## Grade 8 – CA Common Core Standards & Learning Objectives

### 8.8.NS The Number System

#### 8.8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Identify rational and irrational numbers (Eighth grade – D.1)

Convert between decimals and fractions or mixed numbers (Eighth grade – D.6)

#### 8.8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi²).

Estimate positive and negative square roots (Eighth grade – F.16)

Estimate cube roots (Eighth grade – F.21)

### 8.8.EE Expressions and Equations

#### 8.8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.

Understanding exponents (Eighth grade – F.1)

Evaluate exponents (Eighth grade – F.2)

Solve equations with variable exponents (Eighth grade – F.3)

Exponents with negative bases (Eighth grade – F.4)

Exponents with decimal and fractional bases (Eighth grade – F.5)

Understanding negative exponents (Eighth grade – F.6)

Evaluate negative exponents (Eighth grade – F.7)

Multiplication with exponents (Eighth grade – F.8)

Division with exponents (Eighth grade – F.9)

Multiplication and division with exponents (Eighth grade – F.10)

Power rule (Eighth grade – F.11)

Evaluate expressions involving exponents (Eighth grade – F.12)

Identify equivalent expressions involving exponents (Eighth grade – F.13)

Multiply monomials (Eighth grade – Z.6)

Divide monomials (Eighth grade – Z.7)

Multiply and divide monomials (Eighth grade – Z.8)

Powers of monomials (Eighth grade – Z.9)

#### 8.8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that the square root of 2 is irrational.

Identify rational and irrational numbers (Eighth grade – D.1)

Square roots of perfect squares (Eighth grade – F.14)

Positive and negative square roots (Eighth grade – F.15)

Relationship between squares and square roots (Eighth grade – F.17)

Cube roots of perfect cubes (Eighth grade – F.19)

Solve equations involving cubes and cube roots (Eighth grade – F.20)

Evaluate radical expressions (Eighth grade – S.7)

#### 8.8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

Convert between standard and scientific notation (Eighth grade – G.1)

Compare numbers written in scientific notation (Eighth grade – G.2)

#### 8.8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Convert between standard and scientific notation (Eighth grade – G.1)

Multiply numbers written in scientific notation (Eighth grade – G.3)

Divide numbers written in scientific notation (Eighth grade – G.4)

#### 8.8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

Unit rates (Eighth grade – H.5)

Do the ratios form a proportion? (Eighth grade – H.6)

Do the ratios form a proportion: word problems (Eighth grade – H.7)

Solve proportions (Eighth grade – H.8)

Solve proportions: word problems (Eighth grade – H.9)

Find the constant of proportionality from a graph (Eighth grade – I.3)

Graph proportional relationships (Eighth grade – I.5)

Solve problems involving proportional relationships (Eighth grade – I.8)

#### 8.8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Write equations for proportional relationships (Eighth grade – I.4)

Find the slope of a graph (Eighth grade – W.1)

Find the slope from two points (Eighth grade – W.2)

Find the slope of an equation (Eighth grade – W.4)

Graph a linear equation (Eighth grade – W.5)

Write a linear equation from a graph (Eighth grade – W.7)

Graph a line from an equation (Eighth grade – X.9)

#### 8.8.EE.7.a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Find the number of solutions (Eighth grade – U.12)

#### 8.8.EE.7.b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Solve equations involving squares and square roots (Eighth grade – F.18)

Model and solve equations using algebra tiles (Eighth grade – U.3)

Write and solve equations that represent diagrams (Eighth grade – U.4)

Solve one-step equations (Eighth grade – U.5)

Solve two-step equations (Eighth grade – U.6)

Solve multi-step equations (Eighth grade – U.7)

Solve equations involving like terms (Eighth grade – U.8)

Solve equations with variables on both sides (Eighth grade – U.9)

Solve equations: mixed review (Eighth grade – U.10)

Solve equations: word problems (Eighth grade – U.11)

#### 8.8.EE.8.a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Is (x, y) a solution to the system of equations? (Eighth grade – Y.1)

Solve a system of equations by graphing (Eighth grade – Y.2)

Find the number of solutions to a system of equations by graphing (Eighth grade – Y.4)

#### 8.8.EE.8.b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

Find the number of solutions to a system of equations (Eighth grade – Y.5)

Classify a system of equations by graphing (Eighth grade – Y.6)

Classify a system of equations (Eighth grade – Y.7)

Solve a system of equations using substitution (Eighth grade – Y.8)

Solve a system of equations using elimination (Eighth grade – Y.10)

#### 8.8.EE.8.c Solve real-world and mathematical problems leading to two linear equations in two variables.

Solve a system of equations by graphing: word problems (Eighth grade – Y.3)

Solve a system of equations using substitution: word problems (Eighth grade – Y.9)

Solve a system of equations using elimination: word problems (Eighth grade – Y.11)

### 8.8.F Functions

#### 8.8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Identify functions (Eighth grade – X.1)

Complete a table for a linear function (Eighth grade – X.7)

Graph a line from a function table (Eighth grade – X.8)

Evaluate a function graphically (Eighth grade – X.10)

#### 8.8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Graph a line from a function table (Eighth grade – X.8)

Graph a line from an equation (Eighth grade – X.9)

Write a linear function from a table (Eighth grade – X.11)

Identify linear and nonlinear functions (Eighth grade – X.14)

#### 8.8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

Graph a line from an equation (Eighth grade – X.9)

Identify linear and nonlinear functions (Eighth grade – X.14)

#### 8.8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Find the constant of proportionality from a graph (Eighth grade – I.3)

Write equations for proportional relationships (Eighth grade – I.4)

Find the constant of proportionality: word problems (Eighth grade – I.7)

Solve problems involving proportional relationships (Eighth grade – I.8)

Find the slope of a graph (Eighth grade – W.1)

Find the slope from two points (Eighth grade – W.2)

Find a missing coordinate using slope (Eighth grade – W.3)

Write a linear equation from a graph (Eighth grade – W.7)

Write a linear equation from two points (Eighth grade – W.9)

Rate of change (Eighth grade – X.4)

Constant rate of change (Eighth grade – X.5)

Write a linear function from a table (Eighth grade – X.11)

Write linear functions: word problems (Eighth grade – X.12)

#### 8.8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Write linear functions: word problems (Eighth grade – X.12)

### 8.8.G Geometry

#### 8.8.G.1.a Lines are taken to lines, and line segments to line segments of the same length.

Identify reflections, rotations, and translations (Eighth grade – Q.1)

