## 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)

#### 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)

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)

Decimal number lines (Seventh grade – D.3)

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

##### 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)

Decimal number lines (Seventh grade – D.3)

#### 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)

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

##### 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)

#### 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.

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 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)

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)

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.

Factor linear expressions (Seventh grade – R.10)

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

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: 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)

### 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)

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)

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)

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: 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.

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: 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 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).

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 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 – Expressions and Equations

#### Expressions and Equations  6.EE

##### Apply and extend previous understandings of arithmetic to algebraic expressions.

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

reasoning

The ____ students should learn to use with increasing precision with this cluster are: exponents, base, numerical expressions, algebraic expressions, evaluate, sum, term, product, factor, quantity, quotient, coefficient, constant, like terms, equivalent expressions, variables

terms

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

Students demonstrate the meaning of ____ to write and evaluate numerical expressions with whole number exponents.

exponents

The ____ of an exponential term in an expression can be a whole number, positive decimal or a positive fraction.

base

For example, the base ½ raised to the fifth power thereby having the exponent 5 (or ½ ⁵ ) can be written using type vs math symbols as ____.

Write the one-half with numerator-slash-denominator form 1/2 then put in parentheses and use the carat symbol to indicate raising to the fifth power …. (1/2)^5

Find the value of  (1/2)^5

This means  ½ • ½ • ½ •  ½ •  ½ which can be evaluated pairwise in sequence as  (½ • ½) • (½ •  ½ )•  ½     or     [(1/4) (1/4)] (1/2) = (1/16) (1/2) …  which has the same value as 1/32. And so:  ½ ⁵ = 1/32

Students  recognize that an expression with a  represents the same mathematics (ie. x^5 can be written as x • x • x • x • x) and write algebraic expressions from verbal expressions.

variable

Order of operations is introduced throughout elementary grades, including the use of ____ ____, ( ), { }, and [ ] in 5th grade.

grouping symbols

Order of operations with ____ is the focus in 6th grade.

exponents

Example 1: Find the values of expressions with exponential terms

Evaluate terms with exponents after grouping symbols but before multiplication or division

Example 1a: Find value of:  (0.2)^3

(0.2)^3 = (0.2) (0.2) (0.2) = [(0.2) (0,2)] (0.2)    then     (0.04) (0.2) = 0.008

Example 1b: Find the value of  5 + 2^4 • 6

Exponent term first  5 + 16 • 6    multiplication next  5 + 96                   then addition   101

Example 1c: Find the value of  7^2 24 ÷ 3 + 26

Exponent term first  49 – 24 ÷ 3 + 26 … division next  49 – 8 + 26            then addition-subtraction left to right  41 + 26 = 67

Example 2: Area formula expression using substitution with a variable

What is the area of a square with a side length of 3x?

Area of a square with side s is s²  … so let s = 3x   then   s³ = 3x • 3x = 9x²

Example 3: Find the whole-number exponent  given a statement of equality or an equation
4^x = 64

x = 3 because 4 • 4 • 4 = 64

##### 6.EE.2   Write, read, and evaluate expressions in which letters stand for numbers.

Students write expressions from verbal descriptions using letters and numbers, understanding ____ is important in writing subtraction and division problems.

expressions  —  order

Students understand that the expression “5 times any number, n” could be represented with  5n and that a number and letter written together means to multiply.

Al rational numbers may be used in writing expressions when operations are not expected.

Students use appropriate mathematical language to write verbal expressions from algebraic expressions.

##### 6,EE.2a  Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.

It is important for students to read algebraic expressions in a manner that reinforces that the variable represents a number.

