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Common Core State Standards
This is a list of all Virtual Nerd tutorials in Common Core State Standards, organized by topic.
Common Core State Standards Tutorial Browser
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Grade 1
Grade 2
Grade 3
Operations & Algebraic Thinking
Represent and solve problems involving multiplication and division.
Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.
Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
Understand properties of multiplication and the relationship between multiplication and division.
Multiply and divide within 100.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Number & Operations in Base Ten
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Measurement & Data
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
Represent and interpret data.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Geometry
Reason with shapes and their attributes
Grade 4
Operations & Algebraic Thinking
Use the four operations with whole numbers to solve problems
Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Gain familiarity with factors and multiples
Generate and analyze patterns
Number & Operations in Base Ten
Generalize place value understanding for multi-digit whole numbers.
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
- How Do You Compare Two Whole Numbers?
- How Do You Put Whole Numbers in Order From Greatest to Least?
- How Do You Put Whole Numbers in Order From Least to Greatest?
- How Do You Write a Whole Number Given in Expanded Form in Standard Form and in Words?
- How Do You Write a Whole Number Given in Standard Form in Expanded Form and in Words?
- How Do You Write a Whole Number Given in Words in Standard Form and in Expanded Form?
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Fluently add and subtract multi-digit whole numbers using the standard algorithm.
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Number & Operations—Fractions
Extend understanding of fraction equivalence and ordering.
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
Understand decimal notation for fractions, and compare decimal fractions.
Measurement & Data
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.
- How Do You Determine the Best Customary Units to Measure a Capacity?
- How Do You Determine the Best Customary Units to Measure a Weight?
- How Do You Determine the Best Metric Units to Measure a Capacity?
- How Do You Determine the Best Metric Units to Measure a Mass?
- What are the Customary Units of Capacity?
- What are the Customary Units of Length?
- What are the Customary Units of Weight?
- What are the Metric Units of Capacity?
- What are the Metric Units of Length?
- What are the Metric Units of Mass?
Geometric measurement: understand concepts of angle and measure angles
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a 'one-degree angle,' and can be used to measure angles. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
Geometry
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
Grade 5
Operations & Algebraic Thinking
Number & Operations in Base Ten
Understand the place value system.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
Fluently multiply multi-digit whole numbers using the standard algorithm.
- How Do You Do Long Multiplication?
- How Do You Multiply a Three-Digit Number by a One-Digit Number?
- How Do You Multiply a Two-Digit Number by a One-Digit Number?
- How Do You Multiply a Two-Digit Number by a Two-Digit Number Using Partial Products?
- How Do You Multiply a Two-Digit Number by a Two-Digit Number?
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
- How Do You Estimate a Sum of Decimals by Clustering?
- How Do You Multiply a Decimal by a Whole Number Using Estimation to Place the Decimal?
- How Do You Multiply a Decimal by a Whole Number?
- How Do You Regroup Numbers to Add Decimals Mentally?
- How Do You Solve a Word Problem by Dividing with Decimals and Rounding Your Answer Up?
- How Do You Use Front-End Estimation to Estimate a Sum of Several Decimals?
- What are the Associative Properties of Addition and Multiplication?
- What are the Commutative Properties of Addition and Multiplication?
Number & Operations—Fractions
Use equivalent fractions as a strategy to add and subtract fractions.
Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
- How Do You Add Fractions with Different Denominators?
- How Do You Add Mixed Fractions with Different Denominators by Converting to Improper Fractions?
- How Do You Add Two Fractions with Different Denominators?
- How Do You Subtract Mixed Fractions with Different Denominators by Converting to Improper Fractions?
- How Do You Subtract Mixed Fractions with Different Denominators by Regrouping?
- How Do You Subtract Two Fractions with Different Denominators?
- What are Equivalent Fractions?
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Measurement & Data
Convert like measurement units within a given measurement system.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
Geometry
Graph points on the coordinate plane to solve real-world and mathematical problems.
Classify two-dimensional figures into categories based on their properties.
Grade 6
Ratios & Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems.
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. Solve unit rate problems including those involving unit pricing and constant speed. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. - How Do You Estimate a Percent?
- How Do You Put Fractions, Decimals, and Percents in Order?
- How Do You Set Up a Percent Proportion from a Word Problem?
- How Do You Solve a Word Problem Using a Percent Proportion?
- How Do You Turn a Decimal into a Percent?
- How Do You Turn a Fraction Into a Percent?
- How Do You Turn a Percent into a Decimal?
- How Do You Turn a Percent Into a Fraction?
- How Do You Use a Proportion to Find a Whole?
- How Do You Use an Equation to Find a Whole?
- How Do You Use Compatible Numbers to Estimate a Part of a Whole?
- How Do You Use Mental Math to Estimate a Percent?
- What's a Percent Proportion?
- What's a Percent?
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
The Number System
Apply and extend previous understandings of multiplication and division
Compute fluently with multi-digit numbers and find common factors and multiples.
Fluently divide multi-digit numbers using the standard algorithm.
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
- How Do You Add Decimals?
- How Do You Divide a Decimal by a Decimal?
- How Do You Do Long Division with Decimals?
- How Do You Estimate a Sum of Decimals by Clustering?
- How Do You Multiply a Decimal by a Whole Number Using Estimation to Place the Decimal?
- How Do You Multiply a Decimal by a Whole Number?
- How Do You Multiply Decimals?
- How Do You Solve a Word Problem by Dividing with Decimals and Rounding Your Answer Up?
- How Do You Subtract Decimals?
- How Do You Use Front-End Estimation to Estimate a Sum of Several Decimals?
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
- How Do You Distribute a Number into an Addition Problem to Solve?
- How Do You Find All the Factors of a Number?
- How Do You Find the Greatest Common Factor of Two Numbers by Listing Their Factors?
- How Do You Find the Greatest Common Factor of Two Numbers Using Prime Factorization?
- How Do You Use the Distributive Property to Rewrite and Solve an Addition Problem?
- What is a Common Multiple and Least Common Multiple?
- What is a Greatest Common Factor?
- What is a Multiple?
Apply and extend previous understandings of numbers to the system of rational numbers.
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
Understand ordering and absolute value of rational numbers.
Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. Distinguish comparisons of absolute value from statements about order.
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Expressions & Equations
Apply and extend previous understandings of arithmetic to algebraic expressions.
Write and evaluate numerical expressions involving whole-number exponents.
- How Do You Convert a Number in Expanded Form into Exponential Form?
- How Do You Convert a Number in Exponential Form into Expanded Form?
- How Do You Evaluate an Exponent?
- How Do You Simplify an Expression Using the Order of Operations with Exponents?
- How Do You Square a Number?
- How Do You Use Powers to Solve a Word Problem?
- How Do You Write the Prime Factorization of a Number Using Exponents?
- What is an Exponent?
Write, read, and evaluate expressions in which letters stand for numbers.
Write expressions that record operations with numbers and with letters standing for numbers. - How Do You Determine Which Operations to Use in a Word Problem?
- How Do You Translate a Phrase with One Variable and Number Into a Mathematical Expression?
- How Do You Translate Phrases into Numerical Expressions?
