Algebra 1 Grade 8 CA Common Core Review

 

8.A8.N Number and Quantity

8.A8.A Algebra

8.A8.F Functions

8.A8.G Geometry

8.A8.SP Statistics and Probability

8.A8.VA Constructing Viable Arguments

  • 8.A8.VA.VA Constructing Viable Arguments

    • 8.A8.VA.VA.1 Use and know simple aspects of a logical argument.

      • 8.A8.VA.VA.1.a Use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

    • 8.A8.VA.VA.2 Use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements:

      • 8.A8.VA.VA.2.a Use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.

      • 8.A8.VA.VA.2.b Judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.

      • 8.A8.VA.VA.2.c Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, determine whether the statement is true sometimes, always, or never.

How to best learn the math you need?

Use six steps to learn a math skill.  Our new learning materials supports this recommended process.

-1-   EXPLORE   Develop a genuine, enthusiastic curiosity –  become a true mental explorer. Develop this enthusiasm so you will persist in the mental effort to focus attention, understand connections, avoid confusion,  get help for difficulties and overcome inevitable discouragements.

-2-   INQUIRE   Keep formulating inquiry questions –check on what you know and what you don’t know.  Persist in seeking answers to these questions. Use the 5W2H question stems:  Who -, What -, When -, Where -, Why -, How -, How much -?

-3-   CONNECT    your new learning material items to integrate them with the old material items you already learned. Recognize and master all prerequisites first.

-4-   MODEL   Adapt/create mental models to summarize thr new knowledge, contrast similarities and differences and connect knowledge items. Create these models as schema for effective comprehension and improved recall.

-5-   INTERLEAVE   review topics to focus on summary understanding and of differences and similarities.

-6-   SPACE  practice for active recall of these connection models, summaries, concepts, examples and skills.  Do this for long term retention through repeated, directed self-quizzing of your own (for example, Cornell Notes) or our prepared materials.

Grade 7 Math – IXL Skills Practice

 Seventh grade math = Prealgebra

Mathematically Gifted Students

Some students are able to keep up with their class learning.  They are A or B students. They need standard guidance for after-class practice.

Other students are having difficulty keeping up with their class.  These students cannot maintain either an A or B grade.  They need customized guidance for after-class practice.for remediation, reteaching and

Some students simply are not able to keep up with the class pace and are determined to have special education needs.  They need highly customized guidance

There are a few mathematically gifted students who also need highly customized guidance for after-class practice. The goal here is to achieve mastery, maintain it and challenge for advancement.
See more on Mathematically Gifted Students

 

 

 

College Remedial Math – Pre-Algebra Review

This page is a glossary of Pre-Algebra Topics. keyed to a popular communityccollege remedoal text  (Martin-Gay.. Fourth Edition). It is used at our local Antelope Valley College for students who need a remedial course as determined by the ACT College Placement Test.  Mastery of these topics will avoid having to take such an expensive ($ and time) remedial course.

See also a list of mathematical concepts and skills at  IXL Grade  7 Math Practice .

1.0  Whole Numbers
–  1.1 Study Skill Tips for Success in Mathematics

Use our Show-Hide Text Toggling to conveniently:

-1- Read a topic title – generally for a concept definition or procedure name

-2-  Actively recall and recite the definition or procedure steps

-3-  Click the title to check accuracy and completeness of your recall & recitation

-4-  Toggle click on title to repeatedly show-hide the text until you can correctly recall it

-5-  Link to and toggle concrete worked examples for understanding and creating your own examples,

-6-  Link to prior knowledge connections, and then

-7-  Practice answering interleaved review-questions,

-8-  Familiarize yourself with typical questions for each skill topic

-9-  Practice answering skills at IXL Grade 7 Math Skills Practice.

=10- Preview topics for your next-class-lesson work.

–  1.2 Place Value, Names for Numbers, and Reading Tables

Digits

The Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used to write numbers using base ten – our decimal system.

Other important numbers with different bases:The digits 0 and 1 are used to write numbers using base two – our binary system used to represent on/off states in computers and in electrical machinery.

The digits 1 through 59 were used to write numbers using base 60 – the ancient Babylonian numeration system that was convenient for its easier representation of fractions in commerce, etc.

Whole Numbers

The whole numbers for our decimal system are 0, 1, 2, 3, 4, 5 6, 7, 8, 9, 10, 11, … etc.The smallest whole number is 0.

Each whole number is one larger than the previous one.

There is no largest whole number.

Place value in whole numbers

The place value of a digit is the value of a digit’s position in a number determined by its position in the number.

The value of a digit is given by the place value.

Place values are determined from, right to left as: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred million; billions, ten billions, hundred billions, trillions — and so forth.

Standard form for whole numbers

A whole number written in standard form is a string of digits or numerals, separated into groups of threes by commas.

Each group of three digits is called a period.

Beginning from right to left we have the following periods: ones; thousands, millions; billions; trillions; and so forth.

Write a whole number in words

To write a whole number in words, write the number in each period (excepting the ones period).

Read a whole number in words

To read a whole number in words, read the number in each period with the name of the period – from left to right.

Do not name the ones period.

Write a whole number in standard form

To write a number in standard form, write the number in each period followed by a comma.

Expanded form of whole numbers

The expanded form of a whole number shows each digit of the number with its place value.

Write a whole number in expanded form

Write each digit with its place value named.

Whole numbers graphed on the number line

Whole numbers are pictured as equally spaced points on a number line in counting order from 0 to the right.

One whole number greater than another whole number

The whole number graphed to the right of another whole number on the number line is the greater of the two.

One whole number smaller than another whole number

The whole number graphed to the left of another whole number on the number line is the smaller of the two.

Compare whole numbers

We use the symbol “ > ” to state that one number, a, is greater than another, b:
a > b.
We use the symbol “ < “ to state that one number, a is smaller than another, b:
a < b.

–  1.3 Adding Whole Numbers and Perimeter

Sum of whole numbers

The sum is the result of adding the addends when adding whole numbers.

Addends

Addends are the numbers added when you add whole numbers.

Add two or more whole numbers with no carrying

Write the addends so that the digits with the same place value position are vertically aligned.

To add two or more whole numbers with no carrying, we add the digits of each addend in the ones place, then the tens place, then the hundreds place, and so on, from right to left.

No “carrying” is necessary if the sum of each of the digits in its place adds to 9 or less.

Add two whole numbers with carrying

Write the addends so that the digits with the same place value position are vertically aligned.

To add two whole numbers with carrying, we add the digits in each place and carry 1 if the sum of the digits is greater than 9. This means we add 1 when summing the digits in one place to the left of the one we are now adding.

We do this when summing the digits in each place value position, left to right.

Addition property of 0

The sum of 0 and any number is that number.

Commutative property of addition

We can add any two numbers in any order without changing the sum.

Associative property of addition

When adding whole numbers we can change the grouping of the addends without changing the sum.

Polygon

A polygon is a flat figure formed by line segments connected at their ends.

Perimeter of a polygon

The perimeter of a polygon is the distance around the polygon.

The perimeter of a polygon is the sum of the lengths of its sides.

Key words in addition word problems

Key words in addition word problems may include: added to; plus; increased by; more than; total, sum.

–  1.4 Subtracting Whole Numbers

Difference of whole numbers

The difference of two numbers results from subtracting one whole number, the subtrahend, from another, the minuend.

Minuend

The minuend is the number from which one subtracts.

Subtrahend

The subtrahend is the number that one subtracts from the minuend to get the difference.

Subtraction properties of 0

The difference of any number and that same number is 0.

The difference of any number and 0 is that same number.

Subtract whole numbers without borrowing

Write the minuend and subtrahend so that the digits with the same place value position are vertically aligned.

Find the difference between the digits with the same place value position.

Borrowing will not be necessary if the minuend digit is greater than the corresponding subtrahend digit.

Subtract whole numbers with borrowing

Write the minuend and subtrahend so that the digits with the same place value position are vertically aligned.

Find the difference between the digits with the same place value position.

Borrowing is necessary when a digit in the second number (the subtrahend) is larger than the corresponding digit of the first number (the minuend).

We borrow 10 for this place value digit subtraction by subtracting (“borrowing”) 1 from the minuend digit one place value position higher – when next subtracting in that position.

Key words in subtraction word problems

The following key words or phrases may occur in subtraction word problems: subtract; difference; less; take away; decreased by; subtracted from.

–  1.5 Rounding and Estimating

Rounding whole numbers

Whole numbers are rounded or approximated to better use, understand, simplify computation or remember the numbers.

Round whole numbers to a given place value

If this digit is 5 or less, replace it and each digit to its right by 0.

If this digit is 6 or more, add 1 to the digit on left and replace all digits to its right to 0.

Round whole numbers to estimate a sum

Round addends and add to estimate sum and check credibility of exact sums.

Round whole numbers to estimate a difference

Round minuends and subtrahends to estimate differences and check for incorrect determination of them using exact calculation.

–  1.6 Multiplying Whole Numbers and Area

Multiplication of whole numbers

Multiplication is repeated addition but with different notation.

Factors

Factors are the numbers being multiplied to produce the product of a multiplication.

Multiplication property of 0

The product of 0 and any number is 0.

Multiplication property of 1

The product of 1 and any number is that same number.

Commutative property of multiplication

When multiplying two numbers the order of these numbers can be changed without changing the product.

Associative property of multiplication

When multiplying two numbers you can change the grouping of factors without changing the product.

Distributive property of multiplication

Multiplication distributes over addition.

Partial product

The intermediate product of a single digit in the first factor by the digits of the second factor keeping account of the 0’s for the place value of the digit in the first factor.

Partial products are aligned vertically corresponding to place values and summed to form the complete product of a whole number multiplication.

Product of two whole numbers

The product of two whole numbers is the result of multiplying two whole numbers.

Multiply two or more whole numbers

Write each factor to vertically align place value positions.

Multiply each digit of the first factor by the digits of the second factor to form partial products, carrying where necessary.

Align partial products vertically so that place value positions correspond.

Add partial products to determine the product of the factors.

Use Associative property of multiplication to group factors and replace the multiplication of any two by its product and the next remaining factor to be multiplied.

Multiply any whole number by 0

Any whole number multiplied by 0 is 0 according to the Multiplication property of 0

Apply commutative property of multiplication for whole numbers

Change the order of the factors to be multiplied without changing the product.

Apply distributive law of multiplication for whole numbers

Distribute the multiplication over the addition without changing the product.

Apply associative law of multiplication

Change the grouping of factors without changing the product.

Rectangle

A rectangle is a four sided figure with two equal and opposite sides, called the length and width.

The sides at the corners are perpendicular or at right angles to each other.

Area

Area is a measure of the amount of surface of a region. Thus, it is the count of unit square measures covering that region.

It is a special application of multiplication.

Find area of rectangle

The area of a rectangle is the product of its length and its width.

Key words for multiplication word problems

Several words and phrases that indicate the operation of multiplication include:
multiply, product, times.

–  1.7 Dividing Whole Numbers

Division of a whole number by another whole number

Division is the process of separating the quantity into b equal parts and considering a of them.

Division of a whole number a by another whole number b is written as:
a division-sign b or or

a / b     (using parentheses as a most convenient way to write division using a single line of type).

Fraction bar or slash

The bar separating a and b in the division a by b ( or a / b)
is called a fraction bar or the fraction slash – it symbolizes division.

Quotient of two whole numbers

The quotient of two whole numbers is the result of the division process, consisting of another whole number which counts the whole number of multiples of b in a.

Divisor

The divisor is the number that is used to divide into a dividend. It is b in the division a/b.

Dividend

The dividend is a, the whole number to be divided in the division a/b.

Remainder

The remainder of a division q = a/b is the whole number quantity left after determining an even whole number quotient.