Example Set 1: Students read algebraic expressions:

r + 21

“some number plus 21” as well as “r plus 21”

n • 6

“some number times 6” as well as “n times 6”

s ÷ 6

“ some number  s  divided by 6” as well as “s  divided by 6”

Example Set 2: Students write algebraic expressions from word expressions

7 less than 3 times a number

3x – 7

3 times the sum of a number and 5

3 (x + 5)

7 less than the product of 2 and a number

2x – 7

Twice the difference between a number and 5

2(z – 5)

The quotient of the sum of x plus 4 and 2

(x + 4) / 2

##### 6.EE.2b  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. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

Students can describe expressions such as 3 (2 + 6) as the ____ of two factors: 3 and (2 + 6).

product

The quantity (2 + 6) is viewed as one factor consisting of ____ terms.

two

Terms are the parts or addends of a ____.

sum

When the term is an explicit number, it is called a ____.

constant

When the term is a product of a number and a variable, the number is called the ____ of the variable.

coefficient

Students should identify the ____ of an algebraic expression including variables, coefficients, constants, and the ____ of operations (sum, difference, product, and quotient).

parts  —  names

Variables are letters that represent ____. There are various possibilities for the number they can represent.

numbers

Consider the following expression: x² + 5y + 3x + 6

The variables are ____

x and y.

There are 4 terms are,

x², 5y, 3x, and 6.

There are 3 variable terms,

x², 5y, 3x.

The variable terms have coefficients of:

1, 5, and 3 respectively.

The coefficient of x²2 is 1, since

x² = 1x².

The term 5y represents:

5y’s or 5 • y.

There is one constant term:

6.

The expression represents a ____ of all four terms.

sum

##### 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). For example, use the formulas V = s^3 and A = 6 s^2 to find the volume and surface area of a cube with sides of length s = ½.

Students evaluate algebraic expressions, using order of operations as needed.

Problems such as example 1 below require students to understand that multiplication is understood when numbers and variables are written together and to use the order of operations to evaluate.

Order of operations is introduced throughout elementary grades, including the use of grouping symbols, ( ), { }, and [ ] in 5th grade. Order of operations with exponents is the focus in 6th grade.

Example 1: Use substitution to evaluate algebraic expressions.
Evaluate the expression  3x + 2y when x is equal to 4 and y is equal to 2.4.

3 • 4 + 2 • 2.4
12 + 4.8
16.8

Example 2:
Evaluate 5(n + 3) – 7n, when n =

MORE

Example 3:
Evaluate 7xy when x = 2.5 and y = 9
Solution: Students recognize that two or more terms written together indicates multiplication.
7 (2.5) (9)
157.5
In 5th grade students worked with the grouping symbols ( ), [ ], and { }. Students understand that the fraction bar
can also serve as a grouping symbol (treats numerator operations as one group and denominator operations as
another group) as well as a division symbol.
Example 4:
Evaluate the following expression when x = 4 and y = 2

x2 + y3
3
Solution:
(4)2 + (2)3 substitute the values for x and y
3
16 + 8 raise the numbers to the powers
3

24
3
divide 24 by 3
8
Given a context and the formula arising from the context, students could write an expression and then evaluate for
any number.
Example 5:
It costs \$100 to rent the skating rink plus \$5 per person. Write an expression to find the cost for any number (n) of
people. What is the cost for 25 people?
Solution:
The cost for any number (n) of people could be found by the MORE…

Example 6:
The expression c + 0.07c can be used to find the total cost of an item with 7% sales tax, where c is the pre-tax cost
of the item. Use the expression to find the total cost of an item that cost \$25.
Solution: Substitute 25 in for c and use order of operations to simplify
c + 0.07c
25 + 0.07 (25)
25 + 1.75
26.75

##### 6.EE.3 Apply the properties of operations to generate equivalentexpressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Students use the distributive property to write equivalent expressions. Using their understanding of area
models from elementary students illustrate the distributive property with variables.
Properties are introduced throughout elementary grades (3.OA.5); however, there has not been an emphasis on
recognizing and naming the property. In 6th grade students are able to use the properties and identify by name as
used when justifying solution methods (see example 4).
Example 1:
Given that the width is 4.5 units and the length can be represented by x + 2, the area of the flowers below can be
expressed as 4.5(x + 3) or 4.5x + 13.5.

