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Free Math Printable Worksheets with Answer Keys and Activities

Other free resources.

Feel free to download and enjoy these free worksheets on functions and relations. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Long Division with Remainders
• Long Division with Remainders #2 (Zeros in the Quotient)
• Long Division with 2 Digit Divisors
• Whole Number by Unit Fraction
• Equation of Circle
• Simplify Imaginary Numbers
• Adding and Subtracting Complex Numbers
• Multiplying Complex Numbers
• Dividing Complex Numbers
• Dividing Complex Number (Advanced)
• End of Unit, Review Sheet
• Distance Formula
• Simplify Rational Exponents (Algebra 2)
• Solve Equations with Rational Exponents (Algebra 2)
• Solve Equations with variables in Exponents (Algebra 2)
• Exponential Growth (no answer key on this one, sorry)
• Compound Interest Worksheet #1 (No logs)
• Compound Interest Worksheet (Logarithms required)
• Factor Trinomials Worksheet
• Factor by Grouping
• Domain and Range (Algebra 1)
• Functions vs Relations (Distinguish function from relation, state domain etc..) (Algebra 2)
• Evaluating Functions (Algebra 2)
• 1 to 1 Functions (Algebra 2)
• Composition of Functions (Algebra 2)
• Inverse Functions Worksheet (Algebra 2)
• Operations with Functions (Algebra 2)
• Functions Review Worksheet (Algebra 2)
• Logarithmic Equations
• Properties of Logarithms Worksheet
• Product Rule of Logarithms
• Power Rule of Logarithms
• Quotient Rule of Logarithms
• Solve Quadratic Equations by Factoring
• Quadratic Formula Worksheets (3 different sheets)
• Quadratic Formula Worksheet (Real solutions)
• Quadratic Formula (Complex solutions)
• Quadratic Formula (Both real and complex solutions)
• Discriminant and Nature of the Roots
• Solve Quadratic Equations by Completing the Square
• Sum and Product of Roots
• Radical Equations
• Mixed Problems on Writing Equations of Lines
• Slope Intercept Form Worksheet
• Standard Form Worksheet
• Point Slope Worksheet
• Write Equation of Line from the Slope and 1 Point
• Write Equation of Line From Two Points
• Equation of Line Parallel to Another Line and Through a Point
• Equation of Line Perpendicular to Another Line and Through a Point
• Slope of a Line
• Perpendicular Bisector of Segment
• Write Equation of Line Mixed Review
• Word Problems
• Multiplying Monomials Worksheet
• Multiplying and Dividing Monomials Sheet
• Adding and Subtracting Polynomials worksheet
• Multiplying Monomials with Polynomials Worksheet
• Multiplying Binomials Worksheet
• Multiplying Polynomials
• Simplifying Polynomials
• Factoring Trinomials
• Operations with Polynomials Worksheet
• Dividing Radicals
• Simplify Radicals Worksheet
• Adding Radicals
• Multiplying Radicals Worksheet
• Radicals Review (Mixed review worksheet on radicals and square roots)
• Rationalizing the Denominator (Algebra 2)
• Radical Equations (Algebra 2)
• Solve Systems of Equations Graphically
• Solve Systems of Equations by Elimination
• Solve by Substitution
• Solve Systems of Equations (Mixed Review)
• Activity on Systems of Equations (Create an advertisement for your favorite method to Solve Systems of Equations )
• Real World Connections (Compare cell phone plans)
• Identifying Fractions

Trigonomnetry

• Law of Sines and Cosines Worksheet (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle)
• Law of Sines
• Ambiguous Case of the Law of Sines
• Law of Cosines
• Vector Worksheet
• Sine, Cosine, Tangent, to Find Side Length
• Sine, Cosine, Tangent Chart
• Inverse Trig Functions
• Real World Applications of SOHCATOA
• Mixed Review
• Unit Circle Worksheet
• Graphing Sine and Cosine Worksheet
• Sine Cosine Graphs with Vertical Translations
• Sine, Cosine, Tangent Graphs with Phase Shifts
• Sine, Cosine, Tangent Graphs with Change in Period, Amplitude and Phase Shifts (All Translations)
• Tangent Equation, Graph Worksheet
• Graphing Sine, Cosine, Tangent with Change in Period
• Cumulative, Summative Worksheet on Periodic Trig Functions - period, amplitude, phase shift, radians, degrees,unit circle
• Ratio and Proportion
• Similar Polygons
• Area of Triangle
• Interior Angles of Polygons
• Exterior Angles of Polygons

• Identifying Fractions Worksheet
• Associated Powerpoint
• Simplify Fractions Worksheet (Regular Difficulty)
• Associated PowerPoint
• Simplify Fractions Worksheet (Challenging Difficulty level for advanced learners)
• System of Linear Equations Worksheet
• System of Linear Equations - Real World Application
• Compositions of Reflections. Reflections Over Intersecting Lines as Rotations

All of these worksheets and activities are available for free so long as they are used solely for educational, noncommercial purposes and are not distributed outside of a specific teacher's classroom.

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Free Printable Math Worksheets for Algebra 1

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• Writing variable expressions
• Order of operations
• Evaluating expressions
• Number sets
• Adding rational numbers
• Adding and subtracting rational numbers
• Multiplying and dividing rational numbers
• The distributive property
• Combining like terms
• Percent of change
• One-step equations
• Two-step equations
• Multi-step equations
• Absolute value equations
• Solving proportions
• Percent problems
• Distance-rate-time word problems
• Mixture word problems
• Work word problems
• Literal Equations
• Graphing one-variable inequalities
• One-step inequalities
• Two-step inequalities
• Multi-step inequalities
• Compound inequalities
• Absolute value inequalities
• Discrete relations
• Continuous relations
• Evaluating and graphing functions
• Finding slope from a graph
• Finding slope from two points
• Finding slope from an equation
• Graphing lines using slope-intercept form
• Graphing lines using standard form
• Writing linear equations
• Graphing linear inequalities
• Graphing absolute value equations
• Direct variation
• Solving systems of equations by graphing
• Solving systems of equations by elimination
• Solving systems of equations by substitution
• Systems of equations word problems
• Graphing systems of inequalities
• Discrete exponential growth and decay word problems
• Exponential functions and graphs
• Writing numbers in scientific notation
• Operations with scientific notation
• Addition and subtraction with scientific notation
• Naming polynomials
• Adding and subtracting polynomials
• Multiplying polynomials
• Multiplying special case polynomials
• Factoring special case polynomials
• Factoring by grouping
• Dividing polynomials
• Graphing quadratic inequalities
• Completing the square
• By taking square roots
• By factoring
• With the quadratic formula
• By completing the square
• Simplifying radicals
• Adding and subtracting radical expressions
• Multiplying radicals
• Dividing radicals
• Using the distance formula
• Using the midpoint formula
• Simplifying rational expressions
• Finding excluded values / restricted values
• Multiplying rational expressions
• Dividing rational expressions
• Adding and subtracting rational expressions
• Finding trig. ratios
• Finding angles of triangles
• Finding side lengths of triangles
• Visualizing data
• Center and spread of data
• Scatter plots
• Using statistical models

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Free Math Worksheets — Over 100k free practice problems on Khan Academy

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• Addition and subtraction
• Place value (tens and hundreds)
• Addition and subtraction within 20
• Addition and subtraction within 100
• Addition and subtraction within 1000
• Measurement and data
• Counting and place value
• Measurement and geometry
• Place value
• Measurement, data, and geometry
• Add and subtract within 20
• Add and subtract within 100
• Add and subtract within 1,000
• Money and time
• Measurement
• Intro to multiplication
• 1-digit multiplication
• Addition, subtraction, and estimation
• Intro to division
• Understand fractions
• Equivalent fractions and comparing fractions
• More with multiplication and division
• Arithmetic patterns and problem solving
• Quadrilaterals
• Represent and interpret data
• Multiply by 1-digit numbers
• Multiply by 2-digit numbers
• Factors, multiples and patterns
• Add and subtract fractions
• Multiply fractions
• Understand decimals
• Plane figures
• Measuring angles
• Area and perimeter
• Units of measurement
• Decimal place value
• Add decimals
• Subtract decimals
• Multi-digit multiplication and division
• Divide fractions
• Multiply decimals
• Divide decimals
• Powers of ten
• Coordinate plane
• Algebraic thinking
• Converting units of measure
• Properties of shapes
• Ratios, rates, & percentages
• Arithmetic operations
• Negative numbers
• Properties of numbers
• Variables & expressions
• Equations & inequalities introduction
• Data and statistics
• Negative numbers: addition and subtraction
• Negative numbers: multiplication and division
• Fractions, decimals, & percentages
• Rates & proportional relationships
• Expressions, equations, & inequalities
• Numbers and operations
• Solving equations with one unknown
• Linear equations and functions
• Systems of equations
• Geometric transformations
• Data and modeling
• Volume and surface area
• Pythagorean theorem
• Transformations, congruence, and similarity
• Arithmetic properties
• Factors and multiples
• Reading and interpreting data
• Negative numbers and coordinate plane
• Ratios, rates, proportions
• Equations, expressions, and inequalities
• Exponents, radicals, and scientific notation
• Foundations
• Algebraic expressions
• Linear equations and inequalities
• Graphing lines and slope
• Expressions with exponents
• Quadratics and polynomials
• Equations and geometry
• Algebra foundations
• Solving equations & inequalities
• Working with units
• Linear equations & graphs
• Forms of linear equations
• Inequalities (systems & graphs)
• Absolute value & piecewise functions
• Exponents & radicals
• Exponential growth & decay
• Quadratics: Multiplying & factoring
• Quadratic functions & equations
• Irrational numbers
• Performing transformations
• Transformation properties and proofs
• Right triangles & trigonometry
• Non-right triangles & trigonometry (Advanced)
• Analytic geometry
• Conic sections
• Solid geometry
• Polynomial arithmetic
• Complex numbers
• Polynomial factorization
• Polynomial division
• Polynomial graphs
• Rational exponents and radicals
• Exponential models
• Transformations of functions
• Rational functions
• Trigonometric functions
• Non-right triangles & trigonometry
• Trigonometric equations and identities
• Analyzing categorical data
• Displaying and comparing quantitative data
• Summarizing quantitative data
• Modeling data distributions
• Exploring bivariate numerical data
• Study design
• Probability
• Counting, permutations, and combinations
• Random variables
• Sampling distributions
• Confidence intervals
• Significance tests (hypothesis testing)
• Two-sample inference for the difference between groups
• Inference for categorical data (chi-square tests)
• Advanced regression (inference and transforming)
• Analysis of variance (ANOVA)
• Scatterplots
• Data distributions
• Two-way tables
• Binomial probability
• Normal distributions
• Displaying and describing quantitative data
• Inference comparing two groups or populations
• Chi-square tests for categorical data
• More on regression
• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
• Composite functions
• Probability and combinatorics
• Limits and continuity
• Derivatives: definition and basic rules
• Derivatives: chain rule and other advanced topics
• Applications of derivatives
• Analyzing functions
• Parametric equations, polar coordinates, and vector-valued functions
• Applications of integrals
• Differentiation: definition and basic derivative rules
• Differentiation: composite, implicit, and inverse functions
• Contextual applications of differentiation
• Applying derivatives to analyze functions
• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
• AP®︎ Calculus AB Standards mappings
• Infinite sequences and series
• AP Calculus BC solved exams
• AP®︎ Calculus BC Standards mappings
• Integrals review
• Integration techniques
• Thinking about multivariable functions
• Derivatives of multivariable functions
• Applications of multivariable derivatives
• Integrating multivariable functions
• Green’s, Stokes’, and the divergence theorems
• First order differential equations
• Second order linear equations
• Laplace transform
• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