Find quotient of any whole number by itself

a / a = 1 if a is any whole number

Find quotient of any whole number by 1

a / 1 = a where a is any whole number

Find quotient of 0 by any whole number other than 0

0 / b = 0 if b ≠ 0 an db = a whole number

Find quotient of any whole number by 0

a / 0 = undefined

Perform long division of any whole number by another whole number

Write the divisor to the left of the division radical, include the dividend in the radical, and write digits of the long division solution digit by digit over the corresponding place value.

Test each digit as a trial dividend until a whole number quotient can be found. Write the quotient digit above and in alignment with the place values of the rightmost digits.

Multiply the quotient digit by the divisor and write under corresponding place value.
Determine the difference and bring down the next number from the dividend.

Determine a trial quotient using the difference; if no trial quotient is possible, bring down the next digit of the dividend to augment the number of digits in the trial dividend.

Continue until the last digit of dividend is considered.

Write any difference a remainder.

Key words in division word problems

Key words and phrases that may indicate the operation of division include:
divide, quotient, divided by, divided or shared equally among.

Average

The average of a list of numbers is the sum of the numbers divided by the number of listed numbers.
Find average of a list of numbers

The average of a list of numbers is found by adding the numbers in the list and dividing the total by the number of list items.

–  1.8 An Introduction to Problem Solving
–  1.9 Exponents, Square Roots, and Order of Operations

Exponential notation

Exponential notation is shorthand for the repeated multiplication of a number.

Exponent

An exponent is a shorthand notation for the number of repeated multiplications of a factor, also called the base of the exponent.

Base

The base in exponential notation is the number or factor that is to be repeatedly multiplied.

Factor

The factor in exponentiation is the number to be repeatedly multiplied when multiplying.
Write using exponential notation

To write repeated multiplication of a factor using exponential notation – use the factor as the base and use the number of multiplications for the exponent.
Evaluate an exponential expression

To evaluate an exponential expression: write out the repeated multiplication by using the base as the common factor, write out as many multiplications as the exponen, and
then multiply – left to right.

Simplify expressions with whole numbers using order of operations

The order of operations for whole numbers is as follows:
• Do all operations within grouping symbols such as parentheses or brackets.
• Evaluate any expressions with exponents
• Multiply and divide in order from left to right.
• Add or subtract in order from left to right.

Find area of square

A square is a four sided polygon with equal sides perpendicular to each other.

2. Integers and Introduction to Variables
–  2.1 Inroduction to Variables and Algebraic Expressions

Variable

A variable is a letter used to represent a number.

Algebraic expression

An algebraic expression (or an expression) is a combination of operations on variables and numbers.

Expression

See algebraic expression. It is a combination of operations on variables and numbers.

Evaluate (algebraic) expression for the variable

To evaluate an expression for a variable is to replace a variable by a number, translate the verbal phrases into variable expressions and then finding the value of that expression..

–  2.2 Introduction to Integers

Positive numbers

A positive number is a whole number greater than 0.

Negative numbers

A negative number is a whole number less than 0.

Signed numbers

Signed numbers consist of positive numbers, 0, and the negative numbers.

Integers

Integers are signed whole numbers, to include 0.

Graph integers

Use the left to right increase-directed number line to graph the negative numbers to the left of 0 and graph the positive whole numbers to the right of 0.

Compare integers

An integer a is greater than another integer b if the graph position of a on the number line is to the right of b.

We use the notation a > b to say that “a is greater than b”.

We use the notation b < a to say that b is less than a

If you think of “<“ and “ > “ as the points of an arrowhead, notice that the “greater than” or “less than” symbol always points to the smaller integer.

Absolute value of an integer

The absolute number of a number is its distance from 0 on the number line.

The absolute value is always positive.

Opposite integers

Opposite integers are two integers that are the same distance from 0 on the number line and on opposite sides of 0.

The minus sign ” – ” is used to state “opposite of”.

Find opposite of a given integer

The opposite of “a” is “-a”.

If a > 0 then a is to the right of 0 on the number line, “a” units. The opposite of “a” is a units to the left of 0, “a” units.

If a < 0, then a is to the left of 0, ”a” units – and the opposite of “a” is then “a” units to the right of 0.

– 2.3  Adding Integers

Add two integers with the same sign

To add integers on a number line start from 0 on the line and draw an arrow representing the first number.

From the tip of this first arrow draw another arrow representing the second number.

The tip of the second arrow represents their sum.

For larger numbers – add their absolute values and use their common sign as the sign of the sum.

Add two integers with different signs

To add two integers a and b with different signs – first find the larger absolute value minus the smaller absolute value.

Use the sign of the number with the larger absolute value as the sign of the sum.

Apply associative property of integer addition

If a, b and c are integers then

a + (b +c) = (a + b) + c

Apply commutative property of integer addition

If a and b are integers then

a + b = b + a.

Evaluate addition of expressions using integer replacement values

Rewrite expressions showing the positive and negative signs of the numbers.

Add expressions keeping separately the “+” sign for the addition operation.

– 2.4  Subtracting Integers

Subtraction of two integers

We rewrite subtraction problems as integer addition problems.

If a and b are numbers, then a – b = a + ( -b ).

Add and subtract more than two integers

Rewrite the differences as additions of signed numbers and add left to right.

Evaluate adding and subtracting expressions using integer replacement

Substitute integers explicitly for the variables in the expression – being explicitly mindful of each sign.

Rewrite subtractions as additions and add left to right.

– 2.5  Multiplying and Dividing Integers

Multiplication of integers

Multiplication of integers is similar to that for whole numbers – excepting that one must be mindful of the signs of each factor.

The product of two numbers having the same sign is a positive number.

The product of two numbers of different sign is a negative number.

Multiply two like-signed integers

(+a) • (+b) = +(a • b) = a • b
or
(-a) (-a) = + (a • b) = a • b

Multiply two unlike-signed integers

If a and b are whole numbers, then

(-a) • (+b) = – (a • b)

or

(+a) • (-b) = – (a • b)

Division of integers

The quotient of two numbers having the same sign is a positive number.

The quotient of two numbers having different signs is a negative number.

Divide two like-signed integers

If a and b are whole numbers then – if b is not 0 –

-a /-b = +(a / b)
or
+a / +b = +(a / b)

Divide two unlike signed integers

If a and b are whole numbers then – if b is not 0 –
-a / +b = -(a / b)
or
+a / -b = -(a / b)

– 2.6  Order of Operations

Order of Operations Review (GEMDAS)

• Look for and do all operations within a G rouping symbol (such as absolute values, fraction bars, parentheses and bracket)
• Evaluate any expressions with E xponents
M ultiply and D ivide in order from left to right
A dd or S ubtract in order from left to right

3. Fractions and Mixed Numbers
– 3.1  Introduction to Fractions and Mixed Numbers

Fractions 

A fraction is a part of a whole.

It is written in terms of a denominator and a numerator separated by a fraction bar or slash.

The value of the fraction is the quotient of the numerator by the denominator with division signified by the fraction bar or slash.

0  cannot be a denominator because division by  0  is undefined.

The fraction bar is traditionally the horizontal bar – or in single-line math type – the backslash symbol “/” – and it signifies division of the numerator by the denominator.

Numerator

The numerator of a fraction defines how many parts of the whole are being considered. It is the number or expression above the fraction bar.

The denominator defines the total number of equal parts of the whole being considered. It is the number or expression below the fraction bar.

Denominator

The denominator of a fraction defines the total number of equal parts in the whole.

It can never be 0.

Proper fractions

A proper fraction is a fraction whose numerator is less than its denominator.

Improper fractions

An improper fraction is a fraction whose numerator is greater or equal to its denominator.

Write fractions to represent shaded areas of figures

Let a figure be divided into “a” equal parts. Let “b” be the number of parts in the figure that are shaded.

Then the fraction that represents the shaded areas of the figure is the fraction b/a.

This is one way to help one visualize the concept “fraction”.

Graph fractions on the number line

Fractions are graphed on the number line between 0 and 1 or 0 and -1, depending on its sign.

If 1 (one) unit represents the whole, then the fraction “a/b” is plotted “a” units from 0, where a unit is one of “b” equal parts connecting from 0 to 1,if “a” is positive; —– if “a” is negative, it will be “a” units to the left of  0  towards  -1..

Equivalent fractions

Fractions that represent the same proportion of a whole or the same point on the number line are said to be equivalent.

Multiply numerator and denominator by the same non-zero number to get equivalent fraction

If a, b, and c are numbers, then  a / b= (a • c) / (b • c).

This is true because any number multiplied by 1 is equal to that number.

And since c/c = 1,

(c/c) • (a/b) = 1 •(a / b) = a/b = (a • c) / (b • c)

using the distributive rules for multiplication.

Divide numerator and denominator to get the equivalent fraction.

Simplify to equivalent fraction: a/a = 1

If a is any non-zero number then a / a = 1

i.e., a non-zero number, divided by itself is 1,

or,

a non-zero number multiplied by its reciprocal is 1.

Simplify to equivalent fraction: a/1

If a is any number then a/1 = a.

Simplify to equivalent fraction:  0/a

Any non-zero number a divided into 0 is 0;

so 0/a = 0

Simplify a/0?

Not possible.  Division by 0 is undefined.

– 3.2 Factors and Simplest Form

Prime number

A prime number is a whole number greater than 1 whose only divisors are 1 and itself.

The first nine prime numbers are: 2, 3, 5, 11, 13, 17, 19, 23, 29, ….

Composite number

A composite number is a number that is greater than 1 and not prime.

Factor

A factor is any number that divides a number evenly – that is, with a remainder of 0.

Factorization

Factorization is the writing of a number as a product of factors.

Prime factorization

The prime factorization of a number is a factorization in which all the factors, other than 1 and itself, are prime numbers.

Factor tree

A factor tree is a tree with a top node that is the number to be factored and next lower level nodes represent factors in its factorization

Each such lower level node is again subdivided into lower level nodes each again representing a factor in its factorization

The lowest level nodes are factors that are prime numbers, thus halting the factor tree division process. This is so because prime numbers, by definition, cannot be factored further.

Divisibility tests

A whole number is divisible by

  • 2 —  if the ones digit is even (or 0, 2, 4, 6, 8)
  • 3 –- if the sum of the digits is divisible by 3
  • 5 –  if the ones digit is divisible by 0 or 5

Find the prime factorization of a number

Determine the prime factors of a small number (say less than a hundred) by creating a factor tree and using the found prime factors as the factors in the factorization.

For larger numbers – or for showing all work – begin with the lowest prime number to determine if it can be a divisor –

if so, keep dividing by it to determine the largest number of times it divides into the number and using that frequency as its exponent – then, determine how often the remaining factor can be divided by the next highest prime number and using that frequency as its exponent – and so on.

At each step – divisibility tests are useful in determining whether the next higher prime is even a factor.

The prime factorization will be the product of each of the prime numbers raised to exponent numbers that state their frequency as a repetitive divisor.

Fractions in simplest form or lowest terms

A fraction is in simplest form, or lowest terms, when the numerator and the denominator have no common factors other than 1.

Simplify a fraction – write a fraction in simplest form

To write a fraction in simplest form, write the prime factorization of the numerator and the denominator and then divide both by all common factors.

In simplifying a product it may be possible to identify a common factor without writing out its prime factorization.

If the denominator of a fraction contains a variable, it is assumed that the variable is such that the denominator is always non-zero – since division by zero is always undefined.

– 3.3 Multiplying and Dividing Fractions

Multiply two fractions

To multiply two fractions you multiply the numerators to form the numerator of the resulting fraction and you multiply the denominators to form the denominator of the resulting fraction.

If a, b, c and are numbers and b and d are not 0, then

(a / b) • (c / d) = (a • c) / (b • d)

Whenever you can, simplify the fractions before – and after – the multiplication.

Exponential expressions with fractional base

An exponential expression can have a fractional base, so that:

(a / b)^m = (a/b) • (a/b) • … • (a/b) =

(a^m) / (b^m) = (a • a • a • … • a) / (b • b • b • … • b)

— multiplied m times in both numerator and denominator – and as fractions  – where m is a whole number.