- How Do You Turn a Verbal Phrase into a Two-Step Algebraic Expression?
- What Are Numerical and Algebraic Expressions?
- What Are Some Words You Can Use To Write Word Problems?
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). - How Do You Complete a Table By Plugging in Values?
- How Do You Convert a Temperature from Celsius to Fahrenheit?
- How Do You Convert a Temperature from Fahrenheit to Celsius?
- How Do You Evaluate an Algebraic Expression by Plugging In Values?
- How Do You Evaluate an Algebraic Expression with One Variable?
- How Do You Evaluate an Algebraic Expression with Two Variables?
- How Do You Evaluate an Algebraic Expression?
- How Do You Evaluate an Expression with Exponents?
- How Do You Simplify an Expression Using the Order of Operations with Exponents?
- How Do You Simplify an Expression Using the Order of Operations?
- What is a Variable?
- What is the Substitution Property of Equality?
- What's the Order of Operations?
Apply the properties of operations to generate equivalent expressions.
- How Can You Distribute Numbers When You're Adding and Multiplying?
- How Do You Add Like Terms?
- How Do You Add Variables that Are the Same?
- How Do You Distribute a Number into an Addition Problem to Solve?
- How Do You Factor a Greatest Common Factor Out of an Expression?
- How Do You Subtract Like Terms?
- How Do You Use the Distributive Property to Rewrite and Solve an Addition Problem?
- How Do You Use the Distributive Property to Simplify an Expression?
- What is the Distributive Property?
Reason about and solve one-variable equations and inequalities.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
- How Do You Solve a Decimal Equation Using Addition?
- How Do You Solve a Decimal Equation Using Division?
- How Do You Solve a Decimal Equation Using Multiplication?
- How Do You Solve a Decimal Equation Using Subtraction?
- How Do You Solve a Word Problem Using an Equation Where You're Multiplying Fractions?
- How Do You Solve a Word Problem with an Equation Using Addition?
- How Do You Solve a Word Problem with an Equation Using Division?
- How Do You Solve a Word Problem with an Equation Using Multiplication?
- How Do You Solve a Word Problem with an Equation Using Subtraction?
- How Do You Solve an Equation Using Addition?
- How Do You Solve an Equation Using Multiplication?
- How Do You Solve an Equation Using Subtraction?
- How Do You Solve an Equation Where You're Multiplying Fractions?
- How Do You Solve an Equation With Fractions With Different Denominators Using Addition?
- How Do You Solve an Equation With Fractions With Different Denominators Using Subtraction?
- What's the Addition Property of Equality?
- What's the Division Property of Equality?
- What's the Multiplication Property of Equality?
- What's the Subtraction Property of Equality?
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Represent and analyze quantitative relationships between dependent and independent variables.
Geometry
Solve real-world and mathematical problems involving area, surface area, and volume.
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- How Do You Find the Area of a Composite Figure?
- How Do You Find the Area of a Parallelogram?
- How Do You Find the Area of a Trapezoid?
- How Do You Find the Area of a Triangle?
- How Do You Find the Base of a Parallelogram if You Know the Area and Height?
- How Do You Find the Height of a Trapezoid if You Know the Area and Bases?
- How Do You Find the Height of a Triangle if You Know the Area and Base?
- What is the Formula for the Area of a Parallelogram?
- What is the Formula for the Area of a Trapezoid?
- What is the Formula for the Area of a Triangle?
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
- How Do You Draw a Side View of a Three-Dimensional Shape?
- How Do You Find the Surface Area of a Cylinder Using a Net?
- How Do You Find the Surface Area of a Rectangular Prism Using a Net?
- How Do You Identify a Three-Dimensional Figure from a Net?
- What is a Net?
- What is a Prism?
- What is a Pyramid?
- What is a Solid?
- What is the Formula for the Surface Area of a Prism?
Statistics & Probability
Summarize and describe distributions.
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Summarize numerical data sets in relation to their context, such as by:
Reporting the number of observations. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. - How Do You Find an Average?
- How Do You Find the Mean of a Data Set?
- How Do You Find the Median of a Data Set?
- How Do You Find the Minimum, Maximum, Quartiles, and Median of a Data Set?
- How Do You Find the Mode of a Data Set When There is More Than One Mode?
- How Do You Find the Mode of a Data Set Where All The Numbers are Different?
- How Do You Interpret a Bar Graph With a Break in the Scale?
- How Do You Interpret a Circle Graph?
- How Do You Use a Circle Graph to Make Predictions?
- How Do You Use a Line Graph to Make Predictions?
- What is a Weighted Average?
- What is Numerical, or Quantitative, Data?
- What is the Interquartile Range?
- What is the Mean of a Data Set?
- What is the Median of a Data Set?
Grade 7
Ratios & Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
Recognize and represent proportional relationships between quantities.
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. - How Do You Determine if Two Ratios are Proportional Using Cross Products?
- How Do You Determine Whether Values in a Table are Proportional?
- How Do You Find Equivalent Ratios by Making a Table?
- How Do You Find Equivalent Ratios?
- How Do You Know If Two Ratios are Proportional?
- What are Equivalent Ratios?
- What Does Direct Variation Look Like on a Graph?
- What's a Proportion?
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Represent proportional relationships by equations. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Use proportional relationships to solve multistep ratio and percent problems.
- How Do You Estimate a Sale Price?
- How Do You Figure Out a Percent From a Part to Part Ratio?
- How Do You Figure Out a Percent From a Part to Whole Ratio?
- How Do You Figure Out a Percent of Change?
- How Do You Figure Out a Tip?
- How Do You Figure Out How Much Something is Marked Down?
- How Do You Figure Out Sales Tax?
- How Do You Figure Out the Price of a Marked Up Item?
- How Do You Figure Out Whether a Percent of Change is an Increase or a Decrease?
- How Do You Set Up a Percent Proportion from a Word Problem?
- How Do You Set Up a Proportion from a Word Problem?
- How Do You Solve a Proportion by Finding an Equivalent Ratio?
- How Do You Solve a Proportion Using Cross Products?
- How Do You Solve a Proportion Using the Multiplication Property of Equality?
- How Do You Solve a Word Problem Using a Percent Proportion?
- How Do You Solve a Word Problem Using a Proportion?
- How Do You Solve a Word Problem Using Ratios?
- How Do You Use a Proportion to Find a Part of a Whole?
- How Do You Use a Proportion to Find What Percent a Part is of a Whole?
- How Do You Use an Equation to Find a Part of a Whole?
- How Do You Use the Formula for Simple Interest?
- What are the Means and Extremes of Proportions?
- What is the Formula for Simple Interest?
- What's a Percent of Change?
- What's the Means-Extremes Property of Proportions?
The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. - How Do You Add a Negative Number to a Positive Number?
- How Do You Add Decimals?
- How Do You Add Fractions with Different Denominators?
- How Do You Add Fractions with the Same Denominator?
- How Do You Add Integers Using a Number Line?
- How Do You Add Two Negative Numbers?
- What are the Inverse Properties of Addition and Multiplication?
- What are the Rules for Using Absolute Values to Add Integers?
- What is the Opposite, or Additive Inverse, of a Number?
Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - How Do You Rewrite Subtraction as Addition?