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Algebra Worksheets

Welcome to the Algebra worksheets page at Math-Drills.com, where unknowns are common and variables are the norm. On this page, you will find Algebra worksheets for middle school students on topics such as algebraic expressions, equations and graphing functions.

This page starts off with some missing numbers worksheets for younger students. We then get right into algebra by helping students recognize and understand the basic language related to algebra. The rest of the page covers some of the main topics you'll encounter in algebra units. Remember that by teaching students algebra, you are helping to create the future financial whizzes, engineers, and scientists that will solve all of our world's problems.

Algebra is much more interesting when things are more real. Solving linear equations is much more fun with a two pan balance, some mystery bags and a bunch of jelly beans. Algebra tiles are used by many teachers to help students understand a variety of algebra topics. And there is nothing like a set of co-ordinate axes to solve systems of linear equations.

Most Popular Algebra Worksheets this Week

Algebraic Properties, Rules and Laws Worksheets

The commutative law or commutative property states that you can change the order of the numbers in an arithmetic problem and still get the same results. In the context of arithmetic, it only works with addition or multiplication operations , but not mixed addition and multiplication. For example, 3 + 5 = 5 + 3 and 9 × 5 = 5 × 9. A fun activity that you can use in the classroom is to brainstorm non-numerical things from everyday life that are commutative and non-commutative. Putting on socks, for example, is commutative because you can put on the right sock then the left sock or you can put on the left sock then the right sock and you will end up with the same result. Putting on underwear and pants, however, is non-commutative.

• The Commutative Law Worksheets The Commutative Law of Addition (Numbers Only) The Commutative Law of Addition (Some Variables) The Commutative Law of Multiplication (Numbers Only) The Commutative Law of Multiplication (Some Variables)

The associative law or associative property allows you to change the grouping of the operations in an arithmetic problem with two or more steps without changing the result. The order of the numbers stays the same in the associative law. As with the commutative law, it applies to addition-only or multiplication-only problems. It is best thought of in the context of order of operations as it requires that parentheses must be dealt with first. An example of the associative law is: (9 + 5) + 6 = 9 + (5 + 6). In this case, it doesn't matter if you add 9 + 5 first or 5 + 6 first, you will end up with the same result. Students might think of some examples from their experience such as putting items on a tray at lunch. They could put the milk and vegetables on their tray first then the sandwich or they could start with the vegetables and sandwich then put on the milk. If their tray looks the same both times, they will have modeled the associative law. Reading a book could be argued as either associative or nonassociative as one could potentially read the final chapters first and still understand the book as well as someone who read the book the normal way.

• The Associative Law Worksheets The Associative Law of Addition (Whole Numbers Only) The Associative Law of Multiplication (Whole Numbers Only)

Inverse relationships worksheets cover a pre-algebra skill meant to help students understand the relationship between multiplication and division and the relationship between addition and subtraction.

• Inverse Mathematical Relationships with One Blank Addition and Subtraction Easy Addition and Subtraction Harder All Multiplication and Division Facts 1 to 18 in color (no blanks) Multiplication and Division Range 1 to 9 Multiplication and Division Range 5 to 12 Multiplication and Division All Inverse Relationships Range 2 to 9 Multiplication and Division All Inverse Relationships Range 5 to 12 Multiplication and Division All Inverse Relationships Range 10 to 25
• Inverse Mathematical Relationships with Two Blanks Addition and Subtraction (Sums 1-18) Addition and Subtraction Inverse Relationships with 1 Addition and Subtraction Inverse Relationships with 2 Addition and Subtraction Inverse Relationships with 3 Addition and Subtraction Inverse Relationships with 4 Addition and Subtraction Inverse Relationships with 5 Addition and Subtraction Inverse Relationships with 6 Addition and Subtraction Inverse Relationships with 7 Addition and Subtraction Inverse Relationships with 8 Addition and Subtraction Inverse Relationships with 9 Addition and Subtraction Inverse Relationships with 10 Addition and Subtraction Inverse Relationships with 11 Addition and Subtraction Inverse Relationships with 12 Addition and Subtraction Inverse Relationships with 13 Addition and Subtraction Inverse Relationships with 14 Addition and Subtraction Inverse Relationships with 15 Addition and Subtraction Inverse Relationships with 16 Addition and Subtraction Inverse Relationships with 17 Addition and Subtraction Inverse Relationships with 18

The distributive property is an important skill to have in algebra. In simple terms, it means that you can split one of the factors in multiplication into addends, multiply each addend separately, add the results, and you will end up with the same answer. It is also useful in mental math, an example of which should help illustrate the definition. Consider the question, 35 × 12. Splitting the 12 into 10 + 2 gives us an opportunity to complete the question mentally using the distributive property. First multiply 35 × 10 to get 350. Second, multiply 35 × 2 to get 70. Lastly, add 350 + 70 to get 420. In algebra, the distributive property becomes useful in cases where one cannot easily add the other factor before multiplying. For example, in the expression, 3(x + 5), x + 5 cannot be added without knowing the value of x. Instead, the distributive property can be used to multiply 3 × x and 3 × 5 to get 3x + 15.

• Distributive Property Worksheets Distributive Property (Answers do not include exponents) Distributive Property (Some answers include exponents) Distributive Property (All answers include exponents)

Students should be able to substitute known values in for an unknown(s) in an expression and evaluate the expression's value.

• Evaluating Expressions with Known Values Evaluating Expressions with One Variable, One Step and No Exponents Evaluating Expressions with One Variable and One Step Evaluating Expressions with One Variable and Two Steps Evaluating Expressions with Up to Two Variables and Two Steps Evaluating Expressions with Up to Two Variables and Three Steps Evaluating Expressions with Up to Three Variables and Four Steps Evaluating Expressions with Up to Three Variables and Five Steps

As the title says, these worksheets include only basic exponent rules questions. Each question only has two exponents to deal with; complicated mixed up terms and things that a more advanced student might work out are left alone. For example, 4 2 is (2 2 ) 2 = 2 4 , but these worksheets just leave it as 4 2 , so students can focus on learning how to multiply and divide exponents more or less in isolation.

• Exponent Rules for Multiplying, Dividing and Powers Mixed Exponent Rules (All Positive) Mixed Exponent Rules (With Negatives) Multiplying Exponents (All Positive) Multiplying Exponents (With Negatives) Multiplying the Same Exponent with Different Bases (All Positive) Multiplying the Same Exponent with Different Bases (With Negatives) Dividing Exponents with a Greater Exponent in Dividend (All Positive) Dividing Exponents with a Greater Exponent in Dividend (With Negatives) Dividing Exponents with a Greater Exponent in Divisor (All Positive) Dividing Exponents with a Greater Exponent in Divisor (With Negatives) Powers of Exponents (All Positive) Powers of Exponents (With Negatives)

Knowing the language of algebra can help to extract meaning from word problems and to situations outside of school. In these worksheets, students are challenged to convert phrases into algebraic expressions.

• Translating Algebraic Phrases into Expressions Translating Algebraic Phrases into Expressions (Simple Version) Translating Algebraic Phrases into Expressions (Complex Version)

Combining like terms is something that happens a lot in algebra. Students can be introduced to the topic and practice a bit with these worksheets. The bar is raised with the adding and subtracting versions that introduce parentheses into the expressions. For students who have a good grasp of fractions, simplifying simple algebraic fractions worksheets present a bit of a challenge over the other worksheets in this section.

• Simplifying Expressions by Combining Like Terms Simplifying Linear Expressions with 3 terms Simplifying Linear Expressions with 4 terms Simplifying Linear Expressions with 5 terms Simplifying Linear Expressions with 6 to 10 terms
• Simplifying Expressions by Combining Like Terms with Some Arithmetic Adding and simplifying linear expressions Adding and simplifying linear expressions with multipliers Adding and simplifying linear expressions with some multipliers . Subtracting and simplifying linear expressions Subtracting and simplifying linear expressions with multipliers Subtracting and simplifying linear expressions with some multipliers Mixed adding and subtracting and simplifying linear expressions Mixed adding and subtracting and simplifying linear expressions with multipliers Mixed adding and subtracting and simplifying linear expressions with some multipliers Simplify simple algebraic fractions (easier) Simplify simple algebraic fractions (harder)
• Rewriting Linear Equations Rewrite Linear Equations in Standard Form Convert Linear Equations from Standard to Slope-Intercept Form Convert Linear Equations from Slope-Intercept to Standard Form Convert Linear Equations Between Standard and Slope-Intercept Form
• Rewriting Formulas Rewriting Formulas (addition and subtraction; about one step) Rewriting Formulas (addition and subtraction; about two steps) Rewriting Formulas ( multiplication and division ; about one step)

Linear Expressions and Equations

In these worksheets, the unknown is limited to the question side of the equation which could be on the left or the right of equal sign.

