These interactive lessons use dynamic graphing and guided discovery to strengthen and connect symbolic and visual reasoning. They give the student a hands-on visual exposition of all Common Core Algebra 1 topics, reinforced by adaptive exercises and randomly generated tests. All exercises and tests are checked and graded automatically. Hover the mouse over a link below to see an example from that lesson, or click on a test link to see a concise summary of a group of lessons. course glossary.

We also provide aThe Common Core and other standards cover many algebraic topics before the first actual High School Algebra course. Therefore, students may have already completed many of the lessons and exercises below in an earlier grade, or in a summer preparatory course. If so, those assignments will be treated as already completed in this course also.

**Students should have scratch paper available, access to (software or hardware) numeric
calculators except during the Arithmetic Review chapter, and other students or a teacher to ask
for help when they are stuck.** All students are encouraged to both give and receive
mathematical explanations with their peers. Please
send us your comments, questions and suggestions.
(Lessons without links yet are under development.)

- Arithmetic on a Grid: Counting unit squares. Sum, difference, product, quotient.
- Two-Digit Addition: Two-digit numbers shown as
$a(10)+b$ on grid. Addition with regrouping, carrying.

Exercises: addition - Two-Digit Subtraction and
Multiplication: Subtraction with borrowing, multiplication with regrouping.

Exercises: subtraction, multiplication, division - Negative Numbers: Green squares have value
$+1$, pink squares have value $-1$. Sums and differences. Word problems involving debt.

Exercises: addition, subtraction - Multiplication and Division with
Negative Numbers: Products and quotients with negative numbers on the grid,
introduced through analogy and pattern-matching.

Exercises: multiplication, division - Fraction Addition and Subtraction:
Visualizing fractions using pie slices. Reducing to lowest terms. Finding common
denominators.

Exercises: addition, subtraction - Fraction Multiplication and Division:
Using an expanded grid to visualize fractions.

Exercises: multiplication, division - Arithmetic Test

- Variables and Expressions: Expressions as
quantities. Simple linear expressions. Sliders for variables.

Exercises: evaluation 6.EE.2 - Word Problems: Cost of $n$ items. Multi-variable
expressions.

Exercises: expressions 6.EE.6, A.SSE.1a, A.CED.2 - More Complicated Expressions:
Examples involving negative numbers, division, fractions.

Exercises: evaluation, division A.SSE.1a - Equations as Sentences: Solving equations
by trial and error, or by sliding a slider. Inverse problems as word problems.

Exercises: solving 6.EE.5, A.CED.1 - Grouping in Addition and Subtraction
Problems: Parentheses, associative law of addition, order of operations for addition
and subtraction. Simplifying expressions using the associativity of addition to regroup,
including for subtraction.

Exercises: evaluation, simplifying 6.EE.3, A.APR.1 - Solving $x+b=c$: Adding a constant to both sides.

Exercises: solving 6.EE.7, A.REI.1, A.REI.3 - Grouping in Multiplication Problems:
Order of operations including multiplication. Multiplying three numbers, associative law
of multiplication and its use in simplifying expressions.

Exercises: evaluation 1, evaluation 2, multiplication, division 6.EE.3, A.APR.1 - Solving $ax+b=c$: Adding $-b$ and then multiplying by
$$1/a$$, or dividing by $a$.

Exercises: solving 1, solving 2 6.EE.7, A.REI.1, A.REI.3 - Applications of Linear Equations:
Balanced scale, budgeting, temperature conversion.

Exercises: solving 6.EE.7, 7.EE.4a, A.CED.1, F.BF.1a - Variables, Expressions, and Simple Equations Test 1
- The Distributive Law and Combining Like
Terms: Expanding $a(x+c)$, $a(bx+c)$, $-(bx+c)$. Simplifying $ax+bx$ to $[a+b]x$.

Exercises: simplifying 1, simplifying 2, combining like terms 6.EE.3, A.APR.1 - Manipulating Linear Expressions:
Adding, subtracting, scaling, simplifying. Commutativity of addition and multiplication.
Zero and one laws.