- How Do You Subtract a Negative Number from a Positive Number?
- How Do You Subtract a Positive Number from a Negative Number?
- How Do You Subtract Decimals?
- How Do You Subtract Fractions with Different Denominators?
- How Do You Subtract Fractions with the Same Denominator?
- How Do You Subtract Integers Using a Number Line?
- What is the Opposite, or Additive Inverse, of a Number?
Apply properties of operations as strategies to add and subtract rational numbers. - How Do You Add and Subtract a Bunch of Numbers with Different Signs?
- How Do You Add Decimals?
- How Do You Add Fractions with Different Denominators?
- How Do You Add Fractions with the Same Denominator?
- How Do You Subtract a Whole Number from a Fraction?
- How Do You Subtract Decimals?
- How Do You Subtract Fractions with Different Denominators?
- How Do You Subtract Fractions with the Same Denominator?
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. - How Do You Figure Out the Sign of a Product or Quotient?
- How Do You Multiply a Whole Number by a Fraction?
- How Do You Multiply and Divide Numbers with Different Signs?
- How Do You Multiply Decimals?
- How Do You Multiply Fractions?
- What are the Multiplication Properties of 0 and -1?
- What is the Distributive Property?
- What's a Rational Number?
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts. Apply properties of operations as strategies to multiply and divide rational numbers. - How Do You Divide a Decimal by a Decimal?
- How Do You Divide Fractions?
- How Do You Do Long Division with Decimals?
- How Do You Figure Out the Sign of a Product or Quotient?
- How Do You Multiply a Whole Number by a Fraction?
- How Do You Multiply Decimals?
- How Do You Multiply Fractions?
- What is the Distributive Property?
- What's the Reciprocal Rule of Division?
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Solve real-world and mathematical problems involving the four operations with rational numbers.
- How Do You Add Decimals?
- How Do You Add Fractions with Different Denominators?
- How Do You Add Fractions with the Same Denominator?
- How Do You Divide Fractions?
- How Do You Figure Out the Sign of a Product or Quotient?
- How Do You Multiply a Whole Number by a Fraction?
- How Do You Multiply Decimals?
- How Do You Multiply Fractions?
- How Do You Rewrite Subtraction as Addition?
- How Do You Simplify a Fraction Over a Fraction?
- How Do You Simplify a Fraction Over a Whole Number?
- How Do You Subtract Decimals?
- How Do You Subtract Fractions with Different Denominators?
- How Do You Subtract Fractions with the Same Denominator?
- What are the Inverse Properties of Addition and Multiplication?
- What are the Multiplication Properties of 0 and -1?
- What's a Complex Fraction?
- What's the Reciprocal Rule of Division?
Expressions & Equations
Use properties of operations to generate equivalent expressions.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
- How Do You Solve a Two-Step Equation?
- How Do You Solve a Word Problem That Compares Two Fractions?
- How Do You Solve a Word Problem Using a Two-Step Equation with Decimals?
- How Do You Solve a Word Problem Where You Divide Fractions?
- How Do You Solve a Word Problem Where You Multiply and Subtract Whole Numbers and Fractions?
- How Do You Solve a Word Problem Where You Multiply Fractions and Work Backwards?
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. - How Do You Find the Length of a Rectangle if You Know its Width and Area?
- How Do You Find the Width of a Rectangle if You Know its Length and Perimeter?
- How Do You Solve a Word Problem Using a Two-Step Equation with Decimals?
- How Do You Solve a Word Problem Using an Equation Where You're Multiplying Fractions?
- How Do You Solve a Word Problem with an Equation Using Addition?
- How Do You Solve a Word Problem with an Equation Using Division?
- How Do You Solve a Word Problem with an Equation Using Multiplication?
- How Do You Solve a Word Problem with an Equation Using Subtraction?
- How Do You Use an Equation with Consecutive Numbers to Solve a Word Problem?
Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. - How Do You Graph an Inequality or an Infinite Set on a Number Line?
- How Do You Solve a Word Problem Using a Multi-Step Inequality?
- How Do You Solve a Word Problem Using an Inequality Where You're Multiplying Negative Fractions?
- How Do You Solve a Word Problem Using an Inequality Where You're Multiplying Positive Fractions?
- How Do You Solve a Word Problem Using an Inequality With Variables on Both Sides?
- How Do You Use Addition to Solve an Inequality Word Problem?
- How Do You Use Division with Negative Numbers to Solve an Inequality Word Problem?
- How Do You Use Division with Positive Numbers to Solve an Inequality Word Problem?
- How Do You Use Multiplication with Negative Numbers to Solve an Inequality Word Problem?
- How Do You Use Multiplication with Positive Numbers to Solve an Inequality Word Problem?
- How Do You Use Subtraction to Solve an Inequality Word Problem?
Geometry
Draw construct, and describe geometrical figures and describe the relationships between them.
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
- How Do You Figure Out What Size a Model Should Be if You Have a Scale?
- How Do You Find the Scale of a Model?
- How Do You Solve a Scale Model Problem Using a Scale Factor?
- How Do You Solve a Word Problem Using a Proportion?
- How Do You Use the Scale on a Map to Find an Actual Distance?
- What is a Scale Drawing?
- What is a Scale?
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
- How Do You Find the Area of a Circle if You Know the Diameter?
- How Do You Find the Area of a Circle if You Know the Radius?
- How Do You Find the Circumference of a Circle if You Know the Diameter?
- How Do You Find the Circumference of a Circle if You Know the Radius?
- How Do You Find the Radius of a Circle if You Know the Area?
- How Do You Find the Radius of a Circle if You Know the Circumference?
- What is a Circle?
- What is Circumference?
- What is the Formula for the Area of a Circle?
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
- How Do You Find the Area of a Composite Figure?
- How Do You Find the Area of a Parallelogram?
- How Do You Find the Area of a Rectangle?
- How Do You Find the Area of a Trapezoid?
- How Do You Find the Area of a Triangle?
- How Do You Find the Lateral and Surface Areas of a Rectangular Prism?
- How Do You Find the Lateral and Surface Areas of a Triangular Prism?
- How Do You Find the Volume of a Composite Figure?
- How Do You Find the Volume of a Rectangular Prism?
- How Do You Find the Volume of a Triangular Prism?
- What is a Composite Figure?
- What is the Formula for the Area of a Parallelogram?
- What is the Formula for the Area of a Rectangle?
- What is the Formula for the Area of a Trapezoid?
- What is the Formula for the Area of a Triangle?
- What is the Formula for the Surface Area of a Prism?
- What is the Formula for the Volume of a Prism?
Statistics & Probability
Use random sampling to draw inferences about a population.
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
Draw informal comparative inferences about two populations.
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
Investigate chance processes and develop, use, and evaluate probability models. -
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., 'rolling double sixes'), identify the outcomes in the sample space which compose the event. - How Do You Determine All the Possible Outcomes of an Experiment?
- How Do You Find a Number of Combinations Using the Fundamental Counting Principle?
- How Do You Solve a Problem by Making an Organized List?
- How Do You Use a Tree Diagram to Count the Number of Outcomes in a Sample Space?
- What is a Sample Space?