Translations: graph the image (Eighth grade – Q.2)

Reflections: graph the image (Eighth grade – Q.4)

Rotations: graph the image (Eighth grade – Q.6)

#### 8.8.G.1.b Angles are taken to angles of the same measure.

Identify reflections, rotations, and translations (Eighth grade – Q.1)

Translations: graph the image (Eighth grade – Q.2)

Reflections: graph the image (Eighth grade – Q.4)

Rotations: graph the image (Eighth grade – Q.6)

#### 8.8.G.1.c Parallel lines are taken to parallel lines.

Identify reflections, rotations, and translations (Eighth grade – Q.1)

Translations: graph the image (Eighth grade – Q.2)

Reflections: graph the image (Eighth grade – Q.4)

Rotations: graph the image (Eighth grade – Q.6)

#### 8.8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Similar and congruent figures (Eighth grade – N.10)

Congruent figures: side lengths and angle measures (Eighth grade – N.12)

Congruence statements and corresponding parts (Eighth grade – N.13)

#### 8.8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Translations: find the coordinates (Eighth grade – Q.3)

Reflections: find the coordinates (Eighth grade – Q.5)

Rotations: find the coordinates (Eighth grade – Q.7)

Dilations: graph the image (Eighth grade – Q.8)

Dilations: find the coordinates (Eighth grade – Q.9)

#### 8.8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Similar and congruent figures (Eighth grade – N.10)

Similar figures: side lengths and angle measures (Eighth grade – N.11)

#### 8.8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Identify complementary, supplementary, vertical, adjacent, and congruent angles (Eighth grade – N.1)

Find measures of complementary, supplementary, vertical, and adjacent angles (Eighth grade – N.2)

Transversal of parallel lines (Eighth grade – N.3)

Find missing angles in triangles and quadrilaterals (Eighth grade – N.6)

Exterior Angle Theorem (Eighth grade – N.7)

Interior angles of polygons (Eighth grade – N.9)

Congruent triangles: SSS, SAS, and ASA (Eighth grade – N.14)

#### 8.8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

Converse of the Pythagorean theorem: is it a right triangle? (Eighth grade – O.5)

#### 8.8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Pythagorean theorem: find the length of the hypotenuse (Eighth grade – O.1)

Pythagorean theorem: find the missing leg length (Eighth grade – O.2)

Pythagorean theorem: find the perimeter (Eighth grade – O.3)

Pythagorean theorem: word problems (Eighth grade – O.4)

#### 8.8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Distance between two points (Eighth grade – P.4)

#### 8.8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Volume of cylinders and cones (Eighth grade – N.31)

Volume of spheres (Eighth grade – N.32)

### 8.8.SP Statistics and Probability

#### 8.8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

Scatter plots (Eighth grade – AA.14)

Outliers in scatter plots (Eighth grade – BB.8)

#### 8.8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Find the slope of a graph (Eighth grade – W.1)

Constant rate of change (Eighth grade – X.5)

Graph a line from an equation (Eighth grade – X.9)

Write linear functions: word problems (Eighth grade – X.12)

#### 8.8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

Interpret stem-and-leaf plots (Eighth grade – AA.9)

Interpret histograms (Eighth grade – AA.10)

Create histograms (Eighth grade – AA.11)

Create frequency charts (Eighth grade – AA.12)

## Grade 7 – CA Common Core Standards & IXL Practice

##### 7.7.RP.1     Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

Divide fractions and mixed numbers: word problems (Seventh grade – G.14)

Understanding ratios (Seventh grade – J.1)

Unit rates (Seventh grade – J.5)

Unit prices (Seventh grade – M.3)

Unit prices with unit conversions (Seventh grade – M.4)

##### 7.7.RP.2.a     Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

Equivalent ratios (Seventh grade – J.2)

Equivalent ratios: word problems (Seventh grade – J.3)

Do the ratios form a proportion? (Seventh grade – J.6)

Do the ratios form a proportion: word problems (Seventh grade – J.7)

Identify proportional relationships (Seventh grade – K.6)

##### 7.7.RP.2.b     Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

Find the constant of proportionality from a table (Seventh grade – K.1)

Find the constant of proportionality from a graph (Seventh grade – K.3)

Find the constant of proportionality: word problems (Seventh grade – K.7)

##### 7.7.RP.2.c     Represent proportional relationships by equations.

Solve proportions: word problems (Seventh grade – J.9)

Write equations for proportional relationships (Seventh grade – K.4)

Write equations for proportional relationships (Eighth grade – I.4)

##### 7.7.RP.3  Use proportional relationships to solve multistep ratio and percent problems.

Estimate population size using proportions (Seventh grade – J.10)

Estimate percents of numbers (Seventh grade – L.4)

Percents of numbers and money amounts (Seventh grade – L.5)

Percents of numbers: word problems (Seventh grade – L.6)

Solve percent equations (Seventh grade – L.7)

Solve percent equations: word problems (Seventh grade – L.8)

Percent of change (Seventh grade – L.9)

Percent of change: word problems (Seventh grade – L.10)

Unit prices with unit conversions (Seventh grade – M.4)

Unit prices: find the total price (Seventh grade – M.5)

Percent of a number: tax, discount, and more (Seventh grade – M.6)

Find the percent: tax, discount, and more (Seventh grade – M.7)

Sale prices: find the original price (Seventh grade – M.8)

Multi-step problems with percents (Seventh grade – M.9)

Estimate tips (Seventh grade – M.10)

Simple interest (Seventh grade – M.11)

Compound interest (Seventh grade – M.12)

Experimental probability (Seventh grade – CC.3)

##### 7.7.NS.1.a     Describe situations in which opposite quantities combine to make 0.

Absolute value and opposite integers (Seventh grade – B.4)

##### 7.7.NS.1.b       Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

Integers on number lines (Seventh grade – B.2)

Absolute value and opposite integers (Seventh grade – B.4)

Integer inequalities with absolute values (Seventh grade – B.6)

Integer addition and subtraction rules (Seventh grade – C.1)

Add and subtract integers (Seventh grade – C.3)

Complete addition and subtraction equations with integers (Seventh grade – C.4)

Add and subtract integers: word problems (Seventh grade – C.5)

Decimal number lines (Seventh grade – D.3)

Absolute value of rational numbers (Seventh grade – H.3)

Add and subtract rational numbers (Seventh grade – H.6)