Exercises: simplifying 1, simplifying 2 6.EE.3, 7.EE.1, A.APR.1 - Solving $ax+b=cx+d$: Adding $-cx-b$ to both sides.

Exercises: solving 8.EE.7b, A.REI.1, A.REI.3 - Variables, Expressions, and Simple Equations Test 2

- Points and Coordinates:
Connecting the ordered pair $(x, y)$ with the coordinate plane. Quadrants.

Exercises: quadrants, coordinates 6.NS.6bc - Investigating $y=x+b$:
Finding the $y$-intercept $(0, b)$ of a line.

Exercises: $y$-intercepts F.IF.7a, F.BF.3 - Investigating $y=mx$: Lines with slope $m$, positive or
negative.

Exercises: slopes 8.EE.5, 8.F.2, F.LE.1a - Investigating $y=mx+b$: Slope and
$y$-intercept.

Exercises: $y$-intercepts 1, slopes 1, $y$-intercepts 2, slopes 2, equations 8.F.3, F.IF.7a, F.LE.1a - Solving Equations Graphically: Graphing
each side and finding the intersection, e.g. for $ax+b=cx+d$.

Exercises: solving A.REI.11 - Finding Formulas for Approximately Linear Data: Simple statistics application. Uses root-mean-square error informally. 8.SP.2, F.IF.4, S.ID.6ac
- Slopes,
Rates of Change, and Similar Triangles: Calculating the slope of a line from any two
points on it. Derivation of the equation $y=mx$ or $y=mx+b$ from a line’s slope and
$y$-intercept.

Exercises: computing slopes 8.F.4, F.LE.1b, F.LE.2 - Finding Equations for Lines: Point-slope form.

Exercises: point-slope form F.LE.2 - Parallel and Perpendicular
Lines: Slopes $m$ and $$-1/m$$.

Exercises: parallel lines, perpendicular lines 8.EE.8c - Equations for Lines in Standard Form:
$Ax + By = C$. Converting to slope-intercept form. Computing slopes,
$x$- and $y$-intercepts.

Exercises: converting to slope-intercept form, $y$-intercepts, $x$-intercepts A.CED.4, A.REI.10 - Linear Graphs Test

- Linear Inequalities: Half-plane solution
graphs, with or without boundary line.

Exercises: evaluating, identifying inequalities A.CED.1, A.REI.12 - Equivalent Inequalities: Adding a
constant to both sides of an inequality. Multiplying both sides by a positive or
negative number.

Exercises: addition, multiplication - Solving Inequalities
Algebraically: Finding algebraic solutions to $ax+b < c$ and similar
inequalities.

Exercises: solving A.REI.3 - Absolute Value: Definition of absolute value.
Graphing and solving equations which include absolute values. $|ab| = |a||b|$.

Exercises: solving 6.NS.7c, F.IF.7b' - Square Roots: Definition and computation of
square roots. Simplifying square roots via product, quotient, and absolute value
identities.

Exercises: simplifying 8.EE.2, A.REI.4b - Rational and Irrational Numbers:
Definitions. If $q$ and $r$ are rational, then so are $q+r$, $q-r$, $qr$, and
$$q/r$$ if $r ≠ 0$; but $√2$ for instance is not.

Exercises: repeating decimals, rationality 8.NS.1, 8.EE.2, N.RN.3 - Inequalities, Absolute Value, and Square Roots Test

- Summarizing a Collection of
Measurements: Converting to common units. Dot (line) plots. Median and mean.
Histograms.

Exercises: median from a list, median from frequencies, mean 6.SP.3, 6.SP.4, N.Q.1, N.Q.2, N.Q.3, S.ID.1, S.ID.2 - Box Plots: First and third quartiles, interquartile
range. Comparing data collections.

Exercises: quartiles, box plots 6.SP.4, S.ID.1, S.ID.2, S.ID.3 - Measuring Spread in Data: Mean absolute
deviation, standard deviation.

