Algebra 1 Mastered

Define sets, explore subsets and set builder notation, and introduce real numbers, numerical and algebraic expressions; learn to evaluate and translate phrases into algebraic expressions.

Understand that a set is a collection of elements, written with a name, braces, and commas; use ellipses for continuing forever and the element-of symbol to indicate membership or non-membership.

Define a subset and its notation, where every element of A is also in B. Distinguish proper subsets, note the empty set is a subset of every set, with examples.

Define sets using set-builder notation, with braces and the such-that symbol to specify shared properties. Use examples like blue shapes and numbers greater than, less than, or equal to zero.

Explore union and intersection as set operations that combine all elements or only the common ones, with notation and examples, and connect them to algebraic inequalities and overlapping intervals.

Explore the real numbers and their components, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers, and their subset relationships, with common notation Z, Q, and R.

Explore the real number line and interval notation, including square vs round brackets, filled vs hollow dots, and graphical representations for intervals and infinities.

Explore the properties of equality in real numbers and algebraic expressions, including reflexive, symmetric, transitive, and substitution properties, with examples like 2+3=5.

Master the order of operations for numerical expressions, prioritizing parentheses, fractions, powers and roots, and left-to-right multiplication and division, then addition and subtraction.

Absolute value is the distance from a number to zero. It is non-negative; |a| = a if a ≥ 0, and |a| = |-a| if a < 0.

Explore how real numbers behave under addition, subtraction, multiplication, and division, including sign rules, zero properties, and when division by zero is undefined.

Explore the closure properties of real numbers under addition, subtraction, and multiplication, and why division fails due to division by zero. Learn the real-number identity properties with zero and one.

Explore the commutative, associative, and distributive properties of real numbers with examples for addition and multiplication, and note the noncommutativity of subtraction and division.

Explore the additive inverse property, the multiplication property of zero, the multiplication property of negative one, and the multiplicative inverse property for nonzero real numbers.

Explore algebraic expressions as combined variables, constants, and operations, identify terms and literal factors, and simplify by combining like terms and numerical coefficients.

Learn how to evaluate algebraic expressions by substituting variables with constants, simplify by combining like terms, and compute results for chosen x and y values, including powers.

Translate English statements into algebra by recognizing addition, subtraction, multiplication, and division phrases, such as the sum of a number and ten or the quotient of a number and eight.

Download the summary sheet to review the major concepts covered in this algebra 1 section.

Explore solving first degree equations, including those with fractions and decimals, and learn about formulas and inequalities across related topics.

Explore what an equation is and how algebraic equations differ from numerical ones. Learn about open sentences, roots, and the solution set, with a focus on first-degree equations.

Learn to solve first-degree equations by balancing both sides, moving terms, and applying operations such as addition and division, with examples like x equals seven and root verification.

Translate English phrases into algebraic expressions and equations, then solve by forming and using the equal sign, with examples like three times a number minus 20 equals 15.

learn to solve algebraic equations with fractions by clearing denominators with the least common multiple, then multiply through and solve for x.

Master solving algebraic equations with decimals by clearing decimals with a power of ten. Use the smallest decimal to determine multiplier and verify results; apply to discount and selling-price problems.

Expose students to inequalities, distinguish numerical vs algebraic inequalities, and demonstrate solving by isolating x, applying addition and multiplication properties, and flipping signs when multiplying by negatives.

Learn to solve algebra inequalities with fractions and decimals by clearing denominators with the LCM and decimals with powers of ten, and express solutions as interval notation.

Explore compound inequalities on the real number line, join inequalities with and or, and express solutions in interval notation using brackets and unions or intersections.

Learn to solve equations with absolute value by opening the absolute value and using negative and positive cases; examples: |x-1|=5 gives x=6 or -4, and |5x+3|=7 gives x=4/5 or -2.

Explore inequalities involving absolute value, converting |x|<k to -k<x<k and |x|>k to x<-k or x>k, with examples like |3x-1|<8 and |3x-1|>2.

Download the summary sheet containing all major concepts we covered, and use it to review the materials.