• Missing Numbers in Equations with Blanks as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Blanks in Any Position )
• Missing Numbers in Equations with Symbols as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Symbols in Any Position )
• Solving Equations with Addition and Symbols as Unknowns Equalities with Addition (0 to 9) Symbol Unknowns Equalities with Addition (1 to 12) Symbol Unknowns Equalities with Addition (1 to 15) Symbol Unknowns Equalities with Addition (1 to 25) Symbol Unknowns Equalities with Addition (1 to 99) Symbol Unknowns
• Missing Numbers in Equations with Variables as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Variables Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Variables in Any Position )
• Solving Simple Linear Equations Solving Simple Linear Equations with Values from -9 to 9 (Unknown on Left Side) Solving Simple Linear Equations with Values from -99 to 99 (Unknown on Left Side) Solving Simple Linear Equations with Values from -9 to 9 (Unknown on Right or Left Side) Solving Simple Linear Equations with Values from -99 to 99 (Unknown on Right or Left Side)
• Determining Linear Equations from Slopes, y-intercepts and Points Determine a Linear Equation from the Slope and y-intercept Determine a Linear Equation from the Slope and a Point Determine a Linear Equation from Two Points Determine a Linear Equation from Two Points by Graphing

Graphing linear equations and reading existing graphs give students a visual representation that is very useful in understanding the concepts of slope and y-intercept.

• Graphing Linear Equations Graph Slope-Intercept Equations
• Determining Linear Equations from Graphs Determine the Equation from a Graph Determine the Slope from a Graph Determine the y-intercept from a Graph Determine the x-intercept from a Graph Determine the slope and y-intercept from a Graph Determine the slope and intercepts from a Graph Determine the slope, intercepts and equation from a Graph

Solving linear equations with jelly beans is a fun activity to try with students first learning algebraic concepts. Ideally, you will want some opaque bags with no mass, but since that isn't quite possible (the no mass part), there is a bit of a condition here that will actually help students understand equations better. Any bags that you use have to be balanced on the other side of the equation with empty ones.

Probably the best way to illustrate this is through an example. Let's use 3 x + 2 = 14. You may recognize the x as the unknown which is actually the number of jelly beans we put in each opaque bag. The 3 in the 3 x means that we need three bags. It's best to fill the bags with the required number of jelly beans out of view of the students, so they actually have to solve the equation.

On one side of the two-pan balance, place the three bags with x jelly beans in each one and two loose jelly beans to represent the + 2 part of the equation. On the other side of the balance, place 14 jelly beans and three empty bags which you will note are required to "balance" the equation properly. Now comes the fun part... if students remove the two loose jelly beans from one side of the equation, things become unbalanced, so they need to remove two jelly beans from the other side of the balance to keep things even. Eating the jelly beans is optional. The goal is to isolate the bags on one side of the balance without any loose jelly beans while still balancing the equation.

The last step is to divide the loose jelly beans on one side of the equation into the same number of groups as there are bags. This will probably give you a good indication of how many jelly beans there are in each bag. If not, eat some and try again. Now, we realize this won't work for every linear equation as it is hard to have negative jelly beans, but it is another teaching strategy that you can use for algebra.

Despite all appearances, equations of the type a/ x are not linear. Instead, they belong to a different kind of equations. They are good for combining them with linear equations, since they introduce the concept of valid and invalid answers for an equation (what will be later called the domain of a function). In this case, the invalid answers for equations in the form a/ x , are those that make the denominator become 0.

• Solving Linear Equations Combining Like Terms and Solving Simple Linear Equations Solving a x = c Linear Equations Solving a x = c Linear Equations including negatives Solving x /a = c Linear Equations Solving x /a = c Linear Equations including negatives Solving a/ x = c Linear Equations Solving a/ x = c Linear Equations including negatives Solving a x + b = c Linear Equations Solving a x + b = c Linear Equations including negatives Solving a x - b = c Linear Equations Solving a x - b = c Linear Equations including negatives Solving a x ± b = c Linear Equations Solving a x ± b = c Linear Equations including negatives Solving x /a ± b = c Linear Equations Solving x /a ± b = c Linear Equations including negatives Solving a/ x ± b = c Linear Equations Solving a/ x ± b = c Linear Equations including negatives Solving various a/ x ± b = c and x /a ± b = c Linear Equations Solving various a/ x ± b = c and x /a ± b = c Linear Equations including negatives Solving linear equations of all types Solving linear equations of all types including negatives

Algebra rectangles are rectangles that use linear expressions for the side measurements. With a known value (such as the perimeter), students create an algebraic equation that they can solve to determine the value of the unknown (x) and use it to determine the side lengths and area of the rectangle. The terminology in identifying the various options for worksheets use the standard equation y = mx + b where m is the coeffient of x that is generally a known value.

• Algebra Rectangles Algebra Rectangles -- Determining the Value of x, Length, Width and Area Using Algebraic Sides and the Perimeter -- m Range [1,1] Algebra Worksheet -{}- Algebra Rectangles -- Determining the Value of x, Length, Width and Area Using Algebraic Sides and the Perimeter -- m Range [2,9] Algebra Worksheet -{}- Algebra Rectangles -- Determining the Value of x, Length, Width and Area Using Algebraic Sides and the Perimeter -- m Range [2,9] or [-9,-2] Algebra Worksheet -{}- Algebra Rectangles -- Determining the Value of x, Length, Width and Area Using Algebraic Sides and the Perimeter -- m Range [2,9] or [-9,-2] -- Inverse m Possible

Linear Systems

• Solving Systems of Linear Equations Easy Linear Systems with Two Variables Easy Linear Systems with Two Variables including negative values Linear Systems with Two Variables Linear Systems with Two Variables including negative values Easy Linear Systems with Three Variables; Easy Easy Linear Systems with Three Variables including negative values Linear Systems with Three Variables Linear Systems with Three Variables including negative values
• Solving Systems of Linear Equations by Graphing Solve Linear Systems by Graphing (Solutions in first quadrant only) Solve Standard Linear Systems by Graphing Solve Slope-Intercept Linear Systems by Graphing Solve Various Linear Systems by Graphing Identify the Dependent Linear System by Graphing Identify the Inconsistent Linear System by Graphing

Quadratic Expressions and Equations

• Simplifying (Combining Like Terms) Quadratic Expressions Simplifying quadratic expressions with 5 terms Simplifying quadratic expressions with 6 terms Simplifying quadratic expressions with 7 terms Simplifying quadratic expressions with 8 terms Simplifying quadratic expressions with 9 terms Simplifying quadratic expressions with 10 terms Simplifying quadratic expressions with 5 to 10 terms
• Adding/Subtracting and Simplifying Quadratic Expressions Adding and simplifying quadratic expressions. Adding and simplifying quadratic expressions with multipliers. Adding and simplifying quadratic expressions with some multipliers. Subtracting and simplifying quadratic expressions. Subtracting and simplifying quadratic expressions with multipliers. Subtracting and simplifying quadratic expressions with some multipliers. Mixed adding and subtracting and simplifying quadratic expressions. Mixed adding and subtracting and simplifying quadratic expressions with multipliers. Mixed adding and subtracting and simplifying quadratic expressions with some multipliers.
• Multiplying Factors to Get Quadratic Expressions Multiplying Factors of Quadratics with Coefficients of 1 Multiplying Factors of Quadratics with Coefficients of 1 or -1 Multiplying Factors of Quadratics with Coefficients of 1, or 2 Multiplying Factors of Quadratics with Coefficients of 1, -1, 2 or -2 Multiplying Factors of Quadratics with Coefficients up to 9 Multiplying Factors of Quadratics with Coefficients between -9 and 9

The factoring quadratic expressions worksheets in this section provide many practice questions for students to hone their factoring strategies. If you would rather worksheets with quadratic equations, please see the next section. These worksheets come in a variety of levels with the easier ones are at the beginning. The 'a' coefficients referred to below are the coefficients of the x 2 term as in the general quadratic expression: ax 2 + bx + c. There are also worksheets in this section for calculating sum and product and for determining the operands for sum and product pairs.

• Factoring Quadratic Expressions Factoring Quadratic Expressions with Positive 'a' coefficients of 1 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients of 1 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients of 1 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 4 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 4 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 4 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 5 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 5 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 5 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 9 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 9 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 9 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 81 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 81 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 81 with a Common Factor Step Calculating Sum and Product (Operand Range 0 to 9 ) ✎ Calculating Sum and Product (Operand Range 1 to 9 ) ✎ Calculating Sum and Product (Operand Range 0 to 9 Including Negatives ) ✎ Calculating Sum and Product (Operand Range 1 to 9 Including Negatives ) ✎ Calculating Sum and Product (Operand Range -20 to 20 ) ✎ Calculating Sum and Product (Operand Range -99 to 99 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 9 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 9 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 12 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 12 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 9 Including Negatives ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 9 Including Negatives ) ✎ Determining Operands from Sum and Product Pairs (Operand Range -20 to 20 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range -99 to 99 ) ✎

Whether you use trial and error, completing the square or the general quadratic formula, these worksheets include a plethora of practice questions with answers. In the first section, the worksheets include questions where the quadratic expressions equal 0. This makes the process similar to factoring quadratic expressions, with the additional step of finding the values for x when the expression is equal to 0. In the second section, the expressions are generally equal to something other than x, so there is an additional step at the beginning to make the quadratic expression equal zero.