In the above notation the carrot signifies exponentiation, so that  …  a^2  = a • a .

Reciprocal of a fraction

Two numbers are reciprocals of each other if their product is 1.

The reciprocal of the fraction a/b is:    b/a,      since

(a / b) • (b / a) = (a • b) / (b • a) = 1.

Every number has a reciprocal – except 0 – there is no number so that

0 • a = 1.

Find the reciprocal of a fraction

The reciprocal of a fraction “ a/b ” is obtained by exchanging the value of numerator with the denominator – so that it becomes equal to:  “ b/a ”.

Divide two fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction.

If b, c and d are not 0, then:

(a / b) / (c / d) = (a / b) • (d / c) = (a • d) / (b • c)

When dividing by a fraction rewrite the division as a multiplication – THEN look to simplify by canceling common factors.

Solving applications by multiplying and dividing fractions

  1. Understand the problem by reading and re-reading problem.
  2. Determine what is to be found or proved
  3. Determine what is given
  4. Determine how to connect what is given to what is wanted
  5. Formulate an equation by translating key words and word phrases to symbol operations.
  6. Solve for the unknown in the formulated equation.
  7. Interpret the results by checking the work and state conclusions that answers the posed questions.
– 3.4 Adding and Subtracting Like Fractions, Least Common Denominator and Equivalent Fractions

Like fractions

Fractions that have the same denominator are called like fractions.

Unlike fractions

Fractions that have different denominators are called unlike fractions.

Add like fractions

To add two or more like fractions (each having the same denominator as the others) add the sum of the numerators and place the difference over the common denominator.

Subtract like fractions

To subtract two or more like fractions (each having the same denominator as the others) – subtract the numerators, left to right, and place the difference over the common denominator

Least common denominator (LCD)

The least common denominator of a list of fractions is the smallest positive number divisible by all the denominators in the list.

Find the LCD (Divisibility of Multiples of Larger Denominator)

See whether the largest denominator is divisible by smallest.

If so, choose larger denominator as LCD.

If not, check consecutive multiples of larger denominator for divisibility by smallest.

If so, choose larger- denominator- multiple as LCD.

When large multiples are needed, use the algorithm “Find LCD (Division By Primes of All Denominators)”.

Find the LCD (Cancel Common Factors after Prime Factorization of Numerators and Denominators)

Write the numerator as a product of primes.

Write the denominator as a product of primes.

Cancel common factors in numerator and denominator.

Find the LCD (Product of Primes Divisible into all Denominators)

Write all denominators on a line – perform

Begin with 2, smallest prime, to check divisibility into each denominator.

Divide into each number and record multiple factors of primes as above.

Continue the same with 3, 5, 7 etc.

Least common multiple

See Least Common Denominator (LCD).

– 3.5 Adding and Subtracting Unlike Fractions
Add unlike fractions

Find the LCD of the denominators of the fractions.

Write each fraction as an equivalent fraction whose denominator is the LCD.

Add the like fractions.

Write the sum in simplest form.

 Subtract unlike fractions

Find the LCD of the denominators of the fractions.

Write each fraction as an equivalent fraction whose denominator is the LCD.

Subtract the like fractions.

Write the difference in simplest form.

Add and subtract with given fractional replacement values

Substitute fractions as replacement values and add, subtract, multiply and divide as specified.

– 3.6 Complex Fractions, Order of Operations, and Mixed Numbers

Complex fractions

A complex fraction is a fraction whose numerator and denominator is a fraction.

Simplify complex fractions (Multiply reciprocal of denominator)

Simplify numerator and denominator of fraction each to be a single fraction.

Multiply numerator fraction by reciprocal of denominator fraction.

Simplify.

Simplify complex fractions

Simplify by;

finding the common LCD for all of the denominators in both numerator and denominator

then multiply both numerator and denominator to remove all denominators in them.

Order of operations

Start by looking for grouping symbols such as absolute values and fraction bars

Do all operations within grouping symbols such as parentheses and brackets Evaluate any expressions with exponents and find any square roots.

Multiply in order from left to right.

Add or subtract in order from left to right

– 3.7 Operations on Mixed Numbers

Mixed number

A mixed number is a number consisting of a whole number and a proper fraction.

Write a mixed number as an improper fraction

To write a mixed number as an improper fraction:

Multiply the whole number by the denominator of the fraction.

Add the numerator of the fraction to the product above.

Write the sum of the previous step as the numerator of the improper fraction over the denominator of the old fraction which then becomes the denominator of the improper fraction.

Write an improper fraction as a mixed number

To write an improper fraction as a mixed number or a whole number:

Divide the denominator into the numerator

The whole number of the mixed number is the whole number quotient.

The proper fraction part of a mixed number is formed as:

(Remainder) / (Denominator of improper fraction)

When the remainder is 0 the improper fraction is a whole number.

Multiply two mixed numbers

Convert both mixed numbers into improper fractions and multiply the fractions as usual.

Divide two mixed numbers

Convert both mixed numbers into improper fractions and divide the fractions as usual.

You can always write a whole number as a fraction with denominator 1.

Add two mixed numbers

Add the whole number parts to form the whole number portion of an initial mixed number sum.

Add the fractional parts separately.

If the added fractional parts sums to a proper fraction, retain it as the proper fractional part of the mixed number sum. Also retain the whole number portion of the initial mixed number sum for the whole number portion of the final mixed number sum.

If the fractional part sum is an improper fraction, write that as a mixed number, add the whole number part of it to finalize the whole number portion of the mixed number sum, and retain the fractional part as the proper fractional part of the mixed number sum.

In more advanced algebra courses you will be asked to leave the result as a improper fraction properly reduced.

4. Decimals
– 4.1 Introduction to Decimals
– 4.2 Adding and Subtracting Decimals
– 4.3 Multiplying Decimals and Circumference of a Circle
– 4.4 Dividing Decimals
– 4.5 Fractions, Decimals, and Order of Operations
– 4.6 Square Roots and the Pythagorean Theorem
5. Ratio, Proportion, and Measurement
– 5.1 Ratios
– 5.2 Proportions
– 5.3 Proportions and Problem Solving
– 5.4 Length: U.S. and Metric Syst ems of Measurement
– 5.5 Weight and Mass: U.S. and Metric Systems of Measurement
– 5.6 Capacity: U.S. and Metric Systems of Measurement
– 5.7 Conversions Between the U.S. and Metric Systems
6. Percent
– 6.1 Percents, Decimals, and Fractions
– 6.2 Solving Percent Problems Using Equations
– 6.3 Solving Percent Problems Using Proportions
– 6.4 Applications of Percent
– 6.5 Percent and Problem Solving: Sales Tax, Commision, and Discount
– 6.6 Percent and Problem Solving: Interest
7. Statistics and Probability
– 7.1 Reading Pictographs, Bar Graphs, Histograms, and Line Graphs
– 7.2 Reading Circle Graphs
– 7.3 Mean, Median, and Mode
– 7.4 Counting and Introduction to Probability
8. Introduction to Algebra
– 8.1 Introduction to Variables

Variable

A variable is the assignment to a letter symbol of a number selected as a placeholder.

The assignment represents a selection of any one of a collection of numbers – but, once chosen, it represents the same specific number in under consideration.

Terms

The addends of an algebraic expression consisting of a combination of operations with numbers and with letters that represent variables

Constant (term)

A constant term is a term in an expression that is only a number.

Variable term

A variable term is a term that includes a variable.

Numerical coefficient

A numerical coefficient of a term is the number factor of a term.

Like terms

Like terms are terms that have the same variable in common.

Unlike terms

Unlike terms are terms that do not have any variables in common.

Combine like terms

Combine the coefficients of like terms using the rules of integer addition and subtraction and use the resultant sum or difference as the coefficient for the common variable.

Apply distributive property to addition of two variables

Distribute multiplication over addition, so that, if a, b and c are any numbers (but each a specific number for the sake of this discussion), then:
a (b + c) = a • b + a • c

Apply distributive property to subtraction of two variables

Distribute multiplication over addition and subtraction, so that, if a, b and c are any numbers (but each a specific number for the sake of this discussion), then:
a • (b – c) = a • b – a • c

Apply commutative property to addition of two variables

If a and b be any numbers (once chosen, each must be some specific number – it can be any must be some number), then:
a + b = b + a

Apply commutative property to multiplication of two variables

If a and b are any numbers (once chosen, each must be some specific number – usually the same for the discussion at hand – again – it can be any number but must be some specific number), then:
a • b = b • a

Apply associative property to addition of variables

If a, b and c are any numbers (once chosen, each can be any number but it must be some specific number), then their order or grouping in an expression can be changed without altering their sum — so that:
(a + b) + c = a + (b + c)

Apply associative property to multiplication of two variables

If a, b and c are any numbers (once chosen, eachcan be any one but it must be some specific number for the discussion), then their order or grouping in an expression can be changed without altering their product — so that:
(a • b) • c = a • (b • c)

Add algebraic expressions

Add two or more algebraic expressions by adding the terms of each expression.

Multiply algebraic expressions

Multiply two or more algebraic expressions by applying the associative and distributive properties to the operation

Write the terms of each of the algebraic expressions, just as if they were variables or numbers.

Don’t forget to use the rules of “order of operations”.

Simplify expressions

Use the multiplication distributive property for removing parentheses – then multiply expressions and combine like terms.

Order of Operations Review

  1. Start by looking for Grouping symbols such as absolute values and fraction bars
  2. Do all operations within a grouping symbols or nested symbols including  parentheses and bracket
  3. Evaluate any expressions with exponents
  4. Multiply and divide in order from left to right
  5. Add or subtract in order from left to right

Value of an Algebraic Expression

The value of an algebraic expression is equal to the value of its simplified form.

Addition of algebraic identities

Any algebraic expression is evaluated to be equal to the sum of itself and 0.

– 8.2 Solving Equations: The Addition Property

Equation

An equation is a statement of the equality in values of two expressions.

It consists of a left side expression, a right side expression and an equal sign “=” that expresses the equality of values of the expressions.

Equal sign

The equal sign “=” in an equation is used to assert that the values for the two expressions being compared are the same.

Left side of equation

The left side of an equation is an algebraic expression being compared to the right side of the equation.

Right side of equation

The right side of an equation is an algebraic expression being compared to the left side of that equation.

Solution of an equation

A solution to an equation is a value for the variable (or variables) that makes an equation a true statement.

Solving an equation

To solve an equation is to determine the solution of the equation.

Equivalent equations – Simplest equivalent equation

An equation equating expressions of the form:

“x = a number”

or

” ’a number’ = x “.

Both forms correctly express the simplest equivalent equation.

At the very end we conventionally rewrite the equation so that the variable “x“ is on the left side of the equation.

Addition property of equality

The same number may be added to both sides of an equation without changing the solution to an equation.

If a, b and c represent numbers, and, if   a = b   then a + c = b + c.

Simplify an equation by adding the same number to both sides

To simplify an equation seek to get the variable x alone to one side of the equation.

You can add the same number to both sides of the equation without changing the solution value using the Addition property of equality.

Look to add a number on the side of an equation where there is a subtracted number. By adding the same number to both sides you add the two terms to get   0   on this side – which can be then neglected in rewriting the equivalent equation.

Don’t forget to add the number on the other side of the equation.

– 8.3 Solving Equations: The Multiplication Property

Subtraction property of equality

The same number may be subtracted from both sides of an equation without changing the solution to an equation.

If a, b and c represent numbers, and, if a = b — then a – c = b – c.

Simplify an equation by subtracting the same number to both sides

To simplify an equation seek to get the variable x alone to one side of the equation.

You can subtract the same number to both sides of the equation without changing the solution value using the Subtraction property of equality.

Look to subtract a number on the side of an equation where there is an added number. By subtracting the same number to both sides you subtract the two terms to get 0 on this side – which can be then neglected in rewriting the equivalent equation.