##### 7.7.NS.1.c     Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

Understanding integers (Seventh grade – B.1)

Integers on number lines (Seventh grade – B.2)

Integer addition and subtraction rules (Seventh grade – C.1)

Add and subtract integers (Seventh grade – C.3)

Complete addition and subtraction equations with integers (Seventh grade – C.4)

Add and subtract integers: word problems (Seventh grade – C.5)

Decimal number lines (Seventh grade – D.3)

Add and subtract rational numbers (Seventh grade – H.6)

#### 7.7.NS.1.d    Apply properties of operations as strategies to add and subtract rational numbers.

Evaluate numerical expressions involving integers (Seventh grade – C.9)

Add and subtract decimals (Seventh grade – E.1)

Evaluate numerical expressions involving decimals (Seventh grade – E.11)

Add and subtract fractions (Seventh grade – G.1)

Add and subtract mixed numbers (Seventh grade – G.3)

Apply addition and subtraction rules (Seventh grade – H.7)

Properties of addition and multiplication (Seventh grade – S.1)

##### 7.7.NS.2.a     Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

Integer multiplication and division rules (Seventh grade – C.6)

Multiply and divide integers (Seventh grade – C.7)

Complete multiplication and division equations with integers (Seventh grade – C.8)

Multiply and divide rational numbers (Seventh grade – H.8)

Distributive property (Seventh grade – S.2)

##### 7.7.NS.2.b     Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

Multiplicative inverses (Seventh grade – A.3)

Divisibility rules (Seventh grade – A.4)

Integer multiplication and division rules (Seventh grade – C.6)

Multiply and divide integers (Seventh grade – C.7)

Complete multiplication and division equations with integers (Seventh grade – C.8)

Divide decimals by whole numbers: word problems (Seventh grade – E.6)

Understanding fractions: word problems (Seventh grade – F.3)

Divide fractions and mixed numbers: word problems (Seventh grade – G.14)

Multiply and divide rational numbers (Seventh grade – H.8)

##### 7.7.NS.2.c     Apply properties of operations as strategies to multiply and divide rational numbers.

Evaluate numerical expressions involving integers (Seventh grade – C.9)

Multiply decimals (Seventh grade – E.3)

Divide decimals (Seventh grade – E.5)

Evaluate numerical expressions involving decimals (Seventh grade – E.11)

Multiply fractions and whole numbers (Seventh grade – G.7)

Multiply fractions (Seventh grade – G.9)

Multiply mixed numbers (Seventh grade – G.10)

Divide fractions (Seventh grade – G.12)

Divide mixed numbers (Seventh grade – G.13)

Apply multiplication and division rules (Seventh grade – H.9)

Properties of addition and multiplication (Seventh grade – S.1)

#### 7.7.NS.2.d     Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

Classify numbers (Seventh grade – A.10)

Convert between decimals and fractions or mixed numbers (Seventh grade – H.2)

#### 7.7.NS.3     Solve real-world and mathematical problems involving the four operations with rational numbers.

Add and subtract integers (Seventh grade – C.3)

Complete addition and subtraction equations with integers (Seventh grade – C.4)

Add and subtract integers: word problems (Seventh grade – C.5)

Integer multiplication and division rules (Seventh grade – C.6)

Multiply and divide integers (Seventh grade – C.7)

Complete multiplication and division equations with integers (Seventh grade – C.8)

Add and subtract decimals (Seventh grade – E.1)

Add and subtract decimals: word problems (Seventh grade – E.2)

Multiply decimals (Seventh grade – E.3)

Multiply decimals and whole numbers: word problems (Seventh grade – E.4)

Divide decimals (Seventh grade – E.5)

Divide decimals by whole numbers: word problems (Seventh grade – E.6)

Add, subtract, multiply, and divide decimals: word problems (Seventh grade – E.8)

Add and subtract fractions (Seventh grade – G.1)

Add and subtract fractions: word problems (Seventh grade – G.2)

Add and subtract mixed numbers (Seventh grade – G.3)

Add and subtract mixed numbers: word problems (Seventh grade – G.4)

Inequalities with addition and subtraction of fractions and mixed numbers (Seventh grade – G.5)

Multiply fractions and whole numbers (Seventh grade – G.7)

Multiply fractions (Seventh grade – G.9)

Multiply mixed numbers (Seventh grade – G.10)

Multiply fractions and mixed numbers: word problems (Seventh grade – G.11)

Divide fractions (Seventh grade – G.12)

Divide mixed numbers (Seventh grade – G.13)

Divide fractions and mixed numbers: word problems (Seventh grade – G.14)

Add, subtract, multiply, and divide fractions and mixed numbers: word problems (Seventh grade – G.16)

Add and subtract rational numbers (Seventh grade – H.6)

Multiply and divide rational numbers (Seventh grade – H.8)

Add, subtract, multiply, and divide money amounts: word problems (Seventh grade – M.1)

Price lists (Seventh grade – M.2)

##### 7.7.EE.1     Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

Add and subtract like terms (Seventh grade – R.8)

Add, subtract, and multiply linear expressions (Seventh grade – R.9)

Factor linear expressions (Seventh grade – R.10)

Identify equivalent linear expressions (Seventh grade – R.11)

Properties of addition and multiplication (Seventh grade – S.1)

Distributive property (Seventh grade – S.2)

Write equivalent expressions using properties (Seventh grade – S.3)

#### 7.7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

Scientific notation (Seventh grade – A.8)

Compare numbers written in scientific notation (Seventh grade – A.9)

Evaluate numerical expressions involving integers (Seventh grade – C.9)

Round decimals (Seventh grade – D.4)

Estimate sums, differences, and products of decimals (Seventh grade – E.7)

Multi-step inequalities with decimals (Seventh grade – E.9)

Maps with decimal distances (Seventh grade – E.10)

Evaluate numerical expressions involving decimals (Seventh grade – E.11)

Equivalent fractions (Seventh grade – F.1)

Simplify fractions (Seventh grade – F.2)

Compare and order fractions (Seventh grade – F.5)

Compare fractions: word problems (Seventh grade – F.6)

Convert between mixed numbers and improper fractions (Seventh grade – F.7)

Compare mixed numbers and improper fractions (Seventh grade – F.8)

Round mixed numbers (Seventh grade – F.9)

Estimate sums and differences of mixed numbers (Seventh grade – G.6)

Estimate products and quotients of fractions and mixed numbers (Seventh grade – G.15)