- What is an Outcome?
Design and use a simulation to generate frequencies for compound events.
Grade 8
The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π^2).
Expressions & Equations
Work with radicals and integer exponents.
Know and apply the properties of integer exponents to generate equivalent numerical expressions.
- How Do You Evaluate Negative Exponents?
- How Do You Find the Product of Powers with Numbers?
- How Do You Find the Quotient of Powers with Numbers?
- How Do You Take a Power to a Power with Numbers?
- How Do You Take the Power of a Product with Numbers?
- How Do You Take the Power of a Quotient with Numbers?
- What are Exponents?
- What Do You Do With a Negative Exponent?
- What Do You Do With a Zero Exponent?
- What's the Power of a Power Rule?
- What's the Power of a Product Rule?
- What's the Power of a Quotient Rule?
- What's the Product of Powers Rule?
- What's the Quotient of Powers Rule?
Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
Understand the connections between proportional relationships, lines, and linear equations.
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Analyze and solve linear equations and pairs of simultaneous linear equations.
Solve linear equations in one variable.
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). - How Do You Solve a Multi-Step Equation Using Order of Operations in Reverse?
- How Do You Solve a Multi-Step Equation Using the Distributive Property?
- How Do You Solve a Two-Step Equation by Multiplying by -1 First?
- How Do You Solve a Two-Step Equation by Multiplying by -1?
- How Do You Solve a Two-Step Equation?
- How Do You Solve an Equation Using Addition?
- How Do You Solve an Equation Using Division?
- How Do You Solve an Equation Using Multiplication?
- How Do You Solve an Equation Using Subtraction?
- How Do You Solve an Equation with No Solution?
- How Do You Solve an Equation with Variables on Both Sides and Grouping Symbols?
- How Do You Solve an Equation with Variables on Both Sides?
- How Do You Solve an Identity Equation?
- How Do You Solve for a Variable in a Fraction Using Multiplication First?
- What Does It Mean When An Equation Has No Solution?
- What is an Identity Equation?
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - How Do You Solve a Decimal Equation Using Addition?
- How Do You Solve a Decimal Equation Using Division?
- How Do You Solve a Decimal Equation Using Multiplication?
- How Do You Solve a Decimal Equation Using Subtraction?
- How Do You Solve a Multi-Step Equation Using the Distributive Property?
- How Do You Solve a Multi-Step Equation with Fractions by Multiplying Away the Fraction?
- How Do You Solve a Multi-Step Equation with Fractions Using Reverse Order of Operations and Reciprocals?
- How Do You Solve a Two-Step Equation by Combining Like Terms?
- How Do You Solve a Two-Step Equation by Distributing a Fraction First?
- How Do You Solve a Two-Step Equation by Multiplying by a Reciprocal?
- How Do You Solve a Word Problem Using an Equation With Variables on Both Sides?
- How Do You Solve an Equation Where You're Multiplying Fractions?
- How Do You Solve an Equation With Fractions With Different Denominators Using Addition?
- How Do You Solve an Equation With Fractions With Different Denominators Using Subtraction?
- How Do You Solve an Equation With Fractions With The Same Denominators Using Addition?
- How Do You Solve an Equation With Fractions With The Same Denominators Using Subtraction?
- How Do You Solve an Equation with Variables on Both Sides and Fractions?
Analyze and solve pairs of simultaneous linear equations.
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. - How Do You Graph a System of Equations With No Solution?
- How Do You Solve a System of Equations Using the Elimination by Addition Method?
- How Do You Solve a System of Equations Using the Elimination by Subtraction Method?
- How Do You Solve a System of Equations Using the Substitution Method?
- How Do You Solve a Word Problem Using the Elimination by Subtraction Method?
- How Do You Solve Two Equations with Two Variables?
Solve real-world and mathematical problems leading to two linear equations in two variables.
Functions
Define, evaluate, and compare functions.
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
Use functions to model relationships between quantities.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
- How Do You Graph a Linear Equation From a Table?
- How Do You Use the Graph of a Linear Equation to Solve a Word Problem?
- How Do You Write an Equation of a Line in Slope-Intercept Form from a Word Problem?
- How Do You Write an Equation of a Line in Slope-Intercept Form If You Have Two Points?
- How Do You Write the Equation of a Line in Slope-Intercept Form If You Have a Graph?
- How Do You Write the Equation of a Line in Slope-Intercept Form If You Have a Table?
- What's a Function?
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software.
Verify experimentally the properties of rotations, reflections, and translations:
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
- How Do You Make a Figure Larger Using a Dilation?
- How Do You Rotate a Figure 180 Degrees Around the Origin?
- How Do You Rotate a Figure 270 Degrees Clockwise Around the Origin?
- How Do You Rotate a Figure 90 Degrees Around the Origin?
- How Do You Use Coordinates to Reflect a Figure Over the X-Axis?
- How Do You Use Coordinates to Reflect a Figure Over the Y-Axis?
- How Do You Use Coordinates to Translate a Figure Horizontally?
- How Do You Use Coordinates to Translate a Figure Vertically?
- What is a Dilation?
- What is a Line of Reflection?
- What is a Reflection?
- What is a Rotation?
- What is a Translation?
- What Properties of a Figure Stay the Same After a Translation?
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
- How Do You Find a Missing Angle in a Right Triangle?
- How Do You Find a Missing Angle in a Triangle?
- How Do You Find Missing Angles in a Transversal Diagram?
- How Do You Find Missing Angles in a Triangle With Variables?
- How Do You Find Missing Angles in an Isosceles Triangle Using Right Triangles?
- How Do You Find Missing Angles in an Isosceles Triangle?
- How Do You Find the Angles in a Triangle if You Have a Ratio of Their Measures?
- What is a Transversal?
- What is a Triangle?
- What is the Triangle Sum Theorem?
Understand and apply the Pythagorean Theorem.
Explain a proof of the Pythagorean Theorem and its converse.
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
- How Do You Find the Height of a Cylinder if You Know the Volume?
- How Do You Find the Volume of a Cone?
- How Do You Find the Volume of a Cylinder?
- How Do You Find the Volume of a Sphere?
- What is a Cone?
- What is a Cylinder?
- What is a Sphere?
- What is the Formula for the Volume of a Cone?
- What is the Formula for the Volume of a Cylinder?
- What is the Formula for the Volume of a Sphere?
Statistics & Probability
Investigate patterns of association in bivariate data.
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
- How Do You Make a Scatter Plot?
- How Do You Use a Scatter Plot to Find a Negative Correlation?
- How Do You Use a Scatter Plot to Find a Positive Correlation?
- How Do You Use a Scatter Plot to Find No Correlation?
- What Does it Mean To Have No Correlation?
- What's a Scatter Plot?
- What's Negative Correlation?
- What's Positive Correlation?
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.
High School: Number & Quantity
The Real Number System
Quantities*
Reason quantitatively and use units to solve problems.
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.
- How Do You Find the Area of a Composite Figure?
- How Do You Find the Area of a Parallelogram?
- How Do You Find the Area of a Rectangle?
- How Do You Find the Area of a Trapezoid?
- How Do You Find the Area of a Triangle?