Don’t forget to add the number on the other side of the equation.

– 8.4 Solving Equations Using the Arithmetic  Properties

Multiplication property of equality

Both sides of an equation may be multiplied by the same number without changing the solution of the equation.

If a, b, and c are numbers, and if, a = b then a • c = b • c.

Simplify an equation by multiplying both sides by the same number

To simplify an equation seek to get the variable x alone to one side of the equation.

You can multiply the same number on both sides of the equation without changing the solution value using the Multiplication property of equality.

Look to multiply the same number on the side of an equation where there is a division by number. By multiplying the same number on both sides you get 1 as the coefficient of the variable or 1 as a term.

The resultant rewritten equation can then be further simplified by using the Addition or Subtraction properties of equality.

Don’t forget to multiply the same number on the other side of the equation.

Division property of equality

Both sides of an equation may be divided by the same number without changing the solution of the equation.

If a, b, and c are numbers, and if, a = b then a / c = b / c.

Simplify an equation by dividing the same number into both sides

To simplify an equation seek to get the variable x alone to one side of the equation.

You can divide both sides of the equation by the same number without changing the solution value using the Division property of equality.

Look to divide by the same number on the side of an equation where there is a multiplication by a number. By dividing the same number on both sides you get 1 as the coefficient of the variable or 1 as a term.

The resulting rewritten equation can then be further simplified by using the Addition or Subtraction properties of equality.

Don’t forget to add to, subtract from, multiply or divide into the same expression(s) on the other side of the equation.

Check to verify the solution of equation

To check that a number is the solution to an equation – substitute the number in the variable of the original equation and verify that it is a true statement.

Verify the truth of the statement by checking that the values of the original equation’s left and right side expressions are the same.

Write some phrases as expressions

Twice a number, increased by -9

= 2x + (-9)

Three times the difference of a number and 11

= 3(x – 11)

The quotient of 5 times a number and 17

= (5x) / 17

Remember that 5x = 5 • x and that the backslash fraction bar   /  signifies division.

– 8.5 Equations and Problem Solving

Linear equations in one variable

A linear equation in one variable is an equation containing only one variable that is not below the fraction bar of any expression and whose exponent is 1 – whenever it appears.

Steps for solving a linear equation in one variable

  1. If parentheses are present use the distributive property.
  2. Combine like terms on each side of the equation.
  3. Use the addition and subtraction property of equality to rewrite the equation to an equivalent one – in which the variable is on one side of the equation and the constant terms are on the other side.
  4. Combine all terms in x.
  5. Divide both sides of the equation by the coefficient of x.
  6. You have left x = (some simplified number or expression).

Write sentences in word problems as equations

  • Translate “a number” to x.
  • Translate to equal sign words such as: equal, gives, is/was, yields, amounts to, is equal to.
  • Translate into addition words such as: sum, plus, added to, more than, increased by total.
  • Translation into subtraction words such as: difference, minus, subtracted from, less than, decreased by, less.
  • Translation to multiplication words such as: product, times, multiply, twice, of, double.
  • Translation to division words such as: quotient, divided by, into, per.

Steps to solve word problems

  1. Understand the problem by becoming comfortable with it through reading and re-reading it.
  2. Choose a variable to represent the unknown
  3. Construct a drawing if appropriate
  4. Propose a solution and check it
  5. Translate the problem into an equation.
  6. Solve the equation.
  7. Interpret the results.
  8. Check the proposed solution in the stated problem and state your conclusion.
9. Geometry
– 9.1 Lines and Angles
– 9.2 Plane Figures and Solids
– 9.3 Perimeter
– 9.4 Area
– 9.5 Volume and Surface Area
– 9.6 Congruent and Similar Triangles
Appendices
Appendix A: Tables
Appendix B: Exponents and Polynomials
Appendix C: Inductive and Deductive Reasoning

Fraction Skills – Worked Examples and Practice Links

Here is a list of all of the skills that cover fractions!  These skills are organized by grade.  Study the skills

First learn each fraction concept and skill by reviewing, relearning, and self-testing three times per session the following in order:

-1- The Standards and Learning Objectives – set context for determining what is to be “learned”;

-2-  The Glossary – precisely recognize, name and define the needed concepts, skills and rules, forwards and backwards;

-3-  The Worked Examples – become familiar with what is asked, what is given and recognize  similar kinds of questions;

-4-  The Progression – connect each standard, concept and skill with prior knowledge, the current lesson content and future lessons to be learned;

-5-  The Summary Rules – learn a summary of the shortcut rules that identify question types and how to skillfully answer them – to include why the rules are true ; and finally

-6-  The Application of the Rules – to learn when and where to apply them to skillfully answer them.

By working problems for each skill you can practice Application of the Rules for each single skill at IXL Fraction Skills Practice.

Once at the IXL site  and you can move your mouse over any skill name to view a sample question for familiarization and recognition.

Familiarize yourself with each question by determining what is asked and what is given. Ask and answer How are questions similar and different?

Then attempt to answer each question by recognizing it, determine the rules needed fr a solution, find the solution y applying the rules and then checking to see if this is all correct.

Then practice getting answers to the questions by fluently applying the rules. Just click on any skill link.  IXL will then time your session and track your score. The questions will automatically increase in difficulty as you improve!

FINALLY, master application of the Fraction Rule Set for all skills at this Link.

High School Algebra 1 – CA Common Core – Standards & Learning Objectives

9-12.N Number and Quantity

9-12.N-RN The Real Number System

9-12. Extend the properties of exponents to rational exponents.

9-12.N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

Evaluate integers raised to rational exponents (Algebra 1 – V.10)

Evaluate rational exponents (Algebra 2 – M.1)

Evaluate rational exponents (Precalculus – H.4)

 

9-12.N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Simplify radical expressions (Algebra 1 – EE.1)

Simplify radical expressions involving fractions (Algebra 1 – EE.2)

Multiply radical expressions (Algebra 1 – EE.3)

Add and subtract radical expressions (Algebra 1 – EE.4)

Simplify radical expressions using the distributive property (Algebra 1 – EE.5)

Simplify radical expressions: mixed review (Algebra 1 – EE.7)

Simplify radical expressions (Geometry – A.4)

Roots of integers (Algebra 2 – L.1)

Roots of rational numbers (Algebra 2 – L.2)

Nth roots (Algebra 2 – L.4)

Simplify radical expressions with variables I (Algebra 2 – L.5)

Simplify radical expressions with variables II (Algebra 2 – L.6)

Multiply radical expressions (Algebra 2 – L.7)

Divide radical expressions (Algebra 2 – L.8)

Add and subtract radical expressions (Algebra 2 – L.9)

Simplify radical expressions using the distributive property (Algebra 2 – L.10)

Simplify radical expressions using conjugates (Algebra 2 – L.11)

Multiplication with rational exponents (Algebra 2 – M.2)

Division with rational exponents (Algebra 2 – M.3)

Power rule (Algebra 2 – M.4)

Simplify expressions involving rational exponents I (Algebra 2 – M.5)

Simplify expressions involving rational exponents II (Algebra 2 – M.6)

Operations with rational exponents (Precalculus – H.5)

Nth roots (Precalculus – H.6)

Simplify radical expressions with variables (Precalculus – H.7)

Simplify expressions involving rational exponents (Precalculus – H.8)

 

9-12. Use properties of rational and irrational numbers.

9-12.N-RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Classify rational and irrational numbers (Precalculus – Q.1)

Sort rational and irrational numbers (Precalculus – Q.2)

Properties of operations on rational and irrational numbers (Precalculus – Q.3)

 

9-12.N-Q Quantities

 

9-12. Reason quantitatively and use units to solve problems.

 

9-12.N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Scale drawings and scale factors (Algebra 1 – C.7)

Convert rates and measurements: customary units (Algebra 1 – E.1)

Convert rates and measurements: metric units (Algebra 1 – E.2)

Unit prices with unit conversions (Algebra 1 – E.3)

Scale maps and drawings (Geometry – A.2)

Convert rates and measurements: customary units (Geometry – W.1)

Convert rates and measurements: metric units (Geometry – W.2)

Convert square and cubic units of length (Geometry – W.3)

 

9-12.N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.

Interpret bar graphs, line graphs, and histograms (Algebra 1 – N.1)

Create bar graphs, line graphs, and histograms (Algebra 1 – N.2)

Interpret stem-and-leaf plots (Algebra 1 – N.4)

Interpret box-and-whisker plots (Algebra 1 – N.5)

Interpret a scatter plot (Algebra 1 – N.6)

Scatter plots: line of best fit (Algebra 1 – N.7)

 

9-12.N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Precision (Algebra 1 – E.4)

Greatest possible error (Algebra 1 – E.5)

Precision (Geometry – W.4)

Greatest possible error (Geometry – W.5)

Minimum and maximum area and volume (Geometry – W.6)

Percent error (Geometry – W.7)

Percent error: area and volume (Geometry – W.8)

9-12.A. Algebra

9-12.A-SSE Seeing Structure in Expressions

9-12. Interpret the structure of expressions

9-12.A-SSE.1 Interpret expressions that represent a quantity in terms of its context.

 

9-12.A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.

Polynomial vocabulary (Algebra 1 – Z.1)

Polynomial vocabulary (Algebra 2 – K.1)

 

9-12.A-SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.

Factor using a quadratic pattern (Algebra 2 – I.4)

Factor using a quadratic pattern (Precalculus – D.14)

 

9-12.A-SSE.2 Use the structure of an expression to identify ways to rewrite it.

Simplify variable expressions using properties (Algebra 1 – H.3)

Simplify variable expressions involving like terms and the distributive property (Algebra 1 – I.2)

Simplify expressions involving exponents (Algebra 1 – V.8)

Powers of monomials (Algebra 1 – Y.5)

Factor out a monomial (Algebra 1 – AA.2)

Simplify variable expressions using properties (Algebra 2 – A.3)

Pascal’s triangle and the Binomial Theorem (Algebra 2 – K.17)

Binomial Theorem I (Algebra 2 – K.18)

Binomial Theorem II (Algebra 2 – K.19)

Simplify radical expressions with variables I (Algebra 2 – L.5)

Simplify radical expressions with variables II (Algebra 2 – L.6)

Simplify radical expressions using conjugates (Algebra 2 – L.11)

Simplify expressions involving rational exponents I (Algebra 2 – M.5)

Simplify expressions involving rational exponents II (Algebra 2 – M.6)

Simplify rational expressions (Algebra 2 – N.4)

Pascal’s triangle and the Binomial Theorem (Precalculus – D.17)

Binomial Theorem I (Precalculus – D.18)

Binomial Theorem II (Precalculus – D.19)

Simplify radical expressions with variables (Precalculus – H.7)

Simplify expressions involving rational exponents (Precalculus – H.8)

 

9-12. Write expressions in equivalent forms to solve problems

 

9-12.A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

 

9-12.A-SSE.3.a Factor a quadratic expression to reveal the zeros of the function it defines.

Factor quadratics with leading coefficient 1 (Algebra 1 – AA.3)

Factor quadratics with other leading coefficients (Algebra 1 – AA.4)

Factor quadratics: special cases (Algebra 1 – AA.5)

Solve a quadratic equation by factoring (Algebra 1 – BB.6)

Factor quadratics (Algebra 2 – I.2)

Solve a quadratic equation by factoring (Algebra 2 – J.8)

Solve a quadratic equation by factoring (Precalculus – C.6)

 

9-12.A-SSE.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Complete the square (Algebra 1 – BB.7)

Complete the square (Algebra 2 – J.9)

Convert equations of parabolas from general to vertex form (Algebra 2 – T.7)

Find properties of a parabola from equations in general form (Algebra 2 – T.8)

 

9-12.A-SSE.3.c Use the properties of exponents to transform expressions for exponential functions.