PICTURE

When given an expression representing area, students need to find the factors.
Example 2:
The expression 10x + 15 can represent the area of the figure below. Students find the greatest common factor (5) to
represent the width and then use the distributive property to find the length (2x + 3). The factors (dimensions) of
this figure would be 5(2x + 3).

PICTURE

Example 3:
Students use their understanding of multiplication to interpret 3 (2 + x) as 3 groups of (2 + x). They use a model to
represent x, and make an array to show the meaning of 3(2 + x). They can explain why it makes sense that 3(2 + x)
is equal to 6 + 3x.
An array with 3 columns and x + 2 in each column:
Students interpret y as referring to one y. Thus, they can reason that one y plus one y plus one y must be 3y. They
also use the distributive property, the multiplicative identity property of 1, and the commutative property for
multiplication to prove that y + y + y = 3y:
Example 4:
Prove that y + y + y = 3y
Solution:
y + y + y
y • 1 + y • 1 + y • 1 Multiplicative Identity
y • (1 + 1 + 1) Distributive Property
y • 3
3y Commutative Property

##### 6.EE.4 Identify when two expressions are equivalent (i.e., when the twoexpressions name the same number regardless of which value issubstituted into them). For example, the expressions y + y + y and 3y areequivalent because they name the same number regardless of whichnumber y stands for.

Students demonstrate an understanding of like terms as quantities being added or subtracted with the same
variables and exponents. For example, 3x + 4x are like terms and can be combined as 7x; however, 3x + 4×2 are
not like terms since the exponents with the x are not the same.
This concept can be illustrated by substituting in a value for x. For example, 9x – 3x = 6x not 6. Choosing a value
for x, such as 2, can prove non-equivalence.
9(2) – 3(2) = 6(2) however 9(2) – 3(2) = 6
18 – 6 = 12 18– 6 = 6
12 = 12 12 ≠ 6
Students can also generate equivalent expressions using the associative, commutative, and distributive properties.
They can prove that the expressions are equivalent by simplifying each expression into the same form.

Example 1:
4m + 8 4(m+2) 3m + 8 + m 2 + 2m + m + 6 + m

More

##### Reason about and solve one-variable equations and inequalities.

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

The terms students should learn to use with increasing precision with this cluster are: inequalities, equations, greater than, >, less than, <, greater than or equal to, ≥, less than or equal to, ≤, profit, exceed

##### 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.

In elementary grades, students explored the concept of equality. In 6th grade, students explore equations as
expressions being set equal to a specific value. The solution is the value of the variable that will make the equation
or inequality true. Students use various processes to identify the value(s) that when substituted for the variable will
make the equation true.
Example 1:
Joey had 26 papers in his desk. His teacher gave him some more and now he has 100. How many papers did his
teacher give him?
This situation can be represented by the equation 26 + n = 100 where n is the number of papers the teacher gives to
Joey. This equation can be stated as “some number was added to 26 and the result was 100.” Students ask
themselves “What number was added to 26 to get 100?” to help them determine the value of the variable that makes
the equation true. Students could use several different strategies to find a solution to the problem:
 Reasoning: 26 + 70 is 96 and 96 + 4 is 100, so the number added to 26 to get 100 is 74.
 Use knowledge of fact families to write related equations:
n + 26 = 100, 100 – n = 26, 100 – 26 = n. Select the equation that helps to find n easily.
 Use knowledge of inverse operations: Since subtraction “undoes” addition then subtract 26 from 100 to
get the numerical value of n
 Scale model: There are 26 blocks on the left side of the scale and 100 blocks on the right side of the
scale. All the blocks are the same size. 74 blocks need to be added to the left side of the scale to make
the scale balance.
 Bar Model: Each bar represents one of the values. Students use this visual representation to
demonstrate that 26 and the unknown value together make 100.