• Solving Quadratic Equations that Equal Zero Solving Quadratic Equations with Positive 'a' coefficients of 1 Solving Quadratic Equations with Positive or Negative 'a' coefficients of 1 Solving Quadratic Equations with Positive or Negative 'a' coefficients of 1 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 4 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 4 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 4 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 5 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 5 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 5 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 9 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 9 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 9 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 81 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 81 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 81 with a Common Factor Step
• Solving Quadratic Equations that Equal an Integer Solving Quadratic Equations for x ("a" coefficients of 1) Solving Quadratic Equations for x ("a" coefficients of 1 or -1) Solving Quadratic Equations for x ("a" coefficients up to 4) Solving Quadratic Equations for x ("a" coefficients between -4 and 4) Solving Quadratic Equations for x ("a" coefficients up to 81) Solving Quadratic Equations for x ("a" coefficients between -81 and 81)

Other Polynomial and Monomial Expressions & Equations

• Simplifying Polynomials That Involve Addition And Subtraction Addition and Subtraction; 1 variable; 3 terms Addition and Subtraction; 1 variable; 4 terms Addition and Subtraction; 2 variables; 4 terms Addition and Subtraction; 2 variables; 5 terms Addition and Subtraction; 2 variables; 6 terms
• Simplifying Polynomials That Involve Multiplication And Division Multiplication and Division; 1 variable; 3 terms Multiplication and Division; 1 variable; 4 terms Multiplication and Division; 2 variables; 4 terms Multiplication and Division; 2 variables; 5 terms
• Simplifying Polynomials That Involve Addition, Subtraction, Multiplication And Division All Operations; 1 variable; 3 terms All Operations; 1 variable; 4 terms All Operations; 2 variables; 4 terms All Operations; 2 variables; 5 terms All Operations (Challenge)
• Factoring Expressions That Do Not Include A Squared Variable Factoring Non-Quadratic Expressions with No Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Compound Coefficients, and Negative and Positive Multipliers
• Factoring Expressions That Always Include A Squared Variable Factoring Non-Quadratic Expressions with All Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Compound Coefficients, and Negative and Positive Multipliers
• Factoring Expressions That Sometimes Include Squared Variables Factoring Non-Quadratic Expressions with Some Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Compound Coefficients, and Negative and Positive Multipliers
• Multiplying Polynomials With Two Factors Multiplying a monomial by a binomial Multiplying two binomials Multiplying a monomial by a trinomial Multiplying a binomial by a trinomial Multiplying two trinomials Multiplying two random mon/polynomials
• Multiplying Polynomials With Three Factors Multiplying a monomial by two binomials Multiplying three binomials Multiplying two binomials by a trinomial Multiplying a binomial by two trinomials Multiplying three trinomials Multiplying three random mon/polynomials

Inequalities

• Writing The Inequality That Matches The Graph Writing Inequalities for Graphs
• Graphing Inequalities On Number Lines Graphing Inequalities (Basic)
• Solving Linear Inequalities Solving Inequalities Including a Third Term Solving Inequalities Including a Third Term and Multiplication Solving Inequalities Including a Third Term, Multiplication and Division

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6th Grade Algebra Worksheets

6th grade algebra worksheets help students to get a better idea of various concepts related to algebra. The worksheets include questions on solving algebraic expressions, word problems, and other associated questions.

Benefits of Algebra 6th Grade Math Worksheets

The scope of algebra is very wide and hence it can prove to be confusing for some. 6th grade algebra worksheets provide numerous problems that can help children practice, helping them to develop a strong foundation in the topic. The worksheets provide immense flexibility, enabling students to work at their own pace without getting pressured. Another benefit of these 6th grade math worksheets is that they come with an answer key that outlines the step-by-step process of how to correctly solve the questions. If a student gets stuck while solving a question he can refer to this key and immediately solve his doubts.

☛ Practice : Grade 6 Interactive Algebra Worksheets

Printable PDFs for Algebra Worksheets for Grade 6

The algebra 6th grade math worksheets are easy to use and free to download.

• Algebra Math Worksheets for Grade 6
• Sixth Grade Algebra Math Worksheet
• Math Algebra Worksheet for 6th Grade
• Algebra 6th Grade Worksheet

Interactive 6th Grade Algebra Worksheets

• Grade 6 Decoding and Prediction of Patterns Algebra Worksheet
• An Introduction to Variables Algebra Worksheet for Grade 6
• Practice on Conversion of Statements to Algebraic Expressions Worksheet for 6th Grade
• Practice on Variables, Constants, and Terms 6th Grade Algebra Worksheet
• Simplification of Expressions Grade 6 Algebra Worksheet

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Basic Algebra Worksheets

Welcome to the Math Salamanders' Basic Algebra Worksheets. Here you will find a range of algebra worksheets to help you learn about basic algebra, including generating and calculating algebraic expressions and solving simple problems.

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Generate the Expression Worksheets

Calculate the Expression Worksheets

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• Basic Algebra Online Quiz

Want to gain a basic understanding of algebra?

Looking for some simple algebra worksheets?

Do you need a bank of useful algebra resources?

Look no further! The pages you need are below!

Here is our selection of basic algebra sheets to try.

We have split the worksheets up into 3 different sections:

• Generate the algebra - and write your own algebraic expressions;
• Calculate the algebra - work out the value of different expressions;
• Solve the algebra - find the value of the term in the equation.

By splitting the algebra up into sections, you only need to concentrate on one aspect at a time!

Each question sheet comes with its own separate answer sheet.

Want to test yourself to see how well you have understood this skill?.

• Try our NEW quick quiz at the bottom of this page.

What is an algebraic expression?

An expression is a mathematical statement where variables and operations are combined.

• 2a + 5 is an expression involving the variable a
• 5(y 2 - 6) is another expression

What is an algebraic equation?

An equation is where an algebraic expression is equal to something, which might be a number, or another algebraic expression.

• 2a + 5 = 7 is an equation
• 5(y 2 - 6) = 3y + 8 is another equation

How to Generate an Expression

When we are generating an expression, we are taking a rule and turning it into algebra.

• Subtract 6 from n could be written as n - 6.
• Multiply d by 4 could be written as d x 4 or 4d.
• Add 5 to p and then double the result is written as (p + 5) x 2 or 2(p + 5)

How to Calculate an Expression

When we are calculating the value of an expression, we work out the value of the expression when we give a value to the variable.

• p + 5 has a value of 11 when p = 6 because 6 + 5 = 11
• 4(n - 2) has a value of 32 when n = 10 because 4 x (10 - 2) = 4 x 8 = 32
• 4(n - 2) has a value of -8 when n = 0 because 4 x (0 - 2) = 4 x (-2) = -8

How to Solve a Simple Equation

When we are solving an equation, we are finding out the value(s) of the variable in the equation.

• If p + 5 = 9 then p = 4 because 4 + 5 = 9
• then (n - 2) = 28 ÷ 4 = 7
• if (n - 2) = 7 then n = 7 + 2 = 9
• Answer: n = 9
• means that 3f = 12
• so f = 12 ÷ 3 = 4
• Answer: f = 4

Basic Algebra Worksheets for kids

• Generate the Expression 1
• PDF version
• Generate the Expression 2
• Generate the Expression 3

Generate the Expression Word Problems

• Algebra Word Problems 1
• Algebra Word Problems 2
• Algebra Word Problems 3
• Algebra Word Problems 3 UK Version

Algebra Word Problems Walkthrough Video

This short video walkthrough shows the problems from our Algebra Word Problems Worksheet 2 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, please check out the video below!

• Calculate the Expression 1
• Calculate the Expression 2
• Calculate the Expression 3

Solve the Equation Worksheets

• Solve the Equation 1
• Solve the Equation 2
• Solve the Equation 3

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Free Algebra Problem Solver

The Mathway Calculator is a great way to solve algebra problems that you can type into a calculator.

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There are a range of calculators to choose from to meet your needs.

The Mathway problem solver will answer your problem instantly and also give you a link to view each of the steps needed.

If you choose to 'View the steps' you will be directed to the Mathway website where you will be able to see in more detail each of the steps needed to solve the problem. Please note that Mathway may charge you a small fee for this!

• 6th Grade Distributive Property Worksheets

The sheets on this page have been designed to factorize and expand a range of simple expressions using the distributive property..

• Expressions and Equations 6th Grade

The sheets on this page have been specially designed for 6th graders and are a great introduction to expressions and equations.

Input and Output Function Tables

Have a go at these input and output tables.

Test your skills at finding inputs, outputs and rules.

Our easier sheets have rules instead of algebraic functions.

Our harder sheets use algebraic functions.

• Input and Output Tables Worksheets (easier)
• Input and Output Function Tables with Algebraic Functions

Factorising Quadratic Equations

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Take a look at our support pages on quadratic equations where you will hopefully find what you are looking for.

• Factorising Quadratic Equations Support page
• Factoring Quadratic Equations worksheets
• Algebra Math Games

If you are looking for a fun printable algebra game to play then try out our algebra game page.

You will find a range of algebra games that make learning algebra fun and non-threatening.

The only equipment you need is a scientific calculator, some dice, and a few counters!

PEMDAS Worksheets

The sheets in this section involve using parentheses and exponents in simple calculations.

There are also lots of worksheets designed to practice and learn about PEMDAS.

Using these worksheets will help your child to:

• know and understand how parentheses works;
• understand how exponents work in simple calculations.
• understand and use PEMDAS to solve a range of problems.
• PEMDAS Problems Worksheets 5th Grade
• 6th Grade Order of Operations

Interactive Equality Explorer

This interactive equality explorer has been produced by PhET Interactive Simulations at the University of Colorado.

It is a useful tool for exploring different ideas including negative numbers and algebra equations and equality.

Probably the most useful part of the app is to use the 'Solve It' section once you are confident how it works.

You can then select your level of difficulty and start solving some algebraic equations by getting your variables onto one side of the equation and the numerical values on the other, and then multiplying or dividing the equation until you find the value of the required variable.

• Interactive Equality Explorer by PhET

Basic Algebra Quiz

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This quick quiz tests your knowledge and skill at generating and calculating expressions, as well as solving equations.

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Algebra 1 Warm-Ups Inequalities and Systems (11 topics 42 problems)

Subject: Mathematics

Age range: 11-14

Resource type: Worksheet/Activity

Last updated

21 August 2024

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These are 11 Algebra 1 warm-ups, do-nows, bell-ringers, entrance/exit tickets or mini-quizzes/mini homework on inequalities and systems. Each of them is a half-sheet in size and contains 4 problems, only 10th and 11th warm- ups have each 3 text problems.

Topics included:

Multi-Step Inequalities (Variables on Same Side) Multi-Step Inequalities (Variables on Both Sides) Multi-Step Inequalities (Special Cases) Absolute Value Inequalities Compound Inequalities (OR) Compound Inequalities (AND) Systems of Linear Equations (Solving by Graphing) Systems of Linear Equations (Solving by Substitution) Systems of Linear Equations (Solving by Elimination) Systems of Linear Equations (Text Problems) Multi-Step Inequalities (Text Problems)

Answer keys are included.