Negative exponents (Algebra 1 – V.3)

Multiplication with exponents (Algebra 1 – V.4)

Division with exponents (Algebra 1 – V.5)

Multiplication and division with exponents (Algebra 1 – V.6)

Power rule (Algebra 1 – V.7)

Simplify expressions involving exponents (Algebra 1 – V.8)

Evaluate an exponential function (Algebra 1 – X.1)

Match exponential functions and graphs (Algebra 1 – X.2)

Properties of exponents (Geometry – A.3)

Evaluate rational exponents (Algebra 2 – M.1)

Multiplication with rational exponents (Algebra 2 – M.2)

Division with rational exponents (Algebra 2 – M.3)

Power rule (Algebra 2 – M.4)

Simplify expressions involving rational exponents I (Algebra 2 – M.5)

Simplify expressions involving rational exponents II (Algebra 2 – M.6)

Evaluate exponential functions (Algebra 2 – S.2)

Match exponential functions and graphs (Algebra 2 – S.3)

Solve exponential equations using factoring (Algebra 2 – S.4)

Solve exponential equations using factoring (Precalculus – F.9)

 

9-12.A-APR Arithmetic with Polynomials and Rational Expressions

 

9-12. Perform arithmetic operations on polynomials

 

9-12.A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Model polynomials with algebra tiles (Algebra 1 – Z.2)

Add and subtract polynomials using algebra tiles (Algebra 1 – Z.3)

Add and subtract polynomials (Algebra 1 – Z.4)

Add polynomials to find perimeter (Algebra 1 – Z.5)

Multiply a polynomial by a monomial (Algebra 1 – Z.6)

Multiply two polynomials using algebra tiles (Algebra 1 – Z.7)

Multiply two binomials (Algebra 1 – Z.8)

Multiply two binomials: special cases (Algebra 1 – Z.9)

Multiply polynomials (Algebra 1 – Z.10)

Add and subtract polynomials (Algebra 2 – K.2)

Multiply polynomials (Algebra 2 – K.3)

 

9-12.A-CED Creating Equations

 

9-12. Create equations that describe numbers or relationships

 

9-12.A-CED.1 Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.

Write variable equations (Algebra 1 – I.4)

Model and solve equations using algebra tiles (Algebra 1 – J.1)

Write and solve equations that represent diagrams (Algebra 1 – J.2)

Solve linear equations: word problems (Algebra 1 – J.8)

Write inequalities from graphs (Algebra 1 – K.2)

Write compound inequalities from graphs (Algebra 1 – K.13)

Weighted averages: word problems (Algebra 1 – O.5)

Write variable expressions and equations (Geometry – A.5)

Solve linear equations (Geometry – A.6)

Solve linear inequalities (Geometry – A.7)

Solve linear equations (Algebra 2 – B.1)

Solve linear equations: word problems (Algebra 2 – B.2)

Write inequalities from graphs (Algebra 2 – C.3)

Solve linear inequalities (Algebra 2 – C.5)

Solve equations with sums and differences of cubes (Precalculus – D.13)

Solve equations using a quadratic pattern (Precalculus – D.15)

 

9-12.A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Graph a function (Algebra 1 – Q.9)

Write a function rule: word problems (Algebra 1 – Q.10)

Write a rule for a function table (Algebra 1 – Q.12)

Write direct variation equations (Algebra 1 – R.4)

Write inverse variation equations (Algebra 1 – R.7)

Write and solve inverse variation equations (Algebra 1 – R.8)

Find a missing coordinate using slope (Algebra 1 – S.4)

Slope-intercept form: graph an equation (Algebra 1 – S.6)

Slope-intercept form: write an equation from a graph (Algebra 1 – S.7)

Slope-intercept form: write an equation (Algebra 1 – S.8)

Linear function word problems (Algebra 1 – S.10)

Write equations in standard form (Algebra 1 – S.11)

Standard form: graph an equation (Algebra 1 – S.13)

Point-slope form: graph an equation (Algebra 1 – S.16)

Point-slope form: write an equation (Algebra 1 – S.18)

Write linear, quadratic, and exponential functions (Algebra 1 – CC.3)

Graph an absolute value function (Algebra 1 – DD.3)

Graph a linear equation (Geometry – E.3)

Equations of lines (Geometry – E.4)

Graph a linear inequality in the coordinate plane (Algebra 2 – C.2)

Graph a quadratic function (Algebra 2 – J.4)

Write and solve direct variation equations (Algebra 2 – Q.1)

Write and solve inverse variation equations (Algebra 2 – Q.2)

Write joint and combined variation equations I (Algebra 2 – Q.4)

Write joint and combined variation equations II (Algebra 2 – Q.6)

Solve variation equations (Algebra 2 – Q.7)

Graph parabolas (Algebra 2 – T.9)

Graph circles (Algebra 2 – U.7)

Graph sine functions (Algebra 2 – Z.4)

Graph cosine functions (Algebra 2 – Z.8)

Graph sine and cosine functions (Algebra 2 – Z.9)

Graph a quadratic function (Precalculus – C.3)

Graph sine functions (Precalculus – N.4)

Graph cosine functions (Precalculus – N.8)

Graph sine and cosine functions (Precalculus – N.9)

Graph parabolas (Precalculus – P.3)

Graph circles (Precalculus – P.6)

 

9-12.A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

Solve a system of equations by graphing: word problems (Algebra 1 – U.3)

Solve a system of equations using substitution: word problems (Algebra 1 – U.9)

Solve a system of equations using elimination: word problems (Algebra 1 – U.11)

Solve a system of equations using augmented matrices: word problems (Algebra 1 – U.13)

Solve a system of equations using any method: word problems (Algebra 1 – U.15)

Solve systems of linear equations (Geometry – A.8)

Solve a system of equations by graphing: word problems (Algebra 2 – E.3)

Solve a system of equations using substitution: word problems (Algebra 2 – E.7)

Solve a system of equations using elimination: word problems (Algebra 2 – E.9)

Solve a system of equations using any method: word problems (Algebra 2 – E.11)

Solve systems of linear inequalities by graphing (Algebra 2 – F.2)

Solve systems of linear and absolute value inequalities by graphing (Algebra 2 – F.3)

Find the vertices of a solution set (Algebra 2 – F.4)

Linear programming (Algebra 2 – F.5)

Solve a system of equations by graphing (Precalculus – I.1)

Solve a system of equations by graphing: word problems (Precalculus – I.2)

Solve a system of equations using substitution (Precalculus – I.4)

Solve a system of equations using substitution: word problems (Precalculus – I.5)

Solve a system of equations using elimination (Precalculus – I.6)

Solve a system of equations using elimination: word problems (Precalculus – I.7)

Solve systems of linear inequalities by graphing (Precalculus – J.1)

Solve systems of linear and absolute value inequalities by graphing (Precalculus – J.2)

Find the vertices of a solution set (Precalculus – J.3)

Linear programming (Precalculus – J.4)

 

9-12.A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Rate of travel: word problems (Algebra 1 – O.4)

Solve multi-variable equations (Algebra 2 – B.5)

 

9-12.A-REI Reasoning with Equations and Inequalities

 

9-12. Understand solving equations as a process of reasoning and explain the reasoning

 

9-12.A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Properties of equality (Algebra 1 – H.4)

Weighted averages: word problems (Algebra 1 – O.5)

 

9-12. Solve equations and inequalities in one variable

 

9-12.A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Model and solve equations using algebra tiles (Algebra 1 – J.1)

Write and solve equations that represent diagrams (Algebra 1 – J.2)

Solve one-step linear equations (Algebra 1 – J.3)

Solve two-step linear equations (Algebra 1 – J.4)

Solve advanced linear equations (Algebra 1 – J.5)

Solve equations with variables on both sides (Algebra 1 – J.6)

Identities and equations with no solutions (Algebra 1 – J.7)

Solve linear equations: word problems (Algebra 1 – J.8)

Solve linear equations: mixed review (Algebra 1 – J.9)

Identify solutions to inequalities (Algebra 1 – K.3)

Solve one-step linear inequalities: addition and subtraction (Algebra 1 – K.4)

Solve one-step linear inequalities: multiplication and division (Algebra 1 – K.5)

Solve one-step linear inequalities (Algebra 1 – K.6)

Graph solutions to one-step linear inequalities (Algebra 1 – K.7)

Solve two-step linear inequalities (Algebra 1 – K.8)

Graph solutions to two-step linear inequalities (Algebra 1 – K.9)

Solve advanced linear inequalities (Algebra 1 – K.10)

Graph solutions to advanced linear inequalities (Algebra 1 – K.11)

Graph compound inequalities (Algebra 1 – K.12)

Write compound inequalities from graphs (Algebra 1 – K.13)

Solve compound inequalities (Algebra 1 – K.14)

Graph solutions to compound inequalities (Algebra 1 – K.15)

Solve linear equations (Geometry – A.6)

Solve linear inequalities (Geometry – A.7)

Solve linear equations (Algebra 2 – B.1)

Solve linear equations: word problems (Algebra 2 – B.2)

Solve linear inequalities (Algebra 2 – C.5)

Graph solutions to linear inequalities (Algebra 2 – C.6)

 

9-12.A-REI.3.1 Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.

Solve absolute value equations (Algebra 1 – L.1)

Graph solutions to absolute value equations (Algebra 1 – L.2)

Solve absolute value inequalities (Algebra 1 – L.3)

Graph solutions to absolute value inequalities (Algebra 1 – L.4)

Solve absolute value equations (Algebra 2 – B.3)

Graph solutions to absolute value equations (Algebra 2 – B.4)

Solve absolute value inequalities (Algebra 2 – C.7)

Graph solutions to absolute value inequalities (Algebra 2 – C.8)

 

9-12.A-REI.4 Solve quadratic equations in one variable.

 

9-12.A-REI.4.a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.

Complete the square (Algebra 1 – BB.7)

Complete the square (Algebra 2 – J.9)

 

9-12.A-REI.4.b Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Solve a quadratic equation using square roots (Algebra 1 – BB.4)

Solve an equation using the zero product property (Algebra 1 – BB.5)

Solve a quadratic equation by factoring (Algebra 1 – BB.6)

Complete the square (Algebra 1 – BB.7)

Solve a quadratic equation by completing the square (Algebra 1 – BB.8)

Solve a quadratic equation using the quadratic formula (Algebra 1 – BB.9)

Using the discriminant (Algebra 1 – BB.10)

Solve quadratic equations (Geometry – A.9)

Solve a quadratic equation using square roots (Algebra 2 – J.6)

Solve a quadratic equation using the zero product property (Algebra 2 – J.7)

Solve a quadratic equation by factoring (Algebra 2 – J.8)

Solve a quadratic equation using the quadratic formula (Algebra 2 – J.11)

Using the discriminant (Algebra 2 – J.12)

Solve a quadratic equation using square roots (Precalculus – C.5)

Solve a quadratic equation by factoring (Precalculus – C.6)

Solve a quadratic equation by completing the square (Precalculus – C.7)

Solve a quadratic equation using the quadratic formula (Precalculus – C.8)

 

9-12. Solve systems of equations

 

9-12.A-REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Solve a system of equations using elimination (Algebra 1 – U.10)

Solve a system of equations using elimination: word problems (Algebra 1 – U.11)

Solve a system of equations using augmented matrices (Algebra 1 – U.12)

Solve a system of equations using augmented matrices: word problems (Algebra 1 – U.13)

Solve a system of equations using elimination (Algebra 2 – E.8)

Solve a system of equations using elimination: word problems (Algebra 2 – E.9)

Solve a system of equations using elimination (Precalculus – I.6)

Solve a system of equations using elimination: word problems (Precalculus – I.7)

 

9-12.A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Is (x, y) a solution to the system of equations? (Algebra 1 – U.1)

Solve a system of equations by graphing (Algebra 1 – U.2)

Solve a system of equations by graphing: word problems (Algebra 1 – U.3)

Find the number of solutions to a system of equations by graphing (Algebra 1 – U.4)