Maps with fractional distances (Seventh grade – G.17)

Convert between decimals and fractions or mixed numbers (Seventh grade – H.2)

Compare ratios: word problems (Seventh grade – J.4)

Convert between percents, fractions, and decimals (Seventh grade – L.2)

Compare percents to fractions and decimals (Seventh grade – L.3)

Unit prices with unit conversions (Seventh grade – M.4)

Unit prices: find the total price (Seventh grade – M.5)

Estimate to solve word problems (Seventh grade – N.1)

Multi-step word problems (Seventh grade – N.2)

Guess-and-check word problems (Seventh grade – N.3)

Use Venn diagrams to solve problems (Seventh grade – N.4)

Find the number of each type of coin (Seventh grade – N.5)

Elapsed time word problems (Seventh grade – N.6)

##### 7.7.EE.4.a     Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

Solve proportions: word problems (Seventh grade – J.9)

Solve equations using properties (Seventh grade – S.4)

Model and solve equations using algebra tiles (Seventh grade – T.3)

Solve one-step equations (Seventh grade – T.5)

Solve two-step equations (Seventh grade – T.6)

Solve equations: word problems (Seventh grade – T.7)

Solve equations involving like terms (Seventh grade – T.8)

Solve word problems involving two-variable equations (Seventh grade – V.4)

##### 7.7.EE.4.b     Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

Solutions to inequalities (Seventh grade – U.1)

Write inequalities from number lines (Seventh grade – U.2)

Graph inequalities on number lines (Seventh grade – U.3)

Solve one-step inequalities (Seventh grade – U.4)

Graph solutions to one-step inequalities (Seventh grade – U.5)

Solve two-step inequalities (Seventh grade – U.6)

Graph solutions to two-step inequalities (Seventh grade – U.7)

#### 7.7.G.     Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

Scale drawings and scale factors (Seventh grade – J.13)

Similar and congruent figures (Seventh grade – X.12)

Similar figures: side lengths and angle measures (Seventh grade – X.13)

Similar figures and indirect measurement (Seventh grade – X.14)

Congruent figures: side lengths and angle measures (Seventh grade – X.15)

Congruence statements and corresponding parts (Seventh grade – X.16)

Perimeter, area, and volume: changes in scale (Seventh grade – X.30)

#### 7.7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

Front, side, and top view (Seventh grade – X.25)

Names and bases of 3-dimensional figures (Seventh grade – X.26)

#### 7.7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Parts of a circle (Seventh grade – X.21)

Circles: calculate area, circumference, radius, and diameter (Seventh grade – X.22)

Circles: word problems (Seventh grade – X.23)

#### 7.7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Identify complementary, supplementary, vertical, adjacent, and congruent angles (Seventh grade – X.4)

Find measures of complementary, supplementary, vertical, and adjacent angles (Seventh grade – X.5)

#### 7.7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Area of rectangles and parallelograms (Seventh grade – X.18)

Area of triangles and trapezoids (Seventh grade – X.19)

Area and perimeter: word problems (Seventh grade – X.20)

Nets of 3-dimensional figures (Seventh grade – X.27)

Surface area (Seventh grade – X.28)

Volume (Seventh grade – X.29)

### 7.7.SP Statistics and Probability

#### 7.7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

Identify representative, random, and biased samples (Seventh grade – BB.5)

#### 7.7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

Estimate population size using proportions (Seventh grade – J.10)

#### 7.7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

Calculate mean, median, mode, and range (Seventh grade – BB.1)

Interpret charts to find mean, median, mode, and range (Seventh grade – BB.2)

Mean, median, mode, and range: find the missing number (Seventh grade – BB.3)

Changes in mean, median, mode, and range (Seventh grade – BB.4)

##### 7.7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

Experimental probability (Seventh grade – CC.3)

Make predictions (Seventh grade – CC.4)

#### 7.7.SP.7.a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.

Probability of simple events (Seventh grade – CC.1)

#### 7.7.SP.7.b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

Experimental probability (Seventh grade – CC.3)

#### 7.7.SP.8.a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

Probability of opposite, mutually exclusive, and overlapping events (Seventh grade – CC.2)

Identify independent and dependent events (Seventh grade – CC.6)

Probability of independent and dependent events (Seventh grade – CC.7)

#### 7.7.SP.8.b Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.

Compound events: find the number of outcomes (Seventh grade – CC.5)

Factorials (Seventh grade – CC.8)

Permutations (Seventh grade – CC.9)

Counting principle (Seventh grade – CC.10)

Combination and permutation notation (Seventh grade – CC.11)

## Grade 6 CA Common Core – Standards with Learning Objectives

### 6.6.RP Ratios and Proportional Relationships

6 Understand ratio concepts and use ratio reasoning to solve problems.

#### 6.6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

Interpret ratios of two quantities  Write a ratio to describe objects in a picture (Sixth grade – R.1)

Use ratios to solve  word problems (Sixth grade – R.3)

#### 6.6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b is not equal to 0, and use rate language in the context of a ratio relationship.

Interpret and calculate nit rates and equivalent rates (Sixth grade – R.8)

Calculate unit rates.  Unit rates: word problems (Sixth grade – R.9)

#### 6.6.RP.3.a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

Determine ratio tables (Sixth grade – R.2)

Equivalent ratios (Sixth grade – R.4)

Equivalent ratios: word problems (Sixth grade – R.5)

Compare ratios using tables: word problems (Sixth grade – R.6)

Coordinate graphs review (Sixth grade – W.1)

#### 6.6.RP.3.b Solve unit rate problems including those involving unit pricing and constant speed.

Unit rates and equivalent rates (Sixth grade – R.8)

Unit rates: word problems (Sixth grade – R.9)

Unit prices with fractions and decimals (Sixth grade – U.3)

Unit prices with customary unit conversions (Sixth grade – U.4)

#### 6.6.RP.3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Percents of numbers and money amounts (Sixth grade – R.14)

Percents of numbers: word problems (Sixth grade – R.15)

Which is the better coupon? (Sixth grade – U.1)

Unit prices: which is the better buy? (Sixth grade – U.2)

Sale prices (Sixth grade – U.5)

Sale prices: find the original price (Sixth grade – U.6)

Percents – calculate tax, tip, mark-up, and more (Sixth grade – U.7)

#### 6.6.RP.3.d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Convert and compare customary units (Sixth grade – S.3)

Convert, compare, add, and subtract mixed customary units (Sixth grade – S.4)