- How Do You Find the Base of a Parallelogram if You Know the Area and Height?
- How Do You Find the Height of a Cylinder if You Know the Volume?
- How Do You Find the Height of a Trapezoid if You Know the Area and Bases?
- How Do You Find the Height of a Triangle if You Know the Area and Base?
- How Do You Find the Length of a Rectangle if You Know its Width and Area?
- How Do You Find the Volume of a Composite Figure?
- How Do You Find the Volume of a Cone?
- How Do You Find the Volume of a Cylinder?
- How Do You Find the Volume of a Rectangular Prism?
- How Do You Find the Volume of a Rectangular Pyramid?
- How Do You Find the Volume of a Sphere?
- How Do You Find the Volume of a Triangular Prism?
- How Do You Find the Volume of a Triangular Pyramid?
- How Do You Use Dimensional Analysis to Convert Units on Both Parts of a Rate?
- How Do You Use Dimensional Analysis to Convert Units on One Part of a Rate?
- What is Dimensional, or Unit Analysis?
The Complex Number System
Perform arithmetic operations with complex numbers.
Know there is a complex number i such that i^2 = -1, and every complex number has the form a + bi with a and b real.
Use the relation i^2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
- How Do You Add Complex Numbers?
- How Do You Find Higher Powers of i?
- How Do You Find Values for x and y to Make Two Complex Numbers Equal?
- How Do You Multiply Complex Numbers Using FOIL?
- How Do You Multiply Pure Imaginary Numbers?
- How Do You Simplify the Square Root of a Negative Number?
- How Do You Subtract Complex Numbers?
- What are the Associative Properties of Addition and Multiplication?
- What are the Commutative Properties of Addition and Multiplication?
- What is an Imaginary Number?
- What is the Distributive Property?
- What is the Imaginary Unit i?
Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Represent complex numbers and their operations on the complex plane.
Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
Use complex numbers in polynomial identities and equations.
Solve quadratic equations with real coefficients that have complex solutions.
- How Do You Solve a Quadratic Equation with Complex Solutions by Completing the Square?
- How Do You Solve a Quadratic Equation With Complex Solutions by Using the Quadratic Formula?
- How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a≠1?
- How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions?
Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Vector & Matrix Quantities
Represent and model with vector quantities.
Perform operations on matrices and use matrices in applications.
Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
Add, subtract, and multiply matrices of appropriate dimensions.
Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
High School: Algebra
Seeing Structure in Expressions
Interpret the structure of expressions.
Write expressions in equivalent forms to solve problems.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Factor a quadratic expression to reveal the zeros of the function it defines. - How Do You Factor a Common Factor Out Of a Difference of Squares?
- How Do You Factor a Polynomial by Guessing and Checking?
- How Do You Factor a Polynomial Using Difference of Squares?
- How Do You Factor a Polynomial Using the A-C Method?
- How Do You Factor a Trinomial?
- How Do You Figure Out a Template for Factoring a Trinomial?
- How Do You Find All the Possible Factors of a Trinomial?
- How Do You Know if You Have a Difference of Squares?
- How Do You Solve a Quadratic Equation by Factoring?
- How Do You Use a Shortcut to Factor a Perfect Square Trinomial?
- What's the Zero Product Property?
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Use the properties of exponents to transform expressions for exponential functions.
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
Arithmetic with Polynomials & Rational Expressions
Perform arithmetic operations on polynomials.
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.
- How Do You Add Polynomials Vertically?
- How Do You Add Polynomials?
- How Do You Find the Additive Inverse of a Polynomial?
- How Do You Find the Area of a Rectangle Whose Sides are a Monomial and a Polynomial?
- How Do You Find the Area of a Rectangle Whose Sides are Binomials?
- How Do You Find the Volume of a Box Whose Sides are Monomials?
- How Do You Multiply a Monomial by a Polynomial?
- How Do You Multiply Binomials Using FOIL?
- How Do You Multiply Binomials Using the Distributive Property?
- How Do You Multiply Monomials?
- How Do You Multiply Trinomials?
- How Do You Solve a Word Problem by Multiplying Binomials?
- How Do You Solve a Word Problem by Multiplying Trinomials?
- How Do You Solve a Word Problem by Subtracting and Multiplying Polynomials?
- How Do You Solve a Word Problem by Subtracting Polynomials?
- How Do You Solve a Word Problem by Taking a Monomial to a Power?
- How Do You Subtract Polynomials Vertically?
- How Do You Subtract Polynomials?
- How Do You Take a Monomial to a Power?
- How Do You Take the Power of a Monomial?
- How Do You Use the Formula for the Product of a Sum and a Difference?
- What is a Polynomial?
- What's a Monomial?
- What's FOIL?
- What's the Formula for the Product of a Sum and a Difference?
- What's the Formula for the Square of a Difference?
- What's the Formula for the Square of a Sum?
- What's the Grid Method of FOILing?
Understand the relationship between zeros and factors of polynomials.
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Use polynomial identities to solve problems.
Rewrite rational expressions.
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + (r(x)/b(x)), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
- How Do You Divide a Polynomial by a Monomial Using Long Division?
- How Do You Divide Monomials Using Quotient of Powers?
- How Do You Divide Two Polynomials by Factoring and Canceling?
- How Do You Do Long Division With Polynomials?
- How Do You Simplify a Polynomial Over a Polynomial Using Opposite Binomial Factors?
- How Do You Solve a Word Problem by Dividing Monomials?
Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
- How Do You Add Two Rational Expressions with Different Denominators?
- How Do You Add Two Rational Expressions with the Same Denominator?
- How Do You Divide a Rational Expression by a Polynomial?
- How Do You Divide Quotients of Monomials?
- How Do You Divide Two Rational Expressions?
- How Do You Multiply a Rational Expression by a Polynomial?
- How Do You Multiply Quotients of Monomials by Canceling Factors?
- How Do You Multiply Quotients of Monomials?
- How Do You Multiply Two Rational Expressions?
- How Do You Reduce Common Factors in a Rational Expression?
- How Do You Subtract Two Rational Expressions with Different Denominators?
- How Do You Subtract Two Rational Expressions with the Same Denominator?
- What's a Rational Expression?
Creating Equations
Create equations that describe numbers or relationships.
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
- How Do You Figure Out How Much Something is Marked Down?
- How Do You Figure Out Sales Tax?
- How Do You Figure Out the Price of a Marked Up Item?
- How Do You Figure Out the Volume of a Solution Using Percents?
- How Do You Solve a Word Problem by Factoring a Quadratic Equation?
- How Do You Solve a Word Problem Using a Multi-Step Equation?
- How Do You Solve a Word Problem Using a Multi-Step Inequality?
- How Do You Solve a Word Problem Using a Two-Step Equation with Decimals?
- How Do You Solve a Word Problem Using an AND Absolute Value Inequality?
- How Do You Solve a Word Problem Using an AND Compound Inequality?
- How Do You Solve a Word Problem Using an Equation Where You're Multiplying Fractions?
- How Do You Solve a Word Problem Using an Exponential Function?
- How Do You Solve a Word Problem Using an Inequality With Variables on Both Sides?
- How Do You Solve a Word Problem Using the Quadratic Formula?