Find the number of solutions to a system of equations (Algebra 1 – U.5)

Classify a system of equations by graphing (Algebra 1 – U.6)

Classify a system of equations (Algebra 1 – U.7)

Solve a system of equations using substitution (Algebra 1 – U.8)

Solve a system of equations using substitution: word problems (Algebra 1 – U.9)

Solve a system of equations using elimination (Algebra 1 – U.10)

Solve a system of equations using elimination: word problems (Algebra 1 – U.11)

Solve a system of equations using augmented matrices (Algebra 1 – U.12)

Solve a system of equations using augmented matrices: word problems (Algebra 1 – U.13)

Solve a system of equations using any method (Algebra 1 – U.14)

Solve a system of equations using any method: word problems (Algebra 1 – U.15)

Solve systems of linear equations (Geometry – A.8)

Is (x, y) a solution to the system of equations? (Algebra 2 – E.1)

Solve a system of equations by graphing (Algebra 2 – E.2)

Solve a system of equations by graphing: word problems (Algebra 2 – E.3)

Find the number of solutions to a system of equations (Algebra 2 – E.4)

Classify a system of equations (Algebra 2 – E.5)

Solve a system of equations using substitution (Algebra 2 – E.6)

Solve a system of equations using substitution: word problems (Algebra 2 – E.7)

Solve a system of equations using elimination (Algebra 2 – E.8)

Solve a system of equations using elimination: word problems (Algebra 2 – E.9)

Solve a system of equations using any method (Algebra 2 – E.10)

Solve a system of equations using any method: word problems (Algebra 2 – E.11)

Solve a system of equations by graphing (Precalculus – I.1)

Solve a system of equations by graphing: word problems (Precalculus – I.2)

Classify a system of equations (Precalculus – I.3)

Solve a system of equations using substitution (Precalculus – I.4)

Solve a system of equations using substitution: word problems (Precalculus – I.5)

Solve a system of equations using elimination (Precalculus – I.6)

Solve a system of equations using elimination: word problems (Precalculus – I.7)

 

9-12.A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Solve a non-linear system of equations (Algebra 2 – E.15)

 

9-12. Represent and solve equations and inequalities graphically

 

9-12.A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Relations: convert between tables, graphs, mappings, and lists of points (Algebra 1 – Q.1)

Complete a function table (Algebra 1 – Q.6)

Graph a function (Algebra 1 – Q.9)

Find points on a function graph (Algebra 1 – Q.11)

 

9-12.A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Solve a system of equations by graphing (Algebra 1 – U.2)

Solve a system of equations by graphing: word problems (Algebra 1 – U.3)

Find the number of solutions to a system of equations by graphing (Algebra 1 – U.4)

Solve a system of equations by graphing (Precalculus – I.1)

Solve a system of equations by graphing: word problems (Precalculus – I.2)

 

9-12.A-REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Graph a linear inequality in the coordinate plane (Algebra 1 – T.3)

Graph a linear inequality in the coordinate plane (Algebra 2 – C.2)

Solve systems of linear inequalities by graphing (Algebra 2 – F.2)

Solve systems of linear inequalities by graphing (Precalculus – J.1)

9-12.F Functions

 

9-12.F-IF Interpreting Functions

 

9-12. Understand the concept of a function and use function notation

 

9-12.F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Domain and range of relations (Algebra 1 – Q.2)

Identify independent and dependent variables (Algebra 1 – Q.3)

Identify functions (Algebra 1 – Q.4)

Identify functions: vertical line test (Algebra 1 – Q.5)

Domain and range of absolute value functions (Algebra 1 – DD.2)

Domain and range of radical functions (Algebra 1 – FF.2)

Domain and range (Algebra 2 – D.1)

Identify functions (Algebra 2 – D.2)

Domain and range (Precalculus – A.1)

Identify functions (Precalculus – A.2)

Domain and range of exponential and logarithmic functions (Precalculus – F.1)

Domain and range of radical functions (Precalculus – G.1)

 

9-12.F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Complete a function table (Algebra 1 – Q.6)

Evaluate function rules I (Algebra 1 – Q.7)

Evaluate function rules II (Algebra 1 – Q.8)

Evaluate an exponential function (Algebra 1 – X.1)

Complete a function table: quadratic functions (Algebra 1 – BB.2)

Complete a function table: absolute value functions (Algebra 1 – DD.1)

Evaluate a radical function (Algebra 1 – FF.1)

Evaluate functions (Algebra 2 – D.3)

Evaluate logarithms (Algebra 2 – R.4)

Evaluate natural logarithms (Algebra 2 – R.5)

Evaluate logarithms: mixed review (Algebra 2 – R.12)

Evaluate exponential functions (Algebra 2 – S.2)

Evaluate functions (Precalculus – A.5)

 

9-12.F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Identify arithmetic and geometric sequences (Algebra 1 – P.1)

Arithmetic sequences (Algebra 1 – P.2)

Geometric sequences (Algebra 1 – P.3)

Evaluate variable expressions for number sequences (Algebra 1 – P.4)

Write variable expressions for arithmetic sequences (Algebra 1 – P.5)

Write variable expressions for geometric sequences (Algebra 1 – P.6)

Number sequences: mixed review (Algebra 1 – P.7)

Classify formulas and sequences (Algebra 2 – BB.1)

Find terms of an arithmetic sequence (Algebra 2 – BB.2)

Find terms of a geometric sequence (Algebra 2 – BB.3)

Find terms of a recursive sequence (Algebra 2 – BB.4)

Evaluate formulas for sequences (Algebra 2 – BB.5)

Write a formula for an arithmetic sequence (Algebra 2 – BB.6)

Write a formula for a geometric sequence (Algebra 2 – BB.7)

Write a formula for a recursive sequence (Algebra 2 – BB.8)

Sequences: mixed review (Algebra 2 – BB.9)

Find terms of a sequence (Precalculus – W.1)

Find terms of a recursive sequence (Precalculus – W.2)

Identify a sequence as explicit or recursive (Precalculus – W.3)

Find a recursive formula (Precalculus – W.4)

 

9-12. Interpret functions that arise in applications in terms of the context

 

9-12.F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Identify proportional relationships (Algebra 1 – R.1)

Find the constant of variation (Algebra 1 – R.2)

Graph a proportional relationship (Algebra 1 – R.3)

Identify direct variation and inverse variation (Algebra 1 – R.6)

Slope-intercept form: find the slope and y-intercept (Algebra 1 – S.5)

Standard form: find x- and y-intercepts (Algebra 1 – S.12)

Slopes of parallel and perpendicular lines (Algebra 1 – S.19)

Characteristics of quadratic functions (Algebra 1 – BB.1)

Identify linear, quadratic, and exponential functions from graphs (Algebra 1 – CC.1)

Identify linear, quadratic, and exponential functions from tables (Algebra 1 – CC.2)

Graph an absolute value function (Algebra 1 – DD.3)

Rational functions: asymptotes and excluded values (Algebra 1 – GG.1)

Slopes of lines (Geometry – E.2)

Characteristics of quadratic functions (Algebra 2 – J.1)

Graph a quadratic function (Algebra 2 – J.4)

Match quadratic functions and graphs (Algebra 2 – J.5)

Match polynomials and graphs (Algebra 2 – K.14)

Rational functions: asymptotes and excluded values (Algebra 2 – N.1)

Classify variation (Algebra 2 – Q.3)

Find the constant of variation (Algebra 2 – Q.5)

Match exponential functions and graphs (Algebra 2 – S.3)

Linear functions (Precalculus – A.3)

Characteristics of quadratic functions (Precalculus – C.1)

Find the maximum or minimum value of a quadratic function (Precalculus – C.2)

Match quadratic functions and graphs (Precalculus – C.4)

Match polynomials and graphs (Precalculus – D.11)

Rational functions: asymptotes and excluded values (Precalculus – E.1)

 

9-12.F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Domain and range of absolute value functions (Algebra 1 – DD.2)

Domain and range of radical functions (Algebra 1 – FF.2)

Domain and range (Algebra 2 – D.1)

Domain and range of radical functions (Algebra 2 – L.12)

Domain and range of exponential and logarithmic functions (Algebra 2 – S.1)

Domain and range (Precalculus – A.1)

Domain and range of exponential and logarithmic functions (Precalculus – F.1)

Domain and range of radical functions (Precalculus – G.1)

 

9-12.F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Find the constant of variation (Algebra 1 – R.2)

Find the slope of a graph (Algebra 1 – S.2)

Find the slope from two points (Algebra 1 – S.3)

Slope-intercept form: find the slope and y-intercept (Algebra 1 – S.5)

Find the slope of a linear function (Algebra 2 – D.4)

Linear functions (Precalculus – A.3)

 

9-12. Analyze functions using different representations

 

9-12.F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

 

9-12.F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.

Slope-intercept form: graph an equation (Algebra 1 – S.6)

Standard form: graph an equation (Algebra 1 – S.13)

Point-slope form: graph an equation (Algebra 1 – S.16)

Characteristics of quadratic functions (Algebra 1 – BB.1)

Graph a linear equation (Geometry – E.3)

Graph a linear function (Algebra 2 – D.5)

Graph a quadratic function (Algebra 2 – J.4)

Match quadratic functions and graphs (Algebra 2 – J.5)

Characteristics of quadratic functions (Precalculus – C.1)

Find the maximum or minimum value of a quadratic function (Precalculus – C.2)

Graph a quadratic function (Precalculus – C.3)

Match quadratic functions and graphs (Precalculus – C.4)

 

9-12.F-IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph an absolute value function (Algebra 1 – DD.3)

 

9-12.F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Match exponential functions and graphs (Algebra 1 – X.2)

Find properties of sine functions (Algebra 2 – Z.1)

Graph sine functions (Algebra 2 – Z.4)

Find properties of cosine functions (Algebra 2 – Z.5)

Graph cosine functions (Algebra 2 – Z.8)

Graph sine and cosine functions (Algebra 2 – Z.9)

Find properties of sine functions (Precalculus – N.1)

Graph sine functions (Precalculus – N.4)

Find properties of cosine functions (Precalculus – N.5)

Graph cosine functions (Precalculus – N.8)

Graph sine and cosine functions (Precalculus – N.9)

 

9-12.F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

 

9-12.F-IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Characteristics of quadratic functions (Algebra 1 – BB.1)

Solve a quadratic equation by factoring (Algebra 1 – BB.6)

Complete the square (Algebra 1 – BB.7)

Solve a quadratic equation by completing the square (Algebra 1 – BB.8)

Characteristics of quadratic functions (Algebra 2 – J.1)

Solve a quadratic equation by factoring (Algebra 2 – J.8)

Complete the square (Algebra 2 – J.9)

Convert equations of parabolas from general to vertex form (Algebra 2 – T.7)

Find properties of a parabola from equations in general form (Algebra 2 – T.8)

Characteristics of quadratic functions (Precalculus – C.1)

Find the maximum or minimum value of a quadratic function (Precalculus – C.2)

Solve a quadratic equation by factoring (Precalculus – C.6)

Solve a quadratic equation by completing the square (Precalculus – C.7)

 

9-12.F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.

Match exponential functions and graphs (Algebra 1 – X.2)

Match exponential functions and graphs (Algebra 2 – S.3)

 

9-12.F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Match quadratic functions and graphs (Algebra 2 – J.5)

Match polynomials and graphs (Algebra 2 – K.14)

Match quadratic functions and graphs (Precalculus – C.4)

Match polynomials and graphs (Precalculus – D.11)

 

9-12.F-BF Building Functions

 

9-12. Build a function that models a relationship between two quantities

 

9-12.F-BF.1 Write a function that describes a relationship between two quantities.

 

9-12.F-BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.