Multiply and divide mixed customary units (Sixth grade – S.5)

Customary unit conversions involving fractions and mixed numbers (Sixth grade – S.6)

Convert and compare metric units (Sixth grade – S.7)

Convert between customary and metric systems (Sixth grade – S.8)

Unit prices with customary unit conversions (Sixth grade – U.4)

### 6.6.NS The Number System

#### 6.6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

Divide whole numbers by unit fractions using models (Sixth grade – L.1)

Reciprocals (Sixth grade – L.2)

Divide whole numbers and unit fractions (Sixth grade – L.3)

Divide fractions (Sixth grade – L.5)

Estimate quotients when dividing mixed numbers (Sixth grade – L.6)

Divide fractions and mixed numbers (Sixth grade – L.7)

Divide fractions and mixed numbers: word problems (Sixth grade – L.8)

Add, subtract, multiply, or divide two fractions (Sixth grade – O.7)

Add, subtract, multiply, or divide two fractions: word problems (Sixth grade – O.8)

#### 6.6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

Divisibility rules (Sixth grade – C.1)

Division patterns with zeroes (Sixth grade – C.2)

Divide numbers ending in zeroes: word problems (Sixth grade – C.3)

Estimate quotients (Sixth grade – C.4)

Divide whole numbers – 2-digit divisors (Sixth grade – C.5)

Divide whole numbers – 3-digit divisors (Sixth grade – C.6)

Add, subtract, multiply, or divide two whole numbers (Sixth grade – O.1)

Add, subtract, multiply, or divide two whole numbers: word problems (Sixth grade – O.2)

#### 6.6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Add and subtract decimal numbers (Sixth grade – G.1)

Add and subtract decimals: word problems (Sixth grade – G.2)

Estimate sums and differences of decimals (Sixth grade – G.3)

Maps with decimal distances (Sixth grade – G.4)

Multiply decimals (Sixth grade – H.1)

Estimate products of decimal numbers (Sixth grade – H.2)

Inequalities with decimal multiplication (Sixth grade – H.3)

Divide decimals by whole numbers (Sixth grade – H.4)

Divide decimals by whole numbers: word problems (Sixth grade – H.5)

Multiply and divide decimals by powers of ten (Sixth grade – H.6)

Division with decimal quotients (Sixth grade – H.7)

Inequalities with decimal division (Sixth grade – H.8)

Add, subtract, multiply, or divide two decimals (Sixth grade – O.4)

Add, subtract, multiply, or divide two decimals: word problems (Sixth grade – O.5)

Perform multiple operations with decimals (Sixth grade – O.6)

#### 6.6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

Identify factors (Sixth grade – E.4)

Greatest common factor (Sixth grade – E.7)

Least common multiple (Sixth grade – E.8)

GCF and LCM: word problems (Sixth grade – E.9)

#### 6.6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Understanding integers (Sixth grade – M.1)

Working with temperatures above and below zero (Sixth grade – S.9)

#### 6.6.NS.6.a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.

Absolute value and opposite integers (Sixth grade – M.2)

Integers on number lines (Sixth grade – M.3)

#### 6.6.NS.6.b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

Coordinate graphs review (Sixth grade – W.1)

Graph points on a coordinate plane (Sixth grade – W.2)

Reflections: graph the image (Sixth grade – BB.18)

#### 6.6.NS.6.c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

Decimal number lines (Sixth grade – F.9)

Integers on number lines (Sixth grade – M.3)

Graph integers on horizontal and vertical number lines (Sixth grade – M.4)

Rational numbers: find the sign (Sixth grade – P.6)

Coordinate graphs review (Sixth grade – W.1)

Graph points on a coordinate plane (Sixth grade – W.2)

Coordinate graphs as maps (Sixth grade – W.3)

Translations: graph the image (Sixth grade – BB.17)

#### 6.6.NS.7.a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

Write inequalities from number lines (Sixth grade – Z.2)

#### 6.6.NS.7.b Write, interpret, and explain statements of order for rational numbers in real-world contexts.

Compare rational numbers (Sixth grade – P.1)

Put rational numbers in order (Sixth grade – P.2)

#### 6.6.NS.7.c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

Absolute value and opposite integers (Sixth grade – M.2)

Absolute value of rational numbers (Sixth grade – P.3)

#### 6.6.NS.7.d Distinguish comparisons of absolute value from statements about order.

Put rational numbers in order (Sixth grade – P.2)

Absolute value of rational numbers (Sixth grade – P.3)

#### 6.6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Coordinate graphs review (Sixth grade – W.1)

Graph points on a coordinate plane (Sixth grade – W.2)

Coordinate graphs as maps (Sixth grade – W.3)

Distance between two points (Sixth grade – W.4)

Relative coordinates (Sixth grade – W.5)

### 6.6.EE Expressions and Equations

#### 6.6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

Write multiplication expressions using exponents (Sixth grade – D.1)

Evaluate exponents (Sixth grade – D.2)

Find the missing exponent or base (Sixth grade – D.3)

Exponents with decimal bases (Sixth grade – D.4)

Exponents with fractional bases (Sixth grade – D.5)

#### 6.6.EE.2.a Write expressions that record operations with numbers and with letters standing for numbers.

Write variable expressions (Sixth grade – X.1)

Write variable expressions: word problems (Sixth grade – X.2)

Write a two-variable equation (Sixth grade – AA.6)

#### 6.6.EE.2.b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.

Identify terms and coefficients (Sixth grade – X.6)

#### 6.6.EE.2.c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Perform multiple operations with whole numbers (Sixth grade – O.3)

Convert between Celsius and Fahrenheit (Sixth grade – S.10)

Evaluate variable expressions with whole numbers (Sixth grade – X.3)

Evaluate multi-variable expressions (Sixth grade – X.4)

Evaluate variable expressions with decimals, fractions, and mixed numbers (Sixth grade – X.5)

Complete a table for a two-variable relationship (Sixth grade – AA.5)

#### 6.6.EE.3 Apply the properties of operations to generate equivalent expressions.

Properties of addition (Sixth grade – X.7)

Properties of multiplication (Sixth grade – X.8)

Distributive property (Sixth grade – X.9)

Write equivalent expressions using properties (Sixth grade – X.11)

#### 6.6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

Add and subtract like terms (Sixth grade – X.12)

Identify equivalent expressions (Sixth grade – X.13)