Exercises: mean absolute deviations, standard deviations 6.SP.3, S.ID.2 - Analyzing Outliers and Choosing Statistics: Important and unimportant outliers. Choosing the best statistics to describe a collection of data values. 6.SP.5, S.ID.3
- Data Relating Two
Categorizations: Two-way frequency tables. Joint, marginal, and conditional relative
frequencies. Possible associations and trends in the data.

Exercises: joint frequencies, marginal frequencies, conditional relative frequencies 8.SP.4, S.ID.5 - Data Relating Two Measurements:
Scatter plots. Linear model, residuals, root-mean-square error, interpretation of slope
and $y$-intercept. Correlation coefficient, correlation is not causation.

Exercises: root mean square errors, correlation coefficients 8.SP.1, 8.SP.3, S.ID.6abc, S.ID.7, S.ID.8, S.ID.9 - Descriptive Statistics Test

- Solving Systems of Linear Equations by
Graphing: Solutions are intersections of graphs.

Exercises: checking, solving 8.EE.8a, A.REI.6 - Solving Systems of Linear Equations by
Subtraction: Solving for a difference equal to 0.

Exercises: solving 8.EE.8b, A.REI.5, A.REI.6 - Solving Systems of Linear Equations
by Multiplication and Addition: Eliminating a variable in an equation.

Exercises: solving 8.EE.8b, A.REI.5, A.REI.6 - Applications of Systems of Linear Equations: Cost, revenue, profit. Birth and death rates. 8.EE.8c, 8.SP.3, A.CED.2, F.LE.5
- Mixture Problems: Mathematics of mixing
different concentrations.

Exercises: solving A.CED.2, A.REI.6 - Systems of Linear Inequalities: Solution
graphs are intersections of half-planes.

Exercises: solving A.REI.12 - Linear Optimization: Maximizing profit subject to linear inequalities. A.CED.3
- Systems of Linear Equations or Inequalities Test

- Investigating $y=(x-h)^2+k$: Vertex,
axis of symmetry.

Exercises: vertex, axis of symmetry 8.F.3, F.IF.7a', F.IF.9' - Investigating $y=a(x-h)^2+k$: Positive
and negative $a$. Roots.

Exercises: vertex, axis of symmetry, number of roots F.IF.7a' - Quadratic Polynomials: Monomials and
polynomials. Adding, subtracting, scaling quadratics. Multiplying two monic linear
factors to obtain a quadratic polynomial. Uses the algebra grid.

Exercises: addition, subtraction, multiplication 1, multiplication 2 A.APR.1 - More Quadratic Polynomials:
Multiplying non-monic linear polynomials. Performing several simplifications on one
quadratic expression.

Exercises: simplifying A.APR.1 - Factoring $x^2+bx+c$:
Graphing $y=(x-r)(x-s)$. Solving $x^2+bx+c=0$ by factoring.

Exercises: factoring A.SSE.3a, A.REI.4b - Factoring $ax^2+bx+c$:
Constant and linear factors. Clearing denominators. Moving all terms to one side of an
equation.

Exercises: factoring 1, factoring 2, clearing denominators, solving A.SSE.2, A.REI.4b - Quadratics Test 1
- Investigating $y=ax^2+bx$: Roots, axis
of symmetry, vertex.

Exercises: axis of symmetry, vertex F.IF.8a' - Investigating $y=ax^2+bx+c$: Axis of
symmetry, vertex, rewriting in vertex form.

Exercises: axis of symmetry, vertex F.IF.8a' - Solving Quadratic Equations:
Completing the square, solving any quadratic equation.

Exercises: solving, converting vertex form to standard form A.SSE.3b, A.REI.4ab, F.IF.8a' - The Quadratic Formula: Solving any
quadratic equation, using a formula.

Exercises: discriminants, solving 1, solving 2, solving 3 A.REI.4ab - Quadratics Test 2
- Applications of Quadratic
Equations: Objects in free fall, the shape of a suspension bridge, profit
maximization.