- How Do You Solve a Word Problem with a Rational Equation?
- How Do You Solve a Word Problem with an Equation Using Addition?
- How Do You Solve a Word Problem with an Equation Using Division?
- How Do You Solve a Word Problem with an Equation Using Multiplication?
- How Do You Solve a Word Problem with an Equation Using Subtraction?
- How Do You Solve a Word Problem with Exponential Decay?
- How Do You Solve a Word Problem with Exponential Growth?
- How Do You Solve an Opposite-Direction Travel Problem?
- How Do You Use Addition to Solve an Inequality Word Problem?
- How Do You Use an Equation with Consecutive Numbers to Solve a Word Problem?
- How Do You Use Division with Negative Numbers to Solve an Inequality Word Problem?
- How Do You Use Division with Positive Numbers to Solve an Inequality Word Problem?
- How Do You Use Multiplication with Negative Numbers to Solve an Inequality Word Problem?
- How Do You Use Multiplication with Positive Numbers to Solve an Inequality Word Problem?
- How Do You Use Subtraction to Solve an Inequality Word Problem?
- How Do You Write an Absolute Value Inequality from a Word Problem?
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- How Do You Find the Axis of Symmetry for a Quadratic Function?
- How Do You Find the Vertex of a Quadratic Function?
- How Do You Graph a Line If You're Given the Slope and the Intercept?
- How Do You Graph a Quadratic Function?
- How Do You Graph the Parent Quadratic Function y=x2?
- How Do You Make a Table for a Quadratic Function?
- How Do You Solve a Word Problem Using the Direct Variation Formula?
- How Do You Use Point-Slope Form to Write an Equation from a Table?
- How Do You Use the Formula for Direct Variation?
- How Do You Write a Quadratic Equation in Vertex Form if You Have the Vertex and Another Point?
- How Do You Write an Equation For a Quadratic if You Have Three Points?
- How Do You Write an Equation for Direct Variation from a Table?
- How Do You Write an Equation for Direct Variation Given a Point?
- How Do You Write an Equation of a Line in Point-Slope Form and Standard Form If You Have Two Points?
- How Do You Write an Equation of a Line in Point-Slope Form If You Have the Slope and One Point?
- How Do You Write an Equation of a Line in Point-Slope Form If You Have Two Points?
- How Do You Write an Equation of a Line in Slope-Intercept Form from a Word Problem?
- How Do You Write an Equation of a Line in Slope-Intercept Form If You Have the Slope and the Y-Intercept?
- How Do You Write an Equation of a Line in Slope-Intercept Form If You Have Two Points?
- How Do You Write an Equation of a Line in Standard Form if You Have the Slope and One Point?
- How Do You Write and Use a Prediction Equation?
- How Do You Write the Equation of a Line in Slope-Intercept Form If You Have a Graph?
- How Do You Write the Equation of a Line in Slope-Intercept Form If You Have a Table?
- What Are Dependent and Independent Variables?
- What is a Parabola?
- What is the Axis of Symmetry of a Quadratic Function?
- What's a Function?
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
- How Do You Solve a Word Problem Using a System of Inequalities?
- How Do You Solve an Optimization Word Problem?
- How Do You Solve and Graph Inequalities from a Word Problem?
- How Do You Use a System of Linear Equations to Find Coordinates on a Map?
- How Do You Write Inequalities in Set Builder Notation?
- What Are Some Words We Use To Write Inequalities?
- What is Linear Programming?
- What's an Example of a Word Problem That Has a System of Linear Equations with Infinite Solutions?
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Reasoning with Equations & Inequalities
Understand solving equations as a process of reasoning and explain the reasoning.
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.
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
- How Do You Solve a Radical Equation with a Binomial in the Radicand?
- How Do You Solve a Radical Equation?
- How Do You Solve a Rational Equation by Adding Fractions?
- How Do You Solve a Rational Equation by LCD Multiplication?
- How Do You Solve a Rational Equation With an Extraneous Solution?
- How Do You Solve a Rational Equation With Binomials in the Denominator?
- How Do You Solve a Rational Equation With No Solution?
Solve equations and inequalities in one variable.
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
- How Do You Convert an AND Absolute Value Inequality to a Compound Inequality?
- How Do You Figure Out if an Absolute Value Inequality is an AND or OR Compound Inequality?
- How Do You Solve a Decimal Equation Using Addition?
- How Do You Solve a Decimal Equation Using Division?
- How Do You Solve a Decimal Equation Using Multiplication?
- How Do You Solve a Decimal Equation Using Subtraction?
- How Do You Solve a Decimal Inequality Using Addition?
- How Do You Solve a Decimal Inequality Using Division with Negative Numbers?
- How Do You Solve a Decimal Inequality Using Division?
- How Do You Solve a Decimal Inequality Using Subtraction?
- How Do You Solve a Multi-Step Equation Using Order of Operations in Reverse?
- How Do You Solve a Multi-Step Equation Using the Distributive Property?
- How Do You Solve a Multi-Step Equation with Fractions by Multiplying Away the Fraction?
- How Do You Solve a Multi-Step Equation with Fractions Using Reverse Order of Operations and Reciprocals?
- How Do You Solve a Multi-Step Inequality Using Reverse Order of Operations?
- How Do You Solve a Two-Step Equation by Combining Like Terms?
- How Do You Solve a Two-Step Equation by Distributing a Fraction First?
- How Do You Solve a Two-Step Equation by Multiplying by -1 First?
- How Do You Solve a Two-Step Equation by Multiplying by -1?
- How Do You Solve a Two-Step Equation by Multiplying by a Reciprocal?
- How Do You Solve a Two-Step Equation?
- How Do You Solve an AND Absolute Value Inequality and Graph It On a Number Line?
- How Do You Solve an AND Compound Inequality and Graph It On a Number Line?
- How Do You Solve an AND Compound Inequality?
- How Do You Solve an Equation Using Addition?
- How Do You Solve an Equation Using Division?
- How Do You Solve an Equation Using Multiplication?
- How Do You Solve an Equation Using Subtraction?
- How Do You Solve an Equation Where You're Multiplying Fractions?
- How Do You Solve an Equation With Fractions With Different Denominators Using Addition?
- How Do You Solve an Equation With Fractions With Different Denominators Using Subtraction?
- How Do You Solve an Equation With Fractions With The Same Denominators Using Addition?
- How Do You Solve an Equation With Fractions With The Same Denominators Using Subtraction?
- How Do You Solve an Equation with No Solution?
- How Do You Solve an Equation with Variables on Both Sides and Fractions?
- How Do You Solve an Equation with Variables on Both Sides and Grouping Symbols?
- How Do You Solve an Equation with Variables on Both Sides?
- How Do You Solve an Identity Equation?
- How Do You Solve an Inequality By Adding Fractions?
- How Do You Solve an Inequality by Dividing by a Positive Number?
- How Do You Solve an Inequality By Subtracting Fractions?
- How Do You Solve an Inequality Using Addition?
- How Do You Solve an Inequality Using Subtraction?
- How Do You Solve an Inequality Where You're Multiplying Negative Fractions?
- How Do You Solve an Inequality Where You're Multiplying Positive Fractions?