#### 6.6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

Does x satisfy an equation? (Sixth grade – Y.1)

Find the solution from a set (Sixth grade – Y.2)

Solve one-step equations with whole numbers (Sixth grade – Y.6)

Solutions to inequalities (Sixth grade – Z.1)

Solve one-step inequalities (Sixth grade – Z.4)

#### 6.6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Convert between Celsius and Fahrenheit (Sixth grade – S.10)

Write variable expressions: word problems (Sixth grade – X.2)

Write an equation from words (Sixth grade – Y.3)

Solve word problems involving two-variable equations (Sixth grade – AA.4)

#### 6.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Model and solve equations using algebra tiles (Sixth grade – Y.4)

Write and solve equations that represent diagrams (Sixth grade – Y.5)

Solve one-step equations with whole numbers (Sixth grade – Y.6)

Solve one-step equations with decimals, fractions, and mixed numbers (Sixth grade – Y.7)

Solve one-step equations: word problems (Sixth grade – Y.8)

#### 6.6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Write inequalities from number lines (Sixth grade – Z.2)

Graph inequalities on number lines (Sixth grade – Z.3)

#### 6.6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

Solve one-step equations: word problems (Sixth grade – Y.8)

Identify independent and dependent variables (Sixth grade – AA.2)

Find a value using two-variable equations (Sixth grade – AA.3)

Solve word problems involving two-variable equations (Sixth grade – AA.4)

Complete a table for a two-variable relationship (Sixth grade – AA.5)

Write a two-variable equation (Sixth grade – AA.6)

Identify the graph of an equation (Sixth grade – AA.7)

Graph a two-variable equation (Sixth grade – AA.8)

Interpret a graph: word problems (Sixth grade – AA.9)

Write an equation from a graph using a table (Sixth grade – AA.10)

### 6.6.G Geometry

#### 6.6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Area (Sixth grade – BB.23)

Area of compound figures (Sixth grade – BB.24)

Compare area and perimeter of two figures (Sixth grade – BB.28)

#### 6.6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Volume of cubes and rectangular prisms (Sixth grade – BB.36)

#### 6.6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

Coordinate graphs review (Sixth grade – W.1)

Distance between two points (Sixth grade – W.4)

#### 6.6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Nets of 3-dimensional figures (Sixth grade – BB.35)

Surface area of cubes and rectangular prisms (Sixth grade – BB.37)

Volume and surface area of triangular prisms (Sixth grade – BB.38)

### 6.6.SP Statistics and Probability

#### 6.6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

Identify representative, random, and biased samples (Sixth grade – DD.4)

#### 6.6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

Stem-and-leaf plots (Sixth grade – CC.3)

Create line plots (Sixth grade – CC.5)

Interpret box-and-whisker plots (Sixth grade – CC.19)

#### 6.6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Calculate mean, median, mode, and range (Sixth grade – DD.1)

Interpret charts to find mean, median, mode, and range (Sixth grade – DD.2)

Mean, median, mode, and range: find the missing number (Sixth grade – DD.3)

#### 6.6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

Interpret pictographs (Sixth grade – CC.1)

Create pictographs (Sixth grade – CC.2)

Stem-and-leaf plots (Sixth grade – CC.3)

Interpret line plots (Sixth grade – CC.4)

Create line plots (Sixth grade – CC.5)

Create frequency tables (Sixth grade – CC.7)

Interpret bar graphs (Sixth grade – CC.8)

Create bar graphs (Sixth grade – CC.9)

Interpret double bar graphs (Sixth grade – CC.10)

Create double bar graphs (Sixth grade – CC.11)

Create histograms (Sixth grade – CC.13)

Circle graphs with fractions (Sixth grade – CC.14)

Interpret line graphs (Sixth grade – CC.15)

Create line graphs (Sixth grade – CC.16)

Interpret double line graphs (Sixth grade – CC.17)

Create double line graphs (Sixth grade – CC.18)

Interpret box-and-whisker plots (Sixth grade – CC.19)

Choose the best type of graph (Sixth grade – CC.20)

#### 6.6.SP.5.a Reporting the number of observations.

Create frequency tables (Sixth grade – CC.7)

Create histograms (Sixth grade – CC.13)

#### 6.6.SP.5.b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

Identify representative, random, and biased samples (Sixth grade – DD.4)

#### 6.6.SP.5.c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

Calculate mean, median, mode, and range (Sixth grade – DD.1)

Interpret charts to find mean, median, mode, and range (Sixth grade – DD.2)

Mean, median, mode, and range: find the missing number (Sixth grade – DD.3)

## Grade 6 CA Common Core – Statistics and Probability

### Ratios and Proportional Relationships

Understand ratio concepts and use ratio reasoning to solve problems.

### The Number System

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Compute fluently with multi-digit numbers and find common factors and multiples.

Apply and extend previous understandings of numbers to the system of rational numbers.

### Expressions and Equations

Apply and extend previous understandings of arithmetic to algebraic expressions.

Reason about and solve one-variable equations and inequalities.

Represent and analyze quantitative relationships between dependent and independent variables.

### Geometry

Solve real-world and mathematical problems involving area, surface area, and volume.

### 6.SP  Statistics and Probability

Building on and reinforcing their understanding of number, students begin to develop their ability to think ___.

statistically

Students recognize that a data distribution may not have a ____ ____ and that different ways to measure center yield different values.

definite center

The ____ measures center in the sense that it is roughly the middle value.

median

The ____ measures center in the sense that it is the value that each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point.

mean

Students recognize that a ____ ____  (interquartile range or mean absolute deviation) can also be useful for summarizing data. This is because two very different sets of data can have the same mean and median and yet can be distinguished by their ____.

measure of variability  —  variability

Students learn to ____ and ____ numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected.

describe  —  summarize

In Grade 5, students used bar graphs and line plots to represent data and then solved problems using the information presented in the plots (5.MD.B.2). In Grade 6, students move from simply representing data into statistical analysis of data.

Students begin to think and reason statistically, first by recognizing a statistical question as one that can be answered by collecting data (6.SP.A.1).

Students learn that the data collected to answer a statistical question has a distribution that is often summarized in terms of center, variability, and shape (6.SP.A.2).

Students see and represent data distributions using frequency tables, dot plots, histograms and (6.SP.B.4).

Students study quantitative ways to summarize numerical data sets in relation to their context and to the shape of the distribution. The mean and mean absolute deviation (MAD) are used for data distributions that are approximately symmetric, and the median and inter-quartile range (IQR) are used for distributions that are skewed.