Write variable expressions for arithmetic sequences (Algebra 1 – P.5)

Write variable expressions for geometric sequences (Algebra 1 – P.6)

Write inverse variation equations (Algebra 1 – R.7)

Write and solve inverse variation equations (Algebra 1 – R.8)

Write linear, quadratic, and exponential functions (Algebra 1 – CC.3)

Write a formula for an arithmetic sequence (Algebra 2 – BB.6)

Write a formula for a geometric sequence (Algebra 2 – BB.7)

Write a formula for a recursive sequence (Algebra 2 – BB.8)

Find a recursive formula (Precalculus – W.4)

 

9-12.F-BF.1.b Combine standard function types using arithmetic operations.

Add and subtract functions (Algebra 2 – O.1)

Multiply functions (Algebra 2 – O.2)

Divide functions (Algebra 2 – O.3)

Add, subtract, multiply, and divide functions (Precalculus – A.6)

 

9-12.F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Write variable expressions for arithmetic sequences (Algebra 1 – P.5)

Write variable expressions for geometric sequences (Algebra 1 – P.6)

Write a formula for an arithmetic sequence (Algebra 2 – BB.6)

Write a formula for a geometric sequence (Algebra 2 – BB.7)

Write a formula for a recursive sequence (Algebra 2 – BB.8)

Find a recursive formula (Precalculus – W.4)

Find recursive and explicit formulas (Precalculus – W.5)

Convert a recursive formula to an explicit formula (Precalculus – W.6)

Convert an explicit formula to a recursive formula (Precalculus – W.7)

Convert between explicit and recursive formulas (Precalculus – W.8)

 

9-12. Build new functions from existing functions

 

9-12.F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Transformations of quadratic functions (Algebra 1 – BB.3)

Transformations of absolute value functions (Algebra 1 – DD.4)

Translations of functions (Algebra 2 – P.1)

Reflections of functions (Algebra 2 – P.2)

Dilations of functions (Algebra 2 – P.3)

Transformations of functions (Algebra 2 – P.4)

Function transformation rules (Algebra 2 – P.5)

Describe function transformations (Algebra 2 – P.6)

Translations of functions (Precalculus – B.1)

Reflections of functions (Precalculus – B.2)

Dilations of functions (Precalculus – B.3)

Transformations of functions (Precalculus – B.4)

Function transformation rules (Precalculus – B.5)

Describe function transformations (Precalculus – B.6)

 

9-12.F-BF.4 Find inverse functions.

 

9-12.F-BF.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

Find inverse functions and relations (Algebra 2 – O.9)

Solve exponential equations using common logarithms (Algebra 2 – S.5)

Solve exponential equations using natural logarithms (Algebra 2 – S.6)

Solve logarithmic equations I (Algebra 2 – S.7)

Solve logarithmic equations II (Algebra 2 – S.8)

Solve logarithmic equations with one logarithm (Precalculus – F.11)

 

9-12.F-LE Linear, Quadratic, and Exponential Models

 

9-12. Construct and compare linear, quadratic, and exponential models and solve problems

 

9-12.F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

 

9-12.F-LE.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

Describe linear and exponential growth and decay (Algebra 1 – CC.6)

Identify linear and exponential functions (Algebra 2 – S.9)

Describe linear and exponential growth and decay (Algebra 2 – S.11)

Identify linear and exponential functions (Precalculus – F.13)

Describe linear and exponential growth and decay (Precalculus – F.15)

 

9-12.F-LE.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Solve linear equations: word problems (Algebra 1 – J.8)

Linear functions over unit intervals (Algebra 1 – CC.4)

Solve linear equations: word problems (Algebra 2 – B.2)

Linear functions over unit intervals (Algebra 2 – D.7)

Linear functions over unit intervals (Precalculus – A.4)

 

9-12.F-LE.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Exponential growth and decay: word problems (Algebra 1 – X.3)

Identify linear, quadratic, and exponential functions from graphs (Algebra 1 – CC.1)

Identify linear, quadratic, and exponential functions from tables (Algebra 1 – CC.2)

Exponential functions over unit intervals (Algebra 1 – CC.5)

Exponential functions over unit intervals (Algebra 2 – S.10)

Exponential growth and decay: word problems (Algebra 2 – S.12)

Exponential functions over unit intervals (Precalculus – F.14)

Exponential growth and decay: word problems (Precalculus – F.16)

 

9-12.F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Write variable expressions for arithmetic sequences (Algebra 1 – P.5)

Write variable expressions for geometric sequences (Algebra 1 – P.6)

Write a rule for a function table (Algebra 1 – Q.12)

Slope-intercept form: write an equation (Algebra 1 – S.8)

Point-slope form: write an equation from a graph (Algebra 1 – S.17)

Point-slope form: write an equation (Algebra 1 – S.18)

Match exponential functions and graphs (Algebra 1 – X.2)

Write linear, quadratic, and exponential functions (Algebra 1 – CC.3)

Equations of lines (Geometry – E.4)

Equations of parallel and perpendicular lines (Geometry – E.6)

Write the equation of a linear function (Algebra 2 – D.6)

Write a formula for an arithmetic sequence (Algebra 2 – BB.6)

Write a formula for a geometric sequence (Algebra 2 – BB.7)

Linear functions (Precalculus – A.3)

 

9-12.F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

 

9-12. Interpret expressions for functions in terms of the situation they model

 

9-12.F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.

Solve linear equations: word problems (Algebra 1 – J.8)

Exponential growth and decay: word problems (Algebra 1 – X.3)

Solve linear equations: word problems (Algebra 2 – B.2)

Exponential growth and decay: word problems (Algebra 2 – S.12)

Compound interest: word problems (Algebra 2 – S.13)

Continuously compounded interest: word problems (Algebra 2 – S.14)

Exponential growth and decay: word problems (Precalculus – F.16)

Compound interest: word problems (Precalculus – F.17)

 

9-12.F-LE.6 Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.

9-12.S Statistics and Probability

 

9-12.S-ID Interpreting Categorical and Quantitative Data

 

9-12. Summarize, represent, and interpret data on a single count or measurement variable

 

9-12.S-ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

Create bar graphs, line graphs, and histograms (Algebra 1 – N.2)

 

9-12.S-ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Mean, median, mode, and range (Algebra 1 – KK.1)

Quartiles (Algebra 1 – KK.2)

Mean absolute deviation (Algebra 1 – KK.7)

Variance and standard deviation (Algebra 1 – KK.8)

Variance and standard deviation (Algebra 2 – DD.2)

Variance and standard deviation (Precalculus – Z.2)

 

9-12.S-ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Interpret box-and-whisker plots (Algebra 1 – N.5)

Identify an outlier (Algebra 1 – KK.3)

Identify an outlier and describe the effect of removing it (Algebra 1 – KK.4)

Identify an outlier (Algebra 2 – DD.3)

Identify an outlier and describe the effect of removing it (Algebra 2 – DD.4)

Identify an outlier (Precalculus – Z.3)

Identify an outlier and describe the effect of removing it (Precalculus – Z.4)

 

9-12. Summarize, represent, and interpret data on two categorical and quantitative variables

 

9-12.S-ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

 

9-12.S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

Outliers in scatter plots (Algebra 1 – KK.5)

Outliers in scatter plots (Algebra 2 – DD.5)

Outliers in scatter plots (Precalculus – Z.5)

 

9-12.S-ID.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Find the equation of a regression line (Precalculus – Z.8)

Interpret regression lines (Precalculus – Z.9)

Analyze a regression line of a data set (Precalculus – Z.10)

Analyze a regression line using statistics of a data set (Precalculus – Z.11)

 

9-12.S-ID.6.b Informally assess the fit of a function by plotting and analyzing residuals.

Interpret a scatter plot (Algebra 1 – N.6)

 

9-12.S-ID.6.c Fit a linear function for a scatter plot that suggests a linear association.

Scatter plots: line of best fit (Algebra 1 – N.7)

Find the equation of a regression line (Precalculus – Z.8)

 

9-12. Interpret linear models

 

9-12.S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Interpret regression lines (Precalculus – Z.9)

Analyze a regression line using statistics of a data set (Precalculus – Z.11)

 

9-12.S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

Match correlation coefficients to scatter plots (Precalculus – Z.6)

Calculate correlation coefficients (Precalculus – Z.7)

 

9-12.S-ID.9 Distinguish between correlation and causation.

Pre-K – CA Common Core – Standards & Learning Objectives

PK.NO Number Sense

 

PK.1.0 Children expand their understanding of numbers and quantities in their everyday environment.

 

PK.1.1 Recite numbers in order to twenty with increasing accuracy.

Count dots (up to 20) (Pre-K – E.1)

Count shapes (up to 20) (Pre-K – E.2)

Count objects (up to 20) (Pre-K – E.3)

 

PK.1.2 Recognize and know the name of some written numerals.

Represent numbers (up to 10) (Pre-K – D.7)

Represent numbers (up to 20) (Pre-K – E.6)

 

PK.1.3 Identify, without counting, the number of objects in a collection of up to four objects (i.e., subitize).

Count dots (up to 3) (Pre-K – B.1)

Count shapes (up to 3) (Pre-K – B.2)

Count objects (up to 3) (Pre-K – B.3)

Represent numbers (up to 3) (Pre-K – B.7)

Count dots (up to 5) (Pre-K – C.1)

Count shapes (up to 5) (Pre-K – C.2)

Count objects (up to 5) (Pre-K – C.3)

Represent numbers (up to 5) (Pre-K – C.7)

 

PK.1.4 Count up to ten objects, using one-to-one correspondence (one object for each number word) with increasing accuracy.

Count dots (up to 10) (Pre-K – D.1)

Count shapes (up to 10) (Pre-K – D.2)

Count objects (up to 10) (Pre-K – D.3)

 

PK.1.5 Understand, when counting, that the number name of the last object counted represents the total number of objects in the group (i.e., cardinality).

Represent numbers (up to 10) (Pre-K – D.7)

Represent numbers (up to 20) (Pre-K – E.6)

 

PK.2.0 Children expand their understanding of number relationships and operations in their everyday environment.

 

PK.2.1 Compare, by counting or matching, two groups of up to five objects and communicate, “more,” “same as,” or “fewer” (or “less”).

More (Pre-K – F.2)

Compare in a chart (fewer or more) (Pre-K – F.4)

Compare in a mixed group (Pre-K – F.5)

 

PK.2.2 Understand that adding one or taking away one changes the number in a small group of objects by exactly one.

Count up and down – with pictures (Kindergarten – C.10)

 

PK.2.3 Understand that putting two groups of objects together will make a bigger group and that a group of objects can be taken apart into smaller groups.

Addition with pictures – sums up to 5 (Kindergarten – I.1)

 

PK.2.4 Solve simple addition and subtraction problems with a small number of objects (sums up to 10), usually by counting.

Addition with pictures – sums up to 10 (Kindergarten – I.6)

Subtract with pictures – numbers up to 10 (Kindergarten – J.5)

PK.AF Algebra and Functions (Classification and Patterning)

 

PK.1.0 Children expand their understanding of sorting and classifying objects in their everyday environment.

 

PK.1.1 Sort and classify objects by one or more attributes, into two or more groups, with increasing accuracy (e.g., may sort first by one attribute and then by another attribute).

Same (Pre-K – H.1)

Different (Pre-K – H.2)

Same and different (Pre-K – H.3)

Classify by color (Pre-K – H.4)

 

PK.2.0 Children expand their understanding of simple, repeating patterns.

 

PK.2.1 Recognize and duplicate simple repeating patterns.

Similar patterns (Kindergarten – H.1)

Complete missing parts of patterns (Kindergarten – H.2)

 

PK.2.2 Begin to extend and create simple repeating patterns.

Similar patterns (Kindergarten – H.1)

Complete missing parts of patterns (Kindergarten – H.2)

PK.MEA Measurement

 

PK.1.0 Children expand their understanding of comparing, ordering, and measuring objects.

 

PK.1.1 Compare two objects by length, weight, or capacity directly (e.g., putting objects side by side) or indirectly (e.g., using a third object).