Introduce polynomials, define what polynomials are, explore products and quotients, discuss types of factoring, address equations and problem solving, and touch on prime numbers.

Define terms as products of constants and variables, identify monomials as single terms with whole-number exponents, note coefficients and literal factors, and explain that degree equals the sum of exponents.

Add and subtract polynomials by grouping like terms. Use the commutative, associative, and distributive properties to combine coefficients.

Explore how to multiply and divide monomials by adding or subtracting exponents, apply the power rules, distribute exponents over products, and avoid distributing over sums.

Apply the distributive property to multiply polynomials, expanding monomials across binomials and polynomials, and combine like terms while managing exponents and signs for accurate results.

Master shortcut patterns for multiplying polynomials: square a binomial using a^2 ± 2ab + b^2, use a^2 − b^2 for opposite signs, and expand (a+b)^3 via a^3 + 3a^2 b + 3ab^2 + b^3.

Factoring is the opposite of multiplication, shown by turning x^2 - x - 6 into (x+2)(x-3). The lesson uses the distributive property to factor polynomials by pulling out common factors.

Identify the highest common monomial factor from terms to factor polynomials, then express the result in a completely factored form with integral coefficients, and verify no further factorization is possible.

Master factoring by grouping to extract a common binomial factor from polynomials. Group terms with shared factors and rearrange when needed to reveal the binomial and verify by expansion.

Use the difference of two squares to factor A^2 - B^2 as (A - B)(A + B). A sum of squares cannot be factored this way.

Factor the sum and difference of two cubes by creating a small binomial from cube roots and a large binomial using squares and the product term, with consistent signs.

Master factoring simple trinomials by reversing the product of two binomials: find two integers whose product equals the constant term and whose sum equals the middle coefficient, then form (x+a)(x+b).

Learn to factor trinomials with leading coefficients not equal to one by multiplying the leading and constant terms, splitting the middle term, and factoring by grouping, with examples.

Learn how factoring extends solving equations via the zero product property and roots. Apply methods to x^2+6x and 3x^2-5x to verify solutions.

Apply factoring techniques, especially the difference of squares, to solve equations and find roots, illustrated by x^2=16, 7x^2-7=0, and the two-square area problem.

Learn to solve trinomial equations by factoring with two methods based on the leading coefficient, applying the zero-product property, and verify with x^2-11x-12 and 9x^2+9x-4.

Download the summary sheet to review all major concepts covered in this section for algebra 1 mastery.

Introduce rational expressions and their properties, then explore operations—multiplication, division, addition, and subtraction—and apply these to polynomials and fractional equations.

Identify rational numbers as ratios of integers, apply sign rules for same or opposite signs, use fundamental principle of fractions to simplify and view rational expressions as quotients of polynomials.

Learn to simplify rational expressions by factoring and canceling common factors in the numerator and denominator, using prime factorization and the opposite-signed cancellation to reduce expressions.

Multiply and divide rational expressions by multiplying numerators and denominators, canceling common factors, and simplifying. Convert division to multiplication by reciprocals to reveal cancellations and factor when possible.

Learn to add and subtract rational expressions by using same-denominator rules, the least common denominator, and the butterfly method, including grouping strategies for multiple terms.

Explore complex fractions where the numerator and denominator are rational numbers, learn to convert division to multiplication, and simplify using the butterfly technique with practice problems.

Learn to divide polynomials using long division, identifying the dividend, divisor, quotient, and remainder and applying the relation dividend equals divisor times quotient plus remainder.

Master fractional equations by clearing denominators, using least common multiples, and applying restrictions to exclude zero denominators while solving for roots.

Master fractional equations by identifying restricted values that zero denominators, then simplify and clear denominators to solve the examples. Practice problems are included.

Explore ratios and proportions, learn to identify equal ratios, and apply cross multiplication to solve real-world problems such as speed and cost per weight.

Download the attached summary sheet to review all major concepts covered in this section and reinforce your understanding.

