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Common Core Algebra Preparation
The Common Core and other standards cover many algebraic topics before the first actual High
School Algebra course. In practice, not all students learn all these topics thoroughly before
starting High School, and this becomes a major impediment for them. We therefore include the
introductory lessons below in our algebra courses, but ideally students will work through them
in earlier grades, or in this separate preparatory course. This short course is thus designed to
be taken during the summer before High School, or concurrently with the first semester of High
School Algebra, if necessary.
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 many
algebraic 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. We also provide
a course glossary.
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 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.
Two-Digit Subtraction and
Multiplication: Subtraction with borrowing, multiplication with regrouping.
Negative Numbers: Green squares have value
$+1$, pink squares have value $-1$. Sums and differences. Word problems involving debt.
Multiplication and Division with
Negative Numbers: Products and quotients with negative numbers on the grid,
introduced through analogy and pattern-matching.
Fraction Addition and Subtraction:
Visualizing fractions using pie slices. Reducing to lowest terms. Finding common
Fraction Multiplication and Division:
Using an expanded grid to visualize fractions.
Variables, Expressions, and Simple Equations
Variables and Expressions: Expressions as
quantities. Simple linear expressions. Sliders for variables.
6.EE.2 Word Problems: Cost of $n$ items. Multi-variable
6.EE.6, A.SSE.1a, A.CED.2 More Complicated Expressions:
Examples involving negative numbers, division, fractions.
Equations as Sentences: Solving equations
by trial and error, or by sliding a slider. Inverse problems as word problems.
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.
6.EE.3, A.APR.1 Solving $x+b=c$: Adding a constant to both sides.
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.
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,
6.EE.7, A.REI.1, A.REI.3 Applications of Linear Equations:
Balanced scale, budgeting, temperature conversion.
6.EE.7, 7.EE.4a, A.CED.1, F.BF.1a Variables, Expressions, and Simple Equations
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,
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.
6.EE.3, 7.EE.1, A.APR.1 Solving $ax+b=cx+d$: Adding $-cx-b$ to both sides.
8.EE.7b, A.REI.1, A.REI.3 Variables, Expressions, and Simple Equations
Points and Coordinates:
Connecting the ordered pair $(x, y)$ with the coordinate plane. Quadrants.
6.NS.6bc Investigating $y=x+b$:
Finding the $y$-intercept $(0, b)$ of a line.
Investigating $y=mx$: Lines with slope $m$, positive or
F.LE.1a Investigating $y=mx+b$: Slope and
8.F.3, F.IF.7a, F.LE.1a Solving Equations Graphically: Graphing
each side and finding the intersection, e.g. for $ax+b=cx+d$.
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
8.F.4, F.IF.6, F.LE.1b, F.LE.2 Finding Equations for Lines: Point-slope form.
Exercises: point-slope form
Parallel and Perpendicular
Lines: Slopes $m$ and $$-1/m$$.
8.EE.8c Equations for Lines in Standard Form:
$Ax + By = C$. Converting to slope-intercept form. Computing slopes,
$x$- and $y$-intercepts.
converting to slope-intercept
Linear Graphs Test
Summarizing a Collection of
Measurements: Converting to common units. Dot (line) plots. Median and mean.
Exercises: median from a list,
median from frequencies,
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.
6.SP.4, S.ID.1, S.ID.2, S.ID.3 Descriptive Statistics Preparation
Systems of Linear Equations
Solving Systems of Linear Equations by
Graphing: Solutions are intersections of graphs.
8.EE.8a, A.REI.6 Solving Systems of Linear Equations by
Subtraction: Solving for a difference equal to 0.
8.EE.8b, A.REI.5, A.REI.6 Systems of Linear Equations Preparation