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Get this resource as part of a bundle and save up to 20%

A bundle is a package of resources grouped together to teach a particular topic, or a series of lessons, in one place.

ALGEBRA 1 WARM-UPS Entire Year BUNDLE (Growing) (65 topics, 386 problems)

This is a bundle of Algebra 1 warm-ups/bell-ringers/exit/entrance tickets/do-nows/mini quizzes/mini homework for the entire year. Each of warm-ups is a half-sheet in size and contains from 2 to 14 problems. So far the bundle covers the following topics: Multi-Step Equations (Variables on Same Side) Multi-Step Equations (Variables on Both Sides) Multi-Step Equations (Special Case Solutions) Absolute Value Equations Quadratic Equations (Square Root Method) Quadratic Equations (Solving by Factoring) Quadratic Equations (Complete the Square) Quadratic Equations (Quadratic Formula) Literal Equations Proportions (Cross Multiplication) Writing & Solving Equations (Text Problems) Ratios and Proportions (Text Problems) Multi-Step Inequalities (Variables on Same Side) Multi-Step Inequalities (Variables on Both Sides) Multi-Step Inequalities (Special Cases) Absolute Value Inequalities Compound Inequalities (OR) Compound Inequalities (AND) Systems of Linear Equations (Solving by Graphing) Systems of Linear Equations (Solving by Substitution) Systems of Linear Equations (Solving by Elimination) Systems of Linear Equations (Text Problems) Multi-Step Inequalities (Text Problems) Exponent Laws (Product and Power Rules) Exponent Laws (Quotient Rule) Exponent Laws (Zero and Negative Exponents) Scientific Notation and Operations with Scientific Notation Simplifying Radicals (No Variables) Simplifying Radicals with Variables Adding and Subtracting Radicals Multiplying Radicals Dividing Radicals Rationalizing the Denominator Operations with Radicals Combining Like Terms Distributive Property Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying Binomials (FOIL Method) Multiplying Polynomials Dividing a Polynomial by a Monomial Dividing Polynomials (No Remainders) Factoring Out the GCF Factoring Difference of Squares Factoring Sum and Difference of Cubes Factoring Perfect Square Trinomials Factoring Quadratic Trinomial Factoring by Grouping Factoring by Combined Methods Slope of a Line from 2 Points Slope of a Line from a Graph Slope of a Line from a Table Slope of a Line from an Equation Graphing Lines in Slope-Intercept and Standard Forms Writing Equations of Lines from Graphs Writing Equations of Lines from 2 Points Writing Equation of a Line Given the Slope and a Point Writing Equation of a Line from a Table of Values Equations of Parallel and Perpendicular Lines X- and Y- Intercepts Linear Function Applications Inverse Linear Equations Piecewise Linear Functions Graphing Absolute Value Functions Equations of Proportional and Non-Proportional Relationships Answer keys are included.

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How to Excel in Mental Math Practice: A Comprehensive Guide

Boost your mental math skills with our comprehensive guide! Learn practical tips and techniques to excel in mental math and enhance your problem-solving abilities.

Posted August 20, 2024

Featuring Garrett W.

MBB Interviews: Ask Me Anything

Starting friday, august 23.

7:00 PM UTC · 60 minutes

Table of Contents

Ever wondered how some people can solve complex math problems in their heads with lightning speed? The secret lies in mental calculation. It's not just a cool party trick; it's a powerful tool that boosts problem-solving abilities and sharpens cognitive function. Whether you're prepping for case interviews, aiming to excel in consulting, or just looking to improve your mental agility, mastering mental math can give you a significant edge.

In this comprehensive guide, we'll talk about mental calculation, which will provide you with strategies for general practice and techniques to apply these skills in real-world scenarios. You'll also discover how to leverage technology for enhanced learning and get practical exercises to sharpen your abilities. By the end of this article, you'll be well on your way to becoming a mental math whiz, ready to tackle any mathematical challenge with confidence.

What is Mental Math? And What Are Its Techniques?

Mental math , or mental calculation, involves cognitive strategies that enhance flexible thinking and number sense. It requires performing calculations mentally without relying on a calculator or other external aids. This practice improves computational fluency by focusing on efficiency, accuracy, and flexibility. For consultants, strong mental mathematics skills are crucial, as they often need to handle quantitative data and make informed decisions on the spot.

Bringing Mental Math in Consulting Interview

Consulting firms like McKinsey, BCG, and Bain test mental math skills throughout the interview process. Candidates must perform quick calculations under pressure, often focusing on getting "close enough" answers to guide recommendations rather than 100% accuracy.

Developing a Math Mindset

To excel in mental math, practice simplifying large numbers by removing zeros and using labels (m, k, b). Break down complex problems into smaller parts and group numbers into multiples of 10 for easier addition.

Visualization Techniques

Visualizing an abacus in your mind is an effective method for mental math. Other techniques include imagining number lines, grids, and associating numbers with shapes or rhythms. Most people find it challenging to perform mental calculations without a visual aid, so incorporating these visualization techniques can significantly improve your accuracy and speed.

Pattern Recognition

Pattern recognition strengthens your ability to understand shapes, sequences, and systems. Practicing games like chess, sudoku, and puzzles can enhance your pattern recognition skills, which are valuable in mental math and problem-solving.

How to Practice Mental Math Skills

Use math drills.

Math drills are short, timed tests that improve accuracy, speed, and long-term memory in mental calculations. They're essential for case interview preparation, as consulting firms test these skills throughout the interview process. By practicing consistently, you'll build the core skills needed when under pressure.

5 Ways to Use Math Drills Effectively

• Gradually increase difficulty : Begin with simple arithmetic problems to build your confidence, then progressively tackle more complex mathematics exercises. This method allows you to master mental math tricks step by step, ensuring a strong foundation before moving on to tougher challenges. Over time, this approach will improve your ability to handle intricate calculations under pressure.
• Practice consistently : Consistency is key in developing your math skills. Set a regular schedule for your training, starting with shorter sessions and gradually increasing the time and frequency as your interviews approach. By integrating math drills into your daily routine, you'll reinforce your learning and develop the stamina needed to perform well in real scenarios.
• Focus on weak areas : Identify the specific mathematics concepts or simple arithmetic operations that you find challenging, and allocate extra time to practice them. By zeroing in on these weak points, you can turn them into strengths, ensuring a well-rounded skill set that covers all aspects of case interview math.
• Replicate interview conditions : To prepare for the real thing, practice your math drills in a way that mimics the interview environment. Speak through your calculations aloud, explaining your thought process as you work through problems. This not only helps you get comfortable with verbalizing your reasoning but also trains you to stay calm and articulate under pressure.
• Prioritize accuracy : While speed is important, accuracy should always be your top priority. Aim for over 90% accuracy in your drills, and revisit any areas where you fall short. Consistent accuracy in your mental math training will help ensure that you can deliver precise results in actual interview scenarios, making a strong impression on your interviewers.

Remember, frequent, short practice sessions are most effective for retaining math facts.

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Mastering the Fundamentals of Consulting Math Skills

To excel in consulting, you need to master essential math skills . Let's explore key concepts that will boost your problem-solving abilities.

Fractions represent parts of a whole, like 1/2 or 3/4. You should be able to add, subtract, multiply, and divide fractions effortlessly. In consulting, understanding fractions is essential for breaking down and comparing complex data sets, especially when analyzing percentages and distributions. For example, if a company’s revenue is split across different regions, quick mental manipulation of fractions allows you to determine the contribution of each region to the overall performance. Mastering fractions also helps in interpreting financial reports and profitability breakdowns.

Ratios compare quantities, such as 3:4:1 for pens, pencils, and erasers. They're crucial for analyzing relationships between variables, such as in market sizing or operations analysis. Understanding ratios allows you to evaluate proportional relationships, which is often required when comparing growth rates, resource allocation, or revenue distribution across different business segments. Ratios also appear frequently in scenarios like supply chain optimization, where you may need to balance inventory levels across various locations or product lines.

Percentages

Percentages are fractions with a denominator of 100. They're ubiquitous in business, finance, and case studies. Whether you’re calculating profit margins, growth rates, or percentage change in costs, having a strong command of percentages is key to making quick and informed decisions. For instance, in a profitability analysis, you may need to determine the percentage of revenue that comes from a particular product or service, allowing you to quickly assess its relative importance and make data-driven recommendations.

Probability

Probability measures the likelihood of events, ranging from 0 (impossible) to 1 (certain). It's essential for evaluating risks, understanding uncertainty, and making informed decisions. In consulting, probability comes into play when assessing the likelihood of project outcomes, conducting risk assessments, or forecasting future trends based on historical data. For example, you might evaluate the probability of a marketing campaign's success or calculate the risks associated with entering a new market. Probability concepts help consultants think critically about uncertainty and quantify risks in decision-making processes.

Averages, typically referring to the arithmetic mean, measure the "typical" value in a series of numbers. They're fundamental for data analysis, and consultants often use averages to summarize performance metrics, customer behaviors, and market trends. A deep understanding of averages allows you to derive insights from large data sets quickly. For example, calculating the average revenue per customer helps identify opportunities for upselling or increasing customer retention. Being adept at calculating and interpreting averages is key to identifying trends and making informed recommendations.

Rates are ratios with a denominator of 1, like interest rates or exchange rates. They're crucial for financial calculations and comparisons in consulting, especially when dealing with cost analysis, ROI, and financial performance. Understanding rates helps you quantify changes over time, such as the rate of growth in revenue or the rate at which costs are increasing. For instance, analyzing interest rates or inflation rates enables consultants to predict how external factors might impact a client’s profitability or future investments. Rates also play a critical role in pricing strategies and cost optimization efforts.

Optimization

Optimization involves maximizing or minimizing variables to achieve the best possible outcome. While complex in academia, it's straightforward in consulting interviews, focusing on practical problem-solving. Optimization techniques allow you to make data-driven decisions, such as minimizing costs, maximizing revenue, or optimizing resource allocation. For example, you might be tasked with optimizing the production schedule for a manufacturing company to reduce downtime or developing strategies for a retailer to maximize profit margins by optimizing product pricing and inventory levels. Understanding the principles of optimization helps consultants craft practical solutions to improve business performance.

Practical Mental Math Strategies

Breaking down complex problems.