Explore how to use integers that are equal to or less than zero as exponents, work with roots and radicals, simplify radicals containing variables, and solve equations involving radicals.

Explore how zero and negative exponents work, showing x^0 = 1 and x^(-n) = 1/x^n via the multiplicative inverse and exponent addition.

Master the exponent rules across integers and zero: product of powers, power of a power, and power of a product or quotient; practice simplifying with negative exponents.

Explore roots and radicals, defining square roots, principal square roots, cube roots, and nth roots, and distinguish two roots for positive numbers versus one root for cubes and nonreal results.

Explore nth roots as a generalization of square and cube roots, where even indices yield two real roots and odd indices yield one real root regardless of sign.

Understand root properties for real numbers, including the principal square root and cancellation with exponents. Apply distribution of roots over products and quotients under real-number conditions, with odd-root restrictions.

Explore the concept of the simplest radical form by identifying perfect powers and extracting square and cube factors from radicals, then rationalize denominators when needed.

Identify similar radicals with the same index to add or subtract them using the distributive property. Turn different radicands into a common base to combine coefficients.

Learn to combine radicals with variables, using index rules to simplify square and cube roots. Determine simplest radical form and rationalize denominators when needed.

Learn how to multiply radicals by distributing across factors, combine into one big radical when indices match, multiply coefficients, and simplify by factoring out perfect powers.

Rationalize denominators with radicals by multiplying by the conjugate of the binomial, removing radicals from the denominator. Apply the method to expressions like √5+√2 and practice problems are provided.

Explore solving radical equations by isolating the radical, applying exponents to remove radicals, solving the resulting equation, and checking for extraneous solutions.

Master converting between exponents and roots using rational exponents and nth roots; apply numerator-denominator relationships to simplify expressions like x^(6/3) and the third root.

Apply radicals to physics through pendulum period and skid marks. Use pendulum formula for l = 2 m and skid marks formula for d = 20 m, f = 0.35.

Learn how scientific notation expresses large or small numbers as a value between 1 and 10 times ten to a power, by moving the decimal point left or right.

Explore converting numbers between standard and scientific notation, adjust decimal placement to set the ten to the zero, and apply these rules to multiplication and square roots, with practice problems.

Download the summary sheet to review the major concepts covered in this section. Use it to reinforce your understanding of the material.

Explore quadratic equations and inequalities, learn how to solve them with the quadratic formula, examine applications, and see how complex numbers extend real solutions.

Understand how complex numbers extend real numbers, with real numbers as a subset and the imaginary unit i with i^2 = -1, represented as a+bi.

Add and subtract complex numbers by combining real parts and imaginary parts, and write in standard form; illustrated with examples like 4+2i and 5+9i.

Explore how the imaginary unit i works, its powers cycle every four, and how to rewrite sqrt of negatives as i times the root to multiply complex numbers.

Explore multiplying and dividing complex numbers using binomial expansion and conjugates, and learn to rationalize denominators and express quotients in standard form. Includes examples with conjugates and real-number outcomes.

Identify quadratic equations in standard form ax^2 + bx + c = 0 with a ≠ 0 and one variable. Learn to solve by factoring and extend to complex-number solutions.

Learn to recognize and form perfect square trinomials from binomials squared, using first and last term squares and the middle term 2ab, to solve quadratic equations.

Master completing the square to solve any quadratic equation. Divide by a, isolate terms, add B/2 to form a perfect square trinomial, then take square roots to find the roots.

Master quadratic equations using the quadratic formula derived from completing the square for ax^2+bx+c=0. Identify a, b, c and plug into x = (-b ± sqrt(b^2-4ac))/(2a) to find the roots.

Explore the discriminant in the quadratic formula to determine the nature of roots. Learn when the equation has one real root, two real roots, or two non-real complex solutions.

Solve quadratic inequalities by factoring, identifying critical numbers, testing regions, and forming the solution as intervals with endpoints for less than or equal to and greater than or equal to.

Download the summary sheet to review all major concepts covered in this section and previous sections. Use it to review the materials in the Algebra 1 Mastered course.

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