#### Develop an understanding of statistical variability.

Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate ____ language.

mathematical

The terms students should learn to use with increasing ____ with this cluster are: statistics, data, variability, distribution, dot plot, histograms, box plots, median, mean.

precision

##### 6.SP.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a ____ question because one anticipates ____ in students’ ages.

statistical  —  variability

Statistics are ____ relating to a group of individuals; statistics is also the name for the ____ of collecting, analyzing and interpreting such data.

numerical data  —  science

A statistical question anticipates an answer that ____ from one individual to the next and is written to account for the ____ in the data.

varies  —  variability

Data are the ____ produced in response to a statistical question. Data are frequently ____ from surveys or other sources (i.e. documents).

numbers  —  collected

Students ____ between statistical questions and those that are not.

differentiate

A statistical question is one that collects information that addresses ____ in a population.

differences

The question is framed so that the ____ will allow for the differences.

responses

For example, the question, “How tall am I?” is not a statistical question because there is only ____ response; however, the question, “How tall are the students in my class?” is a statistical question since the responses anticipates variability by providing a ____ of possible anticipated responses that have numerical answers.

one  —  variety

Questions can result in a narrow or wide ____ of numerical values.

range

Students might want to know about the fitness of the students at their school so they might want to know about the exercise habits of the students. So . Rather than asking “Do you exercise?” they should ask about the ____ of exercise the students at their school get per week.

amount

A ____ ____ for this study could be: “How many hours per week on average do students at Jefferson Middle School exercise?”

statistical question

##### 6.SP.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

The ____ of the data set is the ordered arrangement of the values of a data set.

distribution

A data distribution can be described using the ____ , ____ and its ____.

center — spread — shape

Data collected can be represented on ____, which will show the shape of the distribution of the data.

graphs

Students examine the distribution of a data set and discuss the center, spread and overall shape with frequency tables, ____ ____, histograms and ____ ____.

dot plots  —  box plots

##### .     Example 1:  Describe a data distribution from a dot plot

The following dot plot shows the writing scores for a group of students on organization. Describe the data.

##### .     Example 1  Problem with Answer

The following dot plot shows the writing scores for a group of students on organization. Describe the data.

Ans:     The values range from 0 to 6. There is a peak at 3. The median is 3, which means 50% of the scores are greater than or equal to 3 and 50% are less than or equal to 3. The mean is 3.68. If all students scored the same, the score would be 3.68. The data is symmetrically distributed about the center.

NOTE: Mode as a measure of center and range as a measure of variability are not addressed in the CCSS and as such are not a focus of instruction. These concepts can be introduced during instruction as needed.

Students learn that the data collected to answer a statistical question has a distribution that is often summarized in terms of center, variability, and shape (6.SP.A.2).

They represent data distributions using dot plots and histograms (6.SP.B.4).

##### 6.SP.3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Data sets contain many numerical values that can be ____ by one number such as a measure of center.

summarized

The measure of center gives a numerical value to represent the ____ of the data (ie. midpoint of an ordered list or the balancing point).

center

Another characteristic of a data set is the ____ (or spread) of the values.

variability

Measures of variability are used to describe this characteristic; these include the Mean Absolute Deviation (____) and the Interquartile Range (____).

##### Example 1: Interpret a dot plot

Consider the data shown in the dot plot of the six trait scores for organization for a group of students.

How many students are represented in the data set?

19  —  count all the x’s

How many students are represented in the data set for less than equal to ?

What are the mean and median of the data set?

What do these values mean?

How do they compare?

What is the range of the data?

What does this value mean?

PICTURE

Ans:
19 students are represented in the data set.

The mean of the data set is 3.5. The median is 3. The mean indicates that is the values were equally distributed, all students would score a 3.5.

The median indicates that 50% of the students scored a 3 or higher;
50% of the students scored a 3 or lower.

• The range of the data is 6, indicating that the values vary 6 points between the lowest and highest scores.

#### Summarize and describe distributions.

Mathematically proficient students ____ by engaging in discussion about their reasoning using appropriate mathematical language.

communicate precisely

The ____ students should learn to use with increasing ____ with this cluster are: box plots, dot plots, histograms, frequency tables, cluster, peak, gap, mean, median, interquartile range, measures of center, measures of variability, data, Mean Absolute Deviation (M.A.D.), quartiles, lower quartile (1st quartile or Q1), upper quartile (3rd quartile or Q3), symmetrical, skewed, summary statistics, outlier

terms   —  precision

##### 6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

Students display data graphically using ____ ____. Dot plots, histograms and box plots are three graphs to be used.

number lines

Students are expected to ____ the appropriate graph as well as ____ ____ from graphs generated by others.

determine  —  read data-

____ ____ are simple plots on a number line where each dot represents a piece of data in the data set.

Dot plots

Dot plots are suitable for small to moderate size data sets and are useful for highlighting the distribution of the data including ____, ____, and ____.

clusters, gaps, and outliers.

A histogram shows the distribution of continuous data using ____ on the number line.

intervals

The ____ of each bar in a histogram represents the  _____ of data values in that interval.

height  —  number

In most real data sets, there is a large amount of data and many numbers will be unique. A graph (such as a ____ ____) that shows how many ones, how many twos, etc. would not be meaningful; however, a ____ can also be used.

dot plot  —  histogram

Students group the data into convenient ranges and use these intervals to generate a ____ ____ and histogram.

frequency table

Note that changing the ____ of the bin changes the appearance of the graph and the conclusions may vary from it.

size

A box plot shows the distribution of values in a data set by ____ the set into ____.

dividing  —  quartiles

A box plot can be graphed either ____ or horizontally.

vertically

The box plot is constructed from the ____ ____ (minimum, lower quartile,
median, upper quartile, and maximum). These values give a summary of the ____ of a distribution.

five-number summary  —  shape

Students understand that the ____ of the box represents the ____ ____ of the data.

size  — middle 50%

Students can use applets to create data displays. Examples of applets include the Box Plot Tool and Histogram Tool on NCTM’s Illuminations.

Box Plot Tool – http://illuminations.nctm.org/ActivityDetail.aspx?ID=77
Histogram Tool — http://illuminations.nctm.org/ActivityDetail.aspx?ID=78

Example 1: Create and Summarize a Data Display

Nineteen students completed a writing sample that was scored on organization. The scores for organization were 0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6. Create a data display. What are some observations that can be made from the data display?

