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 Mathematics 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.

The Common Core and other standards cover many algebraic and related topics before the first actual High School Mathematics 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.

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
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 - 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 - 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.1ab - 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 - 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 - 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.IF.6, 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, G.GPE.5 - 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 - Inequalities 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, A.REI.11 - 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.IF.9, 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

- 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.IF.7e, F.LE.1a, F.LE.2, F.LE.3 - 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 8.F.3, F.IF.1, F.IF.2 - 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 - 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: Compound interest, profit, rice on a chessboard.

Exercises: compound interest 8.F.5, A.SSE.1ab, A.CED.1, F.IF.2, F.IF.5, F.BF.1a, F.BF.3, F.LE.1c, F.LE.5 - Functions Test

- Introduction to Geometry: Some surprising and beautiful geometric facts. Belief through reasoning and proof instead of magical thinking or appeal to authority. Foundations of Western and scientific thought. Postulates and proofs. Basic geometric notions: points, lines, distance, angles, angle measure (degrees). Straightedge, Ruler, and Protractor Postulates. Definitions of line segments, parallel lines, perpendicular lines, and circles. G.CO.1
- Rigid Motions: Translations, rotations, and reflections. These preserve lengths, angles, lines, and parallelism. G.CO.5, G.CO.6
- Congruence: Definition of congruence
using rigid motions. SAS criterion for congruence of triangles.

Exercises: congruent figures G.CO.5, G.CO.6, G.CO.7, G.CO.8 - ASA, Isosceles Triangles, and SSS: Proof of ASA. Definition of an isosceles triangle.
Proof that the base angles of an isosceles triangle have equal measures, and the
converse. Proof of SSS.

Exercises: congruent triangles G.CO.8, G.CO.10 - Symmetries of Polygons: Rigid motions taking a polygon to itself. G.CO.3
- Straightedge and Compass Constructions: Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Verification using triangle congruence criteria and other postulates and theorems. G.CO.12
- Constructing Regular Polygons: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.CO.13
- Rigid Motions, Congruence, and Geometric Constructions 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 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: diagonal lengths, distances using coordinates 8.G.7, 8.G.8 - Proof of the Slope Criteria for Parallel and Perpendicular Lines G.GPE.5
- Prove Geometric Theorems using Coordinates: Determine whether a point lies on a circle. Decide if a quadrilateral is a rectangle. G.GPE.4
- Compute Perimeters and Areas using Coordinates G.GPE.7
- Geometry Using Coordinates Test