To excel in mental math, break larger problems into smaller, manageable parts.

For instance, when calculating 3 x 16, think of it as 3 x 10 = 30, then 3 x 6 = 18, adding 30 to 18 = 48 . This approach simplifies complex calculations, making them easier to solve mentally.

Using Benchmark Numbers

Benchmark numbers, typically ending in zeros or fives, serve as reference points for easier calculations.

For example, when estimating 49 + 23, round to 50 and 20, respectively, for a quick approximation. This technique is particularly useful for comparing numbers and simplifying equations.

Applying the Distributive Property

The distributive property is a powerful tool for mental math. When multiplying 9 by 37, break it down to (9 x 30) + (9 x 7) for easier calculation. This method is especially useful for larger numbers and can significantly speed up mental computations.

Mastering the Basic Charts

Familiarize yourself with basic multiplication tables and number relationships. Practice consistently to build core skills needed under pressure. Remember, frequent, short practice sessions are most effective for retaining math facts.

Integrate Technology in Mental Math Practice

Math apps and games.

To enhance your mental math skills, consider using apps like FastMath. This app offers a variety of exercises, including addition, subtraction, multiplication, and division, with different number types and difficulty levels. You can race against time, set high scores, and compete with friends on leaderboards. These features make practicing mental math engaging and addictive.

Online Tutorials and Courses

Boost your mental arithmetic skills with online resources. YouTube offers numerous mental math tutorials, and platforms that provide free math lessons and exercises. For a more in-depth approach, Stanford University's ' Introduction to Mathematical Thinking ' course can help develop your critical thinking and quantitative reasoning abilities. Each resource or online course serves as an articulate math trainer – offering a conceptual approach to building strong number sense and problem-solving skills.

Virtual Math Communities

Engage with others to boost your mental math prowess. Playing multiplayer mental math games online can make practice more enjoyable and improve your agility. Joining virtual math communities allows you to share strategies and learn from others. Remember, consistent practice using various resources is key to developing strong quantitative skills.

Land Your Dream Consulting Job With the Help of an Expert

Trying to enter into the world of consulting is no small feat. That’s why at Leland, we have a broad network of world-class coaches who know what it takes to get into a consulting job and are ready to help review your resumes, conduct practice interviews, and give you refreshers on key skills needed to land the job. Browse our expert coaches here and find the highest-rated ones below.

Mastering mental math has a significant impact on problem-solving abilities and cognitive function. The techniques and strategies discussed in this guide provide a solid foundation to enhance your skills. By consistently practicing math drills , breaking down complex problems, and using benchmark numbers, you can boost your mental math prowess. This not only prepares you for consulting interviews but also sharpens your mind for everyday challenges.

To keep improving, it's crucial to integrate technology into your practice routine. Math apps, online tutorials, and virtual communities offer engaging ways to hone your skills. Remember, the key to excelling in mental math lies in regular practice and applying these techniques in real-world scenarios. As you continue to develop your mental math abilities, you'll find yourself better equipped to tackle quantitative problems with confidence and ease.

Frequently Asked Questions

What are some effective ways to quickly improve my mental math skills?

• To enhance your mental math abilities swiftly, you can use several techniques such as practicing with flashcards to reinforce repetition, playing number bond games, developing logical thinking, and utilizing apps like DoodleMaths for structured practice.

What strategies can help me excel in a mental math test?

• To prepare for a mental math test, you can adopt several strategies such as breaking down addition and subtraction problems into smaller parts, rounding numbers to simplify calculations, learning to add multiple numbers simultaneously, using left-to-right multiplication methods, applying fast multiplication tricks for numbers between 11 and 19, and simplifying calculations with numbers ending in zero.

How can I improve my skills in the mental math division?

• Improving mental math division involves several strategies like estimating the quotient before beginning the division, rounding numbers to make calculations easier, breaking down complex problems into simpler steps, utilizing multiplication facts to aid division, dividing by powers of 10 for efficiency, adjusting the quotient as needed, practicing with known multiples, and regularly engaging in mental math exercises.

How is mental math used to solve equations?

• Mental math can be effectively used to solve simple equations by translating the equation into a verbal question and then solving it mentally. This often involves using inverse operations to rephrase and simplify the problem if the solution is not immediately apparent.

Preparing for consulting interviews? Here are some additional resources to help:

Case Interview Math Guide: Everything You Need to Know

How to answer the "why consulting" interview question.

• 50+ Case Interview Questions From Top Firms

The Expert Guide to Market Sizing Questions (With Examples)

• Big 4 Consulting Firms vs. MBB: What's the Difference

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The Math You Need, When You Need It

math tutorials for students majoring in the earth sciences

Probability - Practice Problems Solving Earth science problems with probability

Sedimentology.

Hurricane prediction

$P(\text{no landslides per storm}) = 1 = 0.10 = 0.9$

$N = log_{\text{(prob. of non-occurrence)}}(-(\text{prob. of interest}-1)) = N$ .

We want to know how many storms it will take to have a 50% chance of having a landslide, so our probability of interest is 0.5. We can plug this value and our probability of non-occurrence into that equation.

$N = log_{0.9}(-(0.5-1))$

$N = log_{0.9}(-(0.5-1)) = 6.58$

Thus, there is a greater than 50% probability of landsliding over 7 or more storms.

Geochronology

By the exponent rule, the probability of the atom not decaying over 3 successive half-lives is calculated as follows:

$P(\text{not decaying over three half lives}) = P(\text{not decaying over one half life})^3 = 0.5^3 = 0.125$

There is a 12.5% chance that the uranium-238 atom will not decay over 3 half-lives.

By the complement rule, since the probability of not decaying over 3 half-lives is 0.125, the probability of decay over 3 half-lives is 1 - 0.125 = 0.875.  There is an 87.5% chance that the uranium-238 atom will undergo radioactive decay within 3 half-lives, or 13.5 billion years.

$P(\text{no gold}) = 1 - 0.01 = 0.99$

$N = log_{P(\text{non-occurrence outcome})}(1 - P(\geq\text{1 outcome of interest over interval N}))$

Substitute the non-occurrence outcome and the target probability for an outcome over the interval into this equation:

$N = log_{0.99}(1 - 0.5) = log_{0.99}(0.5)$

$N = log_{0.99}(1 - 0.5) = log_{0.99}(0.5) = 69.0$

So, you would need to make mining claims on 69 acres to have an overall >50% likelihood of finding gold.

If you feel comfortable with this topic, you can go on to the assessment . Or you can go back to the XXX explanation page .

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Quizard AI: Homework Helper 4+

Instant math problem solver, quizard ai, inc., designed for iphone.

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Homework & AI Answers Solver. Are you struggling to keep up with your homework and studying? Don't worry, Quizard is here to help! Quizard is a revolutionary AI answer app designed to help students at all levels conquer their studies. Now introducing our new feature - AI answers for math questions! Just snap a photo of your math problem, and Quizard's AI will deliver the solution, along with a comprehensive explanation. This makes Quizard the perfect tool not only for general studies but also for math homework help. Whether you’re a college student, high school student, or even an adult looking to brush up on your knowledge, Quizard is the perfect tool to help you succeed. With Quizard, you can quickly and easily get help with multiple-choice questions and short answer problems. You can quickly and easily prepare for quizzes, tests, and exams, allowing you to confidently ace them. Quizard is free to use! With Quizard, you can get the help you need to understand the material and gain a better understanding of the subject. Quizard is the perfect homework helper and personal tutor, providing you with the answers you need to succeed. With Quizard, you can get help with your homework and studying, so you can get better grades and have more free time. Stop struggling with your homework and start using Quizard today! Fine print: • Payment will be charged to your Apple ID account at the confirmation of purchase. • Offers and pricing are subject to change without notice. Terms of Use: https://lovely-vault-f15.notion.site/Terms-of-Use-18a469738fd74c9bb7c919981de4bbcd Privacy policy: https://lovely-vault-f15.notion.site/Privacy-Policy-4fe9e56db4a04d88af6e1b78f2874a7d Suggestions or questions? Email us at [email protected] TikTok: @quizard.ai Instagram: @quizard.ai

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Quizzard might just be the hidden gem in the app store, assuming you like your learning served with a heaping spoonful of whimsy and a side of technicolor mayhem. The interface could be described as a unicorn's dream, painted in every hue of the rainbow and then some. It's so cheerfully bright, you'll forget whether you're here to learn or to play a round of digital laser tag. But it’s not all about looks; Quizzard delivers content with a zesty twist that keeps you on your toes. Trivia questions that make your brain do somersaults are the norm here. Ever wanted to decode the molecular structure of caffeine in under a minute or guess Shakespeare's favorite cheese? Quizzard makes such intellectual gymnastics delightful, and you’ll chuckle as much as you'll ponder. Plus, the ads, while frequent, are an adventure in themselves, offering you snippets of the outside world at the most unexpected moments. It’s like getting commercial breaks during your personal game show—annoying but part of the fun. So, strap in for a quiz experience that’s as entertaining as it is enlightening, where every question is a surprise party for your neurons!

GREAT LEARNING TOOL MADE BY A GENIUS DUDE!!! Quizard reads your question and like searches the web for similar information to come up with an answer. Quizard provides a paragraph explanation along with the short answer. READ THOROUGHLY bc it is a computer and sometimes the wording makes it come up with “the wrong (short) answer” even though throughout the (longer paragraph) explanation you can figure out the true answer that Quizard happened to word incorrectly or chose a similar but wrong multiple choice option for. If you’re not trying using it for just the short answer not bothering to read and check the answer and dont read the paragraph then ya.. some of your answers are gonna be wrong. But if it doesn’t give you the right answer it’ll at least provide some explanation to put you on the right path.

DO NOT GET THIS APP!

