These interactive lessons use dynamic graphics and guided discovery to strengthen and connect symbolic and visual reasoning. They give the student a hands-on visual exposition of all Common Core Grade 8 Mathematics 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 a**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.

This course is **unfinished**, so the lessons, exercises, and tests here are
currently offered for free to anyone wishing to try them. Lessons without links yet are under
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- 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 - Word Problems: Cost of $n$ items.
Multi-variable expressions.

Exercises: expressions - More Complicated
Expressions: Examples involving negative numbers, division, fractions.

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

Exercises: solving - 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 - Solving $x+b=c$: Adding a constant to both
sides.

Exercises: solving - 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 - Solving $ax+b=c$: Adding $-b$ and then
multiplying by $$1/a$$, or dividing by $a$.

Exercises: solving 1, solving 2 - Applications of Linear
Equations: Balanced scale, budgeting, temperature conversion.

Exercises: solving - 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 - Manipulating Linear
Expressions: Adding, subtracting, scaling, simplifying. Commutativity of addition
and multiplication. Zero and one laws.

Exercises: simplifying 1, simplifying 2 - Solving $ax+b=cx+d$: Adding $-cx-b$ to both
sides.

Exercises: solving 8.EE.7 - Simplifying and Solving
Equations: Expanding and simplifying each side of an equation. Finding the number of
solutions of an equation.

Exercises: simplifying and solving, number of solutions 8.EE.7 - Variables, Expressions, and Simple Equations Test 2

- Rigid Motions: Translations, rotations, and reflections. These preserve lengths, angles, lines, and parallelism. 8.G.1
- Congruence: Definition of congruence using rigid
motions. SAS criterion for congruence of triangles.

Exercises: congruent figures 8.G.2 - Rules of Congruence: ASA and SSS
criteria for congruence of triangles. AAA does not imply congruence, nor does SSSS for
quadrilaterals.

Exercises: congruent triangles 8.G.2 - Related Angles: Acute, obtuse, and right
angles. Supplementary and vertical angles. Translations, parallel lines, and
transversals.

Exercises: related angles 8.G.5 - Angles in Triangles: The angles in a
triangle add up to 180°. The exterior angles of a triangle add up to 360˚.

Exercises: angles in triangles 8.G.5 - Dilations: Definition of a dilation. Dilations
preserve lines and angles, and scale lengths by a single common factor.

Exercises: dilations 8.G.4 - Similarity: Definition of similarity using rigid
motions and dilations. Similar triangles have the same angle measures and proportional
sides, and conversely either of these criteria for two triangles imply similarity.

Exercises: AA rule, proportional sides rule 1, proportional sides rule 2 8.G.4, 8.G.5 - Applications of
Similarity and Related Angles: Length of a ramp. Calculating the height of a
flagpole from shadows. Circumference of the earth.

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

Exercises: quadrants, coordinates - Transformations and
Coordinates: Coordinates of points after translations, reflections, rotations, and
dilations.

Exercises: coordinates after translation, coordinates after reflection, coordinates after rotation, coordinates after dilation 8.G.3 - Congruence, Similarity, and Coordinates Test

- Investigating $y=x+b$:
Finding the $y$-intercept $(0, b)$ of a line.

Exercises: $y$-intercepts - Investigating $y=mx$: Lines with slope $m$,
positive or negative.

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

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

Exercises: solving 8.EE.7b, 8.EE.8a - 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.EE.6, 8.F.4 - Finding Equations for Lines:
Point-slope form.

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

Exercises: parallel lines, perpendicular lines 8.EE.8c - Linear Functions: Linear and nonlinear
functions. Height of a ball.

Exercises: identifying functions, properties of functions 8.F.1, 8.F.3, 8.F.5 - Linear Graphs Test

- Two-Way Frequency Tables: Possible associations between the two variables. 8.SP.4
- Finding Formulas for Approximately Linear Data: Scatter plots, linear models. 8.SP.1, 8.SP.2
- More Scatter Plots: Clustering, outliers, positive or negative association, linear association, and nonlinear association. Interpretation of slope and $y$-intercept of a linear model. 8.SP.1, 8.SP.3

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

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

Exercises: solving 8.EE.8b - Applications of Systems of Linear Equations: Cost, revenue, profit. Birth and death rates. 8.EE.8c, 8.SP.3
- Systems of Linear Equations 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: 8.EE.1 - Scientific Notation and Units 8.EE.3, 8.EE.4
- Square and Cube Roots 8.EE.2
- Rational Numbers and Repeating Decimals: Converting from repeating decimal expansions to rational numbers and vice versa. 8.NS.1
- Irrational Numbers: $π$ and $√2$ are irrational. Using approximations to compare irrational numbers and do arithmetic with them. 8.NS.2, 8.EE.2
- Exponents, Roots, and Real Numbers Test

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

Exercises 8.G.6 - Applications of 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 - Three-Dimensional Geometry: Volume of cylinders, cones, spheres. 8.G.9
- Pythagorean Theorem, Distance, and Volume Test