a normal
distribution and draw conclusions about measurement error.
Another example: When data involving two very different populations is
combined, the resulting histogram may show two distinct peaks, and
fitting the data to a normal distribution is not appropriate.
September 22, 2008 Page 61 of 60
Grade
Strand
Standard
Code
9-11
Data Analysis &
Probability
Explain the uses of data and
statistical thinking to draw
inferences, make predictions and
justify conclusions.
9.4.2.1
source of the data, the design of the study, and the way the data are
analyzed and displayed. Show how graphs and data can be distorted to
support different points of view. Know how to use spreadsheet tables and
graphs or graphing technology to recognize and analyze distortions in
data displays.
For example: Displaying only part of a vertical axis can make differences
in data appear deceptively large.
9-11
Data Analysis &
Probability
Explain the uses of data and
statistical thinking to draw
inferences, make predictions and
justify conclusions.
9.4.2.2
based on data confuse correlation and causation.
9-11
Data Analysis &
Probability
Explain the uses of data and
statistical thinking to draw
inferences, make predictions and
justify conclusions.
9.4.2.3
bias and the phrasing of questions asked during data collection.
9-11
Data Analysis &
Probability
Calculate probabilities and apply
probability concepts to solve real-
world and mathematical problems.
9.4.3.1
addition principles and tree diagrams, to determine the size of a sample
space (the number of possible outcomes) and to calculate probabilities.
For example: If one girl and one boy are picked at random from a class
with 20 girls and 15 boys, there are 20 × 15 = 300 different possibilities,
so the probability that a particular girl is chosen together with a particular
boy is

.
9-11
Data Analysis &
Probability
Calculate probabilities and apply
probability concepts to solve real-
world and mathematical problems.
9.4.3.2
experiments involving a probability model and using relative frequencies
of outcomes.
9-11
Data Analysis &
Probability
Calculate probabilities and apply
probability concepts to solve real-
world and mathematical problems.
9.4.3.3
between the probabilities in a probability model and the experimental
probabilities found by performing simulations or experiments involving
the model.
September 22, 2008 Page 62 of 60
Grade
Strand
Standard
Code
9-11
Data Analysis &
Probability
Calculate probabilities and apply
probability concepts to solve real-
world and mathematical problems.
9.4.3.4
taken from a table, to perform probability simulations and to introduce
fairness into decision making.
For example: If a group of students needs to fairly select one of its
members to lead a discussion, they can use a random number to
determine the selection.
9-11
Data Analysis &
Probability
Calculate probabilities and apply
probability concepts to solve real-
world and mathematical problems.
9.4.3.5
complements of events, and conditional probability and independence, to
calculate probabilities and solve problems.
For example: The probability of tossing at least one head when flipping a
fair coin three times can be calculated by looking at the complement of
this event (flipping three tails in a row).
9-11
Data Analysis &
Probability
Calculate probabilities and apply
probability concepts to solve real-
world and mathematical problems.
9.4.3.6
Venn diagrams. Understand the relationships between these concepts
and the words AND, OR, NOT, as used in computerized searches and
spreadsheets.
9-11
Data Analysis &
Probability
Calculate probabilities and apply
probability concepts to solve real-
world and mathematical problems.
9.4.3.7
unions and complements of events.
For example: If the probability of an event is p, then the probability of the
complement of an event is 1 – p; the probability of the intersection of
two independent events is the product of their probabilities.
Another example: The probability of the union of two events equals the
sum of the probabilities of the two individual events minus the
probability of the intersection of the events.
September 22, 2008 Page 63 of 60
Grade
Strand
Standard
Code
9-11
Data Analysis &
Probability
Calculate probabilities and apply
probability concepts to solve real-
world and mathematical problems.
9.4.3.8
decisions.
For example: Explain why a hockey coach might decide near the end of
the game to pull the goalie to add another forward position player if the
team is behind.
Another example: Consider the role that probabilities play in health care
decisions, such as deciding between having eye surgery and wearing
glasses.
9-11
Data Analysis &
Probability
Calculate probabilities and apply
probability concepts to solve real-
world and mathematical problems.
9.4.3.9
frequencies in contingency tables.
For example: A table that displays percentages relating gender (male or
female) and handedness (right-handed or left-handed) can be used to
determine the conditional probability of being left-handed, given that the
gender is male.