So, I’m in 4th grade, 4TH GRADE! & IT COULDN’T HELP ME CORRECTLY! So, you know in elementary school papers, & the printer will add these little pictures to the side, well, when i took a picture of this paper, quizard told me that “ I’m sorry, there’s a object in this paper, i cannot tell you the correct answer unless you cover it” ….. IM SORRY! IS A SMALL PICTURE DISTRACTING YOU!? & HOW IS PUTTING SOMETHING OVER IT GOING TO HELP!? Mm!? Well, i cover it, & it is so stupid! It can’t even proofread! It told me that the words that were “incorrect” (by the way, i fact checked it, & the words were correct) & it tells me that the correct answer for gymnasium is “gymnasuim” …i’m sorry, did you just make up a word? WHAT KIND OF AI ARE YOU!? So, i tell it that it’s incorrect, it’s actually “gymnasium” & it tells me, “ I’m sorry for the inconvenience, but can you explain the answer’s to me?” ….. WHY DO YOU THINK I GOT THIS APP!? I mean like, it’s not like I wanted YOU to explain it to me! So, yeah, I'm not going to sit there & waist my time trying to explain a ELEMENTARY SCHOOL PAPER TO AN AI! So, I definitely DON’T recommend this app. ( it’s not even worth 1 star )

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How Air-Conditioning Conquered America

Indoor cooling has transformed american life, reshaping homes, skylines and where people choose to live. as the planet warms, is that sustainable.

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

[MUSIC PLAYING]

Today, the story of how air conditioning has become both our answer to a warming planet and a major obstacle to actually confronting it. My colleague, Emily Badger, on the increasingly dangerous paradox of trying to control the temperature.

It’s Friday, August 16.

Emily, I want to start with a very personal question for you. What is your relationship to air conditioning?

So, at this exact moment, I am sitting in no air conditioning and it is kind of uncomfortable. And I’ve turned it off because it’s loud and it’s not very conducive to recording a podcast.

[CHUCKLES]: I didn’t mean right now, I meant in the larger arc of your life. But thank you for turning it off for the purposes of this episode.

Yeah. So I grew up in Chicago in this brick three flat apartment building, this very classic Chicago architecture, you know, built in the early 1900s. And it didn’t have air conditioning, so I didn’t have air conditioning growing up. Hardly anybody I knew had air conditioning growing up because we all lived in buildings like this.

Not even window units, just didn’t happen.

Nope, we didn’t even have a window unit in my family. And it wasn’t that big of a deal, in retrospect. We had, in this apartment, these big open windows that you could open and you’d generate a cross breeze through them. And there’s this kind of lovely breeze that comes off of Lake Michigan in the summer. And when it gets really, really hot, you know, you take a cold shower at night before you get in bed. You eat a lot of ice cream.

I can’t even remember if we had air conditioning in the schools that I went to. But it just wasn’t something that I thought very much about or really even experienced very much.

Right. You didn’t miss it. You didn’t even know it could be.

Yeah, exactly.

And then, the first job that I got out of college, I moved to Orlando, and totally different environment. I mean, living in Florida is the story of moving from one air conditioned box into another. You’re in your air conditioned apartment. You get in your air conditioned car. You don’t walk anywhere. You drive everywhere you go. You drive to your air conditioned office. You go to air conditioned bars. And it’s a really, really integral to life there in a way that was very foreign to me as someone growing up in the North.

So you went from a dearth of air conditioning to suddenly being saturated by it. And was that a happy development?

You know, I don’t think that I really gave it that much thought. I mean, living in Orlando surrounded by air conditioning, it’s just sort of that’s the air that you breathe. That’s the way everyone lives. And I think this is probably true for lots of people. We don’t really give it a lot of thought. It’s just sort of a background part of our environment.

But as I have written for years now about urban policy in cities, and how we live, and how we develop cities, it’s sort of become increasingly clear to me that air conditioning is this incredibly important thing that is shaping everything around us. You know, it’s shaping where Americans live, where they choose to move to. It shapes how our houses look. It shapes what our skylines look like. It’s responsible for saving lives and heat waves. In many ways, it’s really improved our quality of life.

But it’s increasingly clear to us that there are some downsides to this. And one of those downsides is that while we’re all sitting in our air conditioned homes, and offices, and cars, and we’ve set the thermostat to exactly 72 degrees, we’re becoming increasingly detached from what’s happening in the environment outside. It is a lot easier to ignore that it’s 100 degrees outside when you’re sitting inside air conditioning.

And, in some ways, I think we have forgotten how to live with heat. We have forgotten how to live with the climate as it existed before air conditioning. And having forgotten that, it’s probably going to cause some problems for us going forward.

Well, Emily, what did the American landscape look like when people did have to contend with the heat in the days before air conditioning?

So I think about two big things in particular. One is that the buildings that we spend time in looked different. We designed houses and other kinds of buildings in ways that were really sort of thoughtfully trying to contend with the temperature outside.

And so you’ve got these buildings in the Southwest in the United States that have these thick Adobe walls that do a really good job of keeping the sun and its heat out. You’ve got these cottages and bungalows in the Southeast that are raised up off the ground so that they’re not receiving the heat that’s absorbed by the Earth.

They’ve got big windows. They’re thinking a lot about cross ventilation. They’ve got high ceilings so that, as heat rises inside your home, you’re not marinating in it while you’re sitting in your living room. They’ve got front porches where people sit at the end of the day in order to try to cool off.

And then you’ve got the building like the one I grew up in, in Chicago, which I mentioned — these sort of thick brick masonry buildings, which are also designed in a way that is making it possible for me to grow up in the 1980s and ‘90s and be OK with the fact that I don’t have air conditioning.

Because brick kind of retains cool air.

Right. Right. And so part of what results from all of this is that the buildings in Georgia look different from the buildings in Arizona, look different from the buildings in Chicago. Because in each of those places, we’re designing buildings that react to the particular climate in those environments. And so this is the first big change.

Think of a time when you have to design a building to interact with what’s going on outside, with how humid it is, with how hot it gets. But the other thing that was very different in the pre-air conditioning environment is that there were just a lot fewer people living in the parts of the United States that were really hot and swampy. So it’s kind of incredible to think about it, but 1940, there are fewer people living in the state of Florida than living in the state of Arkansas.

There are about 8,000 people total living in the city of Las Vegas. Dallas and Houston are nowhere to be found on the list of the largest cities in America. So fundamentally, before air conditioning, there just aren’t a lot of people living in places where it is uncomfortable if you’re not controlling the temperature in some way.

Right. If it’s too hot, then you just don’t live there.

Right. So climate shapes your decisions about where to live. It shapes your decisions about how to build housing. It shapes your decisions about where to spend your time and your house. Maybe you go onto your front porch in the evening when it’s cooling down. You know, in many ways, our behavior is shaped by the climate. And then air conditioning comes along and it totally changes everything that I’ve been talking about. Because now the outdoor climate doesn’t really affect what your life is like indoors.

Just tell us about that moment, because I don’t think any of us really know the story.

Yeah. So there have been contraptions invented in the 1900s that were trying to do things like blow forced air over big blocks of ice in order to cool it. But the thing that we really think of as air conditioning is just totally a 20th century story.

It starts at the very beginning of the 20th century in 1902, when Willis Carrier invents this machine that’s kind of controlling the temperature, and the humidity, and the purity of air, particularly in an industrial context. The very first use of this in 1902 is in a printing plant, and fundamentally the problem that it’s solving is that the moisture content in the air is really becoming a problem for printing documents.

You’re saying basically, publishing, journalism is responsible for air conditioning.

Yes, everybody can thank us and then later they can blame us.

And so, in the beginning, what air conditioning is doing is it’s solving an industrial problem. The machines are hot, or maybe it’s a textile mill and too much humidity is sort of destroying your textiles. And also, you want your workers to be productive in these manufacturing spaces.

Lots of people in a small space with hot machines. Right.

Yeah. And so in the very beginning of the 20th century, it’s not about providing comfort for people. It’s about conditioning the environments that manufacturing and industry is having. And then it is this very sort of long story that plays out over several decades, where this invention moves from these industrial spaces into these other kinds of spaces.

Yes, you lucky people. Just sit back for a moment, relax, and notice the delightfully clean, cool, and refreshing atmosphere of this scientifically air conditioned theater. Great, isn’t it?

So then it comes into theaters and becomes almost this marketing tool to attract people inside.

You can enjoy great motion picture entertainment all summer long in cool comfort.

Go see a movie and enjoy air conditioning while you’re in there.

Yes, low-cost all-season air conditioning is the right kind for you. And you’re so right to choose a ‘55 Rambler Cross Country, now at all dealers.

And then, at the same time, cars in America that have air conditioning in them — the share of those cars is rising and rising. It moves into office buildings.

Instead of traveling away from business and home to seek relief, you can obtain this same comfort right in your own home or office through air conditioning.

And then, eventually, after decades of refining this technology, and it gets smaller, and it gets more affordable, and it becomes more advanced —

This lucky baby will sleep quietly through the night.

— it reaches the American home and we get the window unit.

This baby’s RCA air conditioner will keep his room filled with cool, dry, fresh air.

And the window unit is this much more affordable, portable, easy to pick up at the store, bring to your house. You don’t need to get a special installer. You stick it in your window, and now all of a sudden you’re getting all of these benefits of humidity controlled, temperature controlled air inside of your home.

Humidity, controlled, dust and pollen filtered. My indoor climate is always perfect.

At that point it’s off to the races. It takes over the American home. And we can see in census data, for instance, that by about the start of the 1970s, about half of all new single family homes that are built in America have air conditioning in them.

And the other thing that we see in census data at this time is that Americans themselves are starting to move to places that are really hot, like Florida, like Texas, like Arizona, like Nevada, places that are kind of uninhabitable before air conditioning. Now they’re booming in population.

And there was this wonderful editorial that was actually published in “The Times” in 1970 about the census that year, and how 1970 was like, the air conditioning census. And it refers to how air conditioning had become this really powerful influence for circulating people as well as air in this country.

And this is a story that continues right up until this day, where air conditioning is sort of extending its reach into every corner of the country, every sort of housing type. And today, about 2/3 of American households in this country have central air, and about 90 percent, so 9 in 10 of them, have some kind of error conditioning if we include things like window units. And if we look just at New housing that’s built in America today, looking back in 2023, about 98 percent of new single family homes in America had air conditioning.

What you’re talking about is basically 200 or so million air conditioning units, condensers, boxes. That’s a lot.

Yeah. And as air conditioning has extended its reach into every corner of the country, into so many of the buildings where we spend time, I think it becomes clear that we’ve really kind of engineered our modern lives entirely around it.

And our reliance on this technology going forward is both unsustainable, and in fact, it’s put a lot of people in a very vulnerable position.

