Common Core Mathematics 1

Arithmetic Review (Optional)

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 6.EE.2
2. Word Problems: Cost of $n$ items. Multi-variable expressions.
   Exercises: expressions 6.EE.6, A.SSE.1a, A.CED.2
3. More Complicated Expressions: Examples involving negative numbers, division, fractions.
   Exercises: evaluation, division A.SSE.1a
4. 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
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.EE.3
6. Solving $x+b=c$: Adding a constant to both sides.
   Exercises: solving 6.EE.7, A.REI.1, A.REI.3
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 6.EE.3
8. 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
9. Applications of Linear Equations: Balanced scale, budgeting, temperature conversion.
   Exercises: solving 6.EE.7, 7.EE.4a, A.CED.1, F.BF.1ab
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 6.EE.3
12. 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
13. Solving $ax+b=cx+d$: Adding $-cx-b$ to both sides.
    Exercises: solving 8.EE.7b, A.REI.1, A.REI.3
14. Variables, Expressions, and Simple Equations Test 2

Linear Graphs

1. Points and Coordinates: Connecting the ordered pair $(x, y)$ with the coordinate plane. Quadrants.
   Exercises: quadrants, coordinates 6.NS.6bc
2. Investigating $y=x+b$: Finding the $y$-intercept $(0, b)$ of a line.
   Exercises: $y$-intercepts F.IF.7a, F.BF.3
3. Investigating $y=mx$: Lines with slope $m$, positive or negative.
   Exercises: slopes 8.EE.5, 8.F.2, F.LE.1a
4. 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
5. Solving Equations Graphically: Graphing each side and finding the intersection, e.g. for $ax+b=cx+d$.
   Exercises: solving A.REI.11
6. Finding Formulas for Approximately Linear Data: Simple statistics application. Uses root-mean-square error informally. 8.SP.2, F.IF.4, S.ID.6ac
7. 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
8. Finding Equations for Lines: Point-slope form.
   Exercises: point-slope form F.LE.2
9. Parallel and Perpendicular Lines: Slopes $m$ and $$-1/m$$.
   Exercises: parallel lines, perpendicular lines 8.EE.8c, G.GPE.5
10. 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
11. Linear Graphs Test

Inequalities

1. Linear Inequalities: Half-plane solution graphs, with or without boundary line.
   Exercises: evaluating, identifying inequalities A.CED.1, A.REI.12
2. Equivalent Inequalities: Adding a constant to both sides of an inequality. Multiplying both sides by a positive or negative number.
   Exercises: addition, multiplication
3. Solving Inequalities Algebraically: Finding algebraic solutions to $ax+b < c$ and similar inequalities.
   Exercises: solving A.REI.3
4. Inequalities Test

Descriptive Statistics

1. 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
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
3. Measuring Spread in Data: Mean absolute deviation, standard deviation.
   Exercises: mean absolute deviations, standard deviations 6.SP.3, S.ID.2
4. 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
5. 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
6. 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
7. Descriptive Statistics Test

Systems of Linear Equations or Inequalities

1. Solving Systems of Linear Equations by Graphing: Solutions are intersections of graphs.
   Exercises: checking, solving 8.EE.8a, A.REI.6, A.REI.11
2. 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
3. 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
4. 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
5. Mixture Problems: Mathematics of mixing different concentrations.
   Exercises: solving A.CED.2, A.REI.6
6. Systems of Linear Inequalities: Solution graphs are intersections of half-planes.
   Exercises: solving A.REI.12
7. Linear Optimization: Maximizing profit subject to linear inequalities. A.CED.3
8. Systems of Linear Equations or Inequalities Test

Functions

1. 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
2. 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
3. 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
4. Shifting Functions: Moving a graph through composition of functions. Periodic functions.
   Exercises: shifting functions F.BF.3
5. 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
6. 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
7. 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
8. Functions Test

Rigid Motions, Congruence, and Geometric Constructions

1. 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
2. Rigid Motions: Translations, rotations, and reflections. These preserve lengths, angles, lines, and parallelism. G.CO.5, G.CO.6
3. 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
4. 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
5. Symmetries of Polygons: Rigid motions taking a polygon to itself. G.CO.3
6. 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
7. Constructing Regular Polygons: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.CO.13
8. Rigid Motions, Congruence, and Geometric Constructions Test

Geometry Using Coordinates

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. Proof of the Slope Criteria for Parallel and Perpendicular Lines G.GPE.5
4. Prove Geometric Theorems using Coordinates: Determine whether a point lies on a circle. Decide if a quadrilateral is a rectangle. G.GPE.4
5. Compute Perimeters and Areas using Coordinates G.GPE.7
6. Geometry Using Coordinates Test