- How Do You Solve an Inequality with Negative Numbers Using Division?
- How Do You Solve an Inequality with Negative Numbers Using Multiplication?
- How Do You Solve an Inequality with Positive Numbers Using Multiplication?
- How Do You Solve an Inequality With Variables on Both Sides?
- How Do You Solve an OR Compound Inequality and Graph It On a Number Line?
- How Do You Solve and Graph a Two-Step Inequality?
- How Do You Solve for a Variable in a Fraction Using Multiplication First?
- What Does It Mean When an Inequality is a Contradiction or Has No Solution?
- What is an Identity Inequality?
- What's a Compound Inequality?
- What's the Addition Property of Inequality?
- What's the Division Property of Inequality?
- What's the Multiplication Property of Inequality?
- What's the Subtraction Property of Inequality?
Solve quadratic equations in one variable.
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. Solve quadratic equations by inspection (e.g., for x^2 = 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. - How Do You Find the a, b, and c Values of a Quadratic Function?
- How Do You Find The Discriminant of a Quadratic Equation With 1 Solution?
- How Do You Find The Discriminant of a Quadratic Equation With 2 Solutions?
- How Do You Graph a Quadratic Equation with No Solution?
- How Do You Solve a Quadratic Equation by Completing the Square?
- How Do You Solve a Quadratic Equation by Factoring?
- How Do You Solve a Quadratic Equation by Using the Quadratic Formula?
- How Do You Solve a Quadratic Equation with Complex Solutions by Completing the Square?
- How Do You Solve a Quadratic Equation With Complex Solutions by Using the Quadratic Formula?
- How Do You Solve a Quadratic Equation with Two Solutions by Graphing?
- How Do You Use The Discriminant to Determine the Number of Real or Complex Solutions to a Quadratic Equation?
- How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a≠1?
- How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions?
- How Do You Use the Square Root Method to Solve a Quadratic Equation with Two Solutions?
Solve systems of equations.
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 systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- How Do You Graph a System of Equations With No Solution?
- How Do You Solve a System of Equations by Graphing?
- How Do You Solve a System of Equations Using the Elimination by Addition Method?
- How Do You Solve a System of Equations Using the Elimination by Multiplication Method?
- How Do You Solve a System of Equations Using the Elimination by Subtraction Method?
- How Do You Solve a System of Equations Using the Substitution Method?
- How Do You Solve a Word Problem Using the Elimination by Subtraction Method?
- How Do You Solve a Word Problem Using Two Equations?
- How Do You Solve Two Equations with Two Variables?
- How Do You Use a System of Linear Equations to Find Coordinates on a Map?
- What Are the Ways You Can Solve a System of Linear Equations?
- What's a Consistent Independent System of Equations?
- What's an Example of a Word Problem That Has a System of Linear Equations with Infinite Solutions?
- What's Another Way of Solving a System of Equations Using the Elimination by Multiplication Method?
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
Represent and solve equations and inequalities graphically.
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).
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.
- How Do You Determine if an Ordered Pair is a Solution to a Linear Inequality?
- How Do You Figure Out Whether a Boundary is a Part of the Graph of an Inequality?
- How Do You Graph a Greater Than Inequality on the Coordinate Plane?
- How Do You Solve a System of Inequalities by Graphing?
- How Do You Solve and Graph Inequalities from a Word Problem?
- What is a Linear Inequality?
- What's a Boundary?
- What's a Half-Plane?
- What's a System of Linear Inequalities?
High School: Functions
Interpreting Functions
Understand the concept of a function and use function notation.
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).
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
Interpret functions that arise in applications in terms of the context.
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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
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.
Analyze functions using different representations.
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Graph linear and quadratic functions and show intercepts, maxima, and minima. - How Do You Determine if the Vertex Will Be a Maximum or Minimum?
- How Do You Find the Zeros of a Quadratic Function on a Graph?
- How Do You Graph a Horizontal Line?
- How Do You Graph a Line If You're Given the Slope and the Intercept?
- How Do You Graph a Linear Equation by Making a Table?
- How Do You Graph a Quadratic Equation in Intercept Form?
- How Do You Graph a Quadratic Function?
- How Do You Graph a Vertical Line?
- How Do You Use X- and Y-Intercepts To Graph a Line In Standard Form?
- What is a Linear Function?
- What is a Quadratic Function?
- What is the Maximum of a Quadratic Function?
- What is the Minimum of a Quadratic Function?
- What is the Vertex of a Quadratic Function?
- What's the X-Intercept?
- What's the Y-Intercept?
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. - How Do You Graph an Exponential Function Using a Table?
- How Do You Graph the Natural Base Exponential Function?
- How Do You Horizontally Translate a Trigonometric Graph?
- What Does the Graph of the Cosine Function Look Like?
- What Does the Graph of the Sine Function Look Like?
- What Does the Graph of the Tangent Function Look Like?
- What is a Periodic Function?
- What is the Natural Base Exponential Function?
- What's an Exponential Function?
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function
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. - How Do You Convert a Quadratic from Standard Form to Vertex Form by Completing the Square if a≠1?
- How Do You Convert a Quadratic from Standard Form to Vertex Form by Completing the Square?
- How Do You Solve a Quadratic Equation by Completing the Square?
- How Do You Solve a Quadratic Equation by Factoring?
- How Do You Solve a Quadratic Equation with Complex Solutions by Completing the Square?
Use the properties of exponents to interpret expressions for exponential functions.
Building Functions
Build a function that models a relationship between two quantities.
Write a function that describes a relationship between two quantities.
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
- How Do You Find the Common Difference in an Arithmetic Sequence?
- How Do You Find the Next Terms in an Arithmetic Sequence?
- How Do You Find the Nth Term in an Arithmetic Sequence?
- How Do You Write a Rule for a Geometric Sequence?
- What is the Explicit Formula for the nth Term in a Geometric Sequence?
- What's an Arithmetic Sequence?
- What's the Common Difference?
- Where Does the Formula for a Term in an Arithmetic Sequence Come From?
Build new functions from existing functions.
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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
- How Do You Graph a Reflection of a Function?
- How Do You Graph a Translation of a Function?
- How Do You Horizontally Translate a Square Root Function?
- How Do You Horizontally Translate a Trigonometric Graph?
- How Do You Reflect a Function?
- How Do You Translate a Function?
- How Do You Translate a Polynomial Function Vertically?
- How Do You Write an Equation for a Translation of an Absolute Value Function?
- What Does the Constant 'h' do in y = |x-h|?
- What Does the Constant 'k' Do in the Function f(x)=[square root of](x)+k?
- What Does the Constant 'k' do in y = |x|+k?
- What Does the Constant 'k' Do to the Graph of f(x)=log(x)+k?
- What Does the Value of 'a' Do in the Exponential Function f(x)= a (bx)?
Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
- How Do You Convert From Exponential Form to Logarithmic Form?
- How Do You Convert From Exponential Form to Natural Logarithmic Form?
- How Do You Convert From Logarithmic Form to Exponential Form?
- How Do You Convert From Natural Logarithmic Form to Exponential Form?
- How Do You Evaluate a Logarithm?
