Common Core Grade 8 Mathematics

Arithmetic Review

1. Arithmetic on a Grid: Counting unit squares. Sum, difference, product, quotient.
2. Two-Digit Addition: Two-digit numbers shown as $a(10)+b$ on grid. Addition with regrouping, carrying.
   Exercises: addition
3. Two-Digit Subtraction and Multiplication: Subtraction with borrowing, multiplication with regrouping.
   Exercises: subtraction, multiplication, division
4. Negative Numbers: Green squares have value $+1$, pink squares have value $-1$. Sums and differences. Word problems involving debt.
   Exercises: addition, subtraction
5. Multiplication and Division with Negative Numbers: Products and quotients with negative numbers on the grid, introduced through analogy and pattern-matching.
   Exercises: multiplication, division
6. Fraction Addition and Subtraction: Visualizing fractions using pie slices. Reducing to lowest terms. Finding common denominators.
   Exercises: addition, subtraction
7. Fraction Multiplication and Division: Using an expanded grid to visualize fractions.
   Exercises: multiplication, division
8. Arithmetic Test

Variables, Expressions, and Simple Equations

1. Variables and Expressions: Expressions as quantities. Simple linear expressions. Sliders for variables.
   Exercises: evaluation
2. Word Problems: Cost of $n$ items. Multi-variable expressions.
   Exercises: expressions
3. More Complicated Expressions: Examples involving negative numbers, division, fractions.
   Exercises: evaluation, division
4. Equations as Sentences: Solving equations by trial and error, or by sliding a slider. Inverse problems as word problems.
   Exercises: solving
5. 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. Solving $x+b=c$: Adding a constant to both sides.
   Exercises: solving
7. 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
8. Solving $ax+b=c$: Adding $-b$ and then multiplying by $$1/a$$, or dividing by $a$.
   Exercises: solving 1, solving 2
9. Applications of Linear Equations: Balanced scale, budgeting, temperature conversion.
   Exercises: solving
10. Variables, Expressions, and Simple Equations Test 1
11. 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
12. Manipulating Linear Expressions: Adding, subtracting, scaling, simplifying. Commutativity of addition and multiplication. Zero and one laws.
    Exercises: simplifying 1, simplifying 2
13. Solving $ax+b=cx+d$: Adding $-cx-b$ to both sides.
    Exercises: solving 8.EE.7
14. 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
15. Variables, Expressions, and Simple Equations Test 2

Congruence, Similarity, and Coordinates

1. Rigid Motions: Translations, rotations, and reflections. These preserve lengths, angles, lines, and parallelism. 8.G.1
2. Congruence: Definition of congruence using rigid motions. SAS criterion for congruence of triangles.
   Exercises: congruent figures 8.G.2
3. 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
4. Related Angles: Acute, obtuse, and right angles. Supplementary and vertical angles. Translations, parallel lines, and transversals.
   Exercises: related angles 8.G.5
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
6. Dilations: Definition of a dilation. A dilation takes a line to the same or a parallel line, preserves angle measures, and scales lengths by a single common factor.
   Exercises: dilations 8.G.4
7. 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
8. 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
9. Points and Coordinates: Connecting the ordered pair $(x, y)$ with the coordinate plane. Quadrants.
   Exercises: quadrants, coordinates
10. 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
11. Congruence, Similarity, and Coordinates Test

Linear Graphs

1. Investigating $y=x+b$: Finding the $y$-intercept $(0, b)$ of a line.
   Exercises: $y$-intercepts
2. Investigating $y=mx$: Lines with slope $m$, positive or negative.
   Exercises: slopes 8.EE.5, 8.F.2
3. Investigating $y=mx+b$: Slope and $y$-intercept.
   Exercises: $y$-intercepts 1, slopes 1, $y$-intercepts 2, slopes 2, equations
4. 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
5. 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
6. Finding Equations for Lines: Point-slope form.
   Exercises: point-slope form
7. Parallel and Perpendicular Lines: Slopes $m$ and $$-1/m$$.
   Exercises: parallel lines, perpendicular lines 8.EE.8c
8. 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
9. Linear Graphs Test

Descriptive Statistics

1. Associations Between Categories: Two-way frequency tables. Possible associations between two ways of dividing up a population. 8.SP.4
2. Finding Formulas for Approximately Linear Data: Introduction to scatter plots and linear models. 8.SP.1, 8.SP.2
3. 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

Systems of Linear Equations

1. 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
2. Solving Systems of Linear Equations by Graphing: Solutions are intersections of graphs.
   Exercises: checking, solving 8.EE.8a
3. Solving Systems of Linear Equations by Subtraction: Solving for a difference equal to 0.
   Exercises: solving 8.EE.8b
4. Applications of Systems of Linear Equations: Cost, revenue, profit. Birth and death rates. 8.EE.8c, 8.SP.3
5. Systems of Linear Equations Test

Exponents, Roots, and Real Numbers

1. Integer Exponents: Basic properties of exponents. Values of exponential expressions with zero or negative exponents.
   Exercises: computing, simplifying 8.EE.1
2. Scientific Notation: Conversion between decimal and scientific notation. Arithmetic with scientific notation. Masses of planetary systems, number of cells in the human body.
   Exercises: conversion, addition and subtraction, multiplication and division 1, multiplication and division 2 8.EE.3, 8.EE.4
3. Square and Cube Roots: Geometric interpretations of second and third powers. Definitions of square and cube roots. 8.EE.2
4. Rational Numbers and Repeating Decimals: Converting from rational numbers to repeating decimal expansions and vice versa. $π$ and $√2$ are irrational. Using approximations to compare irrational numbers and do arithmetic with them.
   Exercises: fractions, repeating decimals 8.NS.1, 8.NS.2, 8.EE.2
5. Exponents, Roots, and Real Numbers Test

The Pythagorean Theorem, Distance, and Volume

1. 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
2. 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
3. Solid Geometry: Volume of prisms, cylinders, pyramids, cones, and spheres.
   Exercises: prisms and pyramids, cylinders, cones, spheres 8.G.9
4. Pythagorean Theorem, Distance, and Volume Test