PICTURE

Solution:
Students recognize the value of 74 would make a true statement if substituted for the variable.
26 + n = 100
26 + 74 = 100
100 = 100 
Example 2:
The equation 0.44 s = 11 where s represents the number of stamps in a booklet. The booklet of stamps costs 11
dollars and each stamp costs 44 cents. How many stamps are in the booklet? Explain the strategies used to
determine the answer. Show that the solution is correct using substitution.
Solution:
There are 25 stamps in the booklet. I got my answer by dividing 11 by 0.44 to determine how many groups of 0.44
were in 11.
By substituting 25 in for s and then multiplying, I get 11.
0.44(25) = 11
11 = 11 
Example 3:
Twelve is less than 3 times another number can be shown by the inequality 12 < 3n. What numbers could possibly
make this a true statement?
Solution:
Since 3 • 4 is equal to 12 I know the value must be greater than 4. Any value greater than 4 will make the
inequality true. Possibilities are 4.13, 6, 5

3
4
, and 200. Given a set of values, students identify the values that make
the inequality true.

##### 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.

Students write expressions to represent various real-world situations.
Example Set 1:
• Write an expression to represent Susan’s age in three years, when a represents her present age.
• Write an expression to represent the number of wheels, w, on any number of bicycles.
• Write an expression to represent the value of any number of quarters, q.
Solutions:
• a + 3
• 2n
• 0.25q

Given a contextual situation, students define variables and write an expression to represent the situation.
Example 2:
The skating rink charges \$100 to reserve the place and then \$5 per person. Write an expression to represent the
cost for any number of people.
n = the number of people
100 + 5n
No solving is expected with this standard; however, 6.EE.2c does address the evaluating of the expressions.
Students understand the inverse relationships that can exist between two variables. For example, if Sally has 3
times as many bracelets as Jane, then Jane has

1
3
the amount of Sally. If S represents the number of bracelets Sally
has, the

1
3
s or

s
3
represents the amount Jane has.
Connecting writing expressions with story problems and/or drawing pictures will give students a context for this
work. It is important for students to read algebraic expressions in a manner that reinforces that the variable
represents a number.
Example Set 3:
• Maria has three more than twice as many crayons as Elizabeth. Write an algebraic expression to represent
the number of crayons that Maria has.
Solution: 2c + 3 where c represents the number of crayons that Elizabeth has
• An amusement park charges \$28 to enter and \$0.35 per ticket. Write an algebraic expression to represent
the total amount spent.
Solution: 28 + 0.35t where t represents the number of tickets purchased
• Andrew has a summer job doing yard work. He is paid \$15 per hour and a \$20 bonus when he completes
the yard. He was paid \$85 for completing one yard. Write an equation to represent the amount of money he
earned.
Solution: 15h + 20 = 85 where h is the number of hours worked
• Describe a problem situation that can be solved using the equation 2c + 3 = 15; where c represents the cost
of an item
Possible solution:
Sarah spent \$15 at a craft store.
• She bought one notebook for \$3.
• She bought 2 paintbrushes for x dollars.
If each paintbrush cost the same amount, what was the cost of one brush?

• Bill earned \$5.00 mowing the lawn on Saturday. He earned more money on Sunday. Write an expression
that shows the amount of money Bill has earned.
Solution: \$5.00 + n

##### 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 non-negative rational numbers.

Students have used algebraic expressions to generate answers given values for the variable. This
understanding is now expanded to equations where the value of the variable is unknown but the outcome is known.
For example, in the expression, x + 4, any value can be substituted for the x to generate a numerical answer;
however, in the equation x + 4 = 6, there is only one value that can be used to get a 6. Problems should be in
context when possible and use only one variable.
Students write equations from real-world problems and then use inverse operations to solve one-step equations
based on real world situations. Equations may include fractions and decimals with non-negative solutions.
Students recognize that dividing by 6 and multiplying by