- How Do You Simplify Logarithms Using the Product Property?
- How Do You Solve a Logarithmic Equation by Exponentiating?
- What is a Logarithm?
- What is a Natural Logarithm?
- What is e?
Linear, Quadratic, & Exponential Models*
Construct and compare linear, quadratic, and exponential models and solve problems.
Distinguish between situations that can be modeled with linear functions and with exponential functions.
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
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).
- How Do You Solve a Word Problem Using an Exponential Function?
- How Do You Use Point-Slope Form to Write an Equation from a Table?
- How Do You Use the Formula for Direct Variation?
- How Do You Write a Rule for a Geometric Sequence?
- How Do You Write an Equation for Direct Variation from a Table?
- How Do You Write an Equation for Direct Variation Given a Point?
- How Do You Write an Equation of a Line in Point-Slope Form and Standard Form If You Have Two Points?
- How Do You Write an Equation of a Line in Point-Slope Form If You Have the Slope and One Point?
- How Do You Write an Equation of a Line in Point-Slope Form If You Have Two Points?
- How Do You Write an Equation of a Line in Slope-Intercept Form If You Have the Slope and the Y-Intercept?
- How Do You Write an Equation of a Line in Slope-Intercept Form If You Have Two Points?
- How Do You Write the Equation of a Line in Slope-Intercept Form If You Have a Graph?
- How Do You Write the Equation of a Line in Slope-Intercept Form If You Have a Table?
- What is a Linear Function?
- What's a Function?
- What's an Exponential Function?
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Interpret expressions for functions in terms of the situation they model.
Trigonometric Functions
High School: Geometry
Congruence
Experiment with transformations in the plane.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
- How Do You Rotate a Figure 180 Degrees Around the Origin?
- How Do You Rotate a Figure 270 Degrees Clockwise Around the Origin?
- How Do You Rotate a Figure 90 Degrees Around the Origin?
- How Do You Use Coordinates to Reflect a Figure Over the X-Axis?
- How Do You Use Coordinates to Reflect a Figure Over the Y-Axis?
- How Do You Use Coordinates to Translate a Figure Horizontally?
- How Do You Use Coordinates to Translate a Figure Vertically?
- How Do You Use Matrices to Translate a Figure?
- What is a Congruence Transformation, or Isometry?
- What is a Dilation?
- What is a Reflection?
- What is a Rotation?
- What is a Transformation?
- What is a Translation?
- What is an Image?
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
- How Do You Graph a Glide Reflection?
- How Do You Identify a Congruence Transformation?
- How Do You Rotate a Figure 180 Degrees Around the Origin?
- How Do You Rotate a Figure 270 Degrees Clockwise Around the Origin?
- How Do You Rotate a Figure 90 Degrees Around the Origin?
- How Do You Use a Graph to Reflect a Figure Over the X-Axis?
- How Do You Use a Graph to Reflect a Figure Over the Y-Axis?
- How Do You Use a Graph to Translate a Figure Diagonally?
- How Do You Use a Graph to Translate a Figure Horizontally?
- How Do You Use a Graph to Translate a Figure Vertically?
Understand congruence in terms of rigid motions.
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
- How Do You Determine if Triangles on the Coordinate Plane are Congruent?
- How Do You Identify Common Parts in Overlapping Triangles?
- How Do You Prove that Two Overlapping Triangles are Congruent?
- How Do You Show that Corresponding Parts of Congruent Triangles are Congruent?
- How Do You Use a Congruence Postulate to Prove Triangles are Congruent?
- What is CPCTC?
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove geometric theorems.
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°
- How Do You Put the Sides of a Triangle in Order According to Size if You Know Two Angles of the Triangle?
- How Do You Use the Centroid to Find Segment Lengths in a Triangle?
- How Do You Use the Hinge Theorem to Compare Side Lengths in Two Triangles?
- How Do You Write an Indirect Proof?
- What is a Median of a Triangle?
- What is the Circumcenter of a Triangle?
- What is the Hinge Theorem?
- What is the Hypotenuse-Leg Congruence Theorem?
- What is the Incenter of a Triangle?
- What is the Triangle Midsegment Theorem?
- What is the Triangle Sum Theorem?
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Make geometric constructions.
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations.
Prove theorems involving similarity
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
- How Do You Determine if Triangles on the Coordinate Plane are Congruent?
- How Do You Use a Congruence Postulate to Prove Triangles are Congruent?
- How Do You Use Similar Figures to Find the Area of a Polygon?
- What is the Angle-Side-Angle Postulate for Triangle Congruence?
- What is the Side-Angle-Side Postulate for Triangle Congruence?
Define trigonometric ratios and solve problems involving right triangles.
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Apply trigonometry to general triangles.
Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
Prove the Laws of Sines and Cosines and use them to solve problems.
Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Circles
Understand and apply theorems about circles.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
- How Do You Determine Whether a Line is Tangent to a Circle?
- How Do You Determine Whether Two Chords are Equidistant From the Center of a Circle?
- How Do You Find the Measure of an Angle Created by Intersecting Chords in a Circle?
- How Do You Find The Measure of an Inscribed Angle When You Know the Measure of the Intercepted Arc?
- How Do You Use Intersecting Chords to Find Arc Measures in a Circle?
- What is a Tangent Line to a Circle?
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Find arc lengths and areas of sectors of circles.
Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section.
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Derive the equation of a parabola given a focus and directrix.
Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
Use coordinates to prove simple geometric theorems algebraically.
Use coordinates to prove simple geometric theorems algebraically.
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
- How Do You Find the Slope of a Line If You Have a Parallel Line?
- How Do You Find the Slope of a Line If You Have a Perpendicular Line?
- How Do You Know if Two Lines are Parallel?
- How Do You Know if Two Lines Are Perpendicular?
- How Do You Write an Equation of a Line in Slope-Intercept Form If You Have One Point and a Parallel Line?
- How Do You Write an Equation of a Line in Slope-Intercept Form If You Have One Point and a Perpendicular Line?
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Geometric Measurement & Dimension
Explain volume formulas and use them to solve problems.
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
- How Do You Find the Height of a Cylinder if You Know the Volume?
- How Do You Find the Volume of a Cone?
- How Do You Find the Volume of a Cylinder?
- How Do You Find the Volume of a Rectangular Pyramid?
- How Do You Find the Volume of a Sphere?
- How Do You Find the Volume of a Triangular Pyramid?
- What is the Formula for the Volume of a Cone?
- What is the Formula for the Volume of a Cylinder?
- What is the Formula for the Volume of a Pyramid?
- What is the Formula for the Volume of a Sphere?
High School: Statistics & Probability
Interpreting Categorical & Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable.
Represent data with plots on the real number line (dot plots, histograms, and box plots).
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.
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Summarize, represent, and interpret data on two categorical and quantitative variables.
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.
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Informally assess the fit of a function by plotting and analyzing residuals. Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models.
Making Inferences & Justifying Conclusions
Understand and evaluate random processes underlying statistical experiments.
Conditional Probability & the Rules of Probability
Understand independence and conditional probability and use them to interpret data.
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or,' 'and,' 'not').
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
Use the rules of probability to compute probabilities of compound events in a uniform probability model.