We’ll be right back.

Emily, walk us through how our reliance on air conditioning is both, as you just said, unsustainable and perhaps even kind of dangerous to us.

So the first obvious thing that it does is it just requires an enormous amount of energy for so many people to be air conditioning so many spaces all the time. And so to think about this in a larger sense, our buildings in the United States are responsible for about 30 percent of the greenhouse gas emissions. And that refers to the fossil fuels that we burn directly to heat and cool buildings, and to cook in them, but also to generate the electricity that then allows us to do things plug in our window units.

So there’s a ton of energy use happening here. But part of what’s also happening is that all of these buildings have been fundamentally designed to consume lots of energy. A lot of these buildings were built during a time, you know, in the ‘50s and the ‘60s and in more recent years, where energy was cheap. The idea that you’re designing a building that demands lots of energy — who cares? We’re not paying a ton of money for the energy.

And in the ‘60s and in the ‘50s, we weren’t particularly thinking about whether or not using energy is going to cause climate change. So because of this, we get this glut of inefficient houses. And this happens not just with houses, but with everything in the built environment.

Think about strip malls, shopping centers, workplaces, even offices — the sort of ubiquitous, tall, boxy, glass-covered office building that we think about in cities all over the country, all over the world — this is a building that is born out of the air conditioning age. That glassy box is designed around air conditioning such that without air conditioning, those kinds of offices don’t make sense.

Right. I’m thinking about the office that you and I call home, the “New York Times” high-rise building in Midtown. That does not feel, for all its virtues, like a building, you’d want to be in without air conditioning.

It’s glass, and tall, and I think it’d be very hot.

Yeah. When you think about tall glass office buildings, they’re basically greenhouses if you’re not controlling the air inside. They’re designed such that not only do you not have to open a window in order to cool off, you couldn’t open a window even if you wanted to. These buildings don’t have windows that open, because they’re designed to be these hermetically sealed environments where we’re going to keep the outside climate out and we’re going to control the climate on the inside. And this idea that the outside doesn’t matter is true in the design of so many of our buildings, our offices, even our homes. And that actually puts people into an incredibly vulnerable situation.

And vulnerable how, exactly?

So let’s assume a storm comes through and the power goes out, or your air conditioning stops working because you’ve been running it all the time, all summer long, or when we have these extreme heat conditions and the electric utility tells you, please try to preserve the amount of air conditioning that you’re using. What happens when, all of a sudden, millions of people who have been living in an environment designed entirely around air conditioning can’t have that air conditioning? We start to see real problems.

And this is an abstract. We have actually seen this happen in the United States even this year, in other recent years, where terrible storms have ripped through the state of Texas and millions of people have been left without power. And when this happens in the middle of a heat wave, people die.

Right. And that seems an example of the multiple ways that air conditioning conspires to make us avoid contending with the realities of heat to return to this idea you introduced earlier on. AC allows more people to go to a place like Texas than they’d ever go if there weren’t AC making them comfortable, and to design and live in homes and offices that become a cauldron without air conditioning when it fails.

Exactly. Air conditioning makes it possible for people to believe that you could be comfortable in Texas in the summer, in Arizona in the summer. And so people move to these places in large numbers. And then, when the air conditioning fails, they’re sort of suddenly thrust into a world where they’re living in the middle of the Arizona desert or they’re living in the middle of Texas on a 110 degree day. And that could be life threatening.

Especially with climate change making it even hotter in these places, it doesn’t really seem sustainable for a lot of people to live in those places without air conditioning, without some kind of artificial tempering of the environment.

Yeah. And it’s not just because of the heat. I mean, is it sustainable for a Metropolitan area of 5 million people to exist in Phoenix in the middle of the desert when there’s also not enough water there for everyone? So air conditioning sort of lulls people into moving to these places, which might be problematic for lots of other reasons, as well. But we’ve sort of convinced ourselves that the climate doesn’t matter. We’re going to control it. We’re going to engineer our way into living with it.

You’re reminding me, Emily, of an episode we did on the show about this very idea. It focused on the water shortage in Arizona and the plans to pipe in — and, as I recall, desalinate ocean water — to deal with the problem of not enough water in Arizona. And it doesn’t really seem fathomable that proposition would ever occur to people if they weren’t living there in the comfort of air conditioning in the first place.

Yeah. So there have been people living in the region of Phoenix for centuries, so it’s not that nobody can ever live there. But what air conditioning does is it enables millions of people to live there who don’t actually want to contend with 100 degree temperatures all summer long. So a place like Phoenix then becomes this perfect example where we now have 5 million people living in the middle of the Arizona desert, and they all have this expectation of comfort there, that any environment that I move into — in my home, in my office, in my car — I should be encased in this cooling, calm, 72 degree humidity controlled environment. And that sense of comfort becomes so deeply entrenched kind of culturally. And this isn’t just about Phoenix. This is about all of us. I think we have set up an expectation or even an entitlement around comfort such that it makes it really difficult to start to ask people, do you really need to turn up your air conditioning today?

So that makes me wonder how people are ever going to get off the air conditioning hamster wheel that we’re describing here. I mean, why would anyone?

Well, we have to figure out how to do something if we want to address climate change. So there are a number of different things that are going to happen here. Air conditioning is going to become more efficient. We’re going to have more renewable energy sources to power it in the future. And I think we’re increasingly going to see architects and builders trying to rediscover these lost ideas that we used to have about how to design buildings with the climate in mind, how to shade them, how to ventilate them in a more natural way.

But I also have talked to some people who say that all of that is not going to be enough. One of them is Daniel Barber, who’s an architectural historian who has thought a lot about life after air conditioning or, as he puts it, after comfort — life in a world where we’re not depending on air conditioning so much. And the point that he makes is that there are difficult things and changes that we would have to do going forward if we know that our buildings are responsible for a lot of greenhouse gas emissions.

Our dependence on air conditioning is responsible for a large share of that, and we have to reduce it in some way. What we all need to do is change our own behavior. We need to think anew about our relationship to comfort. And are we willing to be uncomfortable some of the time? Am I willing to wait until July to turn my air conditioning on? Am I willing to turn it off at night when it’s not really necessary to use it? Am I willing to sleep at 80 degrees instead of 72 degrees?

Or 68 or 65. And he’s talking about asking people to do something really difficult. He is asking people to be uncomfortable.

You are, of course, by conveying this message, putting this problem on individuals, not governments, not states. And lots of people might hear this and think the real solutions have to come from regulators, have to come from institutions, have to come from the people who have a lot more control over how this all works.

I think that there are some ways in which that will happen, too. When we think about new buildings that are being designed or renovated today that are trying to adopt some of these techniques to be less reliant on indoor air conditioning. They’re often institutional buildings you will see cities commit to when we rebuild our schools, when we build a new library, when we build a new civic center, we are going to embody these things that we are asking other people to do, too.

And, obviously, there are government incentives in the United states, for instance, to better insulate your home, to do things that would make your home greener. So there’s certainly a role for government. But what Daniel Barber at least would argue is that we all bear some responsibility. And air conditioning has lulled us into thinking that we’re not impacted by how hot it is outside. But it’s also maybe lulled us into thinking like, I’m not the one who needs to particularly change my behavior in any way.

But, fundamentally, what we’re talking about is people embracing a kind of different cultural idea about what it means to be comfortable. The idea that existing in a room that is artificially cooled to 68 to 72 degrees fahrenheit, that that’s the ideal temperature — that’s not some true fact about the human body. It’s a cultural idea that’s been created over decades by the air conditioning industry, by architects, and builders, and culture, and shopping malls, and movie theaters. And the idea that comfort means this one particular thing is an idea that we have constructed ourselves. And so what if we culturally came up with a different idea about comfort?

What if more people came to accept the idea that going and sitting out on my front porch in the evening is where I get comfort from? And it’s also, by the way, how I interact with my neighbors. And I had stopped doing that when we were all retreating inside to air conditioning. What if we revived the idea that it’s actually quite lovely in the summertime to sleep with an open window and to have fresh air? It’s not impossible to change ideas about this because we created these ideas in the first place.

Well, Emily, thank you very much. We really appreciate it.

Yeah. Thanks, Michael.

Here’s what else you need to know today. On Thursday, the White House said that its newfound authority to use the Medicare program to negotiate prices of prescription drugs with pharmaceutical companies is likely to save taxpayers about $6 billion a year. That power came from President Biden’s Inflation Reduction Act, which became law two years ago. Under it, regulators have now lowered the price of widely used treatments, including blood thinners and medications for arthritis and diabetes, some by up to 79 percent.

And both vice presidential nominees, Minnesota Governor Tim Walz and Ohio Senator JD Vance, have agreed to debate each other on October 1 during a televised face-off hosted by CBS News. That means there will be three debates before election day — one vice presidential debate and two presidential debates between Donald Trump and Kamala Harris.

Finally, remember to catch a new episode of “The Interview” right here tomorrow. This week, David Marchese speaks with the singer Jelly Roll about addiction recovery and putting his whole self into his music.

I think of everything as a going out of business sale, and I give everything I got everything I do every time I do it right now.

Today’s episode was produced by Shannon Lin and Diana Nguyen with help from Michael Simon Johnson. It was edited by Devon Taylor, contains research help from Susan Lee, original music by Marion Lozano, Dan Powell, Rowen Niemisto, and Will Reid, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Michael Barbaro. See you on Monday.

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Hosted by Michael Barbaro

Featuring Emily Badger

Produced by Shannon M. Lin and Diana Nguyen

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Edited by Devon Taylor

Original music by Marion Lozano Dan Powell Rowan Niemisto and Will Reid

Engineered by Alyssa Moxley

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Air-conditioning has become both our answer to a warming planet and a major obstacle to actually confronting it.

Emily Badger, who covers cities and urban policy for The Times, explains the increasingly dangerous paradox of trying to control the temperature.

On today’s episode

Emily Badger , who covers cities and urban policy for The New York Times.

Background reading

From 2017: How air-conditioning conquered America .

Air-conditioning use will surge in a warming world , the U.N. has warned.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

Research help by Susan Lee .

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Michael Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

Emily Badger writes about cities and urban policy for The Times from Washington. She’s particularly interested in housing, transportation and inequality — and how they’re all connected. More about Emily Badger

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