Long and short (Pre-K – I.1)

Tall and short (Pre-K – I.2)

Light and heavy (Pre-K – I.3)

Holds more or less (Pre-K – I.4)

Compare height, weight, and capacity (Pre-K – I.5)

Wide and narrow (Pre-K – I.6)

 

PK.1.2 Order four or more objects by size.

Long and short (Pre-K – I.1)

Tall and short (Pre-K – I.2)

Light and heavy (Pre-K – I.3)

Holds more or less (Pre-K – I.4)

Compare height, weight, and capacity (Pre-K – I.5)

Wide and narrow (Pre-K – I.6)

 

PK.1.3 Measure length using multiple duplicates of the same-size concrete units laid end to end.

PK.G Geometry

 

PK.1.0 Children identify and use a variety of shapes in their everyday environment.

 

PK.1.1 Identify, describe, and construct a variety of different shapes, including variations of a circle, triangle, rectangle, square, and other shapes.

Identify circles, squares, and triangles (Pre-K – A.1)

Identify squares and rectangles (Pre-K – A.2)

Identify cubes and pyramids (Pre-K – A.3)

 

PK.1.2 Combine different shapes to create a picture or design.

 

PK.2.0 Children expand their understanding of positions in space.

 

PK.2.1 Identify positions of objects and people in space, including in/on/ under, up/down, inside/outside, beside/between, and in front/behind.

Inside and outside (Pre-K – G.1)

Above and below (Pre-K – G.2)

Left and right (Pre-K – G.3)

Left, middle, and right (Pre-K – G.4)

Top and bottom (Pre-K – G.5)

PK.MR Mathematical Reasoning

 

PK.1.0 Children expand the use of mathematical thinking to solve problems that arise in their everyday environment.

 

PK.1.1 Identify and apply a variety of mathematical strategies to solve problems in their environment.

Compare in a mixed group (Pre-K – F.5)

Same and different (Pre-K – H.3)

Kindergarten – CA Common Core – Standards & Learning Objectives

K.K.CC Counting and Cardinality

 

K. Know number names and the count sequence.

 

K.K.CC.1 Count to 100 by ones and by tens.

Count to 3 (Kindergarten – A.1)

Count using stickers – up to 3 (Kindergarten – A.2)

Count to 5 (Kindergarten – B.1)

Count using stickers – up to 5 (Kindergarten – B.2)

Count to 10 (Kindergarten – C.1)

Count using stickers – up to 10 (Kindergarten – C.4)

Count to 20 (Kindergarten – D.1)

Show numbers on ten frames – up to 20 (Kindergarten – D.4)

Count tens and ones – up to 20 (Kindergarten – D.16)

Count to 30 (Kindergarten – E.1)

Count to 100 (Kindergarten – E.2)

Counting on the hundred chart (Kindergarten – E.3)

Count groups of ten (Kindergarten – E.4)

Skip-count by tens (Kindergarten – F.4)

 

K.K.CC.2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

Count up – up to 5 (Kindergarten – B.6)

Count up – with pictures (Kindergarten – C.8)

Count up – with numbers (Kindergarten – C.9)

Count forward – up to 10 (Kindergarten – C.16)

Count up – up to 20 (Kindergarten – D.6)

Count forward – up to 20 (Kindergarten – D.11)

 

K.K.CC.3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

Count dots – 0 to 10 (Kindergarten – C.2)

Count dots – 0 to 20 (Kindergarten – D.2)

 

K. Count to tell the number of objects.

 

K.K.CC.4 Understand the relationship between numbers and quantities; connect counting to cardinality.

 

K.K.CC.4.a When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.

Count to 3 (Kindergarten – A.1)

Count to 5 (Kindergarten – B.1)

Count to 10 (Kindergarten – C.1)

Names of numbers – up to 10 (Kindergarten – C.18)

Count to 20 (Kindergarten – D.1)

Show numbers on ten frames – up to 20 (Kindergarten – D.4)

Names of numbers – up to 20 (Kindergarten – D.13)

 

K.K.CC.4.b Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

Count to 3 (Kindergarten – A.1)

Count to 5 (Kindergarten – B.1)

Count to 10 (Kindergarten – C.1)

Count to 20 (Kindergarten – D.1)

 

K.K.CC.4.c Understand that each successive number name refers to a quantity that is one larger.

Count up – up to 5 (Kindergarten – B.6)

Count up and down – with pictures (Kindergarten – C.10)

 

K.K.CC.5 Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.

Count to 3 (Kindergarten – A.1)

Count on ten frames – up to 3 (Kindergarten – A.3)

Show numbers on ten frames – up to 3 (Kindergarten – A.4)

Represent numbers – up to 3 (Kindergarten – A.5)

Count to 5 (Kindergarten – B.1)

Count on ten frames – up to 5 (Kindergarten – B.3)

Show numbers on ten frames – up to 5 (Kindergarten – B.4)

Represent numbers – up to 5 (Kindergarten – B.5)

Count to 10 (Kindergarten – C.1)

Count blocks – up to 10 (Kindergarten – C.3)

Count on ten frames – up to 10 (Kindergarten – C.5)

Show numbers on ten frames – up to 10 (Kindergarten – C.6)

Represent numbers – up to 10 (Kindergarten – C.7)

Count to 20 (Kindergarten – D.1)

Count on ten frames – up to 20 (Kindergarten – D.3)

Show numbers on ten frames – up to 20 (Kindergarten – D.4)

Represent numbers – up to 20 (Kindergarten – D.5)

Count blocks – up to 20 (Kindergarten – D.15)

 

K. Compare numbers.

 

K.K.CC.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.

Are there enough? (Kindergarten – G.1)

Fewer and more – compare by matching (Kindergarten – G.2)

Fewer and more – with charts (Kindergarten – G.3)

Fewer and more – mixed (Kindergarten – G.4)

Fewer, more, and same (Kindergarten – G.5)

 

K.K.CC.7 Compare two numbers between 1 and 10 presented as written numerals.

Compare two numbers – up to 10 (Kindergarten – G.6)

K.K.OA Operations and Algebraic Thinking

 

K. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

 

K.K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

Addition with pictures – sums up to 5 (Kindergarten – I.1)

Add two numbers – sums up to 5 (Kindergarten – I.2)

Addition sentences – sums up to 5 (Kindergarten – I.3)

Addition with pictures – sums up to 10 (Kindergarten – I.6)

Add two numbers – sums up to 10 (Kindergarten – I.7)

Addition sentences – sums up to 10 (Kindergarten – I.8)

Subtract with pictures – numbers up to 5 (Kindergarten – J.1)

Subtraction – numbers up to 5 (Kindergarten – J.2)

Subtraction sentences – numbers up to 5 (Kindergarten – J.3)

Subtract with pictures – numbers up to 10 (Kindergarten – J.5)

Subtraction – numbers up to 9 (Kindergarten – J.6)

Subtraction sentences – numbers up to 10 (Kindergarten – J.7)

 

K.K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Addition with pictures – sums up to 5 (Kindergarten – I.1)

Addition word problems – sums up to 5 (Kindergarten – I.5)

Addition with pictures – sums up to 10 (Kindergarten – I.6)

Addition word problems – sums up to 10 (Kindergarten – I.10)

Subtract with pictures – numbers up to 5 (Kindergarten – J.1)

Subtraction sentences – numbers up to 5 (Kindergarten – J.3)

Subtraction word problems – numbers up to 5 (Kindergarten – J.4)

Subtract with pictures – numbers up to 10 (Kindergarten – J.5)

Subtraction sentences – numbers up to 10 (Kindergarten – J.7)

Subtraction word problems – numbers up to 9 (Kindergarten – J.8)

 

K.K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Addition sentences – sums up to 5 (Kindergarten – I.3)

Ways to make a number – sums up to 5 (Kindergarten – I.4)

Addition sentences – sums up to 10 (Kindergarten – I.8)

Ways to make a number – sums up to 10 (Kindergarten – I.9)

 

K.K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

Count to fill a ten frame (Kindergarten – C.12)

Addition sentences – sums equal to 10 (Kindergarten – I.11)

 

K.K.OA.5 Fluently add and subtract within 5.

Addition with pictures – sums up to 5 (Kindergarten – I.1)

Add two numbers – sums up to 5 (Kindergarten – I.2)

Addition sentences – sums up to 5 (Kindergarten – I.3)

Subtract with pictures – numbers up to 5 (Kindergarten – J.1)

Subtraction – numbers up to 5 (Kindergarten – J.2)

Subtraction sentences – numbers up to 5 (Kindergarten – J.3)

K.K.NBT Number and Operations in Base Ten

 

K. Work with numbers 11–19 to gain foundations for place value.

 

K.K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

Count tens and ones – up to 20 (Kindergarten – D.16)

Write tens and ones – up to 20 (Kindergarten – D.17)

K.K.MD Measurement and Data

 

K. Describe and compare measurable attributes.

 

K.K.MD.1 Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.

Long and short (Kindergarten – Q.1)

Tall and short (Kindergarten – Q.2)

Light and heavy (Kindergarten – Q.3)

Holds more or less (Kindergarten – Q.4)

Compare size, weight, and capacity (Kindergarten – Q.5)

 

K.K.MD.2 Directly compare two objects with a measurable attribute in common, to see which object has “more of”/”less of” the attribute, and describe the difference.

Long and short (Kindergarten – Q.1)

Tall and short (Kindergarten – Q.2)

Light and heavy (Kindergarten – Q.3)

Holds more or less (Kindergarten – Q.4)

Compare size, weight, and capacity (Kindergarten – Q.5)

 

K. Classify objects and count the number of objects in each category.

 

K.K.MD.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

Fewer and more – with charts (Kindergarten – G.3)

Fewer and more – mixed (Kindergarten – G.4)

Same (Kindergarten – N.1)

Different (Kindergarten – N.2)

Same and different (Kindergarten – N.3)

Classify by color (Kindergarten – N.4)

Classify and sort by color (Kindergarten – N.5)

Classify and sort by shape (Kindergarten – N.6)

Classify and sort (Kindergarten – N.7)

Sort shapes into a Venn diagram (Kindergarten – N.8)

Count shapes in a Venn diagram (Kindergarten – N.9)

Put numbers up to 10 in order (Kindergarten – N.10)

Put numbers up to 30 in order (Kindergarten – N.11)

Making graphs (Kindergarten – O.1)

Interpreting graphs (Kindergarten – O.2)

K.K.G Geometry

 

K. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

 

K.K.G.1 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

Inside and outside (Kindergarten – K.1)

Above and below (Kindergarten – K.2)

Above and below – find solid figures (Kindergarten – K.3)

Left, middle, and right (Kindergarten – K.4)

Top, middle, and bottom (Kindergarten – K.5)

Location in a grid (Kindergarten – K.6)

Geometry of everyday objects I (Kindergarten – S.8)

Geometry of everyday objects II (Kindergarten – S.9)

 

K.K.G.2 Correctly name shapes regardless of their orientations or overall size.

Count shapes in a Venn diagram (Kindergarten – N.9)

Identify shapes I (Kindergarten – S.1)

Identify shapes II (Kindergarten – S.2)

Identify solid figures (Kindergarten – S.4)

 

K.K.G.3 Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).

Relate planar and solid figures (Kindergarten – S.5)

 

K. Analyze, compare, create, and compose shapes.

 

K.K.G.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/”corners”) and other attributes (e.g., having sides of equal length).

Identify shapes I (Kindergarten – S.1)

Identify shapes II (Kindergarten – S.2)

Same shape (Kindergarten – S.3)

Identify solid figures (Kindergarten – S.4)

Relate planar and solid figures (Kindergarten – S.5)

Count sides and corners (Kindergarten – S.6)

Compare sides and corners (Kindergarten – S.7)

Symmetry I (Kindergarten – S.10)

Symmetry II (Kindergarten – S.11)

 

K.K.G.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

 

K.K.G.6 Compose simple shapes to form larger shapes.