Exercises: solving 8.F.5, A.SSE.1a', A.CED.1', F.IF.4', F.BF.1a' - Quadratic Systems of Equations: Using
substitution to eliminate a variable.

Exercises: linear systems, number of solutions, solving 1, solving 2 A.REI.7 - Quadratics Test 3

- The Pythagorean Theorem:
Proof that $a^2 + b^2 = c^2$ for a right triangle, and also its converse. 1-1-√2
triangle as a special case.

Exercises: Pythagorean Theorem, converse 8.G.6 - Distances
Using the Pythagorean Theorem: Distance from $(x_1, y_1)$ to $(x_2, y_2)$.
Length of side diagonals and space diagonals in a rectangular box.

Exercises 8.G.7, 8.G.8 - Pythagorean Theorem Test

- The Laws of Exponents: For all positive
integers $c$ and $d$, $a^c a^d = a^{c+d}$, $(ab)^d = a^d b^d$, and $(a^c)^d = a^{cd}$.
If $a ≠ 0$ and $b ≠ 0$, we can extend these laws to all integers $c$ and $d$ using
$a^0 = 1$ and $$a^{- d} = 1/a^d$$.

Exercises: simplifying 1, simplifying 2 8.EE.1, F.LE.1a - Rational Exponents: Given $a > 0$, and
integers $m$ and $n$ with $n > 0$, we define $$a^{m∕n} = √^n{a^m}$$. Then
the laws of exponents still hold for positive bases and rational exponents. Graph of
$y=a^x$ for various $a > 0$.

Exercises: $n$th roots, simplifying N.RN.1, N.RN.2, F.IF.6, F.IF.7e, F.IF.8b', F.LE.2 - Comparing Exponential Graphs: Graphs
of $y = m a^x$ and $y = a^{x+c}$, which coincide when $a^c = m$. Graphs of $y = b^x$ and
$y = a^{rx}$, which coincide when $a^r = b$. Compare linear, quadratic, and exponential
growth, including for large $x$.

Exercises: rewriting 1, rewriting 2, comparing growth rates F.IF.8b', F.BF.3, F.LE.3' - Applications of Exponential
Growth: Compound interest, musical pitches, biological growth.

Exercises: compound interest A.SSE.1ab', A.SSE.3c, A.CED.2', F.LE.1c, F.LE.5 - Applications of Exponential Decay: Carbon dating, cooling objects. A.CED.1', A.REI.11, F.IF.4, F.BF.1b, F.LE.1c, F.LE.5
- Exponentials Test

- Sets, Relations, and Functions:
Sets, relations, domain, range. When is a relation a function?

Exercises: relations from tables, relations from graphs 8.F.1, F.IF.1 - Functions as Transformations:
Function notation $f(x)$. Defining a function by a table, formula, or graph.
Composition.

Exercises: evaluation F.IF.1, F.IF.2 - Inverse Functions:
Definition and computation of the inverse of a function. One-to-one functions. Square
and cube roots. The inverse of a linear function.

Exercises: one-to-one functions, inverting linear functions F.IF.7b', F.BF.4 - Shifting Functions:
Moving a graph through composition of functions. Periodic functions.

Exercises: shifting functions F.BF.3' - Stretching and Flipping
Functions: Dilating or reflecting a graph through composition of
functions. Even and odd functions and polynomials.

Exercises: stretching graphs, even and odd functions F.BF.3' - Piecewise-Defined Functions:
Piecewise-defined functions, step functions, and the floor and ceiling functions.

Exercises: evaluation F.IF.5', F.IF.7b' - Sequences: Arithmetic, geometric, and Fibonacci
sequences. The sum $1+2+...+n$.

Exercises: arithmetic sequences, geometric sequences F.IF.3, F.IF.5, F.BF.2, F.LE.2 - Applications of Functions: Temperature conversion, profit, rice on a chessboard. F.IF.2, F.IF.5, F.BF.1a, F.BF.3
- Functions Test