Ans:

PICTURE

19 students are represented in the data set.

The mean of the data set is 3.5.

The median is 3.

The mean indicates that is the values were equally distributed, all students would score a 3.5.

The median indicates that 50% of the students scored a 3 or higher;
50% of the students scored a 3 or lower.

The range of the data is 6, indicating that the values vary 6 points between the lowest and highest scores

Example 2:

Grade 6 students were collecting data for a math class project. They decided they would survey the other two grade 6 classes to determine how many DVDs each student owns. A total of 48 students were surveyed. The data are shown in the table below in no specific order. Create a data display. What are some observations that can be made from the data display?

PICTURE

Solution:
A histogram using 5 intervals (bins) 0-9, 10-19, …30-39) to organize the data is displayed below.

PICTURE

Most of the students have between 10 and 19 DVDs as indicated by the peak on the graph. The data is pulled to the right since only a few students own more than 30 DVDs.
Example 3:
Ms. Wheeler asked each student in her class to write their age in months on a sticky note. The 28 students in the class brought their sticky note to the front of the room and posted them in order on the white board. The data set is listed below in order from least to greatest. Create a data display. What are some observations that can be made from the data display?

PICTURE

Solution:
Five number summary
Minimum – 130 months
Quartile 1 (Q1) – (132 + 133) ÷ 2 = 132.5 months
Median (Q2) – 139 months
Quartile 3 (Q3) – (142 + 143) ÷ 2 = 142.5 months
Maximum – 150 months

PICTURE

This box plot shows that
• ¼ of the students in the class are from 130 to 132.5 months old
• ¼ of the students in the class are from 142.5 months to 150 months old
• ½ of the class are from 132.5 to 142.5 months old
• The median class age is 139 months.

##### 6.SP.5  Summarize numerical data sets in relation to their context, such as by:

Students summarize numerical data by providing background information about the ____ being measured, methods and unit of ____, the context of data collection activities (addressing random sampling), the number of _____, and summary statistics.

attribute  —  measurement  –  observations

Summary statistics include quantitative measures of ____ (median and median) and ____ (interquartile range and mean absolute deviation) including ____ values (minimum and maximum), mean, median, mode, range, and quartiles.

center  —  variability —  extreme

##### .     6.SP.5a  Reporting the number of observations.

Students record the ____ of observations.

number

Using ____, students determine the number of values between
specified intervals.

histograms

Given a box plot and the total number of data values, students identify the ____ of data points that are represented by the box.

number

Reporting of the number of observations must consider the ____ of the data sets, including ____ of measurement (when applicable).

attribute  —  units

##### .     6.SP.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

Reporting of the number of ____ should include descriptions of the attribute of the data sets, including units of ____ (when applicable).

observations  —  measurement

##### .     6.SP.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

For Measures of Center

Given a set of data values, students summarize the measure of center with the ____ or ____.

median  —  mean

The median is the value in the middle of a ____ ____ of data.

ordered list

The value of the median means that 50% of the data is ____ than or equal to it and that 50% of the data is ____ than or equal to it.

greater  —  less

The mean is the arithmetic average; the ____ of the values in a data set divided by ____ ____ values there are in the data set.

sum  —  how many

The ____ measures center in the sense that it is the value that each data point would take on if the total of the data values were ____ equally, and also in the sense that it is a balance point.

mean  —  redistributed

Students develop these understandings of what the mean represents by ____ data sets to be level or fair (equal distribution) and by observing that the total distance of the data values ____ the is equal to the total distance of the data values ____ the mean (balancing point).

redistributing  —  above  —  below

Students use the concept of ____ to solve problems by calculating it from a data set represented in a frequency table.

mean

Students find a ____ ____ in a data set to produce a specific average.

missing value

Example 1:

Susan has four 20-point projects for math class. Susan’s scores on the first 3 projects are shown below:
Project 1: 18
Project 2: 15
Project 3: 16
Project 4: ??
What does she need to make on Project 4 so that the average for the four projects is 17? Explain your reasoning.

Example 1 Solution:

One possible solution to is calculate the total number of points needed (17 x 4 or 68) to have an average of 17. She has earned 49 points on the first 3 projects, which means she needs to earn 19 points on Project 4 (68 – 49 = 19).

For Measures of Variability

Measures of variability/variation can be described using the inter-quartile range (____) or the Mean Absolute Deviation (____).

describes the variability between the middle 50% of a data set. It is found by ____ the lower quartile from the upper quartile.

subtracting

The inter-quartile range (IQR) represents the ____ of the box in a box plot and is not affected by ____.

length  —  outliers

Students find the IQR from a data set by finding the upper and lower quartiles and taking the ____ or from ____ a box plot.

Example 1:
What is the IQR of the data below:

PICTURE

Solution:
The first quartile is 132.5; the third quartile is 142.5. The IQR is 10 (142.5 – 132.5). This value indicates that the values of the middle 50% of the data vary by 10.

The Mean Absolute Deviation (MAD) describes the variability of the data set by determining the absolute deviation (the distance) of each data piece from the mean and then finding the average of these deviations.

Both the inter-quartile range and the Mean Absolute Deviation are represented by a single numerical value.

Higher values represent a greater variability in the data.

Example 2: Find the Mean Absolute Deviation of a Data Set

The following data set represents the size of 9 families: 3, 2, 4, 2, 9, 8, 2, 11, 4. What is the MAD for this data set?

Solution:
The mean is 5. The MAD is the average variability of the data set. To find the MAD:
1. Find the deviation from the mean.
2. Find the absolute deviation for each of the values from step 1
3. Find the average of these absolute deviations.
The table below shows these calculations:

PICTURE

This value indicates that on average family size varies 2.89 from the mean of 5.

##### .      6.SP.5d  Relating the choice of measures

Students understand how the measures of center and measures of variability are ____ by graphical displays.

represented

Students describe the context of the data, using the shape of the data and are able to use this information to determine an ____ _____ of center and measure of variability.

appropriate measure

The measure of center that a student chooses to describe a data set will depend upon the ____ of the data distribution and ____ of data collection.

shape  —  context

The mode is the value in the data set that occurs ____ ____.

most frequently

The mode is the ____ ____ used as a measure of center because data sets may not have a mode, may have more than one mode, or the mode may not be descriptive of the data set.

least frequently

The mean is a very common measure of center computed by ____ all the numbers in the set and ____ by the number of values.