1
6
produces the same result. For example,

x
6
= 9 and

1
6
x = 9 will produce the same result.
Beginning experiences in solving equations require students to understand the meaning of the equation and the
solution in the context of the problem.
Example 1:
Meagan spent \$56.58 on three pairs of jeans. If each pair of jeans costs the same amount, write an algebraic
equation that represents this situation and solve to determine how much one pair of jeans cost.
Sample Solution:
Students might say: “I created the bar model to show the cost of the three pairs of jeans. Each bar labeled J is the
same size because each pair of jeans costs the same amount of money. The bar model represents the equation 3J =
\$56.58. To solve the problem, I need to divide the total cost of 56.58 between the three pairs of jeans. I know that it
will be more than \$10 each because 10 x 3 is only 30 but less than \$20 each because 20 x 3 is 60. If I start with \$15
each, I am up to \$45. I have \$11.58 left. I then give each pair of jeans \$3. That’s \$9 more dollars. I only have \$2.58
left. I continue until all the money is divided. I ended up giving each pair of jeans another \$0.86. Each pair of jeans
costs \$18.86 (15+3+0.86). I double check that the jeans cost \$18.86 each because \$18.86 x 3 is \$56.58.”

Example 2:
Julie gets paid \$20 for babysitting. He spends \$1.99 on a package of trading cards and \$6.50 on lunch. Write and
solve an equation to show how much money Julie has left.
Solution: 20 = 1.99 + 6.50 + x, x = \$11.51

##### 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.

Many real-world situations are represented by inequalities. Students write inequalities to represent real
world and mathematical situations. Students use the number line to represent inequalities from various contextual
and mathematical situations.
Example 1:
The class must raise at least \$100 to go on the field trip. They have collected \$20. Write an inequality to represent
the amount of money, m, the class still needs to raise. Represent this inequality on a number line.
Solution:
The inequality m ≥ \$80 represents this situation. Students recognize that possible values can include too many
decimal values to name. Therefore, the values are represented on a number line by shading.
A number line diagram is drawn with an open circle when an inequality contains a < or > symbol to show solutions
that are less than or greater than the number but not equal to the number. The circle is shaded, as in the example
above, when the number is to be included. Students recognize that possible values can include fractions and
decimals, which are represented on the number line by shading. Shading is extended through the arrow on a
number line to show that an inequality has an infinite number of solutions.
Example 2:
Graph x ≤ 4.
Solution:
Example 3:
The Flores family spent less than \$200.00 last month on groceries. Write an inequality to represent this amount and
graph this inequality on a number line.

Solution:
200 > x, where x is the amount spent on groceries.

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##### Represent and analyze quantitative relationships between dependent and independnt variables.

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

The terms students should learn to use with increasing precision with this cluster are: dependent variables, independent variables, discrete data, continuous data

##### 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. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

The purpose of this standard is for students to understand the relationship between two variables, which begins with the distinction between dependent and independent variables. The independent variable is the variable that can be changed; the dependent variable is the variable that is affected by the change in the independent variable. Students recognize that the independent variable is graphed on the x-axis; the dependent variable is
graphed on the y-axis.
Students recognize that not all data should be graphed with a line. Data that is discrete would be graphed with
coordinates only. Discrete data is data that would not be represented with fractional parts such as people, tents,
records, etc. For example, a graph illustrating the cost per person would be graphed with points since part of a
person would not be considered. A line is drawn when both variables could be represented with fractional parts.
Students are expected to recognize and explain the impact on the dependent variable when the independent variable
changes (As the x variable increases, how does the y variable change?) Relationships should be proportional with
the line passing through the origin. Additionally, students should be able to write an equation from a word
problem and understand how the coefficient of the dependent variable is related to the graph and /or a table of
values.
Students can use many forms to represent relationships between quantities. Multiple representations include
describing the relationship using language, a table, an equation, or a graph. Translating between multiple
representations helps students understand that each form represents the same relationship and provides a different
perspective.
Example 1:
What is the relationship between the two variables? Write an expression that illustrates the relationship.
Solution:
y = 2.5