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. Relevant standards are listed
after each lesson’s exercises, with a ' (prime) denoting a distinct link. We also provide
a course glossary.
The 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.
Arithmetic Review
(Optional)
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
Negative Numbers: Green squares have value
$+1$, pink squares have value $-1$. Sums and differences. Word problems involving debt.
Exercises: addition,
subtraction
Equations as Sentences: Solving equations
by trial and error, or by sliding a slider. Inverse problems as word problems.
Exercises: solving6.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,
simplifying6.EE.3, A.APR.1
Solving $x+b=c$: Adding a constant to both sides.
Exercises: solving6.EE.7, A.REI.1, A.REI.3
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 slopes8.F.4, F.IF.6, F.LE.1b, F.LE.2
Absolute Value: Definition of absolute value.
Graphing and solving equations which include absolute values. $|ab| = |a||b|$.
Exercises: solving6.NS.7c, F.IF.7b'
Square Roots: Definition and computation of
square roots. Simplifying square roots via product, quotient, and absolute value
identities.
Exercises: simplifying8.EE.2, A.REI.4b
More Quadratic Polynomials:
Multiplying non-monic linear polynomials. Performing several simplifications on one
quadratic expression.
Exercises:
simplifyingA.APR.1
Applications of Quadratic
Equations: Heights of projectiles, the shape of a suspension bridge, profit
maximization.
Exercises:
solving8.F.5, A.SSE.1a', A.CED.1', F.IF.4',
F.BF.1a'
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 28.EE.1, F.IF.7e, F.LE.1a, F.LE.2
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,
simplifyingN.RN.1, N.RN.2, F.IF.8b'
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 ratesF.IF.8b', F.BF.3, F.LE.3'
Functions as Transformations:
Function notation $f(x)$. Defining a function by a table, formula, or graph.
Composition.
Exercises:
evaluationF.